On stochasticity in nearly-elastic systems

TL;DR

This paper demonstrates that nearly-elastic deterministic systems with small energy losses exhibit intrinsic stochastic long-term behavior due to instabilities, with multiple rigorous approaches confirming this stochasticity.

Contribution

It provides a rigorous framework showing that deterministic nearly-elastic systems inherently develop stochastic long-term dynamics due to instabilities.

Findings

Long-term behavior becomes stochastic in nearly-elastic systems.

Multiple methods confirm the intrinsic stochasticity.

Stochasticity is an inherent property of the deterministic system.

Abstract

Nearly-elastic model systems with one or two degrees of freedom are considered: the system is undergoing a small loss of energy in each collision with the "wall". We show that instabilities in this purely deterministic system lead to stochasticity of its long-time behavior. Various ways to give a rigorous meaning to the last statement are considered. All of them, if applicable, lead to the same stochasticity which is described explicitly. So that the stochasticity of the long-time behavior is an intrinsic property of the deterministic systems.