On blow-up solutions of differential equations with poincaré-type compactifications

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We provide explicit criteria for blow-up solutions of autonomous ordinary differential equations. Ideas are based on the quasi-homogeneous desingularization (blowing-up) of singularities and compactifications of phase spaces, which suitably desingularize singularities at infinity. We derive several type of compactifications and show that dynamics at infinity is qualitatively independent of the choice of such compactifications. As a main result, we show that hyperbolic invariant sets, such as equilibria and periodic orbits, at infinity can induce blow-up solutions with specific blow-up rates. In particular, blow-up solutions can be described as trajectories on stable manifolds of invariant sets "at infinity" for vector fields associated with compactifications. Finally, we demonstrate blow-up solutions of several differential equations both analytically and numerically.

Original languageEnglish
Pages (from-to)2249-2288
Number of pages40
JournalSIAM Journal on Applied Dynamical Systems
Volume17
Issue number3
DOIs
Publication statusPublished - Jan 1 2018

Fingerprint

Blow-up of Solutions
Compactification
Differential equations
Infinity
Differential equation
Blow-up Solution
Invariant Set
Blow molding
Ordinary differential equations
Orbits
Trajectories
Singularity
Hyperbolic Set
Desingularization
Blowing-up
Blow-up Rate
Stable Manifold
Periodic Orbits
Phase Space
Vector Field

All Science Journal Classification (ASJC) codes

  • Analysis
  • Modelling and Simulation

Cite this

On blow-up solutions of differential equations with poincaré-type compactifications. / Matsue, Kaname.

In: SIAM Journal on Applied Dynamical Systems, Vol. 17, No. 3, 01.01.2018, p. 2249-2288.

Research output: Contribution to journalArticle

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