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

We discuss in the context of nearly integrable Hamiltonian systems a functional analysis approach to the “splitting of separatrices” and to the “shadowing problem”. As an application we apply our method to the problem of Arnold Diffusion for nearly integrable partially isochronous systems improving known results.

http://www.dma.unina.it/~berti/pubblic/BertiBollePoincare.pdf

A functional analysis approach to Arnold diffusion

For a convex, real analytic, e-close to integrable Hamiltonian system with n≥5 degrees of freedom, we construct an orbit exhibiting Arnold diffusion with the diffusion time bounded by $\exp(C\epsilon^{-\frac{1}{2(n-2)}})$
 . This upper bound of the diffusion time almost matches the lower bound of order $\exp(\epsilon ^{-\frac{1}{2(n-1)}})$
 predicted by the Nekhoroshev-type stability results. Our method is based on the variational approach of Bessi and Mather, and includes a new construction on the space of frequencies.

Speed of Arnold diffusion for analytic Hamiltonian systems

The restricted three body problem models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies called the primaries. When they move along circular Keplerian orbits and the third body moves in the same plane, one has the restricted planar circular three body problem (RPC3BP). In suitable coordinates, it is a Hamiltonian system of two degrees of freedom. The conserved energy is usually called the Jacobi constant. Llibre and Simo [Math Ann 248(2):153–184, 1980] proved the existence of oscillatory motions for this system. That is, orbits which leave every bounded region but which return infinitely often to some fixed bounded region. To prove their existence they had to assume the ratio between the masses of the primaries to be small enough. In this paper we prove the existence of such motions for any value of the mass ratio 
 $$\mu $$
 closing the problem of existence of oscillatory motions in the RPC3BP. To obtain such motions, we restrict ourselves to the level sets of the Jacobi constant. We show that, for any value of the mass ratio and for large values of the Jacobi constant, there exist transversal intersections between the stable and unstable manifolds of infinity in these level sets. These transversal intersections guarantee the existence of a symbolic dynamics that creates the oscillatory orbits. The main achievement is to prove the existence of these orbits without assuming the mass ratio $$\mu $$
 small. When $$\mu $$
 is not small, this transversality can not be checked by means of classical perturbation theory. Since our method is valid for all values of $$\mu $$
 , we are able to detect a curve in the parameter space, formed by $$\mu $$
 and the Jacobi constant, where cubic homoclinic tangencies between the invariant manifolds of infinity appear.

/pdf/oscillatory-motions-for-the-restricted-planar-circular-three-1ic2n00i1i.pdf

Oscillatory motions for the restricted planar circular three body problem

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.

D. Sauzin

Papers

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