Statistical determinism in non-Lipschitz dynamical systems
Statistical determinism in non-Lipschitz dynamical systems
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DOI：
10.1017/etds.2023.74
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发表时间：
2020-04
影响因子：
0.9
通讯作者：
Theodore D. Drivas;A. Mailybaev;Artem Raibekas
中科院分区：
文献类型：
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作者：
Theodore D. Drivas;A. Mailybaev;Artem Raibekas
我们研究一类具有非利普希茨点奇点的常微分方程，它允许通过该点的非唯一解。作为选择标准，我们根据参数 $\nu $ 引入随机正则化：为每个 $\nu> 0$ 全局定义正则化动力学，并且在消失 $\nu $ 的极限内恢复原始奇异系统。我们证明，当确定性系统拥有具有收敛到平衡特性的物理测量的混沌吸引子时，该极限会产生独立于正则化的唯一统计解。在这种情况下，解决方案在通过奇点后变得自发随机：它们是按照内在概率分布随机选择的。
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.