Introduction to the Modern Theory of Dynamical Systems: INTRODUCTION: WHAT IS LOW-DIMENSIONAL DYNAMICS?

We study the evolution, under convex Hamiltonian flows on cotangent bundles of compact manifolds, of certain distinguished subsets of the phase space. These subsets are generalizations of Lagrangian graphs, we call them pseudographs. They emerge in a natural way from Fathi's weak KAM theory. By this method, we find various orbits which connect prescribed regions of the phase space. Our study is inspired by works of John Mather. As an application, we obtain the existence of diffusion in a large class of a priori unstable systems and provide a solution to the large gap problem. We hope that our method will have applications to more examples.

/pdf/the-dynamics-of-pseudographs-in-convex-hamiltonian-systems-1dldam87s1.pdf

The dynamics of pseudographs in convex Hamiltonian systems

We study diffusion phenomena in a priori unstable (initially hyperbolic) Hamiltonian systems These systems are perturbations of integrable ones, which have a family of hyperbolic tori We prove that in the case of two and a half degrees of freedom the action variable generically drifts (ie changes on a trajectory by a quantity of order one) Moreover, there exists a trajectory such that the velocity of this drift is e/loge, where e is the parameter of the perturbation

https://iopscience.iop.org/article/10.1088/0951-7715/17/5/014/pdf

Evolution of slow variables in a priori unstable Hamiltonian systems

Geometric properties of the scattering map of a normally hyperbolic invariant manifold

. – We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose mean speed of drift allows us to check the optimality of the stability theorem.¶Our stability result generalizes those by Lochak-Neishtadt and Poschel, which give precise exponents of stability in the Nekhoroshev Theorem for the quasi-convex case, to the situation in which the Hamiltonian function is only assumed to belong to some Gevrey class instead of being real-analytic. For n degrees of freedom and Gevrey-α Hamiltonians, α ≥ 1, we prove that one can choose a = 1/2nα as an exponent for the time of stability and b = 1/2n as an exponent for the radius of confinement of the action variables, with refinements for the orbits which start close to a resonant surface (we thus recover the result for the real-analytic case by setting α = 1).¶On the other hand, for α > 1, the existence of compact-supported Gevrey functions allows us to exhibit for each n ≥ 3 a sequence of Hamiltonian systems with wandering points, whose limit is a quasi-convex integrable system, and where the speed of drift is characterized by the exponent 1/2(n−2)α. This exponent is optimal for the kind of wandering points we consider, inasmuch as the initial condition is located close to a doubly-resonant surface and the stability result holds with precisely that exponent for such an initial condition. We also discuss the relationship between our example of instability, which relies on a specific construction of a perturbation of a discrete integrable system, and Arnold’s mechanism of instability, whose main features (partially hyperbolic tori, heteroclinic connections) are indeed present in our system.

/pdf/stability-and-instability-for-gevrey-quasi-convex-near-3numsmgzwj.pdf

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

Introduction and some salient features of the model Hamiltonian Symplectic geometry and the splitting of invariant manifolds Estimating the splitting matrix using normal forms The Hamilton-Jacobi method for a simple resonance Appendix. Invariant tori with vanishing or zero torsion Bibliography.

On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems

For a positive integer n and R>0, we set
 
$$B_R^n = \left\{ {x \in \mathbb{R}^n |\left\| x \right\|_\infty< R} \right\}$$

 . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H
 j
 ) of the completely integrable Hamiltonian
 
$$h\left( r \right) = \tfrac{1}{2}r_1^2 + ...\tfrac{1}{2}r_{n - 1}^2 + r_n $$

 on
 
$$\mathbb{T}^n \times B_R^n $$

 , with unstable orbits for which we can estimate the time of drift in the action space. These functions H
 j
 are analytic on a fixed complex neighborhood V of
 
$$\mathbb{T}^n \times B_R^n $$

 , and setting
 
$$\varepsilon _j : = \left\| {h - H_j } \right\|_{C^0 (V)} $$

 the time of drift of these orbits is smaller than (C(1/ɛ
 j
 )1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg's conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.

/pdf/diffusion-times-and-stability-exponents-for-nearly-4lrk8ou6nn.pdf

Diffusion times and stability exponents for nearly integrable analytic systems

We introduce two numerical conjugacy invariants of dynamical systems — the polynomial entropy and the weak polynomial entropy — which are well-suited for the study of “completely integrable” Hamiltonian systems. These invariants describe the polynomial growth rate of the number of balls (for the usual “dynamical” distances) of covers of the ambient space. We give explicit examples of computation of these polynomial entropies for generic Hamiltonian systems on surfaces.

Polynomial entropies and integrable Hamiltonian systems

In this paper, we study the entropy of a Hamiltonian flow in restriction to an energy level where it admits a first integral which is nondegenerate in the sense of Bott. It is easy to see that for such a flow, the topological entropy vanishes. We focus on the polynomial and the weak polynomial entropies hpol and h pol * . We show that, under natural conditions on the critical levels of the Bott first integral and on the Hamiltonian function H, h pol * ∈ {0, 1} and hpol ∈ {0, 1, 2}. To prove this result, our main tool is a semi-global desingularization of the Hamiltonian system in the neighborhood of a polycycle.

Jean-Pierre Marco

Papers

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems

Diffusion times and stability exponents for nearly integrable analytic systems

Polynomial entropies and integrable Hamiltonian systems

Polynomial entropies for Bott integrable Hamiltonian systems