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singular dynamical systems

Ergodic Theory and Dynamical Systems (1)

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Statistical determinism in non-Lipschitz dynamical systems
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THEODORE D. DRIVAS, ALEXEI A. MAILYBAEV, ARTEM RAIBEKAS
Journal:

Ergodic Theory and Dynamical Systems / Volume 44 / Issue 7 / July 2024

Published online by Cambridge University Press:

11 October 2023, pp. 1856-1884

Print publication:

July 2024
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We study a class of ordinary differential equations with a non-Lipschitz point singularity that admits non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on a parameter $\nu $: the regularized dynamics is globally defined for each $\nu> 0$, and the original singular system is recovered in the limit of vanishing $\nu $. We prove that this limit yields a unique statistical solution independent of regularization when the deterministic system possesses a chaotic attractor having a physical measure with the convergence to equilibrium property. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution.