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.
Bibliographical noteFunding 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.
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