ISSN:
1572-9222
Keywords:
global bifurcations
;
homoclinic orbits
;
singularly perturbed systems
;
return maps
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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(ε) perturbation destroys a manifold consisting of a family of integrable homoclinic orbits: it breaks open into two manifolds, W s(Γ) and W u(Γ), the stable and unstable manifolds of a slow manifold Γ. Homoclinic orbits to Γ correspond to intersections W s(Γ)∩W u(Γ); W s(Γ)∩W u(Γ)=∅ for a〈a*, a pair of 1-pulse homoclinic orbits emerges as first intersection of W s(Γ) and W u(Γ) as a〉a*. The bifurcation at a=a* is followed by a sequence of nearby, O(ε 2(logε)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 Γ. The structure of W s(Γ)∩W u(Γ) 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1009050803510
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