We prove, under suitable non-resonance and non-degeneracy "twist" conditions, a Birkhoff-Lewis ty... more We prove, under suitable non-resonance and non-degeneracy "twist" conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricityinclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the "planets". The proofs are based on averaging theory, KAM theory and variational methods 1 .
We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of sm... more We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations.
We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (tri... more We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O(\mu)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $ T_d = O((1/ \mu) \log (1/ \mu))$ by a variational method which does not require the existence of ``transition chains of tori'' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d $ is optimal as a consequence of a general stability result derived from classical perturbation theory.
Celestial Mechanics and Dynamical Astronomy, May 15, 2012
The weak stability boundary (WSB) is the transition region of the phase space where the change fr... more The weak stability boundary (WSB) is the transition region of the phase space where the change from gravitational escape to ballistic capture occurs. Studies on this complicated region of chaotic motion aim to investigate its unique, fuel saving properties to enlarge the frontiers of low energy transfers. This "fuzzy stability" region is characterized by highly sensitive motion, and any analysis of it has been carried out almost exclusively using numerical methods. On the contrary this paper presents, for the planar circular restricted 3 body problem (PCR3BP), 1) an analytic definition of the WSB which is coherent with the known algorithmic definitions; 2) a precise description of the topology of the WSB; 3) analytic estimates on the "stable region" (nearby the smaller primary) whose boundary is, by definition, the WSB.
All the almost periodic solutions for non integrable PDEs found in the literature are very regula... more All the almost periodic solutions for non integrable PDEs found in the literature are very regular (at least C ∞) and, hence, very close to quasi periodic ones. This fact is deeply exploited in the existing proofs. Proving the existence of almost periodic solutions with finite regularity is a main open problem in KAM theory for PDEs. Here we consider the one dimensional NLS with external parameters and construct almost periodic solutions which have only Sobolev regularity both in time and space. Moreover many of our solutions are so only in a weak sense. This is the first result on existence of weak, i.e. non classical, solutions for non integrable PDEs in KAM theory. Contents 39 9. Measure estimates 46 Appendix A. Technicalities 52 Appendix B. Topology, measure and continuous functions on infinite product spaces 58 References 62
Communications in Mathematical Physics, Nov 18, 2019
We study stability times for a family of parameter dependent nonlinear Schrödinger equations on t... more We study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively.
Communications in Mathematical Physics, May 22, 2011
We consider infinite dimensional Hamiltonian systems. First we prove the existence of "Cantor man... more We consider infinite dimensional Hamiltonian systems. First we prove the existence of "Cantor manifolds" of elliptic tori-of any finite higher dimension-accumulating on a given elliptic KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of elliptic tori which are "branching" points of other Cantor manifolds of higher dimensional tori. We also provide a positive answer to a conjecture of Bourgain [8] proving the existence of invariant elliptic KAM tori with tangential frequency constrained to a fixed Diophantine direction. These results are obtained under the natural nonresonance and nondegeneracy conditions. As applications we prove the existence of new kinds of quasi periodic solutions of the one dimensional nonlinear wave equation. The proofs are based on averaging normal forms and a sharp KAM theorem, whose advantages are an explicit characterisation of the Cantor set of parameters, quite convenient for measure estimates, and weaker smallness conditions on the perturbation. Functional setting and notations Phase space. We consider the Hilbert space of complex-valued sequences a,p := z = (z 1 , z 2 ,. . .) : z 2 a,p := j≥1 |z j | 2 j 2p e 2ja < +∞ with a > 0, p > 1/2, and the toroidal phase space (x, y, w) ∈ T n s × C n × a,p b , w := (z,z) ∈ a,p b := a,p × a,p ,
Annales Scientifiques De L Ecole Normale Superieure, 2013
KAM THEORY FOR THE HAMILTONIAN DERIVATIVE WAVE EQUATION ʙʏ Mɪɪʟɪɴ BERTI, L BIASCO ɴ Mɪʜʟ PROCESI ... more KAM THEORY FOR THE HAMILTONIAN DERIVATIVE WAVE EQUATION ʙʏ Mɪɪʟɪɴ BERTI, L BIASCO ɴ Mɪʜʟ PROCESI Aʙʀ.-We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations. R.-Nous prouvons un théorème KAM en dimension infinie, qui implique l'existence de familles de Cantor de tores invariants de petite amplitude, réductibles, elliptiques et analytiques, pour les équations des ondes hamiltoniennes avec dérivées.
In this note we present the new KAM result in [3] which proves the existence of Cantor families o... more In this note we present the new KAM result in [3] which proves the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coe‰cients. In turn, this result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems*.
We discuss the holomorphic properties of the complex continuation of the classical Arnol'd-Liouvi... more We discuss the holomorphic properties of the complex continuation of the classical Arnol'd-Liouville action-angle variables for real analytic 1 degree-of-freedom Hamiltonian systems depending on external parameters in suitable 'generic standard form', with particular regard to the behaviour near separatrices.
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Aug 1, 2006
We prove existence and regularity of periodic in time solutions of completely resonant nonlinear ... more We prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. It turns out that the infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. We solve it via a minimization argument and a-priori estimate methods related to the regularity theory of [P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math. 20 (1967) 145-205]. On prouve l'existence et la régularité de solutions périodiques en temps d'équations des ondes non linéaires forcées, complètement résonnantes, avec des conditions au bord de Dirichlet, pour une grande classe de termes forcants non-monotones. Notre approche est basée sur une réduction de Lyapunov-Schmidt variationnelle. L'équation de bifurcation en dimension infinie présente un manque intrinsèque de compacité. Nous la résolvons par un argument de minimisation et à l'aide d'estimations a priori inspirées de la théorie de la régularité de [P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm.
All the almost periodic solutions for non integrable PDEs found in the literature are very regula... more All the almost periodic solutions for non integrable PDEs found in the literature are very regular (at least C ∞) and, hence, very close to quasi periodic ones. This fact is deeply exploited in the existing proofs. Proving the existence of almost periodic solutions with finite regularity is a main open problem in KAM theory for PDEs. Here we consider the one dimensional NLS with external parameters and construct almost periodic solutions which have only Sobolev regularity both in time and space. Moreover many of our solutions are so only in a weak sense. This is the first result on existence of weak, i.e. non classical, solutions for non integrable PDEs in KAM theory. Contents 39 9. Measure estimates 46 Appendix A. Technicalities 52 Appendix B. Topology, measure and continuous functions on infinite product spaces 58 References 62
We discuss a method for the construction of almost periodic solutions of the one dimensional anal... more We discuss a method for the construction of almost periodic solutions of the one dimensional analytic NLS with only Sobolev regularity both in time and space. This is the first result of this kind for PDEs.
We prove the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions ... more We prove the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. This result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems
We consider a nearly integrable, non-isochronous, a-priori unstable Hamiltonian system with a (tr... more We consider a nearly integrable, non-isochronous, a-priori unstable Hamiltonian system with a (trigonometric polynomial) O(µ)-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time T d = O((1/µ) log(1/µ)) by a variational method which does not require the existence of "transition chains of tori" provided by KAM theory. We also prove that our estimate of the diffusion time T d is optimal as a consequence of a general stability result proved via classical perturbation theory. 1
We study stability times for a family of parameter dependent nonlinear Schrödinger equations on t... more We study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively.
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, 2005
Existence and regularity of periodic solutions of nonlinear, completely resonant, forced wave equ... more Existence and regularity of periodic solutions of nonlinear, completely resonant, forced wave equations is proved for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. The corresponding infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. This difficulty is overcome finding a-priori estimates for the constrained minimizers of the reduced action functional, through techniques inspired by regularity theory as in [R67].
Journal de Mathématiques Pures et Appliquées, Jun 1, 2003
We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (tri... more We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) O(µ)-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time T d = O((1/µ) log(1/µ)) by a variational method which does not require the existence of "transition chains of tori" provided by KAM theory. We also prove that our estimate of the diffusion time T d is optimal as a consequence of a general stability result derived from classical perturbation theory. 1
Uploads
Papers by Luca Biasco