Homoclinic Solutions and Chaos

in Ordinary Differential Equations

with Singular Perturbations

Author:
Joseph Gruendler

Journal:
Trans. Amer. Math. Soc. **350** (1998), 3797-3814

MSC (1991):
Primary 34E15, 34C37, 58F13

DOI:
https://doi.org/10.1090/S0002-9947-98-02211-9

MathSciNet review:
1475684

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Abstract | References | Similar Articles | Additional Information

Abstract: Ordinary differential equations are considered which contain a singular perturbation. It is assumed that when the perturbation parameter is zero, the equation has a hyperbolic equilibrium and homoclinic solution. No restriction is placed on the dimension of the phase space or on the dimension of intersection of the stable and unstable manifolds. A bifurcation function is established which determines nonzero values of the perturbation parameter for which the homoclinic solution persists. It is further shown that when the vector field is periodic and a transversality condition is satisfied, the homoclinic solution to the perturbed equation produces a transverse homoclinic orbit in the period map. The techniques used are those of exponential dichotomies, Lyapunov-Schmidt reduction and scales of Banach spaces. A much simplified version of this latter theory is developed suitable for the present case. This work generalizes some recent results of Battelli and Palmer.

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Additional Information

**Joseph Gruendler**

Affiliation:
Department of Mathematics, North Carolina A&T State University, Greensboro, North Carolina 27411

Email:
gruendlj@ncat.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-02211-9

Keywords:
Ordinary differential equations,
homoclinic solutions,
bifurcations,
singular perturbations

Received by editor(s):
December 28, 1995

Received by editor(s) in revised form:
October 23, 1996

Additional Notes:
This work supported in part by the J. William Fulbright Commission.

Article copyright:
© Copyright 1998
American Mathematical Society