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.
Funder
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro
National Science Foundation
Conselho Nacional de Desenvolvimento Científico e Tecnológico
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,General Mathematics