Existence of infinitely many connecting orbits in a singularly perturbed ordinary differential equation

Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

We consider a one-parameter family of two-dimensional ordinary differential equations with a slow parameter drift. Our equation assumes that when there is no parameter drift, there are two invariant curves consisting of equilibria, one of which is normally hyperbolic and whose stable and unstable manifolds intersect transversely. The slow parameter drift is introduced in a way that it creates two hyperbolic equilibria in the invariant normally hyperbolic curve that is persistent under perturbation. In this situation, we prove that the number of distinct orbits which connects these two equilibria changes from finite to infinite depending on the direction of the slow parameter drift. The proof uses the Conley index theory. The relation to a singular boundary value problem studied by W Kath is also discussed.

Original languageEnglish (US)
Pages (from-to)1263-1280
Number of pages18
JournalNonlinearity
Volume9
Issue number5
DOIs
StatePublished - Dec 1 1996
Externally publishedYes

Fingerprint

Connecting Orbits
Singularly Perturbed
Ordinary differential equations
Boundary value problems
Ordinary differential equation
Orbits
differential equations
orbits
Conley Index
Invariant Curves
Stable and Unstable Manifolds
Index Theory
Singular Boundary Value Problem
curves
Intersect
boundary value problems
Orbit
Perturbation
Distinct
perturbation

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)
  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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Existence of infinitely many connecting orbits in a singularly perturbed ordinary differential equation. / Kokubu, Hiroshi; Mischaikow, Konstantin; Oka, Hiroe.

In: Nonlinearity, Vol. 9, No. 5, 01.12.1996, p. 1263-1280.

Research output: Contribution to journalArticle

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