Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

TL;DR

This paper develops a computer-assisted methodology to analyze saddle-type blow-up solutions of autonomous ODEs, revealing their global structure, dependence on initial conditions, and singular behaviors.

Contribution

It combines dynamical systems techniques with computer-assisted proofs to rigorously validate and extract the global nature of saddle-type blow-up solutions.

Findings

Global trajectories on stable manifolds are characterized.

Blow-up times depend smoothly on initial conditions.

Phase space decomposition acts as separatrices.

Abstract

In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery (e.g., compactifications, time-scale desingularizations of vector fields) with tools from computer-assisted proofs (e.g., rigorous integrators, the parameterization method for invariant manifolds), these blow-up solutions are obtained as trajectories on local stable manifolds of hyperbolic saddle equilibria at infinity. With the help of computer-assisted proofs, global trajectories on stable manifolds, inducing blow-up solutions, provide a global picture organized by global-in-time solutions and blow-up solutions simultaneously. Using the proposed methodology, intrinsic features of saddle-type blow-ups are observed: locally smooth dependence of blow-up times on initial points, level set distribution of blow-up times, and decomposition of the phase space playing a role as separatrixes among solutions, where the magnitude of initial points near those blow-ups does not matter for asymptotic behavior. Finally, singular behavior of blow-up times on initial points belonging to different family of blow-up solutions is addressed.