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The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit. (English) Zbl 0864.58044

Summary: B. Deng [J. Dyn. Differ. Equations 5, No. 3, 417-467 (1993; Zbl 0782.34042)] has demonstrated a mechanism through which a perturbation of a vector field having an inclination-flip homoclinic orbit would have a Smale horseshoe. In this article we prove that if the eigenvalues of the saddle to which the homoclinic orbit is asymptotic satisfy the condition \(2\lambda^u<\min\{-\lambda^s,\lambda^{uu}\}\) then there are arbitrarily small perturbations of the vector field which possess a Smale horseshoe. Moreover we analyze a sequence of bifurcations leading to the annihilation of the horseshoe. This sequence contains, in particular, the points of existence of \(n\)-homoclinic orbits with arbitrary \(n\).

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion

Citations:

Zbl 0782.34042
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References:

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