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Bifurcation of homoclinic orbits with saddle-center equilibrium. (English) Zbl 1127.34022

The paper presents new global perturbation techniques for detecting the persistence of transversal homoclinic orbits in general nondegenerate systems with action-angle variables:

z ˙=f(z,I)+εg z (z,I,θ,λ,ε),I ˙=εg I (z,I,θ,λ,ε),θ ˙=ω,(1)

where (z,I,θ) n × m ×𝕋 l , λ k , 0ε1, |λ|1, and g z ,g I are 2π-periodic in θ. The unperturbed system (ε=0) is assumed to have a saddle-center type equilibrium whose stable and unstable manifolds intersect in a one dimensional manifold, and is not completely integrable or near-integrable. By constructing local coordinate systems near the unperturbed homoclinic orbit, conditions for the existence of a transversal homoclinic orbit are obtained. Conditions for the existence of periodic orbits bifurcating from the homoclinic orbit are also given.

MSC:
34C23Bifurcation (ODE)
34C37Homoclinic and heteroclinic solutions of ODE
37C29Homoclinic and heteroclinic orbits
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