Statistical determinism in non-Lipschitz dynamical systems

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.

28 pages, 7 figures, 1 supplementary video

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Cited by in corpus (4)

• From the butterfly effect to intrinsic randomness: the spontaneous growth of singular shear flows
• RG approach to the inviscid limit for shell models of turbulence
• Spontaneous stochasticity in the fluctuating Navier-Stokes equations on a logarithmic lattice
• Assigning probabilities to non-Lipschitz mechanical systems