Blow-up behavior for ODEs with normally hyperbolic nature in dynamics at infinity

TL;DR

This paper analyzes blow-up behavior in autonomous and nonautonomous ODEs using dynamics at infinity, phase space embeddings, and geometric properties to understand asymptotic behavior and invariant manifolds.

Contribution

It introduces a novel geometric framework for describing blow-ups via dynamics at infinity and invariant manifolds with asymptotic phase.

Findings

Blow-up solutions are characterized by shadowing properties at infinity.

Invariant manifolds with asymptotic phase induce blow-ups.

The approach applies to both autonomous and nonautonomous systems.

Abstract

We describe blow-up behavior for ODEs by means of dynamics at infinity with complex asymptotic behavior in autonomous systems, as well as in nonautonomous systems. Based on preceding studies, a variant of closed embeddings of phase spaces and the time-scale transformation determined by the structure of vector fields at infinity reduce our description of blow-ups to unravel the shadowing property of (pre)compact trajectories on the horizon, the geometric object expressing the infinity, with the specific convergence rates. Geometrically, this description is organized by asymptotic phase of invariant sets on the horizon. Blow-up solutions in nonautonomous systems can be described in a similar way. As a corollary, normally, or partially hyperbolic invariant manifolds on the horizon possessing asymptotic phase are shown to induce blow-ups.