Multi-order asymptotic expansion of blow-up solutions for autonomous ODEs. II -- Dynamical Correspondence

TL;DR

This paper establishes a connection between the eigenstructures of Jacobian matrices of two related systems describing finite-time blow-ups in quasi-homogeneous ODEs, providing a criterion for blow-up existence based on asymptotic expansions.

Contribution

It introduces a natural correspondence of eigenstructures between two systems and links asymptotic blow-up expansions to stability criteria, advancing understanding of blow-up solutions.

Findings

Eigenstructure correspondence between systems

Asymptotic expansions as blow-up criteria

Stability gap structure in blow-up solutions

Abstract

In this paper, we provide a natural correspondence of eigenstructures of Jacobian matrices associated with equilibria for appropriately transformed two systems describing finite-time blow-ups for ODEs with quasi-homogeneity in an asymptotic sense. As a corollary, we see that asymptotic expansions of blow-ups proposed in Part I themselves provide a criterion of the existence of blow-ups with an intrinsic gap structure of stability information among two systems. Examples provided in Part I are revisited to show the above correspondence.