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Jean-Pierre Marco

Bio: Jean-Pierre Marco is an academic researcher from University of Paris. The author has contributed to research in topics: Integrable system & Hamiltonian system. The author has an hindex of 5, co-authored 7 publications receiving 299 citations.

Papers
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TL;DR: In this article, the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems was studied, and it was shown that 1/2nα is the optimal exponent for the time of stability and b = 1 2n as an exponent for radius of confinement of the action variables.
Abstract: . – 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.

119 citations

Journal ArticleDOI
TL;DR: The Hamiltonian Symplectic geometry and the splitting of invariant manifolds have been studied in this article, where the splitting matrix is estimated by the Hamilton-Jacobi method for a simple resonance.
Abstract: 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.

109 citations

Journal ArticleDOI
TL;DR: In this paper, a sequence of analytic perturbations of the completely integrable Hamiltonian was constructed for a positive integer n and R>0, where R>1 and n≥4 were used to estimate the time of drift in the action space.
Abstract: 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.

45 citations

Journal ArticleDOI
TL;DR: Two numerical conjugacy invariants of dynamical systems — the polynomial entropy and the weak polynometric entropy — are introduced which are well-suited for the study of “completely integrable” Hamiltonian systems.
Abstract: 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.

31 citations

Journal ArticleDOI
TL;DR: In this article, the entropy of a Hamiltonian flow in restriction to an energy level where it admits a first integral which is non-degenerate in the sense of Bott is studied.
Abstract: 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.

15 citations


Cited by
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TL;DR: In this paper, the authors studied the evolution of pseudographs under convex Hamiltonian flows on cotangent bundles of compact manifolds, and obtained the existence of diffusion in a large class of a priori unstable systems.
Abstract: 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.

131 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied diffusion phenomena in a priori unstable (initially hyperbolic) Hamiltonian systems and proved that in the case of two and a half degrees of freedom the action variable generically drifts (i.e. changes on a trajectory by a quantity of order one).
Abstract: 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

126 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the scattering map is symplectic (resp. exact symplectic) when f and are symplectic and the primitive function is a variational interpretation as dierence of actions.

124 citations

Journal ArticleDOI
TL;DR: In this article, the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems was studied, and it was shown that 1/2nα is the optimal exponent for the time of stability and b = 1 2n as an exponent for radius of confinement of the action variables.
Abstract: . – 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.

119 citations