Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs

TL;DR

This paper explores geometric methods to analyze finite-time blow-up solutions in autonomous ODEs, focusing on invariant sets at infinity and their dynamics to understand blow-up behavior and rates.

Contribution

It introduces a unified geometric framework using compactifications and center-stable manifolds to characterize blow-up solutions and their rates in autonomous ODEs.

Findings

Dynamics on center-stable manifolds determine blow-up solutions.

Invariant sets at infinity characterize blow-up behavior.

Time-scale desingularizations reveal blow-up rates.

Abstract

Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and dynamics on their center-stable manifolds. In particular, we show that dynamics on center-stable manifolds of invariant sets at infinity with appropriate time-scale desingularizations as well as blowing-up of singularities characterize dynamics of blow-up solutions as well as their rigorous blow-up rates.