Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse

Abstract

We study billiard dynamics inside an ellipse for which the axes lengths are changed periodically in time and an O(δ)-small quartic polynomial deformation is added to the boundary. In this situation the energy of the particle in the billiard is no longer conserved. We show a Fermi acceleration in such system: there exists a billiard trajectory on which the energy tends to infinity. The construction is based on the analysis of dynamics in the phase space near a homoclinic intersection of the stable and unstable manifolds of the normally hyperbolic invariant cylinder Λ, parameterised by the energy and time, that corresponds to the motion along the major axis of the ellipse. The proof depends on the reduction of the billiard map near the homoclinic channel to an iterated function system comprised by the shifts along two Hamiltonian flows defined on Λ. The two flows approximate the so-called inner and scattering maps, which are basic tools that arise in the studies of the Arnold diffusion; the scattering maps defined by the projection along the strong stable and strong unstable foliations W ss,uu of the stable and unstable invariant manifolds W s,u (Λ) at the homoclinic points. Melnikov type calculations imply that the behaviour of the scattering map in this problem is quite unusual: it is only defined on a small subset of Λ that shrinks, in the large energy limit, to a set of parallel lines t = const as δ → 0. *

Arnold Diffusion in Multi-Dimensional Convex Billiards

“…In the context of billiards, an example of a nonautonomous system would be the billiard dynamics inside a table with moving walls, for instance. There have been several results in recent years establishing Fermi acceleration in such circumstances [14][19] [20][21]. In our case the system is autonomous, and so the billiard map conserves energy.…”

Chaotic Motion in the Breathing Circle Billiard

“…Moreover, the existence of unbounded motions is a symptom of complex dynamics. In fact, in [9] it is also proved that the phenomenon of splitting of the separatrices occurs and a scattering map can be defined. We refer to [12] for more insight on the topic of Fermi acceleration in general time-dependent billiards.…”

“…If the boundary is a moving ellipse, then it has been proved in [9] that it is possible to construct orbits that gain energy. Moreover, the existence of unbounded motions is a symptom of complex dynamics.…”

Chaotic motion in the breathing circle billiard

“…Moreover, the existence of unbounded motions is a symptom of complex dynamics. In fact, in [7] it is also proved that the phenomenon of splitting of the separatrices occurs and a scattering map can be defined. We refer to [10] for more insight on the topic of Fermi acceleration in general time-dependent billiards.…”

“…If the boundary is a moving ellipse then it has been proved in [7] that it is possible to construct orbits that gain energy. Moreover, the existence of unbounded motions is a symptom of complex dynamics.…”