### 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 language | English (US) |
---|---|

Pages (from-to) | 1263-1280 |

Number of pages | 18 |

Journal | Nonlinearity |

Volume | 9 |

Issue number | 5 |

DOIs | |

State | Published - Dec 1 1996 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Nonlinearity*,

*9*(5), 1263-1280. https://doi.org/10.1088/0951-7715/9/5/009

}

*Nonlinearity*, vol. 9, no. 5, pp. 1263-1280. https://doi.org/10.1088/0951-7715/9/5/009

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Existence of infinitely many connecting orbits in a singularly perturbed ordinary differential equation

AU - Kokubu, Hiroshi

AU - Mischaikow, Konstantin

AU - Oka, Hiroe

PY - 1996/12/1

Y1 - 1996/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0001398433&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001398433&partnerID=8YFLogxK

U2 - https://doi.org/10.1088/0951-7715/9/5/009

DO - https://doi.org/10.1088/0951-7715/9/5/009

M3 - Article

VL - 9

SP - 1263

EP - 1280

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 5

ER -