'Life after death' in ordinary differential equations with a non-Lipschitz singularity. (17th February 2021)

• Record Type: Journal Article
• Authors: Drivas, Theodore D; Mailybaev, Alexei A
• Is Part Of: Nonlinearity. Volume 34:Number 4(2021)
• Page Start: 2296
• Page End: 2326
• Publication Date: 2021-02-17
• Subjects: selection principle -- non-Lipschitz singularity -- non-uniqueness in ordinary differential equations, blowup; 34A12 37C83; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/abbe60
• Languages: English
• ISSNs: 0951-7715

Abstract

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.