On stochasticity in nearly-elastic systems

Mark Freidlin; Wenqing Hu

doi:10.1142/S0219493711500201

On stochasticity in nearly-elastic systems

Mark Freidlin, Wenqing Hu

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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.

Original language	English (US)
Article number	1150020
Journal	Stochastics and Dynamics
Volume	12
Issue number	3
DOIs	https://doi.org/10.1142/S0219493711500201
State	Published - Sep 2012
Externally published	Yes

Bibliographical note

Funding Information:
This work was supported in part by NSF grants DMS-0803287 and DMS-0854982. We would like to thank the anonymous referee for carefully reading the manuscript.

Keywords

Averaging principle
Hamiltonian flows
Markov processes on graphs
chaotic systems

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AU - Hu, Wenqing

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N2 - 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.

AB - 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.

KW - Averaging principle

KW - Hamiltonian flows

KW - Markov processes on graphs

KW - chaotic systems

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On stochasticity in nearly-elastic systems

Abstract

Bibliographical note

Keywords

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