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Subsidiary bifurcations near bifocal homoclinic orbits

Published online by Cambridge University Press:  04 October 2011

Paul Glendinning
Paul Glendinning
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW
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Abstract

Suppose that an ordinary differential equation in ℝ4 has an orbit Γ bi-asymptotic to a stationary point O of the flow. If the characteristic equation of the linear flow near O has roots (λ0 ± ιω0, λ1±ιω1) with —λ0 > λ1 > 0 and ωi > 0 for i = 0,1, we show that there are sequences of more complicated orbits bi-asymptotic to O in generic one-parameter perturbations of the equations. When ω0 ≠ 2nω1 for all n ∈ ℕ, there are sequences of such orbits on both sides of the bifurcation value of the parameter, in contrast to a similar case for flows in ℝ3.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 597 - 605
Copyright
Copyright © Cambridge Philosophical Society 1989

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