Statistical determinism in non-Lipschitz dynamical systems

Abstract

Abstract 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.