Periodic Motions to Chaos in Pendulum

International Journal of Bifurcation and Chaos, Vol. 26, No. 9 (2016) 1650159 (64 pages)
c World Scientific Publishing Company DOI: 10.1142/S0218127416501595 Periodic Motions to Chaos in Pendulum Albert C. J. Luo∗ Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA [email protected] Yu Guo McCoy School of Engineering, Midwestern State University, Wichita Falls, TX 76308, USA by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com Received March 28, 2016 It is not easy to find periodic motions to chaos in a pendulum system even though the periodically forced pendulum is one of the simplest nonlinear systems. However, the inherent complex dynam- ics of the periodically forced pendulum is much beyond our imaginations through the traditional approach of linear dynamical systems. Until now, we did not know complex motions of pendulum yet. What are the mechanism and mathematics of such complex motions in the pendulum? The results presented herein give a new view of complex motions in the periodically forced pendulum. Thus, in this paper, periodic motions to chaos in a periodically forced pendulum are predicted analytically by a semi-analytical method. The method is based on discretization of differential equations of the dynamical system to obtain implicit maps. Using the implicit maps, mapping structures for specific periodic motions are developed, and the corresponding periodic motions can be predicted analytically through such mapping structures. Analytical bifurcation trees of periodic motions to chaos are obtained, and the corresponding stability and bifurcation analysis of periodic motions to chaos are carried out by eigenvalue analysis. From the analytical pre- diction of periodic motions to chaos, the corresponding frequency-amplitude characteristics are obtained for a better understanding of motions’ complexity in the periodically forced pendulum. Finally, numerical simulations of selected periodic motions are illustrated. The nontravelable and travelable periodic motions on the bifurcation trees are discovered. Through this investigation, the periodic motions to chaos in the periodically forced pendulums can be understood further. Based on the perturbation method, one cannot achieve the adequate solutions presented herein for periodic motions to chaos in the periodically forced pendulum. Keywords: Nonlinear pendulum; bifurcation trees to chaos; periodic motions; dual-helix switch- ing; nontravelable periodic motion; travelable periodic motions. Contents 1. Introduction 2 2. Theory and Methods for Periodic Flows 4 3. Periodic Motions in Pendulums 6 3.1. Implicit discretization 6 3.2. Mapping structure of periodic motions 7 ∗ Author for correspondence 1650159-1  A. C. J. Luo & Y. Guo 4. Bifurcation Trees to Chaos in Pendulum 9 4.1. Period-1 motions to chaos 9 4.2. Period-3 to period-6 motions 21 4.3. Period-5 motions 22 5. Nonlinear Frequency-Amplitude Characteristics 22 5.1. Period-1 motions to chaos 23 5.2. Period-3 to period-6 motions 25 5.3. Period-5 motions 26 6. Bifurcation Trees Varying with Excitation Amplitude 29 6.1. Nontravelable period-1 motions to chaos 29 6.2. Nontravelable period-3 motions to chaos 33 6.3. Travelable period-1 motions to chaos 36 6.4. Travelable period-2 motions to chaos 38 7. Numerical Simulations 40 7.1. Nontravelable period-1 motions 40 7.1.1. Period-1 to period-4 motions 40 by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 7.1.2. Period-3 to period-6 motions 48 7.1.3. Period-5 motions 52 7.2. Travelable periodic motions 54 7.2.1. Travelable period-1 to period-2 motions 54 7.2.2. Travelable period-2 to period-8 motions 56 8. Conclusions 62 References 62 1. Introduction Before doing so, let us overview the history of finding periodic solutions in nonlinear dynamical A periodically forced mathematical pendulum is one of the typical and popular nonlinear oscillators systems. that possess complicated and rich dynamical behav- The brief history of periodic motions in non- iors. It seems that the periodically forced pendulum linear systems can be found in [Lagrange, 1788]. is one of the simplest nonlinear oscillators. However, The method of averaging was introduced for peri- until now, a systematic study of periodic motions odic motions in the three-body problems as a to chaos could not be achieved. To know periodic perturbation of the two-body problems. Poincaré motions in the periodically forced pendulum, per- [1899] extended the Lagrange perturbation ideas turbation method has been adopted. Because the to develop perturbation theory and applied in the sine function is not easy to be handled using the motions of celestial bodies. Since then, there is current mathematical tools, one can use the Taylor interest in perturbation theory for approximation series to expand the sine function to the polyno- of the analytical periodic solutions of nonlinear sys- mial nonlinear terms, and then the traditional per- tems. van der Pol [1920] used the method of aver- turbation method is used for obtaining the periodic aging for the periodic solutions of an oscillation motions of the approximated differential system. It circuit. Fatou [1928] proved the asymptotic valid- is always emphasized that the periodic solutions for ity of such a perturbation method. Krylov and the original pendulum are suitable for the small Bogoliubov [1935] further developed the method variation of equilibrium. Thus, this paper will use a of averaging for nonlinear vibration systems. The semi-analytical method based on the implicit maps perturbation methods including averaging method to investigate periodic motions to chaos in the peri- and principle of harmonic balance were presented odically forced pendulum. The periodic motions in [Hayashi, 1964] for periodic motions in nonlinear and bifurcation analysis of such a nonlinear pendu- physical systems. The Krylov–Bogoliubov method lum have been of great interests for a long time. was extended in [Barkham & Soudack, 1969, However, no significant results are obtained yet. 1970] for the approximate solutions of nonlinear 1650159-2  Periodic Motions to Chaos in Pendulum autonomous second-order differential equations. layers in a periodically driven pendulum with- Nayfeh [1973] employed the multiple-scale pertur- out damping. Chaotic motions in the inner and bation method to develop approximate solutions of outer stochastic layers for the periodically driven periodic motions in nonlinear systems. Nayfeh and pendulum were discussed, and resonant separatrix Mook [1979] used the perturbation method to inves- involved in the stochastic layer were predicted ana- tigate nonlinear structural vibrations. Rand and lytically. The onset and disappearance of chaotic Armbruster [1987] used the perturbation method motions in the resonant separatrix layers were ana- and bifurcation theory for the stability of periodic lytically predicted, and the mechanism of chaotic solutions in nonlinear dynamical systems. Garcia- motions in the stochastic and resonant layers were Margallo and Bejarano [1987] used a generalized discussed. In addition, Luo [2001] investigated the harmonic balance approach for the approximate resonance and stochastic layer of a parametrically solutions of nonlinear oscillations with strong non- excited pendulum, and the analytical and numer- linearity. Yuste and Bejarano [1986] used the ellip- ical predictions of resonance in the stochastic lay- tic functions rather than trigonometric functions to ers were presented. In [Luo, 2002], chaotic motions improve the Krylov–Bogoliubov method (also see, in the resonant separatrix band of the paramet- [Yuste & Bejarano, 1989, 1990]). Coppola and Rand rically forced pendulum were discussed, and the by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com [1990] used the averaging method with elliptic func- onset and destruction of the resonant separatrix tions for the approximation of limit cycles. band were analytically predicted. Chaotic motions For a better understanding of complex motions in the stochastic and resonant layers in the peri- in nonlinear dynamical systems, one always consid- odically forced Hamiltonian systems were system- ered the periodically forced pendulum. Since 1960, atically investigated. However, periodic and chaotic the nonlinear dynamics of a particle in a traveling motion in the periodically forced, dissipated, Hamil- electric field was investigated by a nonlinear pen- tonian systems cannot be discussed because no ade- dulum. For example, Zaslavsky and Chirikov [1972] quate mathematical methods can be used. Thus, discussed the stochastic (chaotic) instability of non- Luo [2012] developed an analytical method for linear oscillation which is based on a periodically obtaining analytical solutions of periodic motions forced pendulum, and the resonance overlap was in nonlinear dynamical systems. Luo and Huang discussed. Ben-Jacob et al. [1982] used the pendu- [2012a] used such a method to obtain approximated lum to investigate intermittent chaos in Josephson solutions of periodic motions in the Duffing oscil- junctions. Kadanoff [1985] investigated the routes of lator. Luo and Huang [2012b] also provided the periodic motions to unbounded chaos through the bifurcation trees of period-m motions to chaos in simple pendulum, and the Chirikov–Taylor model such a Duffing oscillator. Other further discussions (or the standard mapping model) was obtained. of periodic motions to chaos in the Duffing oscilla- The scaling analysis for the onset of chaos was tors can be found in [Luo & Huang, 2012c, 2012d]. completed. Gwinn and Westervelt [1985] discussed Wang and Liu [2015] developed a numerical scheme intermittent chaos and low frequency noise in the to determine the coefficients in the finite Fourier driven damped pendulum through the fourth-order series expression of periodic motions in nonlinear Runge–Kutta method, and the attraction basin was dynamical systems. Such a numerical method can presented. To find the chaos in the periodically give the same solutions of periodic motion as the forced Josephson junction, Salam and Sastry [1985] analytical method. However, the analytical method used the Melnikov function to predict the onset of presented in [Luo, 2012] is very difficult for nonpoly- chaos. de Paula et al. [2006] established an exper- nomial nonlinear dynamical systems. Thus, a new imental nonlinear pendulum to investigate chaotic technique had to be developed. motions. Luo [2015a, 2015b] developed a semi-analytical From the aforementioned studies, one tried to method to determine periodic motions in nonlin- find chaotic motions in nonlinear dynamical sys- ear dynamical systems. The semi-analytical method tems including pendulums, but no significant pre- is based on discrete implicit maps. The discrete dictions of chaotic motions in nonlinear dynamical implicit maps are developed from discretization of systems have been presented. Thus, Luo and Han differential equations of nonlinear dynamical sys- [2000] investigated the resonant and stochastic tems. For specific periodic motions, the mapping 1650159-3  A. C. J. Luo & Y. Guo structures of discrete implicit maps can be devel- better understanding of motion complexity in such oped. From a specific mapping structure, a set of a pendulum. nonlinear algebraic equations given by the implicit maps will be solved for the analytical prediction of a periodic motion in the nonlinear dynami- 2. Theory and Methods for Periodic cal system. Luo and Guo [2015] and Guo and Flows Luo [2015a, 2015b] applied such a semi-analytical From [Luo, 2015a, 2015b], periodic motions in method to predict bifurcation trees of Duffing oscil- continuous dynamical systems will be presented lators. From the semi-analytical method, the bifur- through the corresponding implicit mappings. If a cation trees of periodic motions to chaos were nonlinear system has a periodic flow with a period obtained. Using the finite discrete Fourier series, of T = 2π/Ω, then the period-m motions can be the nonlinear frequency-amplitude characteristics determined by discrete nodes and mappings of the for the bifurcation trees of periodic motions to chaos continuous system. were obtained. From the quantity level analysis, the appropriate analytical expressions of periodic Theorem 1. Consider a nonlinear dynamical motions can be obtained, and such expressions can system as by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com be used for analytical methods based on the gen- ẋ = f (x, t, p) ∈ R n (1) eralized harmonic balance in [Luo, 2012]. Based on the semi-analytical method, periodic motions in the where f (x, t, p) is a C r -continuous nonlinear vec- Duffing oscillator are quite close to the approximate tor function (r ≥ 1). If such a dynamical sys- analytical solutions in [Luo & Huang, 2012a, 2012b, tem has a period-m flow x(m) (t) with finite norm 2012c, 2012d]. Because the periodically forced pen- x(m)  and period mT (T = 2π/Ω), there is a set dulum possesses a nonpolynomial nonlinear func- of discrete time tk (k = 0, 1, . . . , mN ) with (N → tion, it is very difficult to use the analytical method ∞) during m-periods (mT ), and the corresponding in [Luo, 2012] to obtain analytical solutions of solution x(m) (tk ) and vector field f (x(m) (tk ), tk , p) (m) periodic motions in the periodically forced pendu- are exact. Suppose a discrete node xk is on the lum. Until now, no adequate solutions of periodic approximate solutions of the periodic flow under motions to chaos in the periodically forced pen- (m) x(m) (tk ) − xk  ≤ εk with a small εk ≥ 0 and dulum are obtained. One still does not know the (m) motion complexity of the periodically forced pendu- f (x(m) (tk ), tk , p) − f (xk , tk , p) ≤ δk (2) lum. Herein, the purpose of this paper is to give an analytical prediction of periodic motions to chaos with a small δk ≥ 0. During a time interval of t ∈ (m) (m) in such a pendulum system and to determine the [tk−1 , tk ], there is a mapping Pk : xk−1 → xk (k = motion complexity in such a system. 1, 2, . . . , mN ), i.e. In this paper, periodic motions to chaos in (m) (m) (m) (m) xk = Pk xk−1 with gk (xk−1 , xk , p) = 0, the periodically forced pendulum will be studied by implicit mapping. The implicit mapping will k = 1, 2, . . . , mN (3) be derived from the discretization of the differen- tial equation of the periodically forced pendulum where gk is an implicit vector function. Consider a through the midpoint scheme. From the mapping mapping structure as structures of periodic motions, the bifurcation trees (m) (m) P = PmN ◦ PmN −1 ◦ · · · ◦ P1 : x0 → xmN ; of periodic motions to chaos will be obtained. The corresponding stability and bifurcation analysis will (m) with Pk : xk−1 → xk (m) (k = 1, 2, . . . , mN ). be carried out through eigenvalue analysis. Using the discrete Fourier series method, the frequency- (4) amplitude characteristics of periodic motions to (m) (m) (m)∗ For xmN = P x0 , if there is a set of points xk chaos will be presented for a better understand- (k = 0, 1, . . . , mN ) computed by ing of global characteristics of periodic motions in the periodically forced pendulum. From the analyt- (m)∗ (m)∗ gk (xk−1 , xk , p) = 0, (k = 1, 2, . . . , mN ) ical predictions, numerical simulations of periodic (5) (m)∗ (m)∗ motions will be intuitively illustrated, which is for a x0 = xmN , 1650159-4  Periodic Motions to Chaos in Pendulum then the points xk (m)∗ (k = 0, 1, . . . , mN ) are (iii) The boundaries between stable and unstable approximations of points x(m) (tk ) of the period-m period-m flow with higher order singularity give (m)∗ (m) bifurcation and stability conditions. solution. In the neighborhood of xk , with xk = (m)∗ (m) xk + ∆xk , the linearized equation is given by Proof. See [Luo, 2015a, 2015b].
(m) (m) ∆xk = DP k · ∆xk−1 with Consider a nonlinear dynamical system. If such (m)∗ (m) (m)∗ (m) a dynamical system has a period-m flow x(m) (t) gk (xk−1 + ∆xk−1 , xk + ∆xk , p) = 0 with finite norm x(m)  and period mT (T = (k = 1, 2, . . . , mN ). 2π/Ω), then (6) x(m) (t + mT ) = x(m) (t). (11) The resultant Jacobian matrices of the periodic flow From the Fourier series theory of periodic function, are we have the following definition. DP k(k−1)···1 = DP k · DP k−1 · . . . · DP 1 , Definition 1. Consider a nonlinear dynamical sys- by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com (k = 1, 2, . . . , mN ); tem with/without time delay, and such a dynamical (7) system has a period-m flow x(m) (t) with finite norm DP ≡ DP mN (mN −1)···1 x(m)  and period mT (T = 2π/Ω). If x(m) (t) is a continuous flow on t ∈ (0, mT ), there is the finite = DP mN · DP mN −1 · . . . · DP 1 (m) Fourier series TM (t) ∈ R n for the period-m flow where x(m) (t) ∈ R n as
(m)  ∂xk M   DP k = (m) (m)  j (m) ∂xk−1 TM (t) = a0 + bj/m cos Ωt (m)∗ (xk−1 ,xk (m)∗ ) m j=1
−1
   ∂gk ∂gk  j =− . (8) + cj/m sin Ωt (12) (m) (m)  m ∂xk ∂xk−1  (m)∗ (m)∗ (xk−1 ,xk ) which is called a trigonometric polynomial of The eigenvalues of (m)∗ DP(x0 ) and DP k(k−1)···1 for order M. such a periodic flow are determined by From discrete mapping structures, the node |DP k(k−1)···1 − λIn×n | = 0, (k = 1, 2, . . . , mN ); points of period-m flows are computed. Consider (m) (m) the node points of period-m flows as xk = (x1k , |DP − λIn×n | = 0. (m) (m) x2k , . . . , xnk )T for k = 0, 1, 2, . . . , mN in a nonlin- (9) ear dynamical system. The approximate expression Thus, the eigenvalues of DP k(k−1)···1 give the prop- for the period-m flow is determined by the Fourier erties of xk varying with x0 . The stability and bifur- series as cation of the periodic flow can be classified by the M   (m)∗ (m) (m)  j eigenvalues of DP(x0 ) with x (t) ≈ a0 + bj/m cos Ωt m j=1 ([nm o m o 1 , n1 ] : [n2 , n2 ] : [n3 , κ3 ] : [n4 , κ4 ] |   j n5 : n6 : [n7 , l, κ7 ]). (10) + cj/m sin Ωt . (13) m (i) If the magnitudes of all eigenvalues of DP (m) There are (2M + 1) unknown vector coefficients are less than one (i.e. |λi | < 1, i = 1, 2, . . . , n), (m) of a0 , bj/m , cj/m . To determine such unknowns, the approximate period-m solution is stable. (m) (ii) If at least the magnitude of one eigenvalue at least we have the given nodes xk (k = 0, 1, of DP (m) is greater than one (i.e. |λi | > 1, 2, . . . , mN ) with mN + 1 ≥ 2M + 1. In other words, (m) i ∈ {1, 2, . . . , n}), the approximate period-m we have M ≤ mN/2. The node points xk on the solution is unstable. period-m flow can be expressed by the finite Fourier 1650159-5  A. C. J. Luo & Y. Guo series as for tk ∈ [0, mT ] Proof. See [Luo, 2015a, 2015b].
(m) x(m) (tk ) ≡ xk The harmonic amplitudes and harmonic phases mN/2   for period-m motion are (m)  j  = a0 + bj/m cos Ωtk Aj/ms = b2j/ms + c2l/ms , m j=1   cj/ms (18) j ϕj/ms = arctan , (s = 1, 2, . . . , n). + cj/m sin Ωtk bj/ms m Thus the approximate expression for period-m mN/2   (m)  j 2kπ motion in Eq. (13) is determined by = a0 + bj/m cos m N mN/2   j=1  j (m) (m)   x (t) ≈ a0 + bj/m cos Ωt j 2kπ m + cj/m sin j=1 m N   j by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com (k = 0, 1, . . . , mN − 1). (14) + cj/m sin Ωt . (19) m Theorem 2. Consider a nonlinear dynamical sys- The foregoing equation can be expressed as tem with/without time delay, and such a dynami- mN/2   cal system has a period-m flow x(m) (t) with finite (m) (m)  j xs (t) = a0s + Aj/ms cos Ωt − ϕj/ms norm x(m)  and period mT (T = 2π/Ω). If the m j=1 node points of period-m flows in a nonlinear dynam- (m) (m) (m) (m) ical system are xk = (x1k , x2k , . . . , xnk )T for (s = 1, 2, . . . , n). (20) k = 0, 1, 2, . . . , mN with 2kπ T 2π 3. Periodic Motions in Pendulums tk = k∆t = with ∆t = = , (15) ΩN N ΩN In this section, the implicit discretization of pendu- (m) then a trigonometric polynomial TM (t) with min- lum will be presented to obtain the implicit maps. imization of mN (m) (m) (tk ) − TmN/2 (tk )]2 exists From the discrete nodes and mappings, periodic k=0 [(x (m) motions in pendulums can be predicted, and the and x(m) (t) can be approximated by TM (t) (i.e. stability and bifurcation of periodic motions are (m) x(m) (t) ≈ TmN/2 (t)). That is, discussed through eigenvalue analysis. mN   (m)  j 3.1. Implicit discretization x(m) (t) ≈ a0 + bj/m cos Ωt m j=1 A periodically driven pendulum system can be  j  described by + cj/m sin Ωt (16) m ẍ + δẋ + α sin x = Q0 cos Ωt (21) where where δ is the damping coefficient, α is the stiffness, mN 1  (m) Q0 and Ω are excitation amplitude and frequency, (m) a0 = xk , respectively. The system can be expressed in state N k=0 space mN   ẋ = y, ẏ = Q0 cos Ωt − δẋ − α sin x. (22) 2  (m) 2jπ  bj/m = xk cos k ,  mN mN   From [Luo, 2015a, 2015b], such a nonlinear dynam- k=0 (17) mN   ical system can be discretized. Using a midpoint  2  (m) 2jπ   scheme for the time interval t ∈ [tk−1 , tk ], an cj/m = xk sin k   mN mN  implicit map Pk (k = 0, 1, 2, . . .) can be formed as k=0   Pk : (xk−1 , yk−1 ) → (xk , yk ) mN j = 1, 2, . . . , 2 ⇒ (xk , yk ) = Pk (xk−1 , yk−1 ) (23) 1650159-6  Periodic Motions to Chaos in Pendulum with the implicit relations (m) (m) (xmN , y mN ) = (x0 (m) (m) + 2lπ, y 0 ), 1 l = 0, ±1, ±2, . . . ; m = 1, 2, . . . . xk = xk−1 + h(yk−1 + yk ) 2   (28) 1 yk = yk−1 + h Q0 cos Ω tk−1 + h A new vector function is introduced as 2 (m) (m) (m)   gk = (gk1 , gk2 )T , xk = (xk , y k )T . (29) 1 1 − δ(yk−1 + yk ) − α sin (xk−1 + xk ) . The governing equations for Pk in Eq. (7) are rewrit- 2 2 ten as (24) (m) (m) gk (xk−1 , xk , p) = 0 (30) The foregoing discretization experiences an accu- where racy of O(h3 ) for each step. To keep computational 1 (m) (m) (m) (m) accuracy less than 10−9 , h < 10−3 needs to be gk1 = xk − xk−1 − h(y k−1 + y k−1 ), 2 maintained.   1 by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com (m) (m) (m) gk2 = y k − y k−1 − h Q0 cos Ω tk−1 + h 2 3.2. Mapping structure of periodic   motions 1 (m) (m) 1 (m) (m) − δ(y k−1 + y k ) − α sin (xk−1 + xk ) . From the above implicit discretization, the discrete 2 2 implicit map is obtained. Using the implicit map, (31) a period-m motion in such a pendulum system can For a period-1 motion, the corresponding map- be expressed by a mapping structure, i.e. ping structure is P = PmN ◦ PmN −1 ◦ · · · ◦ P2 ◦ P1 : (x0 , y0 ) P = PN ◦ PN −1 ◦ · · · ◦ P2 ◦ P1 : x0 → xN ; with    mN -actions Pk : xk−1 → xk (k = 1, 2, . . . , N ), (32) → (xmN , ymN ) (25) where implicit mapping Pk with gk (xk−1 , xk , p) = 0 (k = 1, 2, . . . , N ) and the periodicity condition is with xN = x0 . Such a mapping structure for period-1 (m) (m) (m) (m) motion is presented in Fig. 1. The node points are Pk : (xk−1 , y k−1 ) → (xk , y k ) depicted by circles on the trajectory and the map- (m) (m) (m) (m) pings are depicted by arrows. The periodicity guar- ⇒ (xk , y k ) = Pk (xk−1 , y k−1 ) antees that the initial and final nodes overlap each (k = 1, 2, . . . , mN ). (26) other. The mapping structure consists of N nodes and N mappings. The N nodes corresponding to a From Eq. (4), the governing algebraic equation for set of points x∗k (k = 0, 1, . . . , N ) are computed by each mapping Pk is the 2(N + 1) implicit vector functions as (m) (m) 1 (m) (m) gk (x∗k−1 , x∗k , p) = 0, (k = 1, 2, . . . , N ) xk = xk−1 + h(y k−1 + y k−1 ), (33) 2   x∗0 = x∗N . (m) (m) (m) 1 yk = yk−1 + h Q0 cos Ω tk−1 + h Through a period doubling of the period-1 2 motion, a period-2 motion could be obtained, and 1 (m) (m) (27) the corresponding mapping structure is − δ(y k−1 + y k ) 2 P = P2N ◦ P2N −1 ◦ · · · ◦ P2 ◦ P1 : x0 → xN ; (2) (2)   1 (m) (m) −α sin (xk−1 + xk ) , (2) with Pk : xk−1 → xk (2) (k = 1, 2, . . . , N ). 2 (34) (k = 1, 2, . . . , mN ). In a similar fashion, a sketch of the mapping Using the periodicity condition, we have structure of such a period-2 motion is presented 1650159-7  A. C. J. Luo & Y. Guo x N , x0 through the eigenvalue analysis. For a small pertur- x N −1 x1 (m)∗ (m) bation in vicinity of xk , xk = xk (m)∗ (m) + ∆xk , one could obtain P1 x2 PN (m) 1  (m) ∆xmN = DP ∆x0 = DP k ∆x0 . (37) k=mN For each mapping Pk , we have Pk ∆xk (m) (m) = DP k ∆xk−1 , (k = 1, 2, . . . , mN ) (38) x k −1 where DP k is the Jacobian matrix of each mapping y
(m)  x k +1 xk ∂xk DP k = (m) , x ∂xk−1 (x(m)∗ ,x(m)∗ ) k k−1 Fig. 1. Mapping structure of period-1 motion with N -nodes. (k = 1, 2, . . . , mN ). (39) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com in Fig. 2. The mapping structure consists of 2N To determine the stability and bifurcations of such nodes and 2N mappings. The 2N node points x∗k a period-m motion, the eigenvalues of the resultant (k = 0, 1, . . . , 2N ) are computed by the following Jacobian matrix are as implicit vector functions. |DP − λI| = 0 (40) (2)∗ (2)∗ gk (xk−1 , xk , p) = 0, (k = 1, 2, . . . , 2N ) where the resultant Jacobian matrix is computed (35) by (2)∗ (2)∗ x0 = x2N
(m)  1 1 Similarly, the solution of a period-m motion can   ∂xk DP = DP k = (m) . be determined through the 2(mN + 1) equations. k=mN k=mN ∂xk−1 (x(m)∗ ,x(m)∗ ) That is, k k−1 (m)∗ (m)∗ (41) gk (xk−1 , xk , p) = 0, (k = 1, 2, . . . , mN ) (36) Since this is a two-dimensional mapping, there are (m)∗ (m)∗ x0 = xmN two eigenvalues. From Luo [2012], the stability con- (m)∗ ditions can be given as follows: If the node points xk (k = 1, 2, . . . , mN ) of the period-m motions are obtained, the stability and (i) If the magnitudes of two eigenvalues are less bifurcation of such a periodic motion can be studied than one (i.e. |λi | < 1, i = 1, 2), the period-m motion is stable. (ii) If one of two eigenvalue magnitudes are greater x2 N x0 x1 than one (i.e. |λi | > 1, i ∈ {1, 2}), the period-m motion is unstable. x 2 N −1 xN x N +1 x2 For the bifurcation conditions, we have the x N −1 following: (i) If λi = 1, i ∈ {1, 2} and |λj | < 1, j ∈ {1, 2} but j = i, the saddle-node bifurcation of period-m motion occurs. (ii) If λi = −1, i ∈ {1, 2} and |λj | < 1, j ∈ {1, 2} x n2 but j = i, the period-doubling bifurcation of x k +1 x k −1 xk period-m motion occurs. For the stable period- doubling bifurcation, the period-doubling peri- x n1 odic motion will be observed. Fig. 2. Mapping structure of period-2 motion with (iii) If |λ1,2 | = 1 with λ1,2 = α ± iβ, the Neimark 2N -nodes. bifurcation of period-m motion occurs. 1650159-8  Periodic Motions to Chaos in Pendulum 4. Bifurcation Trees to Chaos in another saddle-node bifurcation. As Ω → 0, peri- Pendulum odic motions are very complicated, as shown in Figs. 4(a) and 4(b). The two branches of symmetric This section will present the bifurcation trees of period-1 motions frequently switch between stable period-m motions (m = 1, 3, 5, . . .) to chaos in the and unstable motions through saddle-node bifur- periodically forced pendulum. For the periodically cations. The saddle-node bifurcations also occur excited pendulum discussed above, a set of system in both unstable symmetric period-1 motions and parameters are considered as α = 1.5, δ = 0.75, stable asymmetric period-1 motions. In addition, Q0 = 5.0. Simulation of periodic trajectories will be jumping phenomena of symmetric and asymmet- presented in verification of the predicted trajectory. ric period-1 motions also take place frequently. The unstable period-1 motion is enclosed by two jump- 4.1. Period-1 motions to chaos ing phenomena with two saddle-node bifurcations. In this section, analytical prediction of bifurcation Cascaded period-doubling bifurcations leading to trees will be presented for period-1, period-2, and chaos also occur once the asymmetric period-1 period-4 motions to chaos. Stability and bifurcation motions appear. However the asymmetric period-1 analysis will be illustrated through eigenvalues. to period-m motions generated by period-doubling by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com A global view of the analytical prediction of bifurcations are not illustrated herein due to the bifurcation tree from period-1 motions to chaos is very tiny stable ranges. As the excitation frequency presented in Fig. 3 for such a periodically excited increases, the variation of periodic motion changes pendulum. The solid and dashed curves depict sta- slowly and gradually. ble and unstable periodic motions, respectively. The From Ω = 0.25 to 0.7, pairs of asymmetric pairs of asymmetric motions are presented with period-1 motions are plotted in Figs. 4(c) and 4(d). black and red colors, respectively. The acronyms The asymmetric period-1 motions do not possess “SN” and “PD” represent saddle-node and period- any jumping phenomenon. Again, the asymmet- doubling bifurcations, respectively. The displace- ric period-1 motion connects the two branches ment and velocity of the periodic nodes xmod(k,N ) of symmetric period-1 motions together. There and ymod(k,N ) for mod(k, N ) = 0 varying with are four pairs of asymmetric period-1 motions for excitation frequency are presented in Figs. 3(a) Ω ∈ (0.25, 0.7). The four branches are for Ω ∈ and 3(b), respectively. The period-1, period-2, and (0.3112, 0.3485), (0.3634, 0.4118), (0.4358, 0.5015), period-4 motions are labeled by P-1, P-2, and P- and (0.5434, 0.6390). The corresponding cascaded 4, respectively. In order to provide better views, period-doubling bifurcations to chaos in the four five regions, Ω ∈ (0.05, 0.25), (0.25, 0.7), (0.7, 0.9) branches exist for Ω ∈ (0.3118, 0.3485), (0.3644, and (1.1, 1.51) with Ω ∈ (1.28, 1.31) are chosen to 0.4065), (0.4376, 0.4931), and (0.5469, 0.6245), be zoomed and presented in Figs. 4–8, respectively. respectively. Due to the very tiny stable range of The stability range and bifurcation points are listed the period-m motions after period-doubling bifur- in Tables 1–5. cations, the period-m motions (m = 2l , l = 1, 2, . . .) Two branches of symmetric period-1 motions are not illustrated herein. exist for the whole range of Ω ∈ (0, +∞). For Ω ∈ (0.7254, 0.8820), a pair of asym- The two branches of symmetric period-1 motions metric period-1 motions are generated through form a double helix structure together. The dou- saddle-node bifurcations of the symmetric period-1 ble helix structure of the symmetric period-1 motions, as shown in Figs. 4(e) and 4(f). This motions cannot be observed easily from the dis- pair of asymmetric period-1 motions possess placement due to the modulus of displacement into period-doubling bifurcations at Ω ≈ 0.7336 and the range of mod(xmod(k,N ) , 2π) ∈ (0, 2π). How- Ω ≈ 0.8517, and such bifurcation points produce ever, from the velocity plot, such a double helix unstable asymmetric period-1 motions and stable structure of symmetric period-1 motions can be period-2 motions. The period-2 motions are in the observed as in Fig. 3(b). The two branches alterna- range of Ω ∈ (0.7336, 0.8517). Two pairs of cascaded tively become stable and unstable. The asymmet- period-doubling bifurcations and one jumping phe- ric period-1 motions start from one branch of the nomenon exist for such period-2 motions. The symmetric period-1 motions through a saddle-node saddle-node bifurcations associated with the jump- bifurcation, and will join the other branch through ing phenomena exist at Ω ≈ 0.7776 and 0.81512. 1650159-9  A. C. J. Luo & Y. Guo (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 0.30 SN Eigenvalue Imaginary Part, Imλ1,2 1.0 Eigenvalue Real Part, Reλ1,2 0.15 0.0 0.00 -0.15 PD -1.0 -0.30 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Excitation Frequency, Ω Excitation Frequency, Ω (c) (d) 1.0 Eigenvalue Magnitude, |λ1,2| 0.5 0.0 0.0 0.5 1.0 1.5 2.0 Excitation Frequency, Ω (e) Fig. 3. A global view of the analytical prediction of bifurcation trees for period-1 to chaos varying with excitation frequency Ω ∈ (0.0, 2.0): (a) Periodic node displacement mod(xmod(k,N ) , 2π), (b) periodic node velocity ymod(k,N ) , (c) real part of eigenvalues, (d) imaginary part of eigenvalues, (e) magnitude of eigenvalues. (α = 1.5, δ = 0.75, Q0 = 5.0). 1650159-10  Periodic Motions to Chaos in Pendulum 2π 7.0 Periodic Node Displacement, mod(xmod(k,N),2π) SN SN SN SNxn ... Periodic Node Velocity, ymod(k,N) SN 3π/2 SN SN SN SN 6.6 SN SN SN SN SN SN π SN SN SN SN SN SN SN SNxn ... 6.2 SN SN π/2 SN SN SN SN SN SN SN SN SN SN 0 5.8 0.05 0.10 0.15 0.20 0.25 0.05 0.10 0.15 0.20 0.25 Excitation Frequency, Ω Excitation Frequency, Ω (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com PD PD PDPDPD PD PD PD PD PD PD PD PD PD PD PD PD PD PD PD 2π 6.5 Periodic Node Displacement, mod(xmod(k,N),2π) S S S A SN S A Periodic Node Velocity, ymod(k,N) S 3π/2 A 5.5 SN SN A SN A SN S A SN SN SN A A SN S π 4.5 SN SN SN SN S S S A S SN π/2 A S 3.5 SN SN SN 0 2.5 0.25 SN 0.40 0.55 0.70 0.25 0.40 0.55 0.70 Excitation Frequency, Ω Excitation Frequency, Ω (c) (d) SN PD SN SN PD PD SN SN PD SN SN PD PD SN 2π 4.5 Periodic Node Displacement, mod(xmod(k,N),2π) S A S Periodic Node Velocity, ymod(k,N) 3π/2 A 3.5 P-1 π P-2 S 2.5 P-1 P-2 π/2 P-1 A S P-2 A P-2 P-1 0 1.5 0.70 0.75 0.80 0.85 0.90 0.70 0.75 0.80 0.85 0.90 Excitation Frequency, Ω Excitation Frequency, Ω (e) (f) Fig. 4. Periodic node displacement mod(xmod(k,N ) , 2π) and velocity ymod(k,N ) for bifurcation trees of period-1 motions to chaos: (a), (b) Zoom-1 view (Ω ∈ (0.05, 0.25)), (c), (d) Zoom-2 view (Ω ∈ (0.25, 0.7)), (e), (f) Zoom-3 view (Ω ∈ (1.1, 1.51)), (g), (h) Zoom-4 view (Ω ∈ (1.1, 1.51)), (i), (j) detailed view of zoom-4 window (Ω ∈ (1.28, 1.31)). (α = 1.5, δ = 0.75, Q0 = 5.0). 1650159-11  A. C. J. Luo & Y. Guo SN PD PD Zoom-5 PD PD SN SN PD PD Zoom-5 PD PD SN 2π 2.78 Periodic Node Displacement, mod(xmod(k,N),2π) P-4 A P-2 Periodic Node Velocity, ymod(k,N) P-1 3π/2 2.21 P-4 π P-1 1.64 A P-1 P-2 P-1 S π/2 1.07 S SN 0 0.50 1.1 1.2 1.3 1.4 1.5 1.1 1.2 1.3 1.4 1.5 Excitation Frequency, Ω Excitation Frequency, Ω (g) (h) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com SNPD PDSN PD PD PD PD PD PD SN PD PD SN PD PD PD PD PD PD 2π 2.78 Periodic Node Displacement, mod(xmod(k,N),2π) SN SN SN SN Periodic Node Velocity, ymod(k,N) P-4 SN SN P-2 P-4 3π/2 2.21 P-4 P-2 P-4 P-4 SN P-4 P-4 1.64 π P-4 SN P-4 P-4 π/2 1.07 0 0.50 1.28 1.29 1.30 1.31 1.28 1.29 1.30 1.31 Excitation Frequency, Ω Excitation Frequency, Ω (i) (j) Fig. 4. (Continued) SN SNPD PD SN SN SN SN SN SNPD PD SN SN SN SN 3π/2 4.5 Periodic Node Displacement, mod(xmod(k,N),2π) S A SN P-3 P-6 Periodic Node Velocity, ymod(k,N) P-6 P-3 π 3.5 A S π/2 2.5 S 0 1.5 0.735 0.753 0.771 0.789 0.807 0.735 0.753 0.771 0.789 0.807 Excitation Frequency, Ω Excitation Frequency, Ω (a) (b) Fig. 5. Period-3 motions to chaos for Ω ∈ (0.735, 0.807): (a) Periodic node displacement mod(xmod(k,N ) , 2π) and (b) periodic node velocity ymod(k,N ) ; (α = 1.5, δ = 0.75, Q0 = 5.0). 1650159-12  Periodic Motions to Chaos in Pendulum SN SNPD PD SN SN SN SN PD PD SN SN 2π 3.0 Periodic Node Displacement, mod(xmod(k,N),2π) S Periodic Node Velocity, ymod(k,N) P-5 A S 2.0 P-5 A π 1.4 π/2 0 0.6 1.22 1.24 1.26 1.28 1.22 1.24 1.26 1.28 Excitation Frequency, Ω Excitation Frequency, Ω (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com Fig. 6. Period-5 motions to chaos for Ω ∈ (1.22, 1.28): (a) Periodic node displacement mod(xmod(k,N ) , 2π). (b) Periodic node velocity ymod(k,N ) ; (α = 1.5, δ = 0.75, Q0 = 5.0). SN SN SN SN SN SN SN SNPD PD PD SN 2π 0.48 S A SN PD P-4 PD Harmonic Amplitude, A1/4 3π/2 Constant, mod(a0 ,2π) P-2 SN SN A 0.32 (m) P-4 PD SN S π PD 0.16 PD π/2 PD PD PD A S P-1 SN SN SN SN 0 0.00 0.0 0.5 1.0 1.5 2.0 1.14 1.20 1.26 1.32 1.38 Excitation Frequency, Ω Excitation Frequency, Ω (a) (b) 2.0 0.21 PD SN SN PD PD SN SN Harmonic Amplitude, A1/2 Harmonic Amplitude, A3/4 1.5 SN PD SN 0.14 PD PD SN P-4 1.0 P-2 SN SN PD SN P-2 PD 0.07 PD PD 0.5 PD P-4 PD PD PD PD PD SN SN SN SN SN SN SN SN 0.0 0.00 0.6 0.9 1.2 1.5 1.14 1.20 1.26 1.32 1.38 Excitation Frequency, Ω Excitation Frequency, Ω (c) (d) (m) Fig. 7. Frequency-amplitude characteristics for bifurcation trees of period-1 to period-4 motions: (a) a0 (m = 1, 2, 4), (b)–(p) Ak/m (m = 4, k = 1, 2, 3, 4; 6, 8, 12, 16; 80, 81, . . . , 84; 240, 244); parameters: (α = 1.5, δ = 0.75, Q0 = 5.0). 1650159-13  A. C. J. Luo & Y. Guo 130 0.4 10 SN SN 4 SN 104 SN A PD P-1 Harmonic Amplitude, A3/2 0.3 Harmonic Amplitude, A1 PD P-2 SN 78 P-2 PD A PD SN P-1 P-2 2 SN SN PD SN 5 SN PD S S 0.2 52 PD SN 0.5 1.0 1.0 1.5 PD SN P-2 PD PD 0.1 PD PD 26 SNxn PD P-4 PD PD SN PD SN SN PD PD SN SN SN SN SN 0 0.0 0.0 0.5 1.0 1.5 2.0 0.6 0.9 1.2 1.5 Excitation Frequency, Ω Excitation Frequency, Ω (e) (f) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 0.3 3.0 SN PD 0.04 PD SN Harmonic Amplitude, A3 Harmonic Amplitude, A2 0.2 P-4 2.0 P-1 A SNxn PD P-2 S SN SN SN A PD P-1 0.00 PD PD SN SN 0.1 PD 1.0 1 2 PD A P-4 P-2 PD SN PD A SN SN P-1 SN S SN PD SN PD SN SN SN SN SN S PD SN 0.0 0.0 0.0 SN SN SN 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Excitation Frequency, Ω Excitation Frequency, Ω (g) (h) 0.3 0.008 P-2 PD 1e-2 PD P-4 A Harmonic Amplitude, A20 Harmonic Amplitude, A4 SN PD PD 0.2 0.004 P-1 PD 1e-6 PD PD PD PD A S 0.000 0.1 1.1 SN SN 1.6 PD PD PD PD 1e-10 PD PD PD SN PD SN PD P-1 S SN SN SN SN PD SN 0.0 1e-14 0.0 SN SN SN 0.5 1.0 1.5 2.0 0.0 0.4 SN SN 0.8 SN SN 1.2 SN 1.6 Excitation Frequency, W Excitation Frequency, W (i) (j) Fig. 7. (Continued) 1650159-14  Periodic Motions to Chaos in Pendulum 2.4e-12 4e-7 PD 2e-12 SN SN P-4 Harmonic Amplitude, A41/2 Harmonic Amplitude, A81/4 PD 3e-7 1.6e-12 P-4 1e-12 PD 2e-7 PD PD SN P-2 PD PD PD PD 8.0e-13 SN SN1.2 SN1.4 PD PD 1e-7 PD PD PD PD 0.0 0 1.14 SN 1.20 1.26 SN SN 1.32 SN 1.38 0.6 SN SN0.9 1.2 1.5 Excitation Frequency, Ω Excitation Frequency, Ω (k) (l) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 1.5e-12 PD 1e-2 P-1 Harmonic Amplitude, A83/4 Harmonic Amplitude, A21 SN 1.0e-12 P-4 SN SN 1e-6 SN PD PD PD 5.0e-13 PD 1e-10 SN PD S PD PD PD SN 1e-14 PD 0.0 1.14 SN 1.20 1.26 SNSN 1.32 SN 1.38 0.0 0.4 0.8 1.2 1.6 Excitation Frequency, Ω Excitation Frequency, Ω (m) (n) 1e-2 1e-2 SNxn Harmonic Amplitude, A61 Harmonic Amplitude, A60 PD P-1 1e-6 PD 1e-6 PD P-1 PD SN PD SN PD 1e-10 PD 1e-10 SN PD 1e-14 1e-14 0.0 SN SN 0.2 SNSN SN 0.4 SN SN 0.6 0.0 0.2 0.4 0.6 Excitation Frequency, Ω Excitation Frequency, Ω (o) (p) Fig. 7. (Continued) 1650159-15  A. C. J. Luo & Y. Guo SNSN PD PD SN SN SN SN 2π 0.09 P-6 P-3 S P-3 Harmonic Amplitude, A1/6 PD Constant, mod(a0 ,2π) 6.2 0.06 (m) P-6 A 0.1 0.03 PD 0.0 0.00 0.73 0.75 0.77 0.79 0.81 0.744 SN 0.749 0.754 SN 0.759 Excitation Frequency, Ω Excitation Frequency, Ω (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 3.0 SN S 0.09 SN SN P-6 Harmonic Amplitude, A1/3 Harmonic Amplitude, A1/2 SN SN 0.06 PD 2.0 SN PD A PD PD P-6 P-3 0.03 SN PD 1.0 SN SN SN SN 0.00 0.73 0.75 0.77 0.79 0.81 0.744 0.749 0.754 0.759 Excitation Frequency, Ω Excitation Frequency, Ω (c) (d) 0.42 0.09 PD PD Harmonic Amplitude, A2/3 Harmonic Amplitude, A5/6 P-3 P-6 0.28 0.06 PD A P-6 0.14 PD 0.03 PD PD SN SN P-3 S SN SN SN S N 0.00 0.00 0.73 0.75 0.77 0.79 0.81 0.744 SN 0.749 0.754 SN 0.759 Excitation Frequency, Ω Excitation Frequency, Ω (e) (f) (m) Fig. 8. Frequency-amplitude characteristics for bifurcation trees of period-3 to period-6 motions: (a) a0 (m = 3, 6), (b)–(r) Ak/m (m = 6, k = 1, 2, . . . , 6; 12, 18, . . . , 30; 120, 121, . . . , 126); parameters: (α = 1.5, δ = 0.75, Q0 = 5.0). 1650159-16  Periodic Motions to Chaos in Pendulum 6.09 0.04 SN SN PD 0.03 Harmonic Amplitude, A2 Harmonic Amplitude, A1 SN PD 5.86 P-6 PD PD P-3 SN A P-3 0.02 A PD SN P-6 5.63 SN 0.01 PD S PD SN P-3 S SN SN SN SN SN SN 5.40 0.00 0.73 0.75 0.77 0.79 0.81 0.73 0.75 0.77 0.79 0.81 Excitation Frequency, Ω Excitation Frequency, Ω (g) (h) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 0.10 0.016 PD PD SN 0.08 0.012 Harmonic Amplitude, A3 Harmonic Amplitude, A4 P-3 P-3 S A SN PD 0.06 PD 0.008 P-6 A SN SN P-6 PD PD PD 0.04 PD SN 0.004 SN SN S SN SN SN SN SN P-3 SN SN 0.02 0.000 0.73 0.75 0.77 0.79 0.81 0.7 0.8 0.8 0.8 0.8 Excitation Frequency, Ω Excitation Frequency, Ω (i) (j) 0.06 3.2e-7 SN SN P-3 PD PD Harmonic Amplitude, A20 2.4e-7 Harmonic Amplitude, A5 P-6 PD 0.04 SN PD PD PD 1.6e-7 P-6 SN A PD A P-3 SN S PD SN 8.0e-8 0.02 SN SN P-3 S S N SN SN SN SN S N 0.0 0.73 0.75 0.77 0.79 0.81 0.73 0.75 0.77 0.79 0.81 Excitation Frequency, Ω Excitation Frequency, Ω (k) (l) Fig. 8. (Continued) 1650159-17  A. C. J. Luo & Y. Guo 6.0e-8 6e-7 SN SN PD Harmonic Amplitude, A121/6 Harmonic Amplitude, A61/3 SN 4.0e-8 4e-7 PD SN A PD P-6 PD PD SN P-3 SN P-3 2.0e-8 2e-7 PD P-6 SN S SN 0.0 0 0.744 SN 0.749 0.754 SN 0.759 0.73 0.75 0.77 0.79 0.81 Excitation Frequency, Ω Excitation Frequency, Ω (m) (n) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 6.0e-8 3e-7 PD Harmonic Amplitude, A41/2 Harmonic Amplitude, A62/3 PD 4.0e-8 2e-7 PD P-6 P-3 A 2.0e-8 1e-7 P-6 PD PD SN SN SN SN P-3 S SN SN 0.0 0 0.744 SN 0.749 0.754 SN 0.759 0.73 0.75 0.77 0.79 0.81 Excitation Frequency, Ω Excitation Frequency, Ω (o) (p) 2.4e-8 3.2e-7 SN S Harmonic Amplitude, A125/6 P-6 Harmonic Amplitude, A21 2.4e-7 SN 1.6e-8 SN SN PD P-6 P-3 1.6e-7 A SN 8.0e-9 PD PD SN SN 8.0e-8 PD PD 0.0 0.0 0.744 SN 0.749 0.754 SN 0.759 0.73 0.75 0.77 0.79 0.81 Excitation Frequency, Ω Excitation Frequency, Ω (q) (r) Fig. 8. (Continued) 1650159-18  Periodic Motions to Chaos in Pendulum Table 1. Excitation frequencies for unstable symmetric period-1 motions associated with jumping (α = 1.5, δ = 0.75, Q0 = 5.0). Branch 1 Branch 2 Ω SN (L) SN (R) Ω SN (L) SN (R) (0.051916, 0.053579) 0.051916 0.053579 (0.051811, 0.052309) 0.051811 0.052309 (0.053167, 0.053698) 0.053167 0.053698 (0.053288, 0.055007) 0.053288 0.055007 (0.054737, 0.056513) 0.054737 0.056513 (0.054597, 0.055161) 0.054597 0.055161 (0.056105, 0.056707) 0.056105 0.056707 (0.056269, 0.058103) 0.056269 0.058103 (0.057892, 0.059786) 0.057892 0.059786 (0.057698, 0.058341) 0.057698 0.058341 (0.060072, 0.059385) 0.060072 0.059385 (0.059614, 0.061569) 0.059614 0.061569 (0.061448, 0.063461) 0.061448 0.063461 (0.061173, 0.061907) 0.061173 0.061907 (0.063072, 0.063857) 0.063072 0.063857 (0.063404, 0.065474) 0.063404 0.065474 (0.065498, 0.067619) 0.065498 0.067619 (0.065092, 0.065933) 0.065092 0.065933 (0.070175, 0.072361) 0.070175 0.072361 (0.067747, 0.069909) 0.067747 0.069909 (0.072009, 0.073044) 0.072009 0.073044 (0.069546, 0.070511) 0.069546 0.070511 (0.075688, 0.077820) 0.075688 0.077820 (0.072809, 0.074991) 0.072809 0.074991 by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com (0.077495, 0.078687) 0.077495 0.078687 (0.074652, 0.075762) 0.074652 0.075762 (0.082369, 0.084175) 0.082369 0.084175 (0.078858, 0.080872) 0.078858 0.080872 (0.083881, 0.085257) 0.083881 0.085257 (0.080563, 0.081843) 0.080563 0.081843 (0.090539, 0.091671) 0.090539 0.091671 (0.086259, 0.087761) 0.086259 0.087761 (0.091405, 0.092997) 0.091405 0.092997 (0.087483, 0.088962) 0.087483 0.088962 (0.100268, 0.100661) 0.100268 0.100661 (0.095205, 0.095952) 0.095205 0.095952 (0.111332, 0.113474) 0.111332 0.113474 (0.095693, 0.097406) 0.095693 0.097406 (0.124896, 0.127387) 0.124896 0.127387 (0.105762, 0.105876) 0.105762 0.105876 (0.142151, 0.145042) 0.142151 0.145042 (0.105586, 0.107574) 0.105586 0.107574 (0.164797, 0.168131) 0.164797 0.168131 (0.117730, 0.120040) 0.117730 0.120040 (0.195739, 0.199529) 0.195739 0.199529 (0.132975, 0.135660) 0.132975 0.135660 (0.240340, 0.244530) 0.240340 0.244530 (0.152656, 0.155765) 0.152656 0.155765 (0.309673, 0.314011) 0.309673 0.314011 (0.178976, 0.182540) 0.178976 0.182540 (0.431072, 0.434582) 0.431072 0.434582 (0.215837, 0.219842) 0.215837 0.219842 (0.270817, 0.275135) 0.270817 0.275135 (0.360809, 0.364953) 0.360809 0.364953 Note: L and R denote “left” and “right”. SN is for saddle-node bifurcation. Table 2. Excitation frequencies for unstable symmetric period-1 to asymmetric motions (α = 1.5, δ = 0.75, Q0 = 5.0). Branch 1 Branch 2 Ω SN (L) SN (R) Ω SN (L) SN (R) (0.050522, 0.053579) 0.050522 0.053579 (0.050614, 0.050884) 0.050614 0.050884 (0.051917, 0.052190) 0.051917 0.052190 (0.051897, 0.052308) 0.051897 0.052308 (0.053248, 0.053695) 0.053248 0.053695 (0.051811, 0.055007) 0.051811 0.055007 (0.053167, 0.056512) 0.053167 0.056512 (0.053291, 0.053561) 0.053291 0.053561 (0.054742, 0.055002) 0.054742 0.055002 (0.054670, 0.055158) 0.054670 0.055158 (0.056168, 0.056702) 0.056168 0.056702 (0.054597, 0.058103) 0.054597 0.058103 (0.056105, 0.059785) 0.056105 0.059785 (0.056277, 0.056518) 0.056277 0.056518 (0.057908, 0.058110) 0.057908 0.058110 (0.057749, 0.058333) 0.057749 0.058333 (0.059421, 0.060058) 0.059421 0.060058 (0.057698, 0.061568) 0.057698 0.061568 (0.059385, 0.063461) 0.059385 0.063461 (0.059650, 0.059779) 0.059650 0.059779 (0.063071, 0.063824) 0.063071 0.063824 (0.061192, 0.061886) 0.061192 0.061886 (0.063072, 0.067617) 0.063072 0.067617 (0.061173, 0.065473) 0.061173 0.065473 (0.067200, 0.067246) 0.067200 0.067246 (0.065069, 0.065881) 0.065069 0.065881 (0.067246, 0.072357) 0.067246 0.072357 (0.065092, 0.069907) 0.065092 0.069907 (0.072009, 0.077811) 0.072009 0.077811 (0.069484, 0.070381) 0.069484 0.070381 (0.077707, 0.078007) 0.077707 0.078007 (0.069546, 0.074985) 0.069546 0.074985 (Continued) 1650159-19  A. C. J. Luo & Y. Guo Table 2. (Continued) Branch 1 Branch 2 Ω SN (L) SN (R) Ω SN (L) SN (R) (0.077496, 0.084155) 0.077496 0.084155 (0.074652, 0.080859) 0.074652 0.080859 (0.083882, 0.091624) 0.083882 0.091624 (0.080563, 0.087731) 0.080563 0.087731 (0.091407, 0.100543) 0.091407 0.100543 (0.087484, 0.095877) 0.087484 0.095877 (0.102243, 0.100399) 0.102243 0.100399 (0.095695, 0.105683) 0.095695 0.105683 (0.100402, 0.111371) 0.100402 0.111371 (0.105591, 0.117697) 0.105591 0.117697 (0.111339, 0.124771) 0.111339 0.124771 (0.117741, 0.132730) 0.117741 0.132730 (0.124914, 0.141750) 0.124914 0.141750 (0.133002, 0.152040) 0.133002 0.152040 (0.142192, 0.163880) 0.142192 0.163880 (0.152719, 0.177650) 0.152719 0.177650 (0.164893, 0.193830) 0.164893 0.193830 (0.179125, 0.213120) 0.179125 0.213120 (0.195970, 0.236450) 0.195970 0.236450 (0.216199, 0.265220) 0.216199 0.265220 (0.240910, 0.301500) 0.240910 0.301500 (0.271740, 0.348530) 0.271740 0.348530 (0.311200, 0.411900) 0.311200 0.411900 (0.363400, 0.501500) 0.363400 0.501500 (0.435780, 0.639000) 0.435780 0.639000 (0.543300, 0.881900) 0.543300 0.881900 by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com (0.725370, 1.496000) 0.725370 1.496000 Note: L and R denote “left” and “right”. SN is for saddle-node bifurcation. Table 3. Excitation frequencies for unstable asymmetric period-1 to period-2 motions (α = 1.5, δ = 0.75, Q0 = 5.0). Ω PD (L) PD (R) (0.272110, 0.298880) 0.272110 0.298880 (0.311800, 0.344860) 0.311800 0.344860 (0.364430, 0.406460) 0.364430 0.406460 (0.437580, 0.493120) 0.437580 0.493120 (0.546910, 0.624500) 0.546910 0.624500 (0.733600, 0.851700) 0.733600 0.851700 (1.185900, 1.380900) 1.185900 1.380900 Note: L and R denote “left” and “right”. PD is for period-doubling bifurcation. Table 4. Excitation frequencies for unstable period-2 motions (α = 1.5, δ = 0.75, Q0 = 5.0). With Period-4 Motion With Jumping Phenomena Ω PD (L) PD (R) Ω SN (L) SN (R) (0.735946, 0.815082) 0.735946 0.815082 (0.777632, 0.815116) 0.777632 0.815116 (0.777674, 0.844840) 0.777674 0.844840 (1.295148, 1.298300) 1.295148 1.298300 (1.198330, 1.296350) 1.198330 1.296350 (1.298600, 1.352000) 1.298600 1.352000 Note: L and R denote “left” and “right”. SN and PD are for saddle-node and period-doubling bifurcations. Table 5. Excitation frequencies for unstable period-4 motions (α = 1.5, δ = 0.75, Q0 = 5.0). With Period-8 Motion With Jumping Phenomena Ω PD (L) PD (R) Ω SN (L) SN (R) (1.295424, 1.285843) 1.295424 1.285843 (1.285755, 1.291662) 1.285755 1.291662 (1.201350, 1.291596) 1.201350 1.291596 (1.307720, 1.308359) 1.307720 1.308359 (1.300908, 1.307618) 1.300908 1.307618 (1.308240, 1.344570) 1.308240 1.344570 Note: L and R denote “left” and “right”. SN and PD are for saddle-node and period-doubling bifurcations. 1650159-20  Periodic Motions to Chaos in Pendulum One pair of the cascading period-doubling bifur- these cascading period-doubling bifurcations induce cation trees exists at Ω ≈ 0.7359 and 0.81508. period-8, period-16 motions to chaos. Jumping phe- The other pair of the bifurcation trees exists nomena also exist for Ω ∈ (1.3077, 1.3084). The cor- at Ω ≈ 0.8448 and Ω ≈ 0.7777. The cascad- responding saddle-node bifurcations occur at Ω ≈ ing period-doubling bifurcation scenarios gener- 1.3077 and Ω ≈ 1.3084. The period-8, period-16. . . ate period-4, period-8. . . motions to chaos. The motions are not demonstrated herein due to the period-4, period-8. . . motions are not illustrated very tiny ranges of stable motions. herein due to the very tiny range of stable motions. Finally, for Ω ∈ (1.1500, 1.4960), another 4.2. Period-3 to period-6 motions branch of asymmetric period-1 motions exist in pair as shown in Figs. 4(g) and 4(h). The corre- In this section, analytical prediction of bifurca- sponding saddle-node bifurcations for the asymmet- tion trees will be presented for period-3, period-6 ric period-1 motions are at Ω ≈ 1.15 and 1.496. motions to chaos. The bifurcation trees of symmet- The asymmetric period-1 motions possess period- ric period-3 motions to chaos will be presented in doubling bifurcations at Ω ≈ 1.1859 and 1.3809. Fig. 5 for Ω ∈ (0.741784, 0.802574) with two saddle- The period-doubling bifurcations yield unstable node bifurcation points at Ω ≈ 0.741784, and by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com asymmetric period-1 motions and stable period-2 0.802574 for catastrophes and motion appearance. motions. The period-2 motions experience two pairs In addition, for Ω ∈ (0.762588, 0.763558), there of period-doubling bifurcations and a pair of jump- are three solutions of symmetric period-3 motions ing phenomena. The two jumping phenomena are with jumping phenomena at the two saddle-node caused by the saddle-node bifurcations at Ω ≈ bifurcation points of Ω ≈ 0.762588, 0.763558. For 1.2951 and 1.2983, as shown in Figs. 4(i) and 4(j). Ω ∈ (0.763920, 0.801850), the symmetric period-3 The corresponding unstable period-2 motions con- motions become unstable, and at the two saddle- nect the stable motions from Ω ≈ 1.2951 to node bifurcations of Ω ≈ 0.763920 and 0.801850, the 1.2983. The first pairs of period-doubling bifurca- asymmetric period-3 motions will appear. Because tions occur for Ω ≈ 1.1983 and 1.2963, which induce the frequency range for the stable asymmetric unstable period-2 motions and a branch of stable period-3 motion is very short for such a frequency period-4 motions. This period-4 motion has two range, such asymmetric period-3 motion will not pairs of further cascading period-doubling bifur- be presented. For Ω ∈ (0.744290, 0.759830), the cations at Ω ≈ 1.2014, 1.2916 and Ω ≈ 1.2858, symmetric period-3 motions also become unsta- 1.2954. The two pairs of cascading period-doubling ble, and at the two saddle-node bifurcations of bifurcations lead to period-8, period-16 motions to Ω ≈ 0.744290 and 0.759830, the asymmetric chaos. In addition, jumping phenomena occur in period-3 motions will appear and exist in such the range of Ω ∈ (1.2857, 1.2917), with the corre- a frequency range. However, the frequency range sponding saddle-node bifurcations at Ω ≈ 1.2857 for the stable asymmetric period-3 motion is rela- and Ω ≈ 1.2917. Similarly the period-2 motions tively large for such a frequency range, such asym- encounters a second pair of period-doubling bifurca- metric period-3 motion is presented for the bifur- tion at Ω ≈ 1.2986 and 1.3520, which induce unsta- cation tree. The asymmetric period-3 motion has ble period-2 motions and stable period-4 motions. two period-doubling bifurcations at Ω ≈ 0.745990, The period-4 motions encounter two pairs of fur- 0.756100 and the period-6 motion will appear. The ther cascading period-doubling bifurcations at Ω ≈ asymmetric period-3 motion becomes unstable and 1.3009, 1.3076 and Ω ≈ 1.3082, 1.3446. Again period-6 motion exist for Ω ∈ (0.745990, 0.756100). Table 6. Excitation frequencies for unstable symmetric period-3 motions (α = 1.5, δ = 0.75, Q0 = 5.0). With Asymmetric Period-3 Motion With Jumping Phenomena Ω SN (L) SN (R) Ω SN (L) SN (R) (0.763920, 0.801850) 0.763920 0.801850 (0.741784, 0.802574) 0.741784 0.802574 (0.744290, 0.759830) 0.744290 0.759830 (0.762588, 0.763558) 0.762588 0.763558 Note: L and R denote “left” and “right”. SN is for saddle-node bifurcation. 1650159-21  A. C. J. Luo & Y. Guo Table 7. Excitation frequencies for unstable asymmetric period-3 and unstable period-6 motions (α = 1.5, δ = 0.75, Q0 = 5.0). Unstable Period-3 Motion Unstable Period-6 Motions Ω PD (L) PD (R) Ω PD (L) PD (R) (0.745990, 0.756100) 0.745990 0.756100 (0.746450, 0.755716) 0.746450 0.755716 Note: L and R denote “left” and “right”. PD is for period-doubling bifurcation. Table 8. Excitation frequencies for unstable symmetric period-5 motions (α = 1.5, δ = 0.75, Q0 = 5.0). With Asymmetric Motion With Jumping Phenomena Ω SN (L) SN (R) Ω SN (L) SN (R) (1.230530, 1.264160) 1.230530 1.264160 (1.229014, 1.270134) 1.229014 1.270134 Note: L and R denote “left” and “right”. SN is for saddle-node bifurcation. by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com For Ω ∈ (0.746450, 0.755716), the period-6 motions The asymmetric period-5 motion becomes unsta- become unstable with two period-doubling bifur- ble and period-10 motion exists for Ω ∈ (1.231350, cations at Ω ≈ 0.746450, 0.755716. With the 1.262742). Continuously, the bifurcation tree of two period-doubling bifurcations, period-12 motion symmetric period-5 motion to chaos can be appears and exists for such a frequency range. obtained. The frequency ranges for unstable Continuously, the bifurcation tree of symmetric period-5 and period-10 motions and bifurcation period-3 motion to chaos can be obtained. The fre- points are listed in Tables 8 and 9. quency ranges for unstable period-3 and period-6 motions and bifurcation points are listed in Tables 6 5. Nonlinear Frequency-Amplitude and 7. Characteristics (m) From the node points of period-m motions xk = (m) (m) 4.3. Period-5 motions (xk , y k )T (k = 0, 1, 2, . . . , mN ) in the pendu- In this section, analytical prediction of bifurca- lum, the period-m motions can be approximately tion trees will be presented for period-5 motions to expressed by the Fourier series, i.e. chaos. The bifurcation trees of symmetric period-5 M   (m) (m)  k motions to chaos will be presented in Fig. 5 x (t) ≈ a0 + bj/m cos Ωt m for Ω ∈ (1.229014, 1.270134) with two saddle- j=1 node bifurcation points at Ω ≈ 1.229014 and  k  1.270134 for catastrophes and motion appearance. + cj/m sin Ωt . (42) m For Ω ∈ (1.230530, 1.264160), the symmetric (m) period-5 motions become unstable, and at the two The (2M + 1) unknown vector coefficients of a0 , saddle-node bifurcations of Ω ≈ 1.230530 and bj/m , cj/m should be determined from discrete 1.264160, the asymmetric period-5 motions will (m) nodes xk (k = 0, 1, 2, . . . , mN ) with mN + 1 ≥ appear. The asymmetric period-5 motion has two (m) period-doubling bifurcations at Ω ≈ 1.231350, 2M + 1. For M = mN/2, the node points xk 1.262742 and the period-10 motion will appear. on the period-m motion can be expressed for tk ∈ [0, mT ] (m) Table 9. Excitation frequencies for unstable asymmetric x(m) (tk ) ≡ xk period-5 to period-10 motions (α = 1.5, δ = 0.75, Q0 = 5.0). mN/2   Ω PD (L) PD (R) (m)  j = a0 + bj/m cos Ωtk m j=1 (1.231350, 1.262742) 1.231350 1.262742   Note: L and R denote “left” and “right”. PD is for period- j + cj/m sin Ωtk doubling bifurcation. m 1650159-22  Periodic Motions to Chaos in Pendulum mN/2   For the pendulum, we have (m)  j 2kπ = a0 + bj/m cos  (m)   (m)   (m)  m N x (t) x (t) a  1 01 j=1 ≡ ≈ (m) y (t)  (m)   (m)  j 2kπ  x2 (t) a02  + cj/m sin    m N   k   mN/2  A j/m1 cos Ωt − ϕj/m1    (k = 0, 1, . . . , mN − 1),   m  +  . (43)   j=1  k  Aj/m2 cos  Ωt − ϕj/m2     m  where 2π 2kπ (48) T = = N ∆t; Ωtk = Ωk∆t = Ω N To reduce illustrations, only frequency-amplitude mN −1 curves of displacement x(m) (t) for period-m motions (m) 1  (m) are presented. However, the frequency-amplitudes a0 = xk , N for velocity y (m) (t) can also be done in a similar by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com k=0 fashion. Thus the displacement for period-m motion mN −1   2  (m) 2jπ  is given by bj/m = xk cos k ,  (44) mN mN   mN/2   k=1 (m) (m)  k x (t) ≈ a0 + bj/m cos Ωt mN −1 2  (m)  2jπ    m  j=1 cj/m = xk sin k   mN mN  k   k=1 + cj/m sin Ωt (49)  mN  m j = 1, 2, . . . , or 2 mN/2   (m) (m)  k and x (t) ≈ a0 + Aj/m cos Ωt − ϕj/m m j=1 (m) (m) (m) a0 = (a01 , a02 )T , (50) bj/m = (bj/m1 , bj/m2 )T , (45) where  cj/m cj/m = (cj/m1 , cj/m2 ) . T Aj/m = b2j/m + c2j/m , ϕj/m = arctan . (51) bj/m The harmonic amplitudes and harmonic phases for The bifurcation trees of period-m motion to chaos the period-m motions are expressed by can be illustrated by the frequency-amplitude curves. In all plots of frequency-amplitude curves,  cj/m1 the acronyms SN and PD are the saddle-node Aj/m1 = b2j/m1 + c2j/m1 , ϕj/m1 = arctan , bj/m1 and period-doubling bifurcations, respectively. The  cj/m2 unstable and stable solutions of period-m motions Aj/m2 = b2j/m2 + c2j/m2 , ϕj/m2 = arctan . are represented by the dashed and solid curves, bj/m2 respectively. (46) Thus the approximate expression of period-m 5.1. Period-1 motions to chaos motions in Eq. (42) becomes The bifurcation trees of period-1 motions to chaos will be presented through the period-1 to period-4 mN/2   (m) (m)  k motions, as presented in Fig. 7. The given parame- x (t) ≈ a0 + bj/m cos Ωt ters are α = 1.5, δ = 0.75, Q0 = 5.0. The constant m j=1 (m)   term a0 (m = 1, 2, 4) is presented in Fig. 7(a) for k (m) + cj/m sin Ωt . (47) the solution center at mod(a0 , lπ) = 0 (l = 0, 1, m 2, . . .). The bifurcation tree is clearly observed. For 1650159-23  A. C. J. Luo & Y. Guo the asymmetric period-m motion center on the left observed, and the quantity of the harmonic ampli- (m) (m)L tude for period-4 motion is A1/4 ∼ 0.5. In Fig. 7(c), side of mod(a0 , lπ) = 0, mod(lπ − a0 , 2π) = mod(a0 (m)R − lπ, 2π) (l = 0, 1, 2, . . .). The sym- harmonic amplitudes A1/2 for period-4 and period-2 (m) motions are presented. For the first branch, only the metric period-m motion with mod(a0 , lπ) = 0 period-2 motions are presented because the ranges (l = 0, 1, 2, . . .) is labeled by “S”. However, the of stable period-4 motions are very small and more (m) asymmetric period-m motion with mod(a0 , lπ) = discrete nodes are needed to obtain such period-4 0 is labeled by “A”. For the symmetric period-1 motions. The quantity level of the harmonic ampli- motion to an asymmetric period-1 motion, the tude A1/2 is A1/2 ∼ 2.0. In Fig. 7(d), harmonic saddle-node bifurcation will occur, i.e. at Ω ≈ amplitude A3/4 is presented, which is similar to 0.3112, 0.3485, 0.3634, 0.4118, 0.4358, 0.5015, etc. the harmonic amplitude A1/4 . The quantity level For such saddle-node bifurcations, the asymmet- of such harmonic amplitude is A3/4 ∼ 0.2. The ric periodic motions appear, and the symmetric other harmonic amplitudes Ak/4 (k = 4l + 1, 4l + 3, motions change from stable to unstable solutions l = 1, 2, . . .) will not be presented herein for reduc- or from unstable to stable solutions. As Ω → 0, tion of abundant illustrations. In Fig. 7(e), the pri- jumping phenomenon occurs for symmetric motion, mary harmonic amplitudes A1 versus excitation fre- by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com the saddle-node bifurcations corresponding to the quency Ω are presented for the period-1 to period-4 symmetric motion jumping points take place, i.e. motion. The bifurcation trees are clearly observed. at Ω ≈ 0.1643, 0.1679, 0.1789, 0.1827, 0.1956, The entire skeleton of frequency-amplitude for the 0.1997, etc. The symmetric period-1 motion is only symmetric period-1 motion is presented, and the from the stable to unstable solution or from the asymmetric period-1 motions and the correspond- unstable to stable solution. When the asymmet- ing period-2 and period-4 motions are attached ric period-1 motion experiences a period-doubling to the symmetric period-1 motion. The maximum bifurcation, the period-2 motions will appear and quantity level of the primary amplitude is A1 ∼ the asymmetric period-1 motion are from the sta- 130.0, which varies with the excitation frequency. ble to unstable solution. The frequencies of Ω ≈ The bifurcation points are presented as before. In 0.7336, 0.8517, 1.1859, 1.3809 etc. are not only for Fig. 7(f), harmonic amplitudes A3/2 for period-2 the period-doubling bifurcations of the asymmet- and period-4 motions are presented. The bifurca- ric period-1 motions but also for the saddle-node tion trees are similar to the harmonic amplitudes bifurcations of the period-2 motion. The period-2 A1/2 . The quantity level of A3/2 is A3/2 ∼ 0.4. To motions also encounter jumping phenomenon, these reduce abundant illustrations, Ak/2 (k = 2l + 1, saddle-node bifurcations for period-2 motion jump- l = 2, 3, . . .) will not be presented any more. In ing points exist for Ω ≈ 0.7776, 0.81512, 1.2951, Figs. 7(g), 7(i), 7(k) and 7(m), harmonic ampli- 1.2983, etc. When the period-2 motion possesses tudes Ak (k = 2l, l = 1, 2, . . . , 4) are presented a period-doubling bifurcation, the period-4 motion (m) which are similar to constant term a0 . The bifur- appears and the period-2 motion is from the stable cation trees have similar structures for the different to unstable solution. The frequencies of Ω ≈ 0.7359, harmonic amplitudes but the corresponding quan- 0.81508, 0.8448, 0.7777, 1.1983, 1.2963, 1.2986, tity levels of harmonic amplitudes are different. The 1.3520, etc. are for the period-doubling bifurcations overall maximum quantity levels are A2 ∼ 0.3, of period-2 motions and for the saddle-node bifurca- A4 ∼ 0.3, A6 ∼ 0.25, A8 ∼ 0.2, respectively. The tion for the period-4 motions. The period-4 motions quantity levels drop to 10−3 for Ω ∈ (1.0, 2.0) as jumping points are at Ω ≈ 1.2857, 1.2917, 1.3077, the order increases, i.e. A2 ∼ 0.2, A4 ∼ 10−2 , 1.3084. The frequencies of Ω ≈ 1.2014, 1.2916, A6 ∼ 10−4 , A8 ∼ 10−3 . In Figs. 7(h), 7(j), 7(l) 1.2858, 1.2954, 1.3009, 1.3076, 1.3082, 1.3446 are and 7(n), the harmonic amplitudes Ak (k = 2l + 1, for the period-doubling bifurcations of period-4 l = 1, 2, . . . , 4) are presented, which are similar to motions and for the saddle-node bifurcations of the primary harmonic amplitude A1 . The bifurca- the period-8 motions. All period-2 and period-4 tion trees are different for the different harmonic motions are on the branches of asymmetric period-1 amplitudes and the corresponding quantity lev- motions. In Fig. 7(b), harmonic amplitude A1/4 els of harmonic amplitudes are different. That is, is presented. For period-1 and period-2 motions, A3 ∼ 3.0, A5 ∼ 2.0, A7 ∼ 1.4, A9 ∼ 0.8 for A1/4 = 0. The bifurcation points are clearly overall maximum. However, for Ω ∈ (1.0, 2.0), the 1650159-24  Periodic Motions to Chaos in Pendulum quantity level of harmonic amplitudes drop rapidly is from the stable to unstable solution. The fre- to 10−5 , i.e. A3 ∼ 0.2, A5 ∼ 0.05, A7 ∼ 10−4 , quencies of Ω ≈ 0.746450, 0.755716 are for the A9 ∼ 10−5 . To avoid abundant illustrations, har- period-doubling bifurcations of period-6 motions monic amplitudes of A20 and A61 are presented. and for the saddle-node bifurcation of the period-12 The overall maximum quantity levels for A20 and motions. In Fig. 8(b), harmonic amplitude A1/6 A61 are A20 < 0.1 and A61 < 0.024. However for is presented. For period-3 motions, A1/6 = 0. Ω ∈ (1.0, 2.0), A20 ∼ 10−12 and A61 ∼ 10−12 . From The bifurcation points are clearly observed, and the above discussion, the periodic motion, for Ω > the quantity level of the harmonic amplitude for 1.0, one can use about 80 harmonic terms to approx- period-6 motion is A1/6 ∼ 0.07. In Fig. 8(c), har- imate period-1, period-2 and period-4 motions. For monic amplitudes A1/3 for period-6 and period-3 Ω < 1.0 but not close to zero, one can use 250 har- motions are presented. For the branch on the right monic terms to approximate period-1, period-2, and side (Ω ∈ (0.763920, 0.801850)), only the sym- period-4 motions. For Ω ≈ 0.0, infinite harmonic metric period-3 motion is presented because the terms should be adopted to approximate the peri- ranges of stable asymmetric period-3 and period-6 odic motions. For harmonic phases, we have ϕRk/2l = motions are very small. The quantity of the har- monic amplitudes A1/3 is in the range of A1/3 ∈ by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. mod(ϕB + (k(1 + 2r)/2l + 1)π, 2π) ∈ [0, 2π) with Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com k/2l (0.9, 3.0). In Fig. 8(d), harmonic amplitude A1/2 is t0 = rT (r ∈ {0, 1, . . . , 2l − 1}) and m = 2l . presented for period-6 motion only, which is simi- lar to the harmonic amplitude A1/6 . The quantity 5.2. Period-3 to period-6 motions level of such harmonic amplitude is A1/2 ∼ 0.09. The bifurcation trees of period-3 motion to chaos In Fig. 8(e), harmonic amplitudes A2/3 for period-6 will be presented through period-3 to period-6 and period-3 motions are presented, which is sim- (m) motions, as presented in Fig. 8. The given param- ilar to a0 (m = 3, 6). For symmetric period-3 eters are also α = 1.5, δ = 0.75, Q0 = 5.0. motions, A2/3 = 0. For asymmetric period-3 and (m) The constant term a0 (m = 3, 6) is presented in period-6 motions, the harmonic amplitudes of A2/3 (m) possess the quantity level of A2/3 ∼ 0.4. In Fig. 8(f), Fig. 8(a) for the solution center at mod(a0 , lπ) = harmonic amplitude A5/6 is presented for period-6 0 (l = 0, 1, 2, . . .). The bifurcation tree is clearly motion only, which is similar to the harmonic ampli- observed. For the asymmetric period-m motion cen- (m) tude A1/6 . The quantity level of such a harmonic ter on the left side of mod(a0 , lπ) = 0, mod(lπ − amplitude is A5/6 ∼ 0.08. The other harmonic (m)L (m)R a0 , 2π) = mod(a0 − lπ, 2π). The symmetric amplitude Ak/6 (mod(k, 6) = 0, k = 6, 7, . . .) will (m) period-m motion with mod(a0 , lπ) = 0 is labeled not be presented herein for reduction of abundant by “S”. However, the asymmetric period-m motion illustrations. In Fig. 8(g), the primary harmonic (m) amplitudes A1 versus excitation frequency Ω are with mod(a0 , lπ) = 0 is labeled by “A”. For the symmetric period-3 motions to asymmetric period-3 presented for the period-3 to period-6 motion. The motions, the saddle-node bifurcations will occur bifurcation trees are clearly observed. The entire at Ω ≈ 0.763920, 0.801850, 0.744290, 0.759830. skeleton of frequency-amplitude for the symmet- For such saddle-node bifurcations, the asymmet- ric period-3 motion is presented, and the asym- ric period-3 motions appear, and the symmetric metric period-3 and period-6 motions are attached period-3 motions change from stable to unstable to the symmetric period-3 motions. The quan- solutions or from unstable to stable solutions. When tity of the primary amplitudes is in the range of the asymmetric period-3 motions experience period- A1 ∈ (5.40, 6.09), which varies with excitation fre- doubling bifurcations, the period-6 motions will quency. The bifurcation points are presented in appear and the asymmetric period-3 motions are Tables 6 and 7. In Fig. 8(h), harmonic ampli- from stable to unstable solutions. The frequen- tudes A2 for period-3 and period-6 motions are cies of Ω ≈ 0.745990, 0.756100 are not only for presented. The bifurcation trees are similar to the the period-doubling bifurcations of the asymmetric harmonic amplitude A2/3 . The quantity levels of period-3 motions but also for the saddle-node bifur- A2 are A2 ∼ 0.04. In Figs. 8(i) and 8(k), har- cations of the period-6 motion. When the period-6 monic amplitudes A3 and A5 are presented which motion possesses a period-doubling bifurcation, a are similar to the primary harmonic amplitude A1 . period-12 motion appears and the period-6 motion The bifurcation trees for such harmonic amplitudes 1650159-25  A. C. J. Luo & Y. Guo (m) (m)L have similar structures for the different harmonic side of mod(a0 , lπ) = 0, mod(lπ − a0 , 2π) = amplitudes but the corresponding quantity levels (m)R mod(a0 − lπ, 2π). The symmetric period-m of harmonic amplitudes are different. The overall (m) motion with mod(a0 , lπ) = 0 (l = 0, 1, 2, . . .) maximum quantity for such two harmonic ampli- is labeled by “S”. However, asymmetric period-m tudes are A3 ∈ (0.02, 0.10), and A5 ∈ (0.01, 0.06). (m) In Fig. 8(j), harmonic amplitudes A4 are presented motion with mod(a0 , lπ) = 0 is labeled by “A”. similar to the primary harmonic amplitude A2 . For the symmetric period-5 motion to asymmet- The quality level of such a harmonic amplitude is ric period-5 motions, the saddle-node bifurcations A4 ∼ 0.016. To look into the higher order harmonic occur at Ω ≈ 1.230530 and 1.264160. For such effects, harmonic amplitudes A20 are presented in saddle-node bifurcations, the asymmetric period-5 Fig. 8(l), which are similar to the harmonic ampli- motions appear, and the symmetric period-5 tudes of A2 but the corresponding quality level is motions change from stable to unstable solutions A20 ∼ 3 × 10−7 . Further the harmonic amplitudes or from unstable to stable solutions. When the of Ak/6 (k = 121, 122, . . . , 126) similar to Ak/6 (k = asymmetric period-5 motion experiences a period- 1, 2, . . . , 6) are presented in Figs. 8(m)–8(r), and doubling bifurcation, the period-10 motions will appear and the asymmetric period-5 motion are by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com the quantity levels of Ak/6 (k = 121, 122, . . . , 126) from the stable to unstable solution. The frequen- are about 10−7 , which is much lower than the cies of Ω ≈ 1.231350, 1.262742 are not only for quantity levels of Ak/6 (k = 1, 2, . . . , 6). ϕR k/2l m = the period-doubling bifurcations of the asymmetric mod(ϕB k/2l m + (k(m + 2r)/2l m + 1)π, 2π) ∈ [0, 2π) period-5 motions but also for the saddle-node bifur- with t0 = rT (r ∈ {0, 1, . . . , 2l m − 1}) and m = 3. cations of the period-10 motion. For the branch of the right side (Ω ∈ (1.231350, 1.262742)), only the period-5 motion are presented because the range of 5.3. Period-5 motions stable asymmetric period-10 motion is very tiny. In The bifurcation trees of period-5 motion to chaos Fig. 9(b), harmonic amplitude A1/5 is presented. will be presented through the symmetric and asym- The bifurcation points are clearly observed, and metric period-5 motions, as presented in Fig. 9. the quantity level of the harmonic amplitude for The given parameters still are α = 1.5, δ = 0.75, period-5 motion is A1/5 ∈ (1.0, 1.7). In Fig. 9(c), the (m) Q0 = 5.0. The constant term a0 (m = 5) harmonic amplitude A2/5 for period-5 motions are is presented in Fig. 9(a) for the solution center (5) presented, which is similar to the constant a0 . The (m) at mod(a0 , lπ) = 0 (l = 0, 1, 2, . . .). For the quantity level of the harmonic amplitude A2/5 is asymmetric period-m motion center on the left A2/5 ∼ 0.15. In Fig. 9(d), harmonic amplitude A3/5 SN SNPD PD SN SN 3.20 1.8 P-5 SN Harmonic Amplitude, A1/5 Constant, mod(a0 ,2π) 3.16 1.5 SN (m) PD S S π A 3.12 1.2 SN PD P-5 SN A 3.08 0.9 1.22 1.24 1.26 1.28 1.22 1.24 1.26 1.28 Excitation Frequency, Ω Excitation Frequency, Ω (a) (b) (m) Fig. 9. Frequency-amplitude characteristics for bifurcation trees of period-5 motions: (a) a0 (m = 5), (b)–(n) Ak/m (m = 5, k = 1, 2, . . . , 5; 10, 15, . . . , 45); parameters: (α = 1.5, δ = 0.75, Q0 = 5.0). 1650159-26  Periodic Motions to Chaos in Pendulum 0.15 0.8 SN Harmonic Amplitude, A3/5 Harmonic Amplitude, A2/5 PD 0.7 A 0.10 SN 0.6 P-5 PD P-5 0.05 A 0.5 PD S SN SN PD SN SN S SN SN 0.00 0.4 1.22 1.24 1.26 1.28 1.22 1.24 1.26 1.28 Excitation Frequency, Ω Excitation Frequency, Ω (c) (d) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 0.06 2.31 S SN Harmonic Amplitude, A4/5 2.28 PD Harmonic Amplitude, A1 A PD 0.04 SN PD SN A 2.25 0.02 P-5 SN PD 2.22 P-5 SN SN S SN SN 0.00 2.19 1.22 1.24 1.26 1.28 1.22 1.24 1.26 1.28 Excitation Frequency, Ω Excitation Frequency, Ω (e) (f) 1.2e-2 3.0e-2 SN SN PD A 2.5e-2 Harmonic Amplitude, A3 Harmonic Amplitude, A2 PD 8.0e-3 P-5 A 2.0e-2 P-5 4.0e-3 S PD 1.5e-2 SN PD SN SN SN S SN SN 0.0 1.0e-2 1.2 1.3 1.22 1.24 1.26 1.28 Excitation Frequency, Ω Excitation Frequency, Ω (g) (h) Fig. 9. (Continued) 1650159-27  A. C. J. Luo & Y. Guo 6.0e-4 1.1e-3 SN PD P-5 PD A SN Harmonic Amplitude, A5 Harmonic Amplitude, A4 4.0e-4 P-5 9.4e-4 S PD SN A 2.0e-4 7.8e-4 SN PD SN SN S SN SN 0.0 6.2e-4 1.22 1.24 1.26 1.28 1.22 1.24 1.26 1.28 Excitation Frequency, Ω Excitation Frequency, Ω (i) (j) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 6.0e-5 9.0e-5 PD SN SN Harmonic Amplitude, A6 7.0e-5 Harmonic Amplitude, A7 PD 4.0e-5 P-5 P-5 5.0e-5 A A S 2.0e-5 3.0e-5 SN PD PD SN SN SN S SN SN 0.0 1.0e-5 1.22 1.24 1.26 1.28 1.22 1.24 1.26 1.28 Excitation Frequency, Ω Excitation Frequency, Ω (k) (l) 2.4e-6 4.5e-6 PD P-5 Harmonic Amplitude, A8 SN Harmonic Amplitude, A9 1.6e-6 3.0e-6 SN A PD SN SN P-5 8.0e-7 1.5e-6 PD A PD S S 0.0 0.0 1.22 SN SN 1.24 1.26 SN SN 1.28 1.22 1.24 1.26 1.28 Excitation Frequency, Ω Excitation Frequency, Ω (m) (n) Fig. 9. (Continued) 1650159-28  Periodic Motions to Chaos in Pendulum is presented for period-5 motion, which is similar to description of periodic motions in pendulum, the the harmonic amplitude A1/5 . The quantity level nontravelable and travelable periodic motions are of such harmonic amplitude is A3/5 ∈ (0.4, 0.8). defined as follows. In Fig. 9(e), harmonic amplitude A4/5 for period-5 Definition 2. For a period-m motion of dynamical motion is presented, which is similar to A2/5 . For system in Eq. (1) for N -nodes per period where N = symmetric period-5 motion, A4/5 = 0. For asym- T /h with time step h, if metric period-5 motions, the harmonic amplitudes of A4/5 possess the quantity level of A4/5 ∼ 0.06. xk = xk+mN and yk = yk+mN , (52) The other harmonic amplitudes Ak/5 (mod(k, 5) = 0, k = 5, 6, . . .) will not be presented herein for then such a period-m motion is called the nontrav- reducing abundant illustrations. In Fig. 9(f), the elable period-m motion in the dynamical system. primary harmonic amplitudes A1 versus excitation Definition 3. For a period-m motion of dynamical frequency Ω are presented for the period-5 motion. system in Eq. (1) for N -nodes per period where N = The bifurcation trees are clearly observed. The T /h with time step h, if skeleton of frequency-amplitude for the symmet- ric period-5 motion is presented. The quantity of mod(xk , 2π) = mod(xk+mN , 2π) with by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com the primary amplitude is A1 ∈ (2.19, 2.31), which xk = xk+mN and yk = yk+mN , varies with excitation frequency. The bifurcation (53) points are presented in Tables 8 and 9. In Fig. 9(g), harmonic amplitude A2 is presented. The bifurca- then such a period-m motion is called the travelable tion trees are similar to the harmonic amplitude period-m motion in the dynamical system. A2/5 . The quantity level of A2 is A2 ∼ 0.012. In Figs. 9(h), 9(j), 9(l) and 9(n), harmonic ampli- Excitation amplitude effects on bifurcation tudes Ak/5 (k = 15, 25, 35, 45) are presented which trees for nontravelable period-1 motions to chaos are similar to primary harmonic amplitude A1 . and nontravelable period-3 motions to chaos will be The bifurcation trees for such harmonic amplitudes discussed first. The bifurcation trees from travelable have similar structures for the different harmonic period-1 motions to chaos and travelable period-2 amplitudes but the corresponding quantity levels motions to chaos in pendulum are also presented. of harmonic amplitudes are different. The overall The parameters of the periodically forced pendu- maximum quantity for two such harmonic ampli- lum are α = 1.5, δ = 0.75, and Ω = 1.0. tudes are A3 ∼ 2 × 10−2 , A5 ∼ 10−3 , A7 ∼ 10−4 , and A7 ∼ 5 × 10−6 , respectively. In Figs. 9(i), 6.1. Nontravelable period-1 motions 9(k) and 9(m), harmonic amplitudes Ak/5 (k = 20, to chaos 30, 40) are presented which are similar to primary As in [Guo & Luo, 2016], the bifurcation trees harmonic amplitude A2 . The quantity levels of such varying with excitation amplitude for nontravelable harmonic amplitudes are A4 ∼ 6 × 10−4 , A6 ∼ period-1 motions to chaos will be presented through 6 × 10−5 , and A8 ∼ 2.5 × 10−6 . Similarly, the other the analytical predictions of period-1 to period-4 harmonic amplitudes can be similarly presented. motions. Displacement and velocity of the peri- odic nodes mod(xmod(k,N ) , 2π) and ymod(k,N ) for mod(k, N ) = 0 are predicted. A global view of 6. Bifurcation Trees Varying with bifurcation trees of period-1 to period-4 motions Excitation Amplitude in the periodically excited pendulum is illustrated If a period-m motion is determined by xk = in Figs. 10(a) and 10(b). For a better illustra- xk+mN with yk = yk+mN , then such a period-m tion of bifurcations trees, three main zoomed win- motion has a center point and the periodic motion dows of Q0 ∈ (2.0, 5.0), (5.0, 9.0) and (9.5, 12.0) can be called the nontravelable period-m motion. are shown in Figs. 10(c) and 10(d), 10(e) and However, if a period-m motion is determined by 10(f) and 10(g) and 10(h), respectively. The global mod(xk , 2π) = mod(xk+mN , 2π) with yk = yk+mN view of bifurcation trees of period-1 to period-4 but xk = xk+mN , such periodic motion does not motions lies in the range of Q0 ∈ (0, ∞). There have a center. Thus such a period-m motion can be are two branches of symmetric period-1 motions for called the travelable period-m motion. For a better Q0 ∈ (0, ∞). The two symmetric period-1 motions 1650159-29  A. C. J. Luo & Y. Guo 2π 11.0 Periodic Node Displacement, mod(xmod(k,N),2π) P-1 SN P-1 P-1 P-1 PD SN SN PD SN Periodic Node Velocity, ymod(k,N) SN PD P-2 P-1 PD 3π/2 SN PD SN P-2 PD P-2 S SN 7.0 P-1 A PD SN PD SN S SN A A A A P-1 π PD A A SN SN S P-2 A SN SN P-2 PD S SN 3.0 SN SN SN PD PD SN PD SN SN π/2 S SN P-1 S A PD S SN S SN S SN P-1 0 -1.0 0.0 5.0 10.0 15.0 20.0 0.0 5.0 10.0 15.0 20.0 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com SN PD PD SN SN PDPD SN SN PD PD SN SN PD PD SN 2π 4.0 Periodic Node Displacement, mod(xmod(k,N),2π) S Periodic Node Velocity, ymod(k,N) 3π/2 P-1 A 3.0 A SN S π 2.0 P-2 P-1 SN π/2 1.0 SN P-2 π 0.0 2.0 3.0 4.0 SN 5.0 2.0 3.0 4.0 5.0 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (c) (d) SN PD-PD PD-PD SN SN PD-PD PD-PD SN 2π 5.5 Periodic Node Displacement, mod(xmod(k,N),2π) SN SN Periodic Node Velocity, ymod(k,N) 3π/2 4.5 P-1 P-1 π 3.5 SN P-2 P-4 SN A P-2 S S π/2 2.5 P-4 A SN SN SN SN π 1.5 5.5 6.5 7.5 8.5 5.5 6.5 7.5 8.5 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (e) (f) Fig. 10. Bifurcation trees of period-1 to period-4 motions varying with excitation amplitude Q0 through node displacement xmod(k,N ) and node velocity ymod(k,N ) : (a), (b) Global view (Q0 ∈ (0, 20]), (c), (d) Zoom-1 view (Q0 ∈ (2.0, 5.0)), (e), (f) Zoom-2 view (Q0 ∈ (5.5, 8.5)): (g), (h) Zoom-3 view (Q0 ∈ (9.5, 12.0)). (α = 2.0, δ = 0.75, Ω = 1.0). mod(k, N ) = 0. 1650159-30  Periodic Motions to Chaos in Pendulum SN PD PD PD PD PD PD SN SN PD PD PD PD PD PD SN 2π 6.5 Periodic Node Displacement, mod(xmod(k,N),2π) S Periodic Node Velocity, ymod(k,N) SN 3π/2 SN P-1 P-1 SN 5.5 P-2 π SN P-2 A P-4 SN 4.5 P-4 π/2 S A SN SN SN π 3.5 9.5 10.0 10.5 11.0 11.5 12.0 9.5 10.0 10.5 11.0 11.5 12.0 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (g) (h) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com Fig. 10. (Continued) alternatively switch between stable and unstable period-doubling bifurcations at Q0 ∈ {2.474226, motions, and the two branches form a double helix 4.298323} and {2.705312, 4.603100} for period-4 shape together, as observed obviously in Fig. 10(b). motions. The period-4, period-8. . . motions are All the unstable symmetric period-1 motions are not illustrated due to the very tiny stable ranges. enclosed by saddle-node bifurcations. Some of the Continuously, such a bifurcation tree of period-1 unstable symmetric periodic motions are relative motions will reach chaos. Two jumping phenomena to coexisting asymmetric motions, while the oth- also exist for the period-2 motions, and the cor- ers are related to jumping phenomena. The ranges responding unstable motions are enclosed by the for unstable period-1 motions and the correspond- two saddle-node bifurcations at Q0 ≈ 2.705311 ing saddle-node bifurcation points for asymmet- and 4.298326. After the saddle-node bifurcation at ric period-1 motion and jumping phenomena are Q0 ≈ 4.6628, the symmetric period-1 motion is sta- tabulated in Table 10. The unstable asymmetric ble for Q0 ∈ (4.6628, 4.8417). The jumping phenom- period-1, period-2 and period-4 motions and the ena occur at Q0 ≈ 4.7742 and 4.8417. The unstable corresponding bifurcation points are tabulated in period-2 motion enclosed by the two saddle-node Tables 11–14. bifurcations exist for Q0 ∈ (4.7742, 4.8417). The symmetric period-1 motion is stable for After the jumping phenomenon, the symmetric the range of Q0 ∈ (0.0, 2.271). The correspond- period-1 motion is stable for Q0 ∈ (4.7742, 6.0000). ing saddle-node bifurcation occurs at Q0 ≈ 2.271 At Q0 ≈ 6.0000, the saddle-node bifurcation and the symmetric period-1 motion becomes unsta- starts a new pair of asymmetric period-1 motions, ble. This unstable motion is connected to a pair which ends at the saddle-node bifurcation from of stable asymmetric period-1 motions. The two a different branch at Q0 ≈ 8.0800, as shown in asymmetric period-1 motions exist for the range Figs. 10(e) and 10(f). The asymmetric period-1 of Q0 ∈ (2.2710, 4.6628). The asymmetric period-1 motions start from the saddle-node bifurcation at Q0 ≈ 2.2710 from one branch of the symmet- Table 10. Excitation amplitudes for unstable symmetric ric period-1 motion and end at the saddle-node period-1 motions with jumping (α = 2.0, δ = 0.75, Ω = 1.0). bifurcation at Q0 ≈ 4.6628 from the other branch of the symmetric period-1 motion, as shown in Branch 1 Branch 2 Figs. 10(c) and 10(d). The asymmetric period-1 Q0 SN (L) SN (R) Q0 SN (L) SN (R) motions possess a pair of period-doubling bifurca- — N/A N/A (4.7742, 4.84174) 4.7742 4.8417 tions at Q0 ≈ 2.4392 and Q0 ≈ 4.6146. The induced period-2 motions exist for Q0 ∈ (2.4392, 4.6146). Note: L and R denote “left” and “right”. SN is for saddle- The period-2 motions possess two pairs of node bifurcation. 1650159-31  A. C. J. Luo & Y. Guo Table 11. Excitation amplitudes for unstable symmetric to asymmetric period-1 motions (α = 2.0, δ = 0.75, Ω = 1.0). Branch 1 (Jumping) Branch 2 (Jumping) Q0 SN (L) SN (R) Q0 SN (L) SN (R) (2.2710, 8.0800) 2.2710 8.0800 (0, 4.6628) — 4.6628 (9.9950, 15.5920) 9.9950 15.5920 (6.0000, 11.7910) 6.0000 11.7910 (17.9680, 20) 17.9680 (13.9870, 19.4330) 13.9870 19.4330 Note: L and R denote “left” and “right”. SN is for saddle-node bifurcations. Table 12. Excitation amplitudes for unstable asymmetric period-2 motion has two jumping phenomena at period-1 to period-2 motions (α = 2.0, δ = 0.75, Ω = 1.0). the saddle-node bifurcations at Q0 ≈ 6.847542 Q0 PD (L) PD (R) and 7.376994. Before and after the jumping phe- nomena, the period-2 motions encounter two pairs (2.439200, 4.614600) 2.439200 4.614600 of period-doubling bifurcations at Q0 ≈ 6.405792, (6.329000, 7.933000) 6.329000 7.933000 7.376442 and Q0 ≈ 6.847842, 7.893200 for period-4 by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com (10.408000, 11.547000) 10.408000 11.547000 motions. The period-4 motions in the range of (14.498000, 15.237000) 14.498000 15.237000 (18.660000, 18.890000) 18.660000 18.890000 Q0 ∈ (6.405792, 7.376442) have jumping phenom- ena occurring at the saddle-node bifurcations of Note: L and R denote “left” and “right”. PD is for period- Q0 ≈ 6.833938 and 7.362212. Two pairs of fur- doubling bifurcation. ther period-doubling bifurcations are located at Q0 ≈ 6.833940, 7.376155 and Q0 ≈ 6.422600, motions for Q0 ∈ (6.0000, 8.0800) possess a pair 7.362210 introducing period-8 motions and fur- of period-doubling bifurcations at Q0 ≈ 6.329 ther period-doubling bifurcations to period-16. . . and 7.933 that enclose both the period-2 motions motions to chaos. On the other hand, the period-4 and unstable asymmetric period-1 motions. The motions for Q0 ∈ (6.3847842, 7.893200) have Table 13. Excitation amplitudes for unstable period-2 motions (α = 2.0, δ = 0.75, Ω = 1.0). Period-4 Motion Jumping Phenomena Q0 PD (L) PD (R) Q0 SN (L) SN (R) (2.474226, 4.298323) 2.474226 4.298323 (2.705311, 4.298326) 2.705311 4.298326 (2.705312, 4.603100) 2.705312 4.603100 (6.847542, 7.376994) 6.847542 7.376994 (6.405792, 7.376442) 6.405792 7.376442 (10.932196, 10.993312) 10.932196 10.993312 (6.847842, 7.893200) 6.847842 7.893200 (10.526300, 10.983100) 10.526300 10.983100 (10.939600, 11.468000) 10.939600 11.468000 Note: L and R denote “left” and “right”. SN and PD are for saddle-node and period-doubling bifurcations. Table 14. Excitation amplitudes for unstable period-4 motions (α = 2.0, δ = 0.75, Ω = 1.0). With Period-8 Motion With Jumping Phenomena Q0 PD (L) PD (R) Q0 SN (L) SN (R) (6.833940, 7.376155) 6.833940 7.376155 (6.833938, 7.362212) 6.833938 7.362212 (6.422600, 7.362210) 6.422600 7.362210 (6.853289, 7.392316) 6.853289 7.392316 (6.847993, 7.392316) 6.847993 7.392316 (10.886778, 10.936429) 10.886778 10.936429 (6.853290, 7.884400) 6.853290 7.884400 (10.972133, 11.038790) 10.972133 11.038790 (10.887250, 10.977490) 10.887250 10.977490 (10.555200, 10.936050) 10.555200 10.936050 (10.943370, 11.038580) 10.943370 11.038580 (10.972310, 11.448800) 10.972310 11.448800 Note: L and R denote “left” and “right”. SN and PD are for saddle-node and period-doubling bifurcations. 1650159-32  Periodic Motions to Chaos in Pendulum jumping phenomena with the SN bifurcations at Q0 ≈ 13.987 from one branch of the symmetric Q0 ≈ 6.853289 and 7.392316. Again, two pairs period-1 motions, and end with another saddle-node of period-doubling bifurcations of the period-4 bifurcation at Q0 ≈ 15.592 from a different branch. motion exist at Q0 ≈ 6.847993, 7.392316 and These asymmetric period-1 motions encounter a Q0 ≈ 6.85329, 7.88440 which lead to period-8 pair of period-doubling bifurcations at Q0 ≈ 14.498 motions and further period-doubling bifurcations and 15.237. The period-2 motions enclosed by the to period-16. . . motions to chaos. The period-8, two period-doubling bifurcations are stable. Simi- period-16. . . motions are not shown due to the very larly, for Q0 ∈ (17.968, 19.433), a pair of asymmetric tiny stable range. period-1 motions exists. The asymmetric period-1 As excitation amplitude increases, another pair motion start with the saddle-node bifurcation at of asymmetric period-1 motions exist for Q0 ∈ Q0 ≈ 17.968 from one branch of the symmetric (9.995, 11.791), as shown in Figs. 10(g) and 10(h). period-1 motions, and end with another saddle-node The asymmetric period-1 motions start from the bifurcation at Q0 ≈ 19.433 from a different branch. saddle-node bifurcation on one branch at Q0 ≈ Again, the asymmetric period-1 motions encounter 9.995 and end at the saddle-node bifurcation on a pair of period-doubling bifurcations at Q0 ≈ 18.66 the other branch at Q0 ≈ 11.791. The asymmet- and 18.89. The period-2 motions enclosed by these by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com ric period-1 motions have a pair of period-doubling period-doubling bifurcations are completely stable. bifurcations at Q0 ≈ 10.408 and 11.547, which yield both the stable period-2 motions and unsta- 6.2. Nontravelable period-3 motions ble asymmetric period-1 motions. The period-2 motions possess two jumping phenomena at Q0 ≈ to chaos 10.932196 and 10.993312 with the saddle-node As in [Guo & Luo, 2016], the bifurcation trees of bifurcations. Before and after the jumping phenom- period-3 motions to chaos will be presented through ena, there are two pairs of period-doubling bifur- period-3 to period-6 motions. A global view of cations at Q0 ≈ {10.5263, 10.9831} and Q0 ∈ analytical bifurcation trees of period-3 to period-6 {10.9396, 11.4680} where period-4 motions are pro- motions in the periodically excited pendulum is duced. The period-4 motions in the range of Q0 ∈ illustrated in Figs. 11(a) and 11(b). There are three (10.5263, 10.9831) have two jumping phenomena bifurcation trees in the range of Q0 ∈ (3.5, 8.0). For enclosed at Q0 ≈ 10.886778 and 10.936429 with a clear view of bifurcations trees, the zoomed win- the saddle-node bifurcations. Two pairs of fur- dows of Q0 ∈ (3.55, 4.09), (6.2, 7.1) and (7.01, 7.90) ther period-doubling bifurcations are located at are shown in Figs. 11(c) and 11(d), 11(e) and Q0 ∈ {10.88725, 10.97749} and Q0 ∈ {10.55520, 11(f) and 11(g) and 11(h), respectively. The ranges 10.93605}, inducing period-8 motions and fur- for unstable symmetric period-3 motions and the ther period-doubling bifurcations to period-16. . . corresponding saddle-node bifurcations for asym- motions to chaos. On the other hand, the period-4 metric period-3 motion and jumping phenomena motions for Q0 ∈ (10.9396, 11.4680) have jump- are tabulated in Table 15. The ranges for unsta- ing phenomena with the saddle-node bifurcations ble asymmetric period-3 and period-6 motions and at Q0 ≈ 10.972133 and 11.038790. Again, two the corresponding period-doubling and saddle-node pairs of period-doubling bifurcations exist for the bifurcations are listed in Table 16. In Figs. 11(c) period-4 motions at Q0 ≈ 10.94337, 11.03858 and 11(d), the symmetric period-3 motion is in and Q0 ≈ 10.97231, 11.44880, leading to period-8 the range of Q0 ∈ (3.5922, 4.0839) with two motions and further period-doubling bifurcations saddle-node bifurcations at Q0 = 3.5922, 4.0839 to period-16. . . motions to chaos. Once again, the for symmetric period-3 motion appearance. The period-8, period-16. . . motions are not shown due symmetric period-3 motion has two saddle-node to the very tiny stable range. bifurcations at Q0 = 3.7970, 4.0620 for the onset The bifurcation trees become simple with fur- of asymmetric period-3 motion. Thus, the asym- ther increase of excitation amplitude Q0 . For Q0 ∈ metric periodic motion is in the range of Q0 ∈ (13.987, 15.592), a pair of asymmetric period-1 (3.7970, 4.0620) with two period-doubling bifurca- motions exists. Similar to previous asymmetric tions at Q0 ≈ 3.8616, 4.0437 for period-6 motion period-1 motions, the asymmetric period-1 appearance. The period-6 motion exists in the range motions start with the saddle-node bifurcation at of Q0 ∈ (3.8616, 4.0437) with the two saddle-node 1650159-33  A. C. J. Luo & Y. Guo SN PDPD SN SNSN SN PD PD SN SN SN PDPD SN SN SN SN PD PDSN SN 2π 6.0 Periodic Node Displacement, mod(xmod(k,N),2π) P-3 P-6 P-3 P-3 S Periodic Node Velocity, ymod(k,N) 3π/2 P-3 P-6 P-3 P-3 SN 4.0 SN A SN A π S S S 2.0 S π/2 S A A π 0.0 3.5 4.0 7.0 8.0 3.5 4.0 7.0 8.0 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com SN SN PD PD PD PDSN SN SN SN PD PD PD SNSN 4.0 Periodic Node Displacement, mod(xmod(k,N),2π) 6.0 Periodic Node Velocity, ymod(k,N) P-3 P-3 P-6 5.4 P-3 P-3 P-6 2.0 A S 0.8 A S 0.0 0.0 3.55 3.73 3.91 4.09 3.55 3.73 3.91 4.09 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (c) (d) SN SN SN SN SN SN SN SN 2π 6.0 Periodic Node Displacement, mod(xmod(k,N),2π) P-3 Periodic Node Velocity, ymod(k,N) P-3 3π/2 4.0 S S π 2.0 π/2 π 0.0 6.2 6.5 6.8 7.1 6.2 6.5 6.8 7.1 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (e) (f) Fig. 11. Bifurcation trees of period-3 motions to chaos through displacement mod(xmod(k,N ) , 2π) and velocity ymod(k,N ) : (a), (b) Global view for Q0 ∈ (3.5, 8.0), (c), (d) Zoom-1 view for Q0 ∈ (3.55, 4.09), (e), (f) Zoom-2 view for Q0 ∈ (6.2, 7.1), (g), (h) Zoom-3 view for Q0 ∈ (7.01, 7.90). (α = 2.0, δ = 0.75, Ω = 1.0). mod(k, N ) = 0. 1650159-34  Periodic Motions to Chaos in Pendulum SN SN PD PDSN-SN SN SN SN PD PDSN-SN SN 2π 6.0 Periodic Node Displacement, mod(xmod(k,N),2π) P-3 P-3 Periodic Node Velocity, ymod(k,N) 3π/2 P-3 P-3 S A 4.0 S A S S π 2.0 π/2 π 0.0 7.01 7.03 7.70 7.80 7.90 7.01 7.03 7.70 7.80 7.90 Excitation Amplitude, Q0 Excitation Amplitude,Q0 (g) (h) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com Fig. 11. (Continued) Table 15. Excitation amplitudes for unstable symmetric period-3 motions (α = 2.0, δ = 0.75, Ω = 1.0). With Asymmetric P-3 Motion With Jumping Phenomena Q0 SN (L) SN (R) Q0 SN (L) SN (R) (3.7970, 4.0620) 3.7970 4.0620 (3.5922, 4.0839) 3.5922 4.0839 (6.3590, 7.0071) 6.3590 7.0071 (6.3110, 7.0140) 6.3110 7.0140 (7.0270, 7.6562) 7.0270 7.6562 (7.6693, 7.6779) 7.6693 7.6779 (7.0193, 7.8626) 7.0193 7.8626 Note: L and R denote “left” and “right”. SN is for saddle-node bifurcation. bifurcations at Q0 ≈ 3.8745, 4.0382 for the onset bifurcations at Q0 ≈ 6.3590, 7.0071 for asymmetric of period-12 motion. If the period-doubling bifur- period-3 motions. In Figs. 11(g) and 11(h), the cations continue occurring, the period-24 motions symmetric period-3 motion exists in the range of to chaos may exist. The range for stable period-6 Q0 ∈ (7.0193, 7.8626) with two saddle-node bifurca- motions is very short. Thus period-12 and higher tions at Q0 ≈ 7.0193, 7.8626 for its appearance. The order periodic motions on the bifurcation trees symmetric period-3 motion has two saddle-node will not be presented. In Figs. 11(e) and 11(f), a bifurcations at Q0 = 7.6693, 7.6779 for two jump- symmetric period-3 motion exists in the range of ing phenomena. In addition, the symmetric period-3 Q0 ∈ (6.3110, 7.0140) with saddle-node bifurcations motion possesses two period-doubling bifurcations at Q0 ≈ 6.3110, 7.0140 for its appearance. This at Q0 ≈ 7.0270, 7.6562 for the onset of asymmet- symmetric period-3 motion has two saddle-node ric period-3 motions. Thus the asymmetric period-3 Table 16. Excitation amplitudes for unstable asymmetric period-3 and period-6 motions (α = 2.0, δ = 0.75, Ω = 1.0). Asymmetric Period-3 Motion Period-6 Motions Q0 PD (L) PD (R) Q0 PD (L) PD (R) (3.8616, 4.0437) 3.8616 4.0437 (3.8745, 4.0382) 3.8745 4.0382 (7.0309, 7.6470) 7.0309 7.6470 Note: L and R denote “left” and “right”. PD is for period-doubling bifurcation. 1650159-35  A. C. J. Luo & Y. Guo motion exists in the range of Q0 ∈ (7.0270, 7.6562). motions. A global view of analytical bifurcation The asymmetric period-3 motion has two period- trees of travelable period-1 to period-2 motions in doubling bifurcations at Q0 ≈ 7.0309, 7.6470 for the periodically excited pendulum is illustrated in the onset of period-6 motion. Thus, the period-6 Figs. 12(a) and 12(b). This bifurcation tree does not motion exists in the range of Q0 ∈ (7.0309, 7.6470). start from symmetric period-1 motions but starts from the travelable asymmetric period-1 motion. In addition, this bifurcation tree of period-1 motion 6.3. Travelable period-1 motions to to chaos is independent, which will not go from chaos zero to infinite excitation amplitudes. For a clear As presented before, the bifurcation trees of peri- view of bifurcations trees, the zoomed windows odic motions to chaos are based on the nontrav- of Q0 ∈ (4.38, 4.40), (3.18, 3.27), (2.65, 2.77), and elable periodic motions. The independent bifurca- (2.312, 2.320) are shown in Figs. 12(c), 12(d)–12(i), tion trees of travelable period-1 motions to chaos 12(j), respectively. The ranges for unstable asym- will be presented through period-1 to period-2 metric period-1 motions and the corresponding by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com SN SNPD SN SN SN SN PD SNPD SN PD SN PD SN Periodic Node Displacement, mod(xmod(k,N),2π) 2π 4.0 Periodic Node Velocity, ymod(k,N) 3.0 3π/2 P-1 P-2 2.0 π P-2 P-2 1.0 π/2 P-1 0.0 P-2 π -1.0 2.0 2.5 3.0 3.5 4.0 4.5 2.0 2.5 3.0 3.5 4.0 4.5 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (a) (b) PD PD SN PD PD SN Periodic Node Displacement, mod(xmod(k,N),2π) 4.1 2.8 P-2 Periodic Node Velocity, ymod(k,N) P-1 4.0 2.7 3.9 P-2 P-1 1.0 0.8 0.8 0.6 0.6 4.380 4.385 4.390 4.395 4.400 4.380 4.385 4.390 4.395 4.400 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (c) (d) Fig. 12. Independent bifurcation trees of travelable period-1 motions to chaos through displacement mod(xmod(k,N ) , 2π) and velocity ymod(k,N ) : (a), (b) Global view (Q0 ∈ (2.0, 4.5)), (c), (d) Zoom-1 view (Q0 ∈ (4.38, 4.44)), (e), (f) Zoom-2 view (Q0 ∈ (3.18, 3.27)), (g), (h) Zoom-3 view (Q0 ∈ (2.65, 2.77)), (i), (j) Zoom-4 view (Q0 ∈ (2.312, 2.322)). (α = 2.0, δ = 0.75, Ω = 1.0). mod(k, N ) = 0. 1650159-36  Periodic Motions to Chaos in Pendulum SN SN PD SN PD SN PD SN Periodic Node Displacement, mod(xmod(k,N),2π) 6.0 P-2 3.4 P-1 Periodic Node Velocity, ymod(k,N) 4.8 2.2 1.5 P-2 1.0 P-1 0.0 -0.2 3.18 3.21 3.24 3.27 3.18 3.21 3.24 3.27 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (e) (f) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com SN PD PD SN PD PD SN PD PDSN PD PD Periodic Node Displacement, mod(xmod(k,N),2π) 6.0 3.0 Periodic Node Velocity, ymod(k,N) 5.0 2.0 1.6 P-2 P-2 0.0 P-1 P-1 0.0 2.65 2.69 2.73 2.77 2.65 2.69 2.73 2.77 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (g) (h) SN PD PD SN PD PD Periodic Node Displacement, mod(xmod(k,N),2π) 2.2 2.7 Periodic Node Velocity, ymod(k,N) P-2 2.1 P-1 2.4 P-2 0.8 P-1 -0.0 0.4 -0.1 2.312 2.314 2.316 2.318 2.320 2.312 2.314 2.316 2.318 2.320 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (i) (j) Fig. 12. (Continued) 1650159-37  A. C. J. Luo & Y. Guo Table 17. Excitation amplitudes for unstable asymmetric period-1 motions (α = 2.0, δ = 0.75, Ω = 1.0). With Period-2 Motion With Jumping Phenomena Q0 PD (L) PD (R) Q0 SN (L) SN (R) (2.3163, 2.7590) 2.3163 2.7590 (2.3127, 4.3950) 2.3127 4.3950 (3.1900, 4.3894) 3.1900 4.3894 Note: L and R denote “left” and “right”. SN and PD are for saddle-node and period-doubling bifurcations. saddle-node bifurcations for asymmetric period-2 in Figs. 12(c), 12(d)–12(i), 12(j) are for better motions and jumping phenomena are tabulated illustrations. in Table 17. The ranges for unstable asymmet- ric period-2 motions and the corresponding period- 6.4. Travelable period-2 motions to doubling and saddle-node bifurcations are listed in Table 18. The travelable asymmetric period-1 chaos by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com motion is in the range of Q0 ∈ (2.3127, 4.3950) The independent bifurcation trees of travelable with two saddle-node bifurcations at Q0 ≈ 2.3127, period-2 motions to chaos will be presented through 4.3950 for the onset of asymmetric period-1 motion. the travelable period-2 to period-8 motions. A The stable asymmetric period-1 motions are in the global view of analytical bifurcation trees of trav- range of Q0 ∈ (2.3127, 2.3163), (2.7590, 3.1900), elable period-2 to period-8 motions in the periodi- and (4.3894, 4.3950). The asymmetric period-1 cally excited pendulum is illustrated in Figs. 13(a) motion has two pairs of period-doubling bifurca- and 13(b). This bifurcation tree is independent tions at Q0 ∈ {2.3163, 2.7590} and {3.1900, 4.3894} and starts from the travelable asymmetric period-2 for the onset of asymmetric period-2 motions. motion without any period-1 motion. For a clear Thus, the travelable asymmetric period-2 motion view of bifurcations trees, the zoomed in windows is in two ranges of Q0 ∈ (2.3163, 2.7590) and of Q0 ∈ (7.32, 7.40) and (7.22, 7.26) are shown in (3.1900, 4.3894). During the two ranges of exci- Figs. 13(c), 13(d) and 13(e), 13(f), respectively. tation amplitudes, the travelable period-2 motion The ranges for unstable asymmetric period-2 possesses three pairs of period-doubling bifurca- motions and the corresponding saddle-node bifur- tions at Q0 ∈ {2.3183, 2.6883}, {2.6645, 2.7270} cations for the appearance of asymmetric period-2 and {3.2418, 4.3864} for the onset of period-4 motions are tabulated in Table 19. The ranges motions and two pairs of saddle-node bifurcations for unstable asymmetric period-4 and period-8 for jumping phenomena at Q0 ∈ {2.6592, 2.6886} motions and the corresponding period-doubling and {3.2394, 3.2605}. For the jumping phenom- bifurcations are listed in Table 20. The travelable ena, the three solutions of period-2 motions are asymmetric period-2 motion is in the range of observed during the range of Q0 ∈ (2.6592, 2.6886) Q0 ∈ (7.2289, 7.3899) with two saddle-node bifur- and (3.2394, 3.2605) — the two stable and one cations at Q0 ≈ 7.2289, 7.3899 for the appear- unstable solutions of travelable period-2 motions. ance of asymmetric period-2 motions. The stable Since the ranges of stable travelable period-1 and travelable period-2 motions are in the ranges of period-2 motions are very tiny, the zoomed in views Q0 ∈ (7.2289, 7.2427) and (7.3541, 7.3899). The Table 18. Excitation amplitudes for unstable period-2 motions (α = 2.0, δ = 0.75, Ω = 1.0). With Period-4 Motion With Jumping Phenomena Q0 PD (L) PD (R) Q0 SN (L) SN (R) (2.3183, 2.6883) 2.3183 2.6883 (2.6592, 2.6886) 2.6592 2.6886 (2.6645, 2.7270) 2.6645 2.7270 (3.2394, 3.2605) 3.2394 3.2605 (3.2418, 4.3864) 3.2418 4.3864 Note: L and R denote “left” and “right”. SN and PD are for saddle-node and period-doubling bifurcations. 1650159-38  Periodic Motions to Chaos in Pendulum SN PD PDPD PD PD PD SN SN PD PDPD PD PD PD SN 2π 5.0 Periodic Node Displacement, mod(xmod(k,N),2π) Periodic Node Velocity, ymod(k,N) 3π/2 4.2 P-2 P-4 P-8 P-8 P-4 P-2 P-2 P-4 P-8 P-8 P-4 P-2 π 3.4 π/2 2.6 π 1.8 7.20 7.25 7.30 7.35 7.40 7.20 7.25 7.30 7.35 7.40 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com PD PD PD SN PD PD PD SN 2π 5.0 Periodic Node Displacement, mod(xmod(k,N),2π) Periodic Node Velocity, ymod(k,N) 3π/2 4.2 π P-8 P-4 P-2 3.4 P-4 P-2 P-8 π/2 2.6 0 1.8 7.32 7.36 7.40 7.32 7.34 7.36 7.38 7.40 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (c) (d) SN PD PD PD SN PD PD PD 2π 5.0 Periodic Node Displacement, mod(xmod(k,N),2π) Periodic Node Velocity, ymod(k,N) 3π/2 4.2 P-2 P-4 P-8 P-2 P-4 P-8 π 3.4 π/2 2.6 0 1.8 7.22 7.24 7.26 7.22 7.24 7.26 Excitation Amplitude, Q0 Excitation Amplitude, Q0 (e) (f) Fig. 13. Independent bifurcation trees of period-2 motions to chaos through displacement mod(xmod(k,N ) , 2π) and velocity ymod(k,N ) : (a), (b) Global view (Q0 ∈ (7.2, 7.4)), (c), (d) zoom-1 view (Q0 ∈ (7.32, 7.4)), (e), (f) zoom-2 view (Q0 ∈ (7.22, 7.26)). (α = 2.0, δ = 0.75, Ω = 1.0). mod(k, N ) = 0. 1650159-39  A. C. J. Luo & Y. Guo Table 19. Excitation amplitudes for unstable asymmetric period-2 motions (α = 2.0, δ = 0.75, Ω = 1.0). Period-4 Motion Jumping Phenomena Q0 PD (L) PD (R) Q0 SN (L) SN (R) (7.2427, 7.3541) 7.2427 7.3541 (7.2289, 7.3899) 2.3127 7.3899 Note: L and R denote “left” and “right”. SN and PD are for saddle-node and period-doubling bifurcations. Table 20. Excitation amplitudes for unstable period-4 and unstable period-8 motions (α = 2.0, δ = 0.75, Ω = 1.0). Period-4 Motion Period-8 Motion Q0 PD (L) PD (R) Q0 SN (L) SN (R) (7.2524, 7.3364) 7.2524 7.3364 (7.2552, 7.3322) 7.2552 7.3322 Note: L and R denote “left” and “right”. SN and PD are for saddle-node and period-doubling bifurcations. by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com asymmetric period-2 motion has a pair of period- 7.1. Nontravelable period-1 motions doubling bifurcations at Q0 ∈ {7.2427, 7.3541} There are two types of nontravelable period-m for the onset of asymmetric period-4 motions. motions with centers determined by the correspond- The travelable period-4 motion exists in the range (m) (m) of Q0 ∈ (7.2427, 7.3541). The unstable period-4 ing constants mod(a0 , 2π) near mod(a0 , 2π) = (m) motion lies in the range of Q0 ∈ (7.2524, 7.3364). π and mod(a0 , 2π) = 0 or 2π in the finite The stable period-4 motions is in the two segments Fourier series. For nontravelable period-m motions, of Q0 ∈ (7.2427, 7.2524) and (7.3364, 7.3541). In (m) (m) xk = xk+mN . However, for travelable period- the range of Q0 ∈ (7.2524, 7.3364), a travelable (m) (m) (m) m motions, xk = xk+mN , but mod(xk , 2π) = period-8 motion exists. Such a period-8 motion (m) has two period-doubling bifurcations at Q0 ∈ mod(xk+mN , 2π). {7.2552, 7.3322} for the appearance of a travelable period-16 motion. 7.1.1. Period-1 to period-4 motions 7. Numerical Simulations 7.1.1.1. Periodic motion with center near (m) mod(a0 , 2π) = π In this section, numerical and analytical results of periodic motions in periodically forced pendu- Consider parameters of α = 1.5, δ = 0.75, Q0 = lum will be presented with different excitation fre- 5.0 for nontravelable periodic motions with center (m) quencies or amplitudes. For numerical simulations, near mod(a0 , 2π) = 2π. The symmetric period-1 the initial conditions of periodic motions will be motion is considered first, and then asymmetric obtained from the analytical predictions presented period-1, period-2 and period-4 motions will be pre- as before. The numerical simulations will be pre- sented herein. sented with solid curves, while the analytically pre- For Ω = 1.0, a symmetric period-1 motion dicted trajectories will be presented with circular is presented in Fig. 14. The initial condition hollow symbols. The initial node and periodic nodes of this symmetric period-1 motion is (x0 , y0 ) ≈ are indicated by green circular symbols. For a pair of (0.8046, 1.4237) from the analytical prediction. asymmetric motions, the black and red colors indi- Such a symmetric periodic motion is centered at cate that the motions are on black or red branches (π, 0). The time history of displacement for the sym- on the bifurcation trees, respectively. The trave- metric period-1 motion is presented in Fig. 14(a). lable and nontravelable periodic motions will be The trajectory of the symmetric period-1 motion in presented herein and the harmonic amplitudes and phase space is illustrated in Fig. 14(b). The sym- phases will be presented for harmonic effects on metric period-1 motion is symmetric to itself about periodic motions. (π, 0) in phase space. The harmonic amplitude of 1650159-40  Periodic Motions to Chaos in Pendulum 4.0 T 2π 2.0 1T Displacement, mod(x,2π) 3π/2 IC Velocity, y π 0.0 π/2 -2.0 IC 0 -4.0 0.0 2.0 4.0 6.0 8.0 0 π/2 π 3π/2 2π Time, t Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π a0 A1 ϕ19 1e+0 ϕ3 ϕ13 3π/2 Harmonic Amplitude, Ak Harmonic Phase, ϕk ϕ1 ϕ7 1e-4 ϕ17 π ϕ11 ϕ5 1e-8 π/2 ϕ15 A19 ϕ9 1e-12 0 0 5 10 15 20 0 5 10 15 20 Harmonic Order, k Harmonic Order, k (c) (d) Fig. 14. Symmetric period-1 motions (Ω = 1.0): (a) Displacement, (b) trajectory, (c) harmonic amplitude, (d) harmonic phase. Initial condition: (x0 , y0 ) ≈ (0.8046, 1.4237). Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). the nontravelable symmetric period-1 motion is pre- with center near mod(a0 , 2π) = π. Thus, two non- sented in Fig. 14(c). The corresponding constant travelable asymmetric period-1 motions are pre- is a0 = π. If mod(x0 , 2π) ≈ 0.8046 and the same sented in Fig. 15 for Ω = 1.16. The initial con- velocity of y0 ≈ 1.4237, there are infinite solu- ditions of the asymmetric period-1 motions are tions of period-1 motions with mod(a0 , 2π) = π. (x0 , y0 ) ≈ (0.7574, 1.3007) (black) and (x0 , y0 ) ≈ For symmetric period-1 motions, A(2l−1) = 0 but (1.3585, 1.0565) (red) from the analytical pre- A2l = 0 (l = 1, 2, . . .). The main harmonic ampli- diction. The trajectories of the two asymmetric tudes are A1 ≈ 2.5854 and A3 ≈ 0.0631. The other period-1 motion in phase space are illustrated in harmonic amplitudes are A(2l−1) ∈ (10−12 , 10−2 ) Figs. 15(a) and 15(b) on the left and right sides of for l = 3, 4, . . . , 10. In Fig. 14(d), the correspond- mod(x, 2π) = π, respectively. The harmonic ampli- ing harmonic phases lie in ϕ2l−1 ∈ [0, 2π) for l = tudes of the two nontravelable asymmetric period-1 1, 2, . . . , 10. motions are presented in Fig. 15(c). The two con- With the saddle-node bifurcation of the sym- stants of the Fourier series satisfy π−mod(aB 0 , 2π) ≈ metric period-1 motion with mod(a0 , 2π) = π, the 0.2761 ≈ mod(aR 0 , 2π) − π, which is for asymmetry nontravelable asymmetric period-1 motion exists of mod(x, 2π) = π. The harmonic amplitudes for 1650159-41  A. C. J. Luo & Y. Guo 4.0 4.0 2.0 1T 2.0 1T Velocity, y Velocity, y IC 0.0 0.0 IC -2.0 -2.0 -4.0 -4.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π a0 A1 ϕ19 1e+0 ϕ13 ϕ3 3π/2 Harmonic Amplitude, Ak Harmonic Phase, ϕk ϕ1 ϕ7 ϕ17 1e-4 π ϕ5 ϕ11 1e-8 A20 π/2 ϕ15 1e-12 ϕ9 0 0 5 10 15 20 0 5 10 15 20 Harmonic Order, k Harmonic Order, k (c) (d) Fig. 15. Asymmetric period-1 motions (Ω = 1.16): (a) Trajectory of black branch, initial condition of (x0 , y0 ) ≈ (0.7574, 1.3007); (b) trajectory of red branch, initial condition of (x0 , y0 ) ≈ (1.3585, 1.0565). Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). two asymmetric period-1 motions are the same. For two trajectories, harmonic amplitudes and phases the two asymmetric period-1 motions, A(2l−1) = 0 of the asymmetric period-2 motions are presented and A2l = 0 (l = 1, 2, . . .). The main harmonic in Fig. 16. The corresponding initial conditions are amplitudes are A1 ≈ 2.2632, A2 ≈ 0.0586, and A3 ≈ (x0 , y0 ) ≈ (0.6708, 1.3309) for the black branch and 0.0410. The other harmonic amplitudes are Ak ∈ (x0 , y0 ) ≈ (1.5784, 0.9840) for the red branch from (10−14 , 10−2 ) for k = 4, 5, . . . , 20. In Fig. 15(d), har- the analytical prediction. The trajectories of the monic phases for two asymmetric period-1 motions two asymmetric period-2 motion in phase space are are presented and distributed in the range of ϕk ∈ illustrated in Figs. 16(a) and 16(b) on the left and [0, 2π) with ϕR B right sides of mod(x, 2π) = π, respectively. The har- k = mod(ϕk + (k + 1)π, 2π) ∈ [0, 2π) monic amplitudes of the two nontravelable asym- for k = 1, 2, . . . , 20. metric period-2 motions are presented in Fig. 16(c). After the period-doubling bifurcation of the (2)B nontravelable asymmetric period-1 motion, the non- The two constants satisfy π − mod(a0 , 2π) ≈ (2)R travelable period-2 motions will appear. Consider 0.5506 ≈ mod(a0 , 2π) − π, which is for asymme- excitation frequency of Ω = 1.19 for a pair of two try of mod(x, 2π) = π. The harmonic amplitudes for nontravelable asymmetric period-2 motions. The two asymmetric period-2 motions are the same. For 1650159-42  Periodic Motions to Chaos in Pendulum 4.0 4.0 1T 2.0 2.0 2T 2T Velocity, y Velocity, y IC IC 1T 0.0 0.0 -2.0 -2.0 -4.0 -4.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com a0(2) 2π A1/2 ϕ13 A1 ϕ17 1e+0 ϕ3 Harmonic Amplitude, Ak/2 3π,2 Harmonic Phase, ϕk/2 ϕ1 ϕ7 1e-4 ϕ11 π ϕ5 ϕ15 1e-8 ϕ19 π/2 ϕ9 A20 1e-12 0 0 5 10 15 20 0 5 10 15 20 Harmonic Order, k/2 Harmonic Order, k/2 (c) (d) Fig. 16. Asymmetric period-2 motions (Ω = 1.19): (a) Phase portrait of black branch, initial condition: (x0 , y0 ) ≈ (0.6108, 1.3309); (b) phase portrait of red branch, initial condition: (x0 , y0 ) ≈ (1.5784, 0.9840). Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). the two asymmetric period-2 motions, the main har- two trajectories, harmonic amplitudes and phases monic amplitudes are A1/2 ≈ 0.1600, A1 ≈ 2.2761, of the asymmetric period-4 motions are presented A3/2 ≈ 0.0383, A2 ≈ 0.1076, A5/2 ≈ 0.0102, and in Fig. 17. The corresponding initial conditions are A3 ≈ 0.0373. The other harmonic amplitudes are (x0 , y0 ) ≈ (0.7730, 1.1834) for the black branch Ak/2 ∈ (10−14 , 10−2 ) for k = 7, 8, . . . , 40. The har- and (x0 , y0 ) ≈ (1.8227, 1.0649) for the red branch. monic phases distributed in ϕk/2 ∈ [0, 2π) (k = The trajectories of the two asymmetric period-4 1, 2, . . . , 40) for the two period-2 motions are pre- motion in phase space are illustrated in Figs. 17(a) sented in Fig. 16(d), and they satisfy a relation of and 17(b) on the left and right sides of mod(x, 2π) = ϕR B k/2 = mod(ϕk/2 + (k(1 + 2r)/2 + 1)π, 2π), r = 1 π, respectively. The harmonic amplitudes of the with t0 = rT (r ∈ {0, 1}). two nontravelable asymmetric period-2 motions are After the period-doubling bifurcation of the placed in Fig. 17(c). The two constants satisfy (4)B (4)R nontravelable asymmetric period-2 motion, the non- π − mod(a0 , 2π) ≈ 0.5864 ≈ mod(a0 , 2π) − π, travelable period-4 motions will exist. Consider which is for asymmetry of mod(x, 2π) = π. The excitation frequency of Ω = 1.20 for a pair of two harmonic amplitudes for two asymmetric periodic nontravelable asymmetric period-4 motions. The motions are the same. For the two asymmetric 1650159-43  A. C. J. Luo & Y. Guo 4.0 4.0 1T 2.0 4T 2.0 3T 1T 2T 4T 3T Velocity, y Velocity, y IC 2T IC 0.0 0.0 -2.0 -2.0 -4.0 -4.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π a0(4) A1/4 ϕ17 1e+0 A1/2 A1 ϕ3 Harmonic Amplitude, Ak/4 3π/2 Harmonic Phase, ϕk/4 ϕ1 ϕ7 ϕ11 ϕ15 1e-4 π ϕ19 ϕ5 1e-8 ϕ9 A20 π/2 ϕ13 1e-12 0 0 5 10 15 20 0 5 10 15 20 Harmonic Order, k/4 Harmonic Order, k/4 (c) (d) Fig. 17. Asymmetric period-4 motions (Ω = 1.20): (a) Phase portrait of black branch, initial condition: (x0 , y0 ) ≈ (0.7730, 1.1834); (b) phase portrait of red branch, initial condition: (x0 , y0 ) ≈ (1.8227, 1.0649). Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). (m) period-4 motions, the main harmonic amplitudes mod(a0 , 2π) = 2π or 0. The symmetric period-1 are A1/4 ≈ 0.0551, A1/2 ≈ 0.2861, A3/4 ≈ 0.0239, motion is considered first, and then asymmetric A1 ≈ 2.2647, A5/4 ≈ 0.0137, A3/2 ≈ 0.0677, A7/4 ≈ period-1, period-2 and period-4 motions will be pre- 6.0897e-3, A2 ≈ 0.1108, A9/4 ≈ 4.6969e-3, A5/2 ≈ sented herein. 0.0173, A11/4 ≈ 8.6895e-4, and A3 ≈ 0.0352. The For Ω = 1.5, a symmetric period-1 motions is other harmonic amplitudes are Ak/4 ∈ (10−14 , 10−2 ) presented in Fig. 18 for one of the two branches for k = 12, 13, . . . , 80. In Fig. 17(d), harmonic in the bifurcation trees. The initial condition phases are presented with ϕR B of the symmetric period-1 motion is (x0 , y0 ) ≈ k/2 = mod(ϕk/2 + (k(2r + 1)/4 + 1)π, 2π) ∈ [0, 2π), k = 1, 2, . . . , 80 (4.2051, 1.9880) from the analytical prediction. and r = 0 with t0 = rT (r ∈ {0, 1, 2, 3}). Such a symmetric periodic motion is centered at (2π, 0) or (0, 0). The time history of displacement for the symmetric period-1 motion is presented in 7.1.1.2. Periodic motions with center near Fig. 18(a). The trajectory of the symmetric period-1 (m) mod(a0 , 2π) = 2π motion in phase space is illustrated in Fig. 18(b). Consider parameters of α = 1.5, δ = 0.75, Q0 = 5.0 The symmetric period-1 motion is symmetric to for nontravelable periodic motions with center near itself about (2π, 0) or (0, 0) in phase space. The 1650159-44  Periodic Motions to Chaos in Pendulum 5.0 T 2π 1T 2.5 Displacement, mod(x,2π) 3π/2 IC Velocity, y IC π 0.0 π/2 -2.5 0 -5.0 0.0 2.0 4.0 6.0 8.0 0 π/2 π 3π/2 2π Time, t Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π a0 A1 ϕ5 ϕ19 1e+0 ϕ13 3π/2 Harmonic Amplitude, Ak ϕ7 Harmonic Phase, ϕk ϕ1 1e-4 ϕ15 π ϕ9 ϕ3 1e-8 ϕ17 π/2 A19 ϕ11 1e-12 0 0 5 10 15 20 0 5 10 15 20 Harmonic Order, k Harmonic Order, k (c) (d) Fig. 18. Symmetric period-1 motions (Ω = 1.50): (a) Displacement, (b) trajectory, (c) harmonic amplitude, (d) harmonic phases. Initial condition: (x0 , y0 ) ≈ (4.2051, 1.9880). Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). harmonic amplitudes of the nontravelable symmet- mod(a0 , 2π) near 2π. Thus, two nontravelable ric period-1 motion are presented in Fig. 18(c). The asymmetric period-1 motions are presented in corresponding constant is a0 = 2π. If mod(x0 , 2π) ≈ Fig. 19 for Ω = 1.40. The initial conditions 4.2051 and the same velocity of y0 ≈ 1.9880, there of the two asymmetric period-1 motions are are also infinite solutions of period-1 motions with (x0 , y0 ) ≈ (5.4339, 2.1523) (black) and (x0 , y0 ) ≈ mod(a0 , 2π) = 2π. For the symmetric period-1 (2.9642, 1.3703) (red) from the analytical pre- motion, A(2l−1) = 0 but A2l = 0 (l = 1, 2, . . .). diction. The trajectories of the two asymmetric The main harmonic amplitudes are A1 ≈ 2.5039 period-1 motions in phase space are illustrated and A3 ≈ 0.0317. The other harmonic amplitudes in Figs. 19(a) and 19(b) on the left and right are A(2l−1) ∈ (10−12 , 10−2 ) for l = 3, 4, . . . , 10. The sides of mod(x, 2π) = 2π, respectively. The har- harmonic phases for the symmetric period-1 motion monic amplitudes of the two nontravelable asym- are in ϕ2l−1 ∈ [0, 2π) for l = 1, 2, . . . , 10, as shown metric period-1 motions are presented in Fig. 19(c). in Fig. 18(d). The two constants satisfy 2π − mod(aB 0 , 2π) ≈ With the saddle-node bifurcation of the 1.2726 ≈ mod(aR 0 , 2π) − 2π, which is for asymme- symmetric period-1 motion, the nontravelable try of mod(x, 2π) = 2π. The harmonic amplitudes asymmetric period-1 motion exists with center for two asymmetric period-1 motions are the same. 1650159-45  A. C. J. Luo & Y. Guo 5.0 5.0 1T 2.5 IC 2.5 1T IC Velocity, y Velocity, y 0.0 0.0 -2.5 -2.5 -5.0 -5.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com a0 2π A1 ϕ9 1e+0 ϕ13 3π/2 Harmonic Amplitude, Ak Harmonic Phase, ϕk ϕ1 ϕ19 1e-4 ϕ5 π ϕ15 ϕ11 1e-8 ϕ3 π/2 ϕ7 A20 ϕ17 1e-12 0 0 5 10 15 20 0 5 10 15 20 Harmonic Order, k Harmonic Order, k (c) (d) Fig. 19. Asymmetric period-1 motions (Ω = 1.40): (a) Trajectory of black branch (right), initial condition: (x0 , y0 ) ≈ (5.4339, 2.1523); (b) trajectory of red branch (left), initial condition: (x0 , y0 ) ≈ (2.9642, 1.3703), (c) harmonic amplitudes, (d) harmonic phases. Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). The main harmonic amplitudes are A1 ≈ 2.4410, corresponding initial conditions of the two period-2 A2 ≈ 0.1534, and A3 ≈ 0.0410. The other har- motions are (x0 , y0 ) ≈ (5.9797, 1.7619) (black) and monic amplitudes are Ak ∈ (10−14 , 10−2 ) for k = (x0 , y0 ) ≈ (2.5288, 1.0577) (red) from the analytical 4, 5, . . . , 20. In Fig. 19(d), harmonic phases for the prediction. The trajectories of the two asymmet- two symmetric period-1 motions satisfy a relation ric period-2 motion in phase space are illustrated of ϕR B k = mod(ϕk + (k + 1)π, 2π) ∈ [0, 2π) for in Figs. 20(a) and 20(b) on the left and right k = 1, 2, . . . , 20. sides of mod(x, 2π) = 2π, respectively. The har- Similarly, after the period-doubling bifurca- monic amplitudes of the two nontravelable asym- tion of the asymmetric period-1 motions, there metric period-2 motions are presented in Fig. 20(c). is an asymmetric period-2 motion in the bifurca- The two constants in the Fourier series satisfy tion tree. Consider excitation frequency of Ω = (2)B (2)R mod(a0 , 2π) ≈ 1.2726 ≈ 2π − mod(a0 , 2π), 1.36 for a pair of two nontravelable asymmet- which is for asymmetry of mod(x, 2π) = 2π. For ric period-2 motions. The two trajectories, har- the two asymmetric period-2 motions, the main monic amplitudes and phases of the asymmetric harmonic amplitudes are A1/2 ≈ 0.5206, A1 ≈ period-2 motions are presented in Fig. 20. The 2.4334, A3/2 ≈ 0.0766, A2 ≈ 0.1558, A5/2 ≈ 0.0144, 1650159-46  Periodic Motions to Chaos in Pendulum 5.0 5.0 2.5 1T 2T 2.5 2T IC Velocity, y Velocity, y 1T IC 0.0 0.0 -2.5 -2.5 -5.0 -5.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π ϕ9 a0(2) A1/2 ϕ19 A1 1e+0 Harmonic Amplitude, Ak/2 3π/2 ϕ11 Harmonic Phase, ϕk/2 ϕ15 ϕ1 ϕ3 1e-4 ϕ5 π 1e-8 ϕ17 π/2 ϕ7 A20 ϕ13 1e-12 0 0 5 10 15 20 0 5 10 15 20 Harmonic Order, k/2 Harmonic Order, k/2 (c) (d) Fig. 20. Asymmetric period-2 motions (Ω = 1.36): (a) Trajectory of black branch (right), initial condition: (x0 , y0 ) ≈ (5.9797, 1.7619); (b) trajectory of red branch (left), initial condition: (x0 , y0 ) ≈ (2.5288, 1.0577); (c) harmonic amplitudes, (d) harmonic phases. Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). and A3 ≈ 1.3072e-3. The other harmonic ampli- motions are presented in Fig. 21. The correspond- tudes are Ak/2 ∈ (10−14 , 10−2 ) for k = 7, 8, . . . , 40. ing initial conditions of the two period-4 motions In Fig. 20(d), the harmonic phases for the two are (x0 , y0 ) ≈ (6.0373, 1.7324) and (x0 , y0 ) ≈ period-2 motions are presented and they are dis- (2.4152, 0.9947) for the black and red branches, tributed in ϕk/2 ∈ [0, 2π) for k = 1, 2, . . . , 40. respectively. The trajectories of the two asymmet- ϕR B k/2 = mod(ϕk/2 + (k(1 + 2r)/2 + 1)π, 2π), r = 1 ric period-4 motion in phase space are illustrated in with t0 = rT (r ∈ {0, 1}). Figs. 21(a) and 21(b) on the left and right sides On the bifurcation trees of nontravelable of mod(x, 2π) = 2π, respectively. The harmonic period-1 motions to chaos, after the period- amplitudes of the two nontravelable asymmetric doubling bifurcation of the nontravelable asym- period-2 motions are placed in Fig. 21(c). The two metric period-2 motion, the nontravelable period-4 constants in the Fourier series expression satisfy (4)B (4)R motions will exist. Consider excitation frequency of mod(a0 , 2π) ≈ 1.4732 ≈ 2π − mod(a0 , 2π), Ω = 1.35 for a pair of two nontravelable asymmetric which is for asymmetry of mod(x, 2π) = 2π. The period-4 motions. The two trajectories, harmonic harmonic amplitudes for two asymmetric period-4 amplitudes and phases of the asymmetric period-4 motions are the same. For the two asymmetric 1650159-47  A. C. J. Luo & Y. Guo 5.0 5.0 3T 2.5 4T 2.5 1T 4T IC 1T Velocity, y Velocity, y 2T IC 3T 2T 0.0 0.0 -2.5 -2.5 -5.0 -5.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π a0(4) A1/4 ϕ19 ϕ9 1e+0 A1/2 A1 Harmonic Amplitude, Ak/4 3π/2 ϕ15 ϕ11 Harmonic Phase, ϕk/4 ϕ1 1e-4 ϕ3 π ϕ5 1e-8 ϕ17 ϕ13 A20 π/2 ϕ7 1e-12 0 0 5 10 15 20 0 5 10 15 20 Harmonic Order, k/4 Harmonic Order, k/4 (c) (d) Fig. 21. Asymmetric period-4 motions (Ω = 1.35): (a) Trajectory of black branch (right), initial condition: (x0 , y0 ) ≈ (6.0373, 1.7324); (b) trajectory of red branch (left), initial condition: (x0 , y0 ) ≈ (2.4152, 0.9947), (c) harmonic amplitudes, (d) harmonic phases. Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). period-4 motions, the main harmonic amplitudes for illustration of complexity in different bifurcation are A1/4 ≈ 0.0888, A1/2 ≈ 0.6255, A3/4 ≈ 0.0424, trees. Such a bifurcation tree is closed and indepen- A1 ≈ 2.4353, A5/4 ≈ 0.0117, A3/2 ≈ 0.0924, A7/4 ≈ dent. The periodic motion on the bifurcation tree 6.6020e-3, A2 ≈ 0.1530, A9/4 ≈ 2.8235e-3, A5/2 ≈ is nontravelable. The center for periodic motions of 0.0166, A11/4 ≈ 3.4517e-3, and A3 ≈ 2.5941e-3. The mod(x, 2π) in such bifurcation tree is near (2π, 0) other harmonic amplitudes are Ak/4 ∈ (10−14 , 10−2 ) or (π, 0). for k = 12, 13, . . . , 80. In Fig. 21(d), the correspond- For Ω = 0.762, a symmetric period-3 motions ing harmonic phases are presented which are dis- is presented in Fig. 22. The initial condition of tributed in the range [0, 2π) for ϕR B k/4 = mod(ϕk/4 + this motion is (x0 , y0 ) ≈ (3.7268, 2.9670) from the (k(1 + 2r)/4 + 1)π, 2π), k = 1, 2, . . . , 80 and r = 1 analytical prediction. Such a symmetric period-3 with t0 = rT (r ∈ {0, 1, 2, 3}). motion is centered at (2π, 0) or (0, 0). The time history of displacement for the symmetric period-3 motion is presented in Fig. 22(a). Compared to 7.1.2. Period-3 to period-6 motions the period-1 motion, period-3 motion becomes In this section, symmetric and asymmetric period-3 much complicated. The trajectory of the symmet- motions and periodic-6 motion will be presented ric period-3 motion in phase space is illustrated 1650159-48  Periodic Motions to Chaos in Pendulum 6.0 3T 2π 3.0 2T 3T Displacement, mod(x,2π) 3π/2 1T IC Velocity, y IC π 0.0 π/2 -3.0 0 -6.0 0.0 10.0 20.0 30.0 0 π/2 π 3π/2 2π Time, t Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π A1/3 ϕ1/3 ϕ7 ϕ21 1e+0 A1 ϕ13 ϕ23 ϕ29 ϕ37 Harmonic Amplitude, Ak/3 3π/2 Harmonic Phase, ϕk/3 ϕ1 ϕ 3 ϕ15 1e-4 ϕ25 ϕ31 ϕ39 ϕ17 π ϕ9 1e-8 ϕ33 ϕ19 A40 π/2 ϕ27 ϕ11 ϕ5 1e-12 ϕ35 0 0 8 16 24 32 40 0 10 20 30 40 Harmonic Order, k/3 Harmonic Order, k/3 (c) (d) Fig. 22. Symmetric period-3 motions (Ω = 0.762): (a) Displacement, (b) trajectory, (c) harmonic amplitudes, (d) harmonic phases. Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). Initial condition: (x0 , y0 ) ≈ (3.7268, 2.9670). (3) in Fig. 22(b). The symmetric period-3 motion is motions with mod(a0 , 2π) = 2π or 0. For symmet- symmetric to itself about (2π, 0) in phase space. ric period-3 motions, A(2l−1)/3 = 0 but A(2l)/3 = 0 Three cycles around the center of (2π, 0) or (0, 0) (l = 1, 2, . . .). The main harmonic amplitudes are are formed. The harmonic amplitude of the nontrav- A1/3 ≈ 2.0410, A1 ≈ 5.9571, A5/3 ≈ 0.2446, elable symmetric period-3 motion is presented in A7/3 ≈ 0.1183, A3 ≈ 0.0271, A11/3 ≈ 0.0545, Fig. 22(c). The displacement gives from x0 ≈ 3.7268 A5 ≈ 0.0222, A17/3 ≈ 0.0193, A19/3 ≈ 0.0239, and to 2π, 2π to 4π, 4π up and returning back to 4π, A7 ≈ 3.7577e-3. The other harmonic amplitudes 4π to 2π; 2π down to almost zero and returning are A(2l−1)/3 ∈ (10−12 , 10−2 ) for l = 22, 23, . . . , 60. back to 2π, 2π up and returning back to 2π, 2π The harmonic phases are distributed in ϕ(2l−1)/3 ∈ down to zero, zero down and return back to zero, [0, 2π) for l = 1, 2, . . . , 60, as shown in Fig. 22(d). zero to 2π, 2π up almost to 4π and returning back With the saddle-node bifurcation of sym- to 2π, finally from 2π to x3T ≈ 3.7268. The corre- metric period-3 motion, the nontravelable asym- (3) metric period-3 motion exists with center near sponding constant in the Fourier series is a0 = 2π. (3) If mod(x0 , 2π) ≈ 3.7268 and the same velocity of mod(a0 , 2π) = 2π. Thus, a pair of two nontrav- y0 ≈ 2.9670, there are infinite solutions of period-3 elable asymmetric period-3 motions is presented in 1650159-49  A. C. J. Luo & Y. Guo 6.0 6.0 3T IC 3.0 3.0 IC 3T 2T 1T 1T 2T Velocity, y Velocity, y 0.0 0.0 -3.0 -3.0 -6.0 -6.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π a0(3) A1/3 1e+0 A2/3 A1 Harmonic Amplitude, Ak/3 3π/2 Harmonic Phase, ϕk/3 ϕ1/3 1e-4 π 1e-8 ϕ39 A40 π/2 1e-12 0 0 10 20 30 40 0 10 20 30 40 Harmonic Order, k/3 Harmonic Order, k/3 (c) (d) Fig. 23. Asymmetric period-3 motions (Ω = 0.758): (a) Trajectory of black branch (right), initial condition: (x0 , y0 ) ≈ (1.3009, 3.4547); (b) trajectory of red branch (left), initial condition: (x0 , y0 ) ≈ (1.1077, 3.9509), (c) harmonic amplitudes, (d) harmonic phases. Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). Fig. 23 for Ω = 0.758. The initial conditions of period-3 motions, A(2l−1)/3 = 0 and A(2l)/3 = 0 the asymmetric period-3 motions are (x0 , y0 ) ≈ (l = 1, 2, . . .). The main harmonic amplitudes are (1.3009, 3.4547) (black) and (x0 , y0 ) ≈ (1.1077, A1/3 ≈ 1.8145, A2/3 ≈ 0.2944, A1 ≈ 5.8967, A4/3 ≈ 3.9509) (red) from the analytical prediction. The 0.0936, A5/3 ≈ 0.2235, A2 ≈ 0.0243, A7/3 ≈ 0.1138, trajectories of the two asymmetric period-3 motion A8/3 ≈ 0.0204, A3 ≈ 0.0309, A10/3 ≈ 7.2685e-3, in phase space are illustrated in Figs. 23(a) A11/3 ≈ 0.0574, A4 ≈ 0.0112, A13/3 ≈ 0.0752, and 23(b) on the left and right sides of mod(x, 2π) = A14/3 ≈ 0.0116, A15 ≈ 0.0298, A16/3 ≈ 8.4148e-3, 2π, respectively. The harmonic amplitudes of the A17/3 ≈ 0.0160, A6 ≈ 6.3730e-3, A19/3 ≈ 0.0215, two nontravelable asymmetric period-3 motions A20/3 ≈ 3.3613e-3, and A7 ≈ 5.2751e-3. The other are presented in Fig. 23(c). The two constants harmonic amplitudes are Ak/3 ∈ (10−14 , 10−2 ) for for two asymmetric period-3 motions satisfy k = 21, 22, . . . , 120. Harmonic phases are presented (3)B (3)R for ϕk/3 ∈ [0, 2π) with ϕR = mod(ϕB + mod(a0 , 2π) ≈ 0.1137 ≈ 2π − mod(a0 , 2π), k/(2l m) k/(2l m) which is for asymmetry of mod(x, 2π) = 2π. The k((m + 2r)/(2l m) + 1)π, 2π) for l = 0, m = 3, harmonic amplitudes for two asymmetric period-3 r = 0 and t0 = rT for r ∈ {0, 1, . . . , 2l m − 1} in motions are the same. For the two asymmetric Fig. 23(d). 1650159-50  Periodic Motions to Chaos in Pendulum After the period-doubling bifurcation of the asymmetric period-6 motions are presented in nontravelable asymmetric period-3 motion, the non- Fig. 24(c). The two period-6 motions are close to travelable asymmetric period-6 motions will be the corresponding asymmetric period-3 motions. obtained. Consider excitation frequency of Ω = The two constants for the asymmetric period-6 0.756 for a pair of two nontravelable asymmet- (6)B motions satisfy mod(a0 , 2π) ≈ 0.0905 ≈ 2π − ric period-6 motions. The two trajectories, har- (6)R mod(a0 , 2π). The harmonic amplitudes for two monic amplitudes and phases of the asymmetric asymmetric period-6 motions are the same. For period-6 motions are presented in Fig. 24. The the two asymmetric period-6 motions, the main corresponding initial conditions of two period-6 harmonic amplitudes are A1/6 ≈ 0.0403, A1/3 ≈ motions are (x0 , y0 ) ≈ (1.3241, 3.2092) (black) 1.6297, A1/2 ≈ 0.0328, A2/3 ≈ 0.3681, A5/6 ≈ and (x0 , y0 ) ≈ (1.2328, 3.7937) (red) from the 0.0318, A1 ≈ 5.8296, A7/6 ≈ 0.0215, A4/3 ≈ analytical prediction. The trajectories of the two 0.1164, A3/2 ≈ 3.7209e-3, A5/3 ≈ 0.2085, A11/6 ≈ asymmetric period-6 motions in phase space are 2.8707e-3, A2 ≈ 0.0305, A13/6 ≈ 1.8433e-3, illustrated in Figs. 24(a) and 24(b) on the left A7/3 ≈ 0.1083, A5/2 ≈ 2.9228e-3, A8/3 ≈ 0.0321, and right sides of mod(x, 2π) = 2π, respectively. A17/6 ≈ 3.3253e-3, A3 ≈ 0.0399, A19/6 ≈ 2.8363e-3, by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. The harmonic amplitudes of the two nontravelable Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com A10/3 ≈ 0.0122, A7/2 ≈ 6.4339e-4, A11/3 ≈ 0.0598, 6.0 6.0 3T 3T 6T IC 3.0 3.0 IC 6T 4T 5T 2T 1T 4T 5T 2T 1T Velocity, y Velocity, y 0.0 0.0 -3.0 -3.0 -6.0 -6.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) a0(6) 2π A1/6 A1/3 1e+0 A1/2 A2/3 Harmonic Amplitude, Ak/6 A5/6 3π/2 Harmonic Phase, ϕk/6 A1 1e-4 π 1e-8 A40 π/2 1e-12 0 0 10 20 30 40 0 10 20 30 40 Harmonic Order, k Harmonic Order, k/6 (c) (d) Fig. 24. Asymmetric period-6 motions (Ω = 0.756): (a) Trajectory of black branch (right), initial condition: (x0 , y0 ) ≈ (1.3241, 3.2092); (b) trajectory of red branch (left), initial condition: (x0 , y0 ) ≈ (1.2328, 3.7937), (c) harmonic amplitudes, (d) harmonic phases. Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). 1650159-51  A. C. J. Luo & Y. Guo A23/6 ≈ 1.2937e-3, A4 ≈ 1.2937e-3, A25/6 ≈ 7.1.3. Period-5 motions 1.0601e-3, A13/3 ≈ 0.0711, A9/2 ≈ 2.5821e-3, In this section, symmetric and asymmetric period-5 A14/3 ≈ 0.0139, A29/6 ≈ 3.6914e-4, A5 ≈ 0.0353, motions will be presented for illustration of com- A31/6 ≈ 1.5019e-3, A16/3 ≈ 7.1537e-3, A11/2 ≈ plexity in different bifurcation trees. Such a bifur- 7.1515e-4, A17/3 ≈ 0.0142, A35/6 ≈ 8.6713e-4, cation tree is closed and independent. The periodic A6 ≈ 7.0971e-3, A37/6 ≈ 4.8717e-4, A19/3 ≈ 0.0193, motion on the bifurcation tree is nontravelable. The A13/2 ≈ 7.3324e-4, A20/3 ≈ 4.3342e-3, A41/6 ≈ center for periodic motions of mod(x, 2π) in such 7.5179e-5, and A7 ≈ 6.2794e-3. The other har- bifurcation tree is near (2π, 0) or (π, 0). monic amplitudes are Ak/6 ∈ (10−13 , 10−2 ) for k = For Ω = 1.265, a symmetric period-5 motions 43, 44, . . . , 240. In Fig. 24(d), the harmonic phases is presented in Fig. 25. The initial condition of of the two period-6 motions are presented and dis- this motion is (x0 , y0 ) ≈ (0.4253, 1.2591) from the tributed in ϕk/6 ∈ [0, 2π) for k = 1, 2, . . . , 240. analytical prediction. Such a symmetric period-5 ϕRk/(2l m) = mod(ϕB k/(2l m) + k((m + 2r)/(2l m) + motion is centered at (π, 0). The time history of 1)π, 2π) for r = 0, m = 3, l = 1 with t0 = rT displacement for the symmetric period-5 motion (r ∈ {0, 1, . . . , 2l m − 1}). is presented in Fig. 25(a). Compared to period-1 by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 5.0 5T 2π 2.5 1T Displacement, mod(x,2π) IC 3π/2 3T Velocity, y 5T 2T 4T π 0.0 π/2 -2.5 IC0 -5.0 0 10 20 30 2 π/2 π 3π/2 2π Time, t Displacement, mod(x,2π) (a) (b) 2π a0(5) A1/5 ϕ3/5 A3/5 1e+0 A1 ϕ3 Harmonic Amplitude, Ak/5 3π/2 Harmonic phase, ϕk/5 ϕ1 ϕ7 ϕ13 ϕ15 ϕ19 ϕ9 1e-4 ϕ5 π ϕ1/5 ϕ11 1e-8 π/2 ϕ17 A20 1e-12 0 0 5 10 15 20 0 5 10 15 20 Harmonic Order, k/5 Harmonic Order, k/5 (c) (d) Fig. 25. Symmetric period-5 motions (Ω = 1.265): (a) Displacement, (b) trajectory, (c) harmonic amplitudes and (d) harmonic phases. Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). Initial condition: (x0 , y0 ) ≈ (0.4253, 1.2591). 1650159-52  Periodic Motions to Chaos in Pendulum and period-3 motions, the period-5 motion becomes the symmetric period-5 motion, A(2l−1)/5 = 0 but complicated. The trajectory of the symmetric A(2l)/5 = 0 (l = 1, 2, . . .). The main harmonic ampli- period-5 motion in phase space is illustrated in tudes are A1/5 ≈ 1.5831, A3/5 ≈ 0.6387, A1 ≈ Fig. 25(b). The symmetric period-5 motion is sym- 2.2893, A7/5 ≈ 0.1963, A9/5 ≈ 0.1186, A11/5 ≈ metric to itself about (π, 0) in phase space. Three 0.0839, A13/5 ≈ 7.8925e-3, and A3 ≈ 0.0151. cycles around the center of (π, 0) are formed. The The other harmonic amplitudes are A(2l−1)/5 ∈ harmonic amplitudes of the nontravelable symmet- (10−12 , 10−2 ) for l = 16, 17, . . . , 100. The harmonic ric period-5 motion are presented in Fig. 25(c). The phases are distributed in ϕ(2l−1)/5 ∈ [0, 2π) for displacement gives: from x0 ≈ 0.4253 to zero, zero l = 1, 2, . . . , 100, as shown in Fig. 25(d). down and returning back to zero, zero to 2π, 2π up With the saddle-node bifurcation of symmet- and returning back to 2π; 2π down to x5T ≈ 0.4253. ric period-5 motion, the nontravelable asymmetric The corresponding constant in the Fourier series is period-5 motion exists with center near mod(a0 , (5) a0 = π. If mod(x0 , 2π) ≈ 0.4253 and the same 2π) = π. Thus, a pair of two nontravelable asym- velocity of y0 ≈ 1.2591, there are infinite solutions metric period-5 motions is presented in Fig. 26 for (5) of period-5 motions with mod(a0 , 2π) = π. For Ω = 1.263. The initial conditions of the asymmetric by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 5.0 5.0 2T 2.5 2.5 4T 1T 5T 5T Velocity, y Velocity, y 1T 4T 3T 2T 3T 0.0 IC 0.0 IC -2.5 -2.5 -5.0 -5.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) 2π a0(5) A1/5 ϕ9 ϕ13 A2/5 ϕ1/5 1e+0 ϕ3 Harmonic Amplitude, Ak/5 3π/2 Harmonic Phase, ϕk/5 ϕ1 ϕ15 ϕ19 1e-4 ϕ5 π ϕ11 ϕ7 1e-8 A20 π/2 ϕ17 1e-12 0 0 5 10 15 20 0 5 10 15 20 Harmonic Order, k/5 Harmonic Order, k/5 (c) (d) Fig. 26. Asymmetric period-5 motions (Ω = 1.263): (a) Trajectory of black branch (right), initial condition: (x0 , y0 ) ≈ (1.7478, 0.9721); (b) trajectory of red branch (left), initial condition: (x0 , y0 ) ≈ (1.4191, 0.7131), (c) harmonic amplitudes, (d) harmonic phases. Parameters (α = 1.5, δ = 0.75, Q0 = 5.0). 1650159-53  A. C. J. Luo & Y. Guo period-5 motions are (x0 , y0 ) ≈ (1.7478, 0.9721) we cannot see the rotation complexity of displace- (black) and (x0 , y0 ) ≈ (1.4191, 0.7131) (red) from ment x. The coordinates X, Y in physical model the analytical prediction. The trajectories of the are locations of the pendulum. Using such ficti- two asymmetric period-5 motion in phase space are tious exponential functions, we can easily watch the illustrated in Figs. 26(a) and 26(b) on the left and motion complexity of angular displacement, and the right sides of mod(x, 2π) = π, respectively. The har- coefficient a > 0 in the fictitious exponential func- monic amplitudes of the two nontravelable asym- tion is arbitrarily chosen. For higher order periodic metric period-5 motions are presented in Fig. 26(c). motions, choose a to be smaller. Otherwise, it is (5)B very difficult to illustrate. Without loss of general- The two constants satisfy π − mod(a0 , 2π) ≈ (5)R ity, for the Fourier series of velocity, the symbols 0.0155 ≈ mod(a0 , 2π) − π, which is for asymme- for harmonic amplitudes and phases used will be try of mod(x, 2π) = π. The harmonic amplitudes for the same as for displacement for the nontravelable two asymmetric period-5 motions are the same. For periodic motions. The periodic motions in pendu- the two asymmetric period-5 motions, A(2l−1)/5 = 0 lum can be characterized by the rotation and libra- and A(2l)/5 = 0 (l = 1, 2, . . .). The main harmonic tion numbers as follow: amplitudes are A1/5 ≈ 1.5329, A2/5 ≈ 0.1013, by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com A3/5 ≈ 0.6173, A4/5 ≈ 0.0410, A1 ≈ 2.2816, A6/5 ≈ (R+ : R− : L) (55) 0.0124, A7/5 ≈ 0.1906, A8/5 ≈ 0.0148, A9/5 ≈ 0.1187, A2 ≈ 7.7201e-3, A11/5 ≈ 0.0838, A12/5 ≈ where R+ is the number of positive rotations, R− 1.8895e-3, A13/5 ≈ 8.7981e-3, A14/5 ≈ 2.5159e-3, is the number of negative rotations, and L is the and A3 ≈ 0.0161. The other harmonic amplitudes number of librations. The libration number consists are Ak/5 ∈ (10−14 , 10−2 ) for k = 16, 17, . . . , 100. of positive librations and negative librations. L = In Fig. 26(d), harmonic phases are presented for L+ + L− and L+ = L− for periodic motions. For ϕk/3 ∈ [0, 2π) with ϕR k/(2l m) = mod(ϕB k/(2l m) + nontravelable period-1 motions, we have R+ = R− . k((m + 2r)/(2l m) + 1)π, 2π) for l = 0, m = 5, r = 2 Definition 4. If a periodic motion in pendulum and t0 = rT with r ∈ {0, 1, . . . , 2l m − 1}. possesses R+ positive rotations, R− negative rota- tions and L librations, then the periodic motion 7.2. Travelable periodic motions is called the (R+ : R− : L)-periodic motion in pendulum. In this section, travelable asymmetric period-1 and period-2 motions to chaos will be presented for For the following illustrations, consider the illustration of complexity in different bifurcation parameters trees. Such bifurcation trees are closed and inde- pendent. The periodic motions on the bifurcation α = 1.5, δ = 0.75, Ω = 1. (56) trees are travelable. There is no center for such peri- odic motions, and the symmetric periodic motions 7.2.1. Travelable period-1 to period-2 do not exist. The travelable period-m motion has motions mod(x0 , 2π) = mod(xmT , 2π) but x0 = xmT with y0 = ymT . Thus the Fourier series of displacement On the bifurcation trees of travelable period-1 cannot exist. Herein, the Fourier series of velocity motion to chaos, a pair of two nontravelable will be presented to show harmonic effects on such asymmetric period-1 motions is presented in travelable period-m motion. To demonstrate trav- Fig. 27 for Q0 = 3.0. The initial conditions of elable periodic motions, the following formula for the travelable asymmetric period-1 motions are coordinates in the physical model are expressed by (x0 , y0 ) ≈ (0.7380, 2.8316) (black) and (x0 , y0 ) ≈ the displacement as (0.0665, 0.8547) (red) from the analytical pre- diction. The trajectories of the two asymmetric X = l0 eat cos x, Y = l0 eat sin x (54) period-1 motion in phase space are illustrated in Figs. 27(a) and 27(b). The velocity harmonic ampli- where l0 is the pendulum length, eat is the ficti- tudes of the two travelable asymmetric period-1 tious exponential function for illustration of pen- motions are presented in Fig. 27(c). The two veloc- dulum rotation motions in the physical model. For ity constants satisfy aB R 0 = −a0 = 1, which is deter- the real physical model, we have eat = 1. However, mined by angular velocity ẋ = y and y = Ω = 1. 1650159-54  Periodic Motions to Chaos in Pendulum 4.0 2.0 1T 1T IC IC 2.0 0.0 Velocity, y Velocity, y 0.0 -2.0 -2.0 -4.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π ϕ27 1e+0 A0 ϕ1 ϕ25 ϕ29 ϕ23 Velocity Harmonic Amplitude, Ak ϕ21 Velocity Harmonic Phase, ϕk 3π/2 ϕ19 1e-4 ϕ17 ϕ15 ϕ13 π ϕ11 1e-8 ϕ7 ϕ9 ϕ5 A30 π/2 1e-12 ϕ3 0 0 10 20 30 0 10 20 30 Harmonic Order, k Harmonic Order, k (c) (d) 4.0 4.0 xΤ=7.0212 xΤ=−6.2167 x0=0.7380 x0=0.0665 2.0 x2Τ 2.0 Coordinate, Y Coordinate, Y xΤ x0 x0 0.0 0.0 -2.0 -2.0 -4.0 -4.0 -4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0 Coordinate, X Coordinate, X (e) (f) Fig. 27. Travelable asymmetric period-1 motions (Q0 = 3.0): (a) Trajectory of black branch (right), initial condition: (x0 , y0 ) ≈ (0.7380, 2.8316); (b) trajectory of red branch (left), initial condition: (x0 , y0 ) ≈ (0.0665, 0.8547), (c) harmonic amplitudes, (d) harmonic phases; (e) pendulum rotation pattern (black), (f) pendulum rotation pattern (red). Parameters (α = 1.5, δ = 0.75, Ω = 1). X = e0.2t cos x, Y = e0.2t sin x. 1650159-55  A. C. J. Luo & Y. Guo The velocity harmonic amplitudes for two trave- A1 ≈ 2.3728, A3/2 ≈ 0.0674, A2 ≈ 0.4764, A5/2 ≈ lable asymmetric period-1 motions are the same. 0.1398, A3 ≈ 0.3130, A7/2 ≈ 0.0367, A4 ≈ 0.0502, The main velocity harmonic amplitudes are A1 ≈ A9/2 ≈ 0.0121, and A5 ≈ 0.0113. The other velocity 2.0164, A2 ≈ 0.6047, A3 ≈ 0.2348, A4 ≈ 0.0289, and harmonic amplitudes are Ak/2 ∈ (10−13 , 10−2 ) for A5 ≈ 0.0150. The other velocity harmonic ampli- k = 11, 12, . . . , 60. In Fig. 28(d), velocity harmonic tudes are Ak ∈ (10−14 , 10−2 ) for k = 6, 7, . . . , 30. In phases are presented and distributed in ϕk/2 ∈ Fig. 27(d), velocity harmonic phases are presented [0, 2π) for k = 1, 2, . . . , 60. ϕR B k/2 = mod(ϕk/2 + k(1+ for ϕk ∈ [0, 2π) with ϕR B k = mod(ϕk + (k + 1)π, 2π) 2r)/2 + 1)π, 2π) for r = 0 with t0 = rT (r ∈ {0, 1}). for k = 1, 2, . . . , 30. To illustrate displacement com- To illustrate displacement complexity of the trav- plexity of the travelable period-1 motions, the tra- elable period-2 motions, the trajectories of ficti- jectories of fictitious coordinates in (X, Y )-space are tious coordinates in (X, Y )-space are presented in presented in Figs. 27(e) and 27(f) for the black and Figs. 28(e) and 28(f) for the black and red branches red branches of the travelable asymmetric period-1 of the travelable period-2 motions. The parame- motions. The parameters in the fictitious exponen- ters in fictitious exponential function are l0 = 1, tial function are l0 = 1, a = 0.2. We have x0 = a = 0.1. We have x0 = 6.1621 and x2T = 18.7285 for by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. 0.7380 and xT = 7.0212 for the black branch. The Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com the black branch. The travelable period-2 motion travelable asymmetric period-1 motion with one with two cycles (4π) including two positive rota- cycle (2π) including one positive rotation plus two tions plus four librations is on the positive direction librations is on the positive direction of x(t). This of x(t). The positive travelable period-1 motion is of pendulum rotation is not simple positive and neg- (R+ : R− : L) = (2 : 0 : 4)-kind. For the red branch, ative rotations. Such a positive travelable period-1 we have x0 = 0.0977 and x2T = −12.4687 for the motion is of (R+ : R− : L) = (1 : 0 : 2)-kind. red branch. The travelable period-2 motion with For the red branch, we have x0 = 0.0665 and the two negative cycles (−4π) including two nega- xT = −6.2167. The travelable asymmetric period-1 tive rotations plus four librations is on the negative motion with one negative cycle (−2π) including direction of x(t). The negative travelable period-1 one negative rotation plus two librations is on the motion is of (R+ : R− : L) = (0 : 2 : 4)-kind. negative direction of x(t). The negative travelable period-1 motion is of (R+ : R− : L) = (0 : 1 : 2)- kind. 7.2.2. Travelable period-2 to period-8 After the period-doubling bifurcation of the motions travelable asymmetric period-1 motion, the trave- On the bifurcation trees of travelable period-2 lable asymmetric period-2 motions will be obtained. motion to chaos, a pair of two nontravelable Consider excitation amplitude of Q0 = 3.2 for a asymmetric period-2 motions is presented in pair of two travelable asymmetric period-2 motions. Fig. 29 for Q0 = 7.37. The initial conditions of The two trajectories, velocity harmonic amplitudes the travelable asymmetric period-2 motions are and phases of the travelable asymmetric period-2 (x0 , y0 ) ≈ (1.2414, 2.2053) (black) and (x0 , y0 ) ≈ motions are presented in Fig. 28. The correspond- (0.1376, 4.7541) (red) from the analytical pre- ing initial conditions are (x0 , y0 ) ≈ (6.1621, 3.3869) diction. The trajectories of the two asymmetric (black) and (x0 , y0 ) ≈ (0.0977, 0.9019) (red) from period-2 motion in phase space are illustrated in the analytical prediction. The trajectories of the Figs. 29(a) and 29(b), which are different from two asymmetric period-2 motion in phase space are the travelable period-2 motions in Figs. 28(a) illustrated in Figs. 28(a) and 28(b). The velocity and 28(b). The velocity harmonic amplitudes of harmonic amplitudes of the two travelable asym- the two travelable asymmetric period-2 motions metric period-2 motions are presented in Fig. 28(c). are presented in Fig. 29(c). The two velocity con- The two velocity constants for the travelable asym- (2)B (2)R stants satisfy −a0 = a0 = 0.5, which is (2)B (2)R metric period-2 motions are a0 = −a0 = 1, determined by angular velocity ẋ = y and y = determined by angular velocity ẋ = y and y = Ω = Ω/2 = 0.5. The velocity harmonic amplitudes for 1. The velocity harmonic amplitudes for two asym- two travelable asymmetric period-2 motions are the metric period-2 motions are the same. For the two same. The main velocity harmonic amplitudes are travelable asymmetric period-2 motions, the main A1/2 ≈ 0.4784, A1 ≈ 5.9068, A3/2 ≈ 0.2423, A2 ≈ velocity harmonic amplitudes are A1/2 ≈ 0.1581, 0.3507, A5/2 ≈ 0.0574, A3 ≈ 0.0720, A7/2 ≈ 0.1281, 1650159-56  Periodic Motions to Chaos in Pendulum 5.0 3.0 2T 1T IC 3.0 1.0 1T 2T IC Velocity, y Velocity, y 1.0 -1.0 -1.0 -3.0 -3.0 -5.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π ϕ7 ϕ13 ϕ ϕ1 ϕ11 1e+0 A0 15 ϕ17 ϕ19 ϕ ϕ A1/2 21 23 ϕ25 Velocity Harmonic Amplitude, Ak/2 A1 ϕ27 ϕ 29 Velocity Harmonic Phase, ϕk/2 3π/2 1e-4 π 1e-8 A30 π/2 1e-12 ϕ9 ϕ3 ϕ 5 0 0 10 20 30 0 10 20 30 Harmonic Order, k/2 Harmonic Order, k/2 (c) (d) 4.0 4.0 x2Τ=18.7285 x2Τ=−12.4687 x0=6.1621 x0=0.0977 2.0 2.0 Coordinate, Y Coordinate, Y x2Τ x0 0.0 0.0 x0 x2Τ -2.0 -2.0 -4.0 -4.0 -4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0 Coordinate, X Coordinate, X (e) (f) Fig. 28. Travelable asymmetric period-2 motions (Q0 = 3.2): (a) Trajectory of black branch (right), initial condition: (x0 , y0 ) ≈ (6.1621, 3.3869); (b) trajectory of red branch (left), initial condition: (x0 , y0 ) ≈ (0.0977, 0.9019), (c) velocity har- monic amplitudes, (d) velocity harmonic phases; (e) pendulum rotation pattern (black), (f) pendulum rotation pattern (red). Parameters (α = 1.5, δ = 0.75, Ω = 1). X = e0.1t cos x, Y = e0.1t sin x. 1650159-57  A. C. J. Luo & Y. Guo 8.0 8.0 1T 2T 4.0 4.0 IC 2T 1T Velocity, y Velocity, y IC 0.0 0.0 -4.0 -4.0 -8.0 -8.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π 1e+0 A0 ϕ1 ϕ5 ϕ13 ϕ25 ϕ31 A1/2 ϕ37 Velocity Harmonic Amplitude, Ak/2 A1 ϕ21 Velocity Harmonic Phase, ϕk/2 3π/2 1e-4 π ϕ3 1e-8 A40 π/2 1e-12 0 0 10 20 30 40 0 10 20 30 40 Harmonic Order, k/2 Harmonic Order, k/2 (c) (d) 4.0 4.0 x2Τ x2Τ=−5.0418 x2Τ=6.4208 x0=1.2414 x0=0.1376 2.0 2.0 x2Τ Coordinate, Y Coordinate, Y x0 0.0 0.0 x0 -2.0 -2.0 -4.0 -4.0 -4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0 Coordinate, X Coordinate, X (e) (f) Fig. 29. Travelable asymmetric period-2 motions (Q0 = 7.37): (a) Trajectory of black branch (right), initial condition: (x0 , y0 ) ≈ (1.2414, 2.2053); (b) trajectory of red branch (left), initial condition: (x0 , y0 ) ≈ (0.1376, 4.7541), (c) velocity har- monic amplitudes, (d) velocity harmonic phases; (e) pendulum rotation pattern (black), (f) pendulum rotation pattern (red). Parameters (α = 1.5, δ = 0.75, Ω = 1). X = e0.1t cos x, Y = e0.1t sin x. 1650159-58  Periodic Motions to Chaos in Pendulum A4 ≈ 0.0875, A9/2 ≈ 0.2367, A5 ≈ 0.1149, A11/2 ≈ period-4 motions are also the same. For the two 0.0469, A6 ≈ 0.0863, A13/2 ≈ 0.0895, A7 ≈ 0.0307, travelable asymmetric period-4 motions, the main A15/2 ≈ 6.0042e-3, A8 ≈ 0.0175, A17/2 ≈ 0.0128, velocity harmonic amplitudes are A1/4 ≈ 0.0205, and A9 ≈ 6.0044e-3. The other velocity har- A1/2 ≈ 0.4829, A3/4 ≈ 0.0285, A1 ≈ 5.8727, monic amplitudes are Ak/2 ∈ (10−13 , 10−2 ) for A5/4 ≈ 0.0223, A3/2 ≈ 0.2398, A7/4 ≈ 0.0103, k = 19, 20, . . . , 80. In Fig. 29(d), velocity harmonic A2 ≈ 0.3567, A9/4 ≈ 4.1114e-3, A5/2 ≈ 0.0564, phases are presented for ϕk/2 ∈ [0, 2π). ϕR k/2 = A11/4 ≈ 3.0961e-3, A3 ≈ 0.0759, A13/4 ≈ 3.3763e-3, B mod(ϕk/2 + (k(1 + 2r)/2 + 1)π, 2π) for r = 1 with A7/2 ≈ 0.1306, A15/4 ≈ 7.3856e-3, A4 ≈ 0.0946, t0 = rT (r ∈ {0, 1}) and k = 1, 2, . . . , 80. To A17/4 ≈ 8.6219e-3, A9/2 ≈ 0.2366, A19/4 ≈ 0.0108, illustrate displacement complexity of the travelable A5 ≈ 0.1123, A21/4 ≈ 0.0123, A11/2 ≈ 0.0423, period-2 motions, the trajectories of fictitious coor- A23/4 ≈ 6.9401e-3, A6 ≈ 0.0865, A25/4 ≈ 1.5894e-3, dinates in (X, Y )-space are presented in Figs. 29(e) A13/2 ≈ 0.0876, A27/4 ≈ 2.5433e-3, A7 ≈ 0.0301, and 29(f) for the black and red branches of the A29/4 ≈ 3.3848e-3, A15/2 ≈ 5.2033e-3, A31/4 ≈ travelable period-2 motions. The parameters in fic- 1.8303e-3, A8 ≈ 0.0173, A33/4 ≈ 2.6111e-4, A17/2 ≈ titious exponential function are l0 = 1, a = 0.1. We 0.0122, A35/4 ≈ 1.5399e-4, and A9 ≈ 6.0248e-3. by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com have x0 = 1.2414 and x2T = −5.0418 for the black Expect for A(4l−3)/4 and A(4l−1)/4 , the velocity har- branch. The travelable period-2 motion with one monic amplitudes A(4l−4)/4 and A(4l−2)/4 for the negative cycle (−2π), including two positive rota- period-4 motion are close to the velocity harmonic tions, three negative rotations plus four librations, amplitudes A2(l−1)/2 and A(2l−1)/2 . The other veloc- is on the negative direction of x(t). The negative ity harmonic amplitudes are Ak/4 ∈ (10−13 , 10−2 ) travelable period-2 motion is of (R+ : R− : L) = for k = 37, 38, . . . , 120. In Fig. 30(d), velocity (2 : 3 : 4)-kind. For the red branch, we have x0 = harmonic phases are presented and distributed in 0.1376 and xT = 6.4208. The travelable period-2 ϕk/4 ∈ [0, 2π). ϕR B k/4 = mod(ϕk/4 + k(1 + 2r)/4 + motion with one positive cycle (2π), including 1)π, 2π) with r = 0 with t0 = rT (r ∈ {0, 1, 2, 3}) three positive rotations, two negative rotations for k = 1, 2, . . . , 120. To illustrate displacement plus four librations, is on the positive direction of complexity of the travelable period-4 motions, the x(t). The positive travelable period-2 motion is of trajectories of fictitious coordinates in (X, Y )-space (R+ : R− : L) = (3 : 2 : 4)-kind. are still used for the black and red branches of On the bifurcation tree of travelable period-2 the travelable period-4 motions, as presented in motion to chaos, after the period-doubling bifurca- Figs. 30(e) and 30(f). The parameters in fictitious tion of the travelable asymmetric period-2 motion, exponential function are l0 = 1, a = 0.05. We the travelable asymmetric period-4 motions will be have x0 = 1.2883 and x4T = −11.2781 for the obtained. Consider excitation amplitude of Q0 = black branch. The travelable period-4 motion with 7.34 for a pair of two travelable asymmetric period-4 two negative cycles (−4π), including four positive motions. The two trajectories, velocity harmonic rotations, six negative rotations plus eight libra- amplitudes and phases of the travelable asymmet- tions is on the negative direction of x(t). The neg- ric period-4 motions are presented in Fig. 30. The ative travelable period-4 motion is of (R+ : R− : initial conditions are (x0 , y0 ) ≈ (1.2883, 2.1487) L) = (4 : 6 : 8)-kind. For the red branch, we have (black) and (x0 , y0 ) ≈ (4.8056, 4.5306) (red) from x0 = 4.8056 and x4T = 17.3720 for the red branch. the analytical prediction. The trajectories of the The travelable period-4 motion with two positive two asymmetric period-4 motions in phase space are cycles (4π), including six positive rotations, four illustrated in Figs. 30(a) and 30(b). The trajectories negative rotations plus eight librations, is on the of period-4 motions are like the doubled period-2 positive direction of x(t). The negative travelable motions. The velocity harmonic amplitudes of the period-1 motion is of (R+ : R− : L) = (6 : 4 : 8)- two travelable asymmetric period-4 motions are kind. presented in Fig. 30(c). The two velocity constants On the bifurcation tree of travelable period-2 for the travelable asymmetric period-4 motions are motion to chaos, after the period-doubling bifurca- (4)B (4)R −a0 = a0 = 0.5 which is determined by tion of the travelable asymmetric period-4 motion, angular velocity ẋ = y and y = Ω/2 = 0.5. The the travelable asymmetric period-8 motions will velocity harmonic amplitudes for two asymmetric be obtained. Consider excitation amplitude of 1650159-59  A. C. J. Luo & Y. Guo 8.0 8.0 3T 2T 4.0 4.0 4T 2T 4T 1T IC 3T IC Velocity, y Velocity, y 1T 0.0 0.0 -4.0 -4.0 -8.0 -8.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π ϕ5 A0 1e+0 A1/4 ϕ1 ϕ25 Velocity Harmonic Amplitude, Ak/4 A1/2 Velocity Harmonic Phase, ϕk/4 A1/8 A1 3π/2 1e-4 ϕ0 π ϕ39 1e-8 A40 π/2 1e-12 0 0 10 20 30 40 0 10 20 30 40 Harmonic Order, k/4 Harmonic Order, k/4 (c) (d) 4.0 4.0 x4Τ x4Τ=−11.2781 x4Τ=17.3720 x0=1.2883 x0=4.8056 2.0 2.0 Coordinate, Y Coordinate, Y x0 0.0 0.0 x0 -2.0 -2.0 x4Τ -4.0 -4.0 -4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0 Coordinate, X Coordinate, X (e) (f) Fig. 30. Travelable asymmetric period-4 motions (Q0 = 7.34): (a) Trajectory of black branch (right), initial condition: (x0 , y0 ) ≈ (1.2883, 2.1487); (b) trajectory of red branch (left), initial condition: (x0 , y0 ) ≈ (4.8056, 4.5306), (c) velocity har- monic amplitudes, (d) velocity harmonic phases; (e) pendulum rotation pattern (black), (f) pendulum rotation pattern (red). Parameters (α = 1.5, δ = 0.75, Ω = 1). X = e0.05t cos x, Y = e0.05t sin x. 1650159-60  Periodic Motions to Chaos in Pendulum 8.0 8.0 4T 3T 2T 1T 4.0 4.0 IC 8T 7T 2T 5T 7T 6T 5T 6T 1T 3T IC 4T Velocity, y Velocity, y 8T 0.0 0.0 -4.0 -4.0 -8.0 -8.0 0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π Displacement, mod(x,2π) Displacement, mod(x,2π) (a) (b) by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 2π A0 1e+0 A1/8 A1/4 Velocity Harmonic Amplitude, Ak/8 A3/8 Velocity Harmonic Phase, ϕk/8 A1/2 3π/2 A5/8 1e-4 A3/4 A7/8 A1 π 1e-8 A40 π/2 ϕ9 ϕ3 ϕ 1e-12 5 0 0 10 20 30 40 0 10 20 30 40 Harmonic Order, k/8 Harmonic Order, k/8 (c) (d) 4.0 4.0 x8Τ x8Τ=−23.8258 x8Τ=25.3838 x0=1.3069 x0=0.2511 2.0 2.0 x8Τ Coordinate, Y Coordinate, Y x0 0.0 0.0 x0 -2.0 -2.0 -4.0 -4.0 -4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0 Coordinate, X Coordinate, X (e) (f) Fig. 31. Travelable asymmetric period-8 motions (Q0 = 7.33): (a) Trajectory of black branch (right), initial condition: (x0 , y0 ) ≈ (1.3069, 2.1298); (b) trajectory of red branch (left), initial condition: (x0 , y0 ) ≈ (0.2511, 4.6304), (c) velocity har- monic amplitudes, (d) velocity harmonic phases; (e) pendulum rotation pattern (black), (f) pendulum rotation pattern (red). Parameters (α = 1.5, δ = 0.75, Ω = 1). X = e0.025t cos x, Y = e0.025t sin x. 1650159-61  A. C. J. Luo & Y. Guo Q0 = 7.33 for a pair of two travelable asym- rotations, 12 negative rotations plus 16 librations is metric period-8 motions. The two trajectories, on the negative direction of x(t). The negative trave- velocity harmonic amplitude and phase of the lable period-8 motion is of (R+ : R− : L) = (8 : 12 : travelable asymmetric period-8 motions are pre- 16)-kind. For the red branch, we have x0 = −0.2511 sented in Fig. 31. The initial conditions are and x8T = 25.3838 for the red branch. The trave- (x0 , y0 ) ≈ (1.3069, 2.1298) (black) and (x0 , y0 ) ≈ lable period-4 motion with two positive cycle (8π), (0.2511, 4.6304) (red) from the analytical pre- including 12 positive rotations, eight negative rota- diction. The trajectories of the two asymmetric tions plus 16 librations, is on the positive direction period-8 motion in phase space are illustrated in of x(t). The negative travelable period-1 motion is Figs. 31(a) and 31(b). The trajectories of period-8 of (R+ : R− : L) = (12 : 8 : 16)-kind. motions are like the doubled period-4 motions and they are also close to the corresponding period-2 8. Conclusions motions. The velocity harmonic amplitudes of the two travelable asymmetric period-4 motions are In this paper, bifurcation trees of nontravelable and presented in Fig. 31(c). The two velocity constants travelable periodic motions to chaos in a periodi- for the travelable asymmetric period-8 motions are cally forced pendulum were obtained by a semi- by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com (8)B (8)R analytical method. The bifurcation trees varying −a0 = a0 = 0.5. The velocity harmonic ampli- tudes for two asymmetric period-8 motions are the with excitation frequency and amplitude were pre- same. For the two travelable asymmetric period-8 sented. The method is based on discretization of motions, the velocity harmonic amplitudes are of differential equations of the dynamical system to four types: (i) A1 ≈ 5.8666, A2 ≈ 0.3576, A3 ≈ obtain implicit maps. Using the implicit maps, 0.0767, A4 ≈ 0.0956, A5 ≈ 0.1126, A6 ≈ 0.0865, mapping structures for specific periodic motions A7 ≈ 0.0302, A8 ≈ 0.0172, A9 ≈ 6.0406e-3; (ii) were developed, and the corresponding periodic A1/2 ≈ 0.4851, A3/2 ≈ 0.2407, A5/2 ≈ 0.0568, motions were determined analytically through such A7/2 ≈ 0.1312, A9/2 ≈ 0.2359, A11/2 ≈ 0.0422, mapping structures. Analytical bifurcation trees of A13/2 ≈ 0.0871, A15/2 ≈ 5.0871e-3, A17/2 ≈ 0.0121; periodic motions were presented, and the corre- (iii) A1/4 ≈ 0.0219, A3/4 ≈ 0.0306, A5/4 ≈ sponding stability and bifurcation analysis of peri- 0.0239, A7/4 ≈ 0.0110, A9/4 ≈ 4.4229e-3, A11/4 ≈ odic motion were carried out through eigenvalue 3.3290e-3, A13/4 ≈ 3.6368e-3, A15/4 ≈ 7.9437e-3, analysis. The nontravelable and travelable periodic A17/4 ≈ 9.2623e-3, A19/4 ≈ 0.0116, A21/4 ≈ 0.0132, motions on the bifurcation trees were discovered. A23/4 ≈ 7.4045e-3, Ak/4 < 10−2 (k = 4l − 1, 4l − 3 From the analytical predictions of periodic motions, the frequency-amplitude characteristic curves were and l = 7, 8 . . .); (iv) Ak/8 < 10−2 (k = 8l−7, 8l−5, obtained for a better understanding of motions 8l − 3, 8l − 1 and l = 1, 2 . . .). The other velocity complexity in the periodically forced pendulum. harmonic amplitudes are Ak/8 ∈ (10−13 , 10−2 ) for Finally, numerical simulations of selected periodic k = 73, 74, . . . , 320. In Fig. 31(d), velocity harmonic motions were illustrated. Through this investiga- phases are presented and distributed in ϕk/8 ∈ tion, one can better understand the complexity of [0, 2π). ϕR B k/8 = mod(ϕk/8 + k(1 + 2r)/8 + 1)π, 2π) nontravelable and travelable periodic motions to for r = 1 with t0 = rT (r ∈ {0, 1, . . . , 7}) for chaos in the periodically forced pendulums. Based k = 1, 2, . . . , 320. To illustrate displacement com- on the traditional analytical methods, one can- plexity of the travelable period-8 motions, the tra- not achieve the adequate solutions presented herein jectories of fictitious coordinates in (X, Y )-space for periodic motions of the periodically forced are still used for the black and red branches of pendulum. the travelable period-1 motions, as presented in Figs. 31(e) and 31(f). The parameters in fictitious exponential function are l0 = 1, a = 0.05. We References have x0 = 1.3069 and x4T = −23.8258 for the Barkham, P. G. D. & Soudack, A. C. [1969] “An exten- black branch. The travelable period-4 motion with sion to the method of Krylov and Bogoliubov,” Int. two negative cycles (−8π), including eight positive J. Contr. 10, 377–392. 1650159-62  Periodic Motions to Chaos in Pendulum Barkham, P. G. D. & Soudack, A. C. [1970] “Approxi- Luo, A. C. J. [2001] “Resonance and stochastic layer in a mate solutions of nonlinear, non-autonomous second- parametrically excited pendulum,” Nonlin. Dyn. 26, order differential equations,” Int. J. Contr. 11, 355–367. 763–767. Luo, A. C. J. [2002] “Resonant layer in a parametrically Ben-Jacob, E., Goldhirsch, I., Imry, Y. & Fishman, S. excited pendulum,” Int. J. Bifurcation and Chaos 12, [1982] “Intermittent chaos in Josephson junctions,” 409–419. Phys. Rev. Lett. 49, 1599–1602. Luo, A. C. J. [2012] Continuous Dynamical Systems Coppola, V. T. & Rand, R. H. [1990] “Averaging using (HEP/L&H Scientific, Beijing/Glen Carbon). elliptic functions: Approximation of limit cycle,” Acta Luo, A. C. J. & Huang, J. Z. [2012a] “Approximate solu- Mech. 81, 125–142. tions of periodic motions in nonlinear systems via a de Paula, A. S., Savi, M. A. & Pereira-Pinto, F. H. I. generalized harmonic balance,” J. Vibr. Contr. 18, [2006] “Chaos and transient chaos in an experimental 1661–1871. nonlinear pendulum,” J. Sound Vibr. 294, 585–595. Luo, A. C. J. & Huang, J. Z. [2012b] “Analytical dynam- Fatou, P. [1928] “Sur le mouvement d’un systeme soumis ics of period-m flows and chaos in nonlinear systems,” ‘a des forces a courte periode,” Bull. Soc. Math. 56, Int. J. Bifurcation and Chaos 22, 1250093-1–29. 98–139. Luo, A. C. J. & Huang, J. Z. [2012c] “Analytical routines Garcia-Margallo, J. & Bejarano, J. D. [1987] “A general- of period-1 motions to chaos in a periodically forced by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com ization of the method of harmonic balance,” J. Sound Duffing oscillator with twin-well potential,” J. Appl. Vibr. 116, 591–595. Nonlin. Dyn. 1, 73–108. Guo, Y. & Luo, A. C. J. [2015a] “Periodic motions in Luo, A. C. J. & Huang, J. Z. [2012d] “Unstable and a double-well Duffing oscillator under periodic excita- stable period-m motions in a twin-well potential tion through discrete implicit mappings,” Int. J. Dyn. Duffing oscillator,” Discont. Nonlin. Compl. 1, 113– Contr., DOI: 10.1007/s40435-015-0161-6. 145. Guo, Y. & Luo, A. C. J. [2015b] “Bifurcation tree Luo, A. C. J. & Guo, Y. [2015] “A semi-analytical predic- of period-1 motion to chaos in a Duffing oscilla- tion of periodic motions in Duffing oscillator through tor with double-well potential,” ASME 2015 Int. mapping structures,” Discont. Nonlin. Compl. 4, Design Engineering Technical Conference & Com- 13–44. puters and Information in Engineering Conf., Paper Luo, A. C. J. [2015a] “Periodic flows to chaos based No. IDETC/CIE2015-48104. on discrete implicit mappings of continuous nonlinear Guo, Y. & Luo, A. C. J. [2016] “Routes of periodic systems,” Int. J. Bifurcation and Chaos 25, 1550044- motions to chaos in a periodically forced pendu- 1–62. lum,” Int. J. Dyn. Contr., DOI: 10.1007/S40435-016- Luo, A. C. J. [2015b] Discretization and Implicit Map- 0249-7. ping Dynamics (HEP/Springer, Beijing/Dordrecht). Gwinn, E. G. & Westervelt, R. M. [1985] “Intermittent Nayfeh, A. H. [1973] Perturbation Methods (Wiley, NY). chaos and low-frequency noise in the driven damped Nayfeh, A. H. & Mook, D. T. [1979] Nonlinear Oscilla- pendulum,” Phys. Rev. Lett. 54, 1613–1616. tion (Wiley, NY). Hayashi, C. [1964] Nonlinear Oscillations in Physical Poincaré, H. [1899] Methodes Nouvelles de la Mecanique Systems (McGraw-Hill Book Company, NY). Celeste, Vol. 3 (Gauthier-Villars, Paris). Kadanoff, L. P. [1985] “From periodic motion to Rand, R. H. & Armbruster, D. [1987] Perturbation unbounded chaos: Investigations of the simple pen- Methods, Bifurcation Theory, and Computer Algebra, dulum,” Phys. Scripta T9, 5–10. Applied Mathematical Sciences, Vol. 65 (Springer- Krylov, N. M. & Bogoliubov, N. N. [1935] Meth- Verlag, NY). odes Approchees de la Mecanique Non-Lineaire dans Salam, F. M. A. & Sastry, S. S. [1985] “Dynamics leurs Application a l’Aeetude de la Perturbation des of forced Josephson junction circuit: The regions of Mouvements Periodiques de Divers Phenomenes de chaos,” IEEE Trans. Circuit Syst. 32, 784–796. Resonance s’y Rapportant (Academie des Sciences van der Pol, B. [1920] “A theory of the amplitude of free d’Ukraine, Kiev). (in French). and forced triode vibrations,” Radio Rev. 1, 701–710, Lagrange, J. L. [1788] Mecanique Analytique, Vol. 2 (edi- 754–762. tion Albert Blanchard, Paris, 1965). Wang, Y. F. & Liu, Z. W. [2015] “A matrix-based com- Luo, A. C. J. & Han, R. P. S. [2000] “The dynamics of putational scheme of generalized harmonic balance stochastic and resonant layers in a periodically driven method for periodic solutions of nonlinear vibratory pendulum,” Chaos Solit. Fract. 11, 2349–2359. systems,” J. Appl. Nonlin. Dyn. 4, 379–389. 1650159-63  A. C. J. Luo & Y. Guo Yuste, S. B. & Bejarano, J. D. [1986] “Construction of Yuste, S. B. & Bejarano, J. D. [1990] “Improvement of approximate analytical solutions to a new class of a Krylov–Bogoliubov method that uses Jacobi elliptic non-linear oscillator equations,” J. Sound Vibr. 110, functions,” J. Sound Vibr. 139, 151–163. 347–350. Zaslavsky, G. M. & Chirikov, B. V. [1972] “Stochastic Yuste, S. B. & Bejarano, J. D. [1989] “Extension and instability of nonlinear oscillations,” Sov. Phys. Usp. improvement to the Krylov–Bogoliubov method that 14, 549–672. use elliptic functions,” Int. J. Contr. 49, 1127–1141. by CITY UNIVERSITY OF HONG KONG on 08/30/16. For personal use only. Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com 1650159-64