Perturbed homoclinic solutions in reversible 1:1 resonance vector fields

Abstract : We consider a smooth reversible vector field in R^4, such that the origin is a fixed point. The differential at the origin has two double pure imaginary eigenvalues ±iq for the critical value 0 of the parameter µ. We show, by a normal form analysis, that the vector field can be approximated by an integrable field in R^4, for which we know all solutions. Specially interesting ones are the homoclinics to 0, and homoclinics to periodic solutions, depending on the sign of a leading nonlinear coefficient. We prove in particular the persistence of these homoclinics for the full vector field.
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https://hal.univ-cotedazur.fr/hal-01271158
Contributor : Gerard Iooss <>
Submitted on : Tuesday, February 9, 2016 - 3:09:31 PM
Last modification on : Thursday, April 4, 2019 - 10:18:05 AM

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  • HAL Id : hal-01271158, version 1

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Gérard Iooss, Marie-Christine Pérouème. Perturbed homoclinic solutions in reversible 1:1 resonance vector fields. Journal of Differential Equations, Elsevier, 1993, 102 (1), pp.27. ⟨hal-01271158⟩

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