‘Life after death’ in ordinary differential equations with a non-Lipschitz singularity

Abstract

We consider a class of ordinary differential equations featuring a non-Lipschitz singularity at the origin. Solutions exist globally and are unique up until the first time they hit the origin. After ‘blowup’, infinitely many solutions may exist. To study continuation, we introduce physically motivated regularizations: they consist of smoothing the vector field in a ν-ball. We show that the limit ν → 0 can be understood using a certain autonomous dynamical system obtained by a solution-dependent renormalization. This procedure maps the pre-blowup dynamics to the solution ending at infinitely large renormalized time. The asymptotic behavior near blowup is described by an attractor. The post-blowup dynamics is mapped to a different renormalized solution starting infinitely far in the past and, consequently, it is associated with another attractor. The regularization establishes a relation between these two different ‘lives’ of the renormalized system and generically selects a restricted family of solutions, not depending on the regularization.