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arXiv · 2004.03075

Statistical determinism in non-Lipschitz dynamical systems

Theodore D. Drivas·Alexei A. Mailybaev·Artem Raibekas
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Abstract

We study a class of ordinary differential equations with a non-Lipschitz point singularity, which admit non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on the 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 certain chaotic properties. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution.

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Theodore D. Drivas, Alexei A. Mailybaev, Artem Raibekas. 2020-04-07. Statistical determinism in non-Lipschitz dynamical systems. https://doi.org/10.1017/etds.2023.74

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