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

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.

45 pages

References in corpus (3)

• Numerical validation of blow-up solutions with quasi-homogeneous compactifications
• Rigorous numerics of finite-time singularities in dynamical systems - methodology and applications
• On blow-up solutions of differential equations with Poincaré-type compactifications