Multi-order asymptotic expansion of blow-up solutions for autonomous ODEs. I -- Method and Justification

TL;DR

This paper introduces a systematic method for deriving multi-order asymptotic expansions of blow-up solutions in autonomous ODEs, providing algebraic tools and concrete examples for precise approximation near blow-up points.

Contribution

It presents a novel systematic methodology and algebraic framework for calculating detailed multi-order asymptotic expansions of blow-up solutions in autonomous ODEs.

Findings

Algebraic objects determine all possible expansion orders.

Method applied to concrete examples of blow-up solutions.

Provides a systematic approach for asymptotic analysis near blow-up.

Abstract

In this paper, we provide a systematic methodology for calculating multi-order asymptotic expansion of blow-up solutions near blow-up for autonomous ordinary differential equations (ODEs). Under the specific form of the principal term of blow-up solutions for a class of vector fields, we extract algebraic objects determining all possible orders in the asymptotic expansions. Examples for calculating concrete multi-order asymptotic expansions of blow-up solutions are finally collected.