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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 1475684
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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: http://dx.doi.org/10.1090/S0002-9947-98-02211-9
PII: S 0002-9947(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