Chaos (Physics)

A system in a spatially (quasi-)degenerate ground state responds in a qualitatively different way to a change in the external potential. Consequently, the usual method for computing the Fukui function, namely, taking the difference between the electron densities of the N- and N ± 1 electron systems, cannot be applied directly. It is shown how the Fukui matrix, and thus also the Fukui function, depends on the nature of the perturbation. One thus needs to use degenerate perturbation theory for the given perturbing potential to generate the density matrix whose change with respect to a change in the number of electrons equals the Fukui matrix. Accounting for the degeneracy in the case of nitrous oxide reveals that an average over the degenerate states differs significantly from using the proper density matrix. We further show the differences in Fukui functions depending on whether a Dirac delta perturbation is used or an interaction with a true point charge (leading to the Fukui potent...

In this paper we study quasiperiodically forced systems exhibiting fractal and Wada basin boundaries. Specifically, by utilizing a class of representative systems, we analyze the dynamical origin of such basin boundaries and we characterize them. Furthermore, we find that basin boundaries in a quasiperiodically driven system can undergo a unique type of bifurcation in which isolated ''islands'' of basins of attraction are created as a system parameter changes. The mechanism for this type of basin boundary bifurcation is elucidated.

This paper presents a unifying model-the Human (Bio)Aetheric Temple-linking biological, scalar, and photonic processes in the human body. Derived from the Unified Biofield Model (UBM) and the Britt-Howard Unified Aether Theory (BHUAT), it describes the transition of human biology from carbon-based chemistry toward a crystalline, photonic, and holographic architecture. Through integration of biophoton coherence, structured water phase transitions, and scalar wave entrainment, the study proposes that the human organism operates as a living light resonator-a coherent interface between matter and consciousness. Simulations and comparative analyses demonstrate how exceeding an 85 % biophotonic coherence threshold catalyzes molecular reorganization, initiating a "Carbon-to-Crystal" cascade that couples DNA fractal resonance, melatonin τ-phase modulation, and nested toroidal field stability. The results synthesize empirical biophysics with metaphysical field theory, proposing experimental pathways toward measurable consciousness-field coupling.

Traditional cosmology and information theory have long regarded "ex nihilo" (creation out of nothing) and the "emergence of complexity" as inexplicable a priori initial conditions or random statistical outcomes. Grounded in the "0/∞ Null Model" and the "Self-Consistency of the 0=∞ Paradox," this study proposes a pure evolutionary framework based on the topological dynamics of the absolute vacuum, aiming to thoroughly deconstruct the geometric self-organization mechanisms of the physical universe from pure nothingness to high-order complexity.
Rejecting phenomenological assumptions such as "supernatural creation" or "random quantum fluctuations," this paper establishes the First Cause of the physical universe as the "zero-mass fall of a future information deadlock particle." Through this causality-breaking topological closed loop, this study deduces the six major dynamical stages of the emergence of universal complexity:
Establishment of the First Cause: A future wave function undergoes a phase transition into a zero-mass deadlocked entity, escapes the arrow of time, and falls into the absolute vacuum.
Static Infinity and the Quantum Vacuum: The mutual repulsion between the deadlocked particle and the absolute vacuum instantaneously excites an infinite number of linear wave functions under the 0-state, reaching full degrees of freedom (Ω→1≡0) and forming a static quantum vacuum filled with linear interaction forces.
Wave Function Curling and the Big Bang: Full degrees of freedom prevent space from expanding ex nihilo. To dissipate the continuous topological repulsion, the system forces all infinitely long straight lines to curl inward into a "singularity." The many-body repulsion of virtual particles inside the singularity triggers the Big Bang; the wave functions undergo a phase transition from parallel straight lines to an intricately interwoven mesh, forcing the emergence of the time metric (dt > 0).
Reverse Generation of Future Particles: Evolving wave functions encounter a polarity transition (from white to black) in the underlying phase space, losing their historical trajectories while retaining limit interaction forces. This triggers an infinite recursion (0123...0), forming a new deadlock that falls back into the absolute vacuum.
Exponential Inflation of Complexity: The number of particles falling back into the absolute vacuum increases with infinite recursive cycles. The many-body repulsion forces superimpose exponentially, driving an endless escalation in the information mesh and dimensional complexity of the physical universe.
Dynamic Transcendence of the Riemann Sphere: Unlike the static quantum vacuum, which is ultimately destined for topological erasure, this universe operates as a 4D Riemann sphere. Through its infinite-speed inside-out folding and solid sphere projection mechanisms, it transforms the full degrees of freedom into the eternal expansion of its outer shell, establishing the universe's topological status as the only True Infinity.
This study demonstrates that the genesis and complexification of the physical universe are purely the inevitable geometric curling, phase space evolution, and retrocausal closed loops exhibited by the absolute vacuum when processing "limit repulsion forces" and "information deadlocks."

By analytically continuing the coupling constant g of a coupled quantum theory, one can, at least in principle, arrive at a state whose energy is lower than the ground state of the theory. The idea is to begin with the uncoupled g = 0 theory in its ground state, to analytically continue around an exceptional point (square-root singularity) in the complexcoupling-constant plane, and finally to return to the point g = 0. In the course of this analytic continuation, the uncoupled theory ends up in an unconventional state whose energy is lower than the original ground state energy. However, it is unclear whether one can use this analytic continuation to extract energy from the conventional vacuum state; this process appears to be exothermic but one must do work to vary the coupling constant g.

We consider a random matching model where heterogeneous agents endogenously choose to invest time and real resources in education. Generically, there is a steady state equilibrium, where some agents, but not all of them, choose to invest. Regular steady state equilibria are constrained Pareto inefficient in a strong sense. The Hosios (1990) condition is neither necessary, nor sufficient, for constrained Pareto optimality. We also provide restrictions on the fundamentals sufficient to guarantee that equilibria are characterized by overeducation (or undereducation), and present some results on their comparative statics properties.

We state and solve a set of relatively harder problems in classical mechanics, generally through the construction of the Lagrangian for each case.  Undergraduate students having Physics as their Major ( or Honours) subject may find the solutions helpful in developing their skills.

The effects of the relativistic corrections on the energy spectra are analyzed. Effective simulations based on manipulations of operators in the Sturmian basis are developed. Discrete and continuous energy spectra of a hydrogen atom with realistic nucleus mass in a strong magnetic field are computed. The transition from regularity to chaos in diamagnetic problem with the effect of the nucleus recoil energy is explored. Anticrossing of energy levels is observed for strong magnetic field.

We study the statistical properties of generalized intensities (squared matrix elements of Hermitian operators) for the hydrogen atom in strong magnetic fields in a range of parameters where the classical analogue of the system exhibits completely chaotic dynamics. In this way we extend previous work by Prosen and Robnik on the statistics of generalized intensities in billiard systems in the transition region with mixed classical dynamics. We observe deviations from the statistics found in that work, and demonstrate that these are due to the effect of scarring of wave functions by unstable periodic orbits.

We evaluate the Gutzwiller trace formula for the level density of classically chaotic systems by considering the level density in a bounded energy range and truncating its Fourier integral. This results in a limiting procedure which comprises a convergent semiclassical approximation to a well defined spectral quantity at each stage. We test this result on the spectrum of zeros of the Riemann zeta function, obtaining increasingly good approximations to the level density. The Fourier approach also explains the origin of the convergence problems encountered by the orbit truncation scheme.

The lattice limit-cycle (LLC) model is introduced as a minimal mean-field scheme which can model reactive dynamics on lattices (low dimensional supports) producing non-linear limit cycle oscillations. Under the influence of an external periodic force the dynamics of the LLC may be drastically modified. Synchronization phenomena, bifurcations and transitions to chaos are observed as a function of the strength of the force. Taking advantage of the drastic change on the dynamics due to the periodic forcing, it is possible to modify the output/product or the production rate of a chemical reaction at will, simply by applying a periodic force to it, without the need to change the support properties or the experimental conditions.

A novel Random Matrix Ensemble is introduced which mimics the global structure inherent in the Hamiltonian matrices of autonomous, ergodic systems. Changes in its parameters induce a transition between a Poisson and a Wigner distribution for the level spacings, P(s). The intermediate distributions are uniquely determined by a single scaling variable. Semiclassical constraints force the ensemble to be in a regime with Wigner P(s) for systems with more than two freedoms.

Quantal (E, 'T) plots are constructed from the eigenvalues of the quantum system. We demonstrate that these representations display the periodic orbits of the classical system, including bifurcations and the transition from stable to unstable.

Quantal (E,τ) plots are constructed from the eigenvalues of the quantum system. We demonstrate that these representations display the periodic orbits of the classical system, including bifurcations and the transition from stable to unstable.

In this paper, a generalized scheme is proposed for designing multistable continuous dynamical systems. The scheme is based on the concept of partial synchronization of states and the concept of constants of motion. The most important observation is that by coupling two mdimensional dynamical systems, multistable nature can be obtained if i number of variables of the two systems are completely synchronized and j number of variables keep a constant difference between them i.e., their differences are constants of motion, where i + j = m and 1 ≤ i, j ≤ m − 1. The proposed scheme is illustrated by taking coupled Lorenz systems and coupled chaotic Lorenzlike systems. According to the scheme, two coupled systems reduce to single modified system with some initial condition-dependent parameters. Time evolution plots, phase diagrams, variation of maximum Lyapunov exponent and bifurcation diagrams of the systems are presented to show the multistable nature of the coupled systems.

This work is aiming to show the advantage of using the Lie algebraic decomposition technique to solvefor Schrödinger’s wave equation for a quantum model, compared with the direct method of solution. The advantageis a two-fold: one is to derive general form of solution, and, two is relatively manageable to deal with the case oftime-dependent system Hamiltonian. Specifically, we consider the model of 2-level optical atom and solve for thecorresponding Schrödinger’s wave equation using the Lie algebraic decomposition technique. The obtained formof solution for the wave function is used to examine computationally the atomic localization in the coordinate space.For comparison, the direct method of solution of the wave function is analysed in order to show its complicationwhen dealing with time-dependent Hamiltonian.The possibility of using the Lie algebraic method for a qubit model(a driven quantum dot model) is briery discussed, if Schrödinger’s wave function is to be examined for the q...

We study the deterministic spin dynamics of an anisotropic magnetic particle in the presence of a magnetic field with a constant longitudinal and a time-dependent transverse component using the Landau-Lifshitz-Gilbert equation. We characterize the dynamical behavior of the system through calculation of the Lyapunov exponents, Poincaré sections, bifurcation diagrams, and Fourier power spectra. In particular we explore the positivity of the largest Lyapunov exponent as a function of the magnitude and frequency of the applied magnetic field and its direction with respect to the main anisotropy axis of the magnetic particle. We find that the system presents multiple transitions between regular and chaotic behaviors. We show that the dynamical phases display a very complicated structure of intricately intermingled chaotic and regular phases.

Phase synchronization of chaos is studied using a modified Rössler system. By employing a lift of the phase variable (i.e., phase points separated by 2p are not considered as the same), the transition to phase synchronization is viewed as a boundary crisis mediated by an unstable-unstable pair bifurcation on a branched manifold, and the accompanying basin boundary structure is found to be of a new type. [S0031-9007(98)05362-9]

We study the dispersion of the "temporally stable" coherent states for the hydrogen atom introduced by Klauder. These are states which under temporal evolution by the hydrogen atom Hamiltonian retain their coherence properties. We show that in the hydrogen atom such wave packets do not move quasiclassically; i.e., they do not follow with no or little dispersion the Keplerian orbits of the classical electron. The poor quantum-classical correspondence does not improve in the semiclassical limit.

Recent findings on the dynamical analysis of human locomotion characteristics such as stride length signal have shown that this process is intrinsically a chaotic behavior. The passive walking has been defined as walking down a shallow slope without using any muscular contraction as an active controller. Based on this definition, some knee-less models have been proposed to present the simplest possible models of human gait. To maintain stability, these simple passive models are compelled to show a wide range of different dynamics from order to chaos. Unfortunately, based on simplifications, for many years the cyclic period-one behavior of these models has been considered as the only stable response. This assumption is not in line with the findings about the nature of walking. Thus, this paper proposes a novel model to demonstrate that the knee-less passive dynamic models also have the ability to model the chaotic behavior

The synchronization of nonlinear oscillators is well-known and is a traditional topic in complex dynamical system theory. The synchronization of chaotic attractors is less well-known, but is of obvious interest in many applications to the sciences: physical, biological, and social. In a recent experimental study of coupled lattices of Rössler attractors, (jointly with Michael Nivala) we were surprised to discover global periodic behavior in large regimes of the parameter space. This emergent periodicity in a field of chaos may be of significance in the origin of life, and in many life processes. In this talk we will explore the emergence of global periodicity, and also the periodic windows in the bifurcation diagram of the Rössler attractor, which may be the local cause of this global behavior.

Sturmians are a discrete set of eigenfunctions of a Sturm-Liouville equation. They form a complete set in terms of which the solution of numerous equations as a Schrödinger equation can be expanded, both for positive and negative energies. The distinguishing feature of these functions is that they embody the appropriate boundary conditions and already solve a portion of the more complicated Schrödinger equation. This expansion method provides an alternative form to the perturbation theory traditionally used to obtain the solution to a Schrödinger equation in the presence of perturbations. We show how to numerically obtain the Sturmian functions, and how their accuracy can be improved by using spectral methods based on Chebyshev expansions. We also show how these functions serve to obtain a separable representation of a general operator (for example of a non-local potential), and we describe the iterative corrections of the truncation error in a Sturmian expansion.

A system in a spatially (quasi-)degenerate ground state responds in a qualitatively different way to a change in the external potential. Consequently, the usual method for computing the Fukui function, namely, taking the difference between the electron densities of the N-and N ± 1 electron systems, cannot be applied directly. It is shown how the Fukui matrix, and thus also the Fukui function, depends on the nature of the perturbation. One thus needs to use degenerate perturbation theory for the given perturbing potential to generate the density matrix whose change with respect to a change in the number of electrons equals the Fukui matrix. Accounting for the degeneracy in the case of nitrous oxide reveals that an average over the degenerate states differs significantly from using the proper density matrix. We further show the differences in Fukui functions depending on whether a Dirac delta perturbation is used or an interaction with a true point charge (leading to the Fukui potential).

In this paper, we investigate global chaos synchronization of unidirectionally coupled and periodically modulated Josephson junctions (PMJJs) based on the stability theory of linear time-varied systems and Lyapunov's direct method. Some necessary and sufficient criteria for global asymptotic stability are derived from which an estimate of the threshold coupling for the onset of complete synchronization is obtained. The analytical and numerical results reveal that the synchronization criteria based on Lyapunov's direct method are sharper than the criteria based on the stability theory of linear time-varied systems. Based on the criteria obtained via Lyapunov direct method, we showed that the method can be applied to secure communication. The numerical simulations results presented verify the effectiveness of theoretical analysis and the proposed scheme's success in secure communication.

It is argued that, if a regular Hamiltonian is perturbed by a term that produces chaos, the onset of chaos is shifted towards larger values of the perturbation parameter if the unperturbed spectrum is degenerate and the lifting of the degeneracy is of second order in this parameter. The argument is based on the behaviour of the exceptional points of the full problem.

We show that the statistics of tunnelling can be dramatically affected by scarring and derive distributions quantifying this effect. Strong deviations from the prediction of random matrix theory can be explained quantitatively by modifying the Gaussian distribution which describes wavefunction statistics. The modified distribution depends on classical parameters which are determined completely by linearised dynamics around a periodic orbit. This distribution generalises the scarring theory of Kaplan [Phys. Rev. Lett. 80, 2582 (1998)] to describe the statistics of the components of the wavefunction in a complete basis, rather than overlaps with single Gaussian wavepackets. In particular it is shown that correlations in the components of the wavefunction are present, which can strongly influence tunnellingrate statistics. The resulting distribution for tunnelling rates is tested successfully on a two-dimensional double-well potential.

En este trabajo se presentan los grupos de cohomología de DeRham como los grupos duales a ciertos grupos que son invariantes topológicos, los grupos de homología singular. Se definirán estos grupos (de cohomología) como conjuntos de formas diferenciales de una variedad M. La dualidad que se establece, se hace mediante el uso del Teorema de DeRham, y para ello debemos considerar nociones que nos introduzcan a la homología singular y al Teorema de Stokes

In this article, using the principles of Random Matrix Theory (RMT) with Gaussian Unitary Ensemble (GUE), we give a measure of quantum chaos by quantifying Spectral From Factor (SFF) appearing from the computation of two point Out of Time Order Correlation function (OTOC) expressed in terms of square of the commutator bracket of quantum operators which are separated in time scale. We also provide a strict model independent bound on the measure of quantum chaos, −1/N (1 − 1/π) ≤ SFF ≤ 0 and 0 ≤ SFF ≤ 1/πN , valid for thermal systems with large and small number of degrees of freedom respectively. We have studied both the early and late behaviour of SFF to check the validity and applicability of our derived bound. Based on the appropriate physical arguments we give a precise mathematical derivation to establish this alternative strict bound of quantum chaos. Finally, we provide an example of integrability from GUE based RMT from Toda Lattice model to explicitly show the application of our derived bound on SFF to quantify chaos.

We study a system of N qubits with a random Hamiltonian obtained by drawing coupling constants from Gaussian distributions in various ways. This results in a rich class of systems which include the GUE and the fixed q SYK theories. Our motivation is to understand the system at large N. In practice most of our calculations are carried out using exact diagonalisation techniques (up to N = 24). Starting with the GUE, we study the resulting behaviour as the randomness is decreased. While in general the system goes from being chaotic to being more ordered as the randomness is decreased, the changes in various properties, including the density of states, the spectral form factor, the level statistics and out-of-time-ordered correlators, reveal interesting patterns. Subject to the limitations of our analysis which is mainly numerical, we find some evidence that the behaviour changes in an abrupt manner when the number of non-zero independent terms in the Hamiltonian is exponentially large in N. We also study the opposite limit of much reduced randomness obtained in a local version of the SYK model where the number of couplings scales linearly in N , and characterise its behaviour. Our investigation suggests that a more complete theoretical analysis of this class of systems will prove quite worthwhile.

A novel demonstration of chaos in the double pendulum is discussed. Experiments to evaluate the sensitive dependence on initial conditions of the motion of the double pendulum are described. For typical initial conditions, the proposed experiment exhibits a growth of uncertainties which is exponential with exponent λ=7.5±1.5 s−1. Numerical simulations performed on an idealized model give good agreement, with the value λ=7.9±0.4 s−1. The exponents are positive, as expected for a chaotic system.

This paper investigates the fundamental dynamical mechanism responsible for transition to chaos in periodically modulated Duffing-Van der Pol oscillator. It is shown that a modulationally unstable pattern appears into an initially stable motionless state. A further spatiotemporal transition occurs with a sharp interface from the selected stable pattern to a stabilized pattern or chaotic state. Also, the synchronization of the chaotic state of the model is investigated. The results are discussed in the context of generalized synchronization. The main idea is to construct an augmented dynamical system from the synchronization error system, which is itself uncertain. The advantage of this method over existing results is that the synchronization time is explicitly computed. Numerical simulations are provided to verify the operation of the proposed algorithm.

Ülkelerin sahip oldukları jeopolitik durum, siyasi yapısı, ekonomik durumu, rejimi, diğer ülkelerle olan ilişkilerinde önemli rol oynamaktadır. Gelişen teknoloji ile ülkeler arası bağlantı daha kolay sağlanmakta ve fikirler sınır ötesine geçmektedir. Aynı zamanda teknolojik gelişmelerle belirsizlik üretilmekte ve bir güvensizlik ortamı meydana gelmektedir. Bir ülkede yaşanan herhangi bir problem/sorun kelebek etkisiyle diğer ülkelerde baş gösterebilmektedir. Dolayısıyla günümüzde yaşanan sorunlara bakıldığında Lorenz’in ifade ettiği gibi “kelebek etkisi” belirgin bir şekilde ortaya çıkmaktadır. Bu durum başlangıç noktasına hassas bağlılığı ifade eder. Başlangıç noktasında olan en küçük değişiklik sonuçlarda da değişikliklere yol açmaktadır. Kaos, hayatımızın her alanında vardır ve bununla yaşamak yeni normallik olarak ortaya çıkmaktadır. Bu noktada önemli olan oluşabilecek kaos durumlarını yönetebilmektir. Bu çalışmada, kaos teorisinden bahsedilmiştir. Aslında her olayın bir başlang...

More than four centuries after the Copernican Revolution and the consequent dismissal of Aristotelian Cosmology, the modern model of the cosmos has reached a similar if not superior level of a satisfactory understanding of physical reality. This extraordinary feat was achieved by using the Galilean scientific method of investigation, which was demonstrated to be extremely powerful in modeling cosmic physical phenomena. Unexpectedly, the main global characteristic of the cosmos was found to be its evolution in time; the universe’s history passes through very different phases, all linked together by a subtle fil rouge. This very fact, by now incontrovertible, is challenging our interpretation of reality by the sole use of the scientific method. The time may have come to reconnect, in a collaborative and constructive way, science, philosophy, and theology, which for too long have proceeded along independent, parallel, or even divergent paths. Only in this way may we hope to reach a mor...

Quantum mechanics of one-dimensional time-independent Hamiltonian system whose energy level statistics obeys the Gaussian ensemble is numerically studied by the inverse scattering method. It has been known that the system must be integrable. However, at least numerically, it can be shown that we can construct a potential for the Schrödinger equation that reproduces a finite number of given energy levels of chaotic regime, i.e. given by the random matrix theory (RMT). Contrary to usual classical Hamiltonian systems, there cannot exit the shortest period from the consequence of the semi-classical argument, if the system completely reproduces the chaotic levels of the RMT. If its density of states is unfolded and its average density is independent of energy, the coarse averaged shape of the potential is just like a harmonic oscillator and the potential has peculiar ripples on it. The more energy levels of chaotic regime we take, the more intricate and the finer these ripples become.

The concepts of integrability, non-integrability and chaos in quantum mechanics are examined, and it is indicated that they all are sensibly de®nable only in connection with the corresponding properties of their classical analogues. The concrete examples concern the quantum and classical properties of dynamic systems on SU3 algebra and classically integrable but quantum nonintegrable system with two degrees of freedom.

We study the spectral fluctuations of the 208 Pb nucleus using the complete experimental spectrum of 151 states up to excitation energies of 6.20 MeV recently identified at the Maier-Leibnitz-Laboratorium at Garching, Germany. For natural parity states the results are very close to the predictions of Random Matrix Theory (RMT) for the nearest-neighbor spacing distribution. A quantitative estimate of the agreement is given by the Brody parameter ω, which takes the value ω = 0 for regular systems and ω ≃ 1 for chaotic systems. We obtain ω = 0.85 ± 0.02 which is, to our knowledge, the closest value to chaos ever observed in experimental bound states of nuclei. By contrast, the results for unnatural parity states are far from RMT behavior. We interpret these results as a consequence of the strength of the residual interaction in 208 Pb, which, according to experimental data, is much stronger for natural than for unnatural parity states. In addition our results show that chaotic and non-chaotic nuclear states coexist in the same energy region of the spectrum.

We atend the semiclassical theory ol spectral fluctuations UI include mmposite systems. These systems are characterized hy mme number of weakly mmmunicaling. highly chaotic subsystems. They mntain new and longer time scales, depending on the mnneding fluxes, in addition to those associaled with the rapid inlernal mixings. The new time scales produce discernible modifications in the nature of the level fluctuations of the corresponding quantum systems. The theory is applied to two coupled quartic cscillators.

A complete quantum solution provides all possible knowledge of a system, whereas semiclassical theory provides at best approximate solutions in a limited region. Nevertheless, semiclassical methods based on the work of Martin Gutzwiller can provide stunning physical insights in regimes where quantum solutions are opaque. Furthermore, they can provide a unique bridge between the quantum and classical worlds. We illustrate these ideas with an account of a theoretical and experimental attack on the paradigm problem of the hydrogen atom in strong magnetic and electric fields.

Bifurcation diagrams and phase diagrams of two coupled periodically driven identical DuKng oscillators are presented. It is shown that the global pattern of bifurcation curves in parameter space consists of repeated subpatterns similar to the superstructure observed for single, periodically driven, strictly dissipative oscillators. The subpattern itself, however, is different from that of a single DufBng oscillator due, in particular, to Hopf bifurcations that are newly added to the bifurcation scenario.

e present a theory of quantum chaos of the Hadamard-Gutzwiller model, a quantum mechanical system which describes the motion of a particle on a surface of constant negative curvature. The theory is based on periodic-orbit sum rules that can be rigorously derived from the Selberg trace formula and which provide an exact substitute, appropriate for our strongly chaotic system, for the Bohr-Sommerfeld-Einstein quantization rules. Our recent enumeration of the classical periodic orbits enables us to evaluate the sum rules numerically and to demonstrate thereby that the theory provides also a practical method to study the quantum chaos of spectra.

The coexistence of infinitely many attractors is called extreme multistability in dynamical systems. In coupled systems, this phenomenon is closely related to partial synchrony and characterized by the emergence of a conserved quantity. We propose a general design of coupling that leads to partial synchronization, which may be a partial complete synchronization or partial antisynchronization and even a mixed state of complete synchronization and antisynchronization in two coupled systems and, thereby reveal the emergence of extreme multistability. The proposed design of coupling has wider options and allows amplification or attenuation of the amplitude of the attractors whenever it is necessary. We demonstrate that this phenomenon is robust to parameter mismatch of the coupled oscillators.

We analyze protein-protein interaction networks for six different species under the framework of random matrix theory. Nearest neighbor spacing distribution of the eigenvalues of adjacency matrices of the largest connected part of these networks emulate universal Gaussian orthogonal statistics of random matrix theory. We demonstrate that spectral rigidity, which quantifies long range correlations in eigenvalues, for all protein-protein interaction networks follow random matrix prediction up to certain ranges indicating randomness in interactions. After this range, deviation from the universality evinces underlying structural features in network.