Homoclinic Saddle-Node Bifurcations in Singularly Perturbed Systems |
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Authors: | Arjen Doelman Geertje Hek |
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Institution: | (1) Korteweg-de Vries Instituut, Universiteit van Amsterdam, Plantage Muidergracht 24, 1028 TV Amsterdam, The Netherlands;(2) Mathematisch Instituut, Universiteit Utrecht, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands |
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Abstract: | In this paper we study the creation of homoclinic orbits by saddle-node bifurcations. Inspired on similar phenomena appearing in the analysis of so-called localized structures in modulation or amplitude equations, we consider a family of nearly integrable, singularly perturbed three dimensional vector fields with two bifurcation parameters a and b. The O() perturbation destroys a manifold consisting of a family of integrable homoclinic orbits: it breaks open into two manifolds, W
s() and W
u(), the stable and unstable manifolds of a slow manifold . Homoclinic orbits to correspond to intersections W
s()W
u(); W
s()W
u()= for a<a*, a pair of 1-pulse homoclinic orbits emerges as first intersection of W
s() and W
u() as a>a*. The bifurcation at a=a* is followed by a sequence of nearby, O(
2(log)2) close, homoclinic saddle-node bifurcations at which pairs of N-pulse homoclinic orbits are created (these orbits make N circuits through the fast field). The second parameter b distinguishes between two significantly different cases: in the cooperating (respectively counteracting) case the averaged effect of the fast field is in the same (respectively opposite) direction as the slow flow on . The structure of W
s()W
u() becomes highly complicated in the counteracting case: we show the existence of many new types of sometimes exponentially close homoclinic saddle-node bifurcations. The analysis in this paper is mainly of a geometrical nature. |
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Keywords: | global bifurcations homoclinic orbits singularly perturbed systems return maps |
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