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Zbl 1127.34022
Liu, Xingbo; Fu, Xianlong; Zhu, Deming
Bifurcation of homoclinic orbits with saddle-center equilibrium. (English)
[J] Chin. Ann. Math., Ser. B 28, No. 1, 81-92 (2007). ISSN 0252-9599; ISSN 1860-6261/e

The paper presents new global perturbation techniques for detecting the persistence of transversal homoclinic orbits in general nondegenerate systems with action-angle variables: $$ \aligned \dot z&=f(z,I)+\varepsilon g^z(z,I,\theta ,\lambda ,\varepsilon ),\\ \dot I&=\varepsilon g^I(z,I,\theta ,\lambda ,\varepsilon ),\\ \dot \theta &=\omega, \endaligned \tag 1$$ where $(z,I,\theta )\in\Bbb R^n\times\Bbb R^m\times\Bbb T^l$, $\lambda\in\Bbb R^k$, $0\le\varepsilon\ll 1$, $\vert \lambda\vert \ll 1$, and $g^z$,$g^I$ are $2\pi$-periodic in $\theta$. The unperturbed system ($\varepsilon =0$) 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.
[Eugene Ershov (St. Petersburg)]

MSC 2000:: *34C23 Bifurcation (periodic solutions)
34C37 Homoclinic and heteroclinic solutions of ODE
37C29 Homoclinic and heteroclinic orbits

Keywords: bifurcation; transversal homoclinic orbits; saddle-center equilibrium; multi-pulse orbits

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