Pinball dynamics: unlimited energy growth in switching Hamiltonian systems

TL;DR

This paper investigates a family of discontinuous symplectic maps modeling nonsmooth Hamiltonian systems, revealing conditions for unbounded energy growth and connecting to classical maps, with explicit examples of unbounded orbits.

Contribution

It introduces a new family of discontinuous symplectic maps, analyzes their dynamics, and constructs explicit unbounded orbits, advancing understanding of energy growth in switching Hamiltonian systems.

Findings

Unbounded orbits can exist in the studied maps.

The maps relate to classical Kesten maps.

Explicit unbounded orbit constructed in a special case.

Abstract

A family of discontinuous symplectic maps on the cylinder is considered. This family arises naturally in the study of nonsmooth Hamiltonian dynamics and in switched Hamiltonian systems. The transformation depends on two parameters and is a canonical model for the study of bounded and unbounded behavior in discontinuous area-preserving mappings due to nonlinear resonances. This paper provides a general description of the map and points out its connection with another map considered earlier by Kesten. In one special case, an unbounded orbit is explicitly constructed.