‘Life after death’ in ordinary differential equations with a non-Lipschitz singularity
| Autor: | Theodore D. Drivas, Alexei A. Mailybaev |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Applied Mathematics 010102 general mathematics FOS: Physical sciences General Physics and Astronomy Statistical and Nonlinear Physics Mathematical Physics (math-ph) Dynamical Systems (math.DS) Lipschitz continuity 01 natural sciences Regularization (mathematics) 010101 applied mathematics Renormalization Singularity Mathematics - Classical Analysis and ODEs Ordinary differential equation Attractor Classical Analysis and ODEs (math.CA) FOS: Mathematics Limit (mathematics) Mathematics - Dynamical Systems 0101 mathematics Dynamical system (definition) Mathematical Physics Mathematics |
| Zdroj: | Nonlinearity. 34:2296-2326 |
| ISSN: | 1361-6544 0951-7715 |
| DOI: | 10.1088/1361-6544/abbe60 |
| Popis: | We consider a class of ordinary differential equations in $d$-dimensions featuring a non-Lipschitz singularity at the origin. Solutions of such systems exist globally and are unique up until the first time they hit the origin, $t = t_b$, which we term `blowup'. However, infinitely many solutions may exist for longer times. To study continuation past blowup, we introduce physically motivated regularizations: they consist of smoothing the vector field in a $\nu$--ball around the origin and then removing the regularization in the limit $\nu\to 0$. We show that this limit can be understood using a certain autonomous dynamical system obtained by a solution-dependent renormalization procedure. This procedure maps the pre-blowup dynamics, $t < t_b$, to the solution ending at infinitely large renormalized time. In particular, the asymptotic behavior as $t \nearrow t_b$ is described by an attractor. The post-blowup dynamics, $t > t_b$, is mapped to a different renormalized solution starting infinitely far in the past. Consequently, it is associated with another attractor. The $\nu$-regularization establishes a relation between these two different "lives" of the renormalized system. We prove that, in some generic situations, this procedure selects a unique global solution (or a family of solutions), which does not depend on the details of the regularization. We provide concrete examples and argue that these situations are qualitatively similar to post-blowup scenarios observed in infinite-dimensional models of turbulence. Comment: 27 pages, 8 figures |
| Databáze: | OpenAIRE |
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