Statistical determinism in non-Lipschitz dynamical systems
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866929596437364736 |
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| author | Drivas, Theodore D. Mailybaev, Alexei A. Raibekas, Artem |
| author_facet | Drivas, Theodore D. Mailybaev, Alexei A. Raibekas, Artem |
| contents | 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 $ν$: the regularized dynamics is globally defined for each $ν> 0$, and the original singular system is recovered in the limit of vanishing $ν$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2004_03075 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Statistical determinism in non-Lipschitz dynamical systems Drivas, Theodore D. Mailybaev, Alexei A. Raibekas, Artem Dynamical Systems Classical Analysis and ODEs Probability 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 $ν$: the regularized dynamics is globally defined for each $ν> 0$, and the original singular system is recovered in the limit of vanishing $ν$. 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. |
| title | Statistical determinism in non-Lipschitz dynamical systems |
| topic | Dynamical Systems Classical Analysis and ODEs Probability |
| url | https://arxiv.org/abs/2004.03075 |