Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A simple description of blow-up solutions through dynamics at infinity in nonautonomous ODEs

Kaname Matsue

TL;DR

This work develops a simple, universal criterion for type-I blow-up in nonautonomous ODEs by linking dynamics at infinity to multi-order asymptotic expansions. By extending the autonomous framework of asymptotically quasi-homogeneous vector fields to time-dependent settings through the desingularized vector field on a horizon, the authors establish a one-to-one correspondence between roots of balance laws and horizon equilibria, and they map the associated eigenstructures to derive a practical blow-up criterion. The key contributions include a detailed analysis of transversal and tangential eigenstructures, a robust correspondence between the matrices $A^{\rm ext}$ and $Dg^{\rm ext}_\ast$, and explicit criteria ensuring the existence and characterization of type-I blow-ups in several nonautonomous examples (Painlevé I, self-similar diffusion, and a time-variant Hamiltonian system). The results provide a computationally accessible pathway to predict blow-up behavior from leading-order terms and horizon dynamics, with explicit blow-up rates and $t_{\max}$ expressions under mild spectral and non-resonance conditions. This framework sharpens the understanding of how asymptotic expansions and dynamics at infinity govern nonautonomous blow-up phenomena and paves the way for analyzing more complex blow-up scenarios.

Abstract

A simple criterion of the existence of (type-I) blow-up solutions for nonautonomous ODEs is provided. In a previous study [Matsue, SIADS, 24(2025), 415-456], geometric criteria for characterizing blow-up solutions for nonautonomous ODEs are provided by means of dynamics at infinity. The basic idea towards the present aim is to correspond such criteria to leading-term equations associated with blow-up ansatz characterizing multiple-order asymptotic expansions, which originated from the corresponding study developed in the framework of autonomous ODEs. Restricting our attention to constant coefficients of leading terms of blow-ups, results involving the simple criterion of blow-up characterizations in autonomous ODEs can be mimicked to nonautonomous ODEs.

A simple description of blow-up solutions through dynamics at infinity in nonautonomous ODEs

TL;DR

This work develops a simple, universal criterion for type-I blow-up in nonautonomous ODEs by linking dynamics at infinity to multi-order asymptotic expansions. By extending the autonomous framework of asymptotically quasi-homogeneous vector fields to time-dependent settings through the desingularized vector field on a horizon, the authors establish a one-to-one correspondence between roots of balance laws and horizon equilibria, and they map the associated eigenstructures to derive a practical blow-up criterion. The key contributions include a detailed analysis of transversal and tangential eigenstructures, a robust correspondence between the matrices and , and explicit criteria ensuring the existence and characterization of type-I blow-ups in several nonautonomous examples (Painlevé I, self-similar diffusion, and a time-variant Hamiltonian system). The results provide a computationally accessible pathway to predict blow-up behavior from leading-order terms and horizon dynamics, with explicit blow-up rates and expressions under mild spectral and non-resonance conditions. This framework sharpens the understanding of how asymptotic expansions and dynamics at infinity govern nonautonomous blow-up phenomena and paves the way for analyzing more complex blow-up scenarios.

Abstract

A simple criterion of the existence of (type-I) blow-up solutions for nonautonomous ODEs is provided. In a previous study [Matsue, SIADS, 24(2025), 415-456], geometric criteria for characterizing blow-up solutions for nonautonomous ODEs are provided by means of dynamics at infinity. The basic idea towards the present aim is to correspond such criteria to leading-term equations associated with blow-up ansatz characterizing multiple-order asymptotic expansions, which originated from the corresponding study developed in the framework of autonomous ODEs. Restricting our attention to constant coefficients of leading terms of blow-ups, results involving the simple criterion of blow-up characterizations in autonomous ODEs can be mimicked to nonautonomous ODEs.

Paper Structure

This paper contains 22 sections, 34 theorems, 216 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Blow-up description for nonautonomous systems through dynamics at infinity: Short review
  3. Asymptotically quasi-homogeneous vector fields
  4. Quasi-parabolic embeddings
  5. Dynamics at infinity in nonautonomous systems
  6. Type-I nonautonomous blow-up
  7. Eigenstructure of dynamics at infinity and a simplified blow-up criterion
  8. A system associated with multi-order asymptotic expansion of type-I blow-up solutions
  9. Balance law and equilibria on the horizon
  10. Structure of $Dg^{\rm ext}_\ast$ and technical assumptions
  11. Eigenstructure in '' transversal" direction; case 1: $k>1$
  12. Eigenstructure in '' tangent spaces"
  13. Eigenstructure in '' transversal" direction; case 2: $k=1$
  14. Correspondence of eigenstructures and simplified blow-up criterion
  15. Examples
  16. ...and 7 more sections

Key Result

Lemma 2.4

A quasi-homogenous function $f_0$ of type $(\alpha_1,\ldots,\alpha_n)$ and order $k$ satisfies the following differential equation: equivalently

Figures (1)

  • Figure 1: Trajectory for (\ref{['WWL_system_k1']})

Theorems & Definitions (67)

  • Definition 2.1: Homogeneity index and admissible domain. cf. Mat2018Mat2025_NHIM
  • Definition 2.2: Asymptotically quasi-homogeneous vector fields, cf. D1993Mat2018
  • Remark 2.3
  • Lemma 2.4: asym2
  • Lemma 2.5: asym2
  • Remark 2.6
  • Definition 2.7: Quasi-parabolic embedding, cf. MT2020_1
  • Remark 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 57 more