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.

我们考虑一类在原点具有非利普希茨奇点的常微分方程。解决方案在全球范围内都存在，并且在它们第一次到达原点之前都是独一无二的。在“爆炸”之后，可能存在无限多个解决方案。为了研究连续性，我们引入了物理激励的正则化：它们包括平滑ν球中的矢量场。我们证明极限ν→ 0 可以使用通过依赖于解决方案的重整化获得的某个自主动力系统来理解。此过程将爆破前动力学映射到以无限大的重整化时间结束的解决方案。爆炸附近的渐近行为由吸引子描述。爆破后的动力学被映射到一个不同的重整化解，从无限远的过去开始，因此，它与另一个吸引子相关联。正则化在重整化系统的这两个不同“生命期”之间建立了关系，并且一般选择了一个受限的解决方案系列，而不依赖于正则化。