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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:

$\begin{array}{cc}\hfill \stackrel{˙}{z}& =f\left(z,I\right)+\epsilon {g}^{z}\left(z,I,\theta ,\lambda ,\epsilon \right),\hfill \\ \hfill \stackrel{˙}{I}& =\epsilon {g}^{I}\left(z,I,\theta ,\lambda ,\epsilon \right),\hfill \\ \hfill \stackrel{˙}{\theta }& =\omega ,\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $\left(z,I,\theta \right)\in {ℝ}^{n}×{ℝ}^{m}×{𝕋}^{l}$, $\lambda \in {ℝ}^{k}$, $0\le \epsilon \ll 1$, $|\lambda |\ll 1$, and ${g}^{z}$,${g}^{I}$ are $2\pi$-periodic in $\theta$. The unperturbed system ($\epsilon =0$) 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:
 34C23 Bifurcation (ODE) 34C37 Homoclinic and heteroclinic solutions of ODE 37C29 Homoclinic and heteroclinic orbits
##### References:
 [1] Wiggins, S., Global Bifurcation and chaos, Springer-Verlag, New York, 1988. [2] Wiggins, S. and Holmes, P., Homoclinic orbits in slowly varying oscillators, SIAM J. Math. Anal., 18, 1987, 612–629. · Zbl 0622.34041 · doi:10.1137/0518047 [3] Yagasaki, K., The method of Melnikov for perturbations of multi-degree-of-freedom Hamiltonian systems, Nonlinearity, 12, 1999, 799–822. · Zbl 0967.34042 · doi:10.1088/0951-7715/12/4/304 [4] Huang, D., Liu, Z. and Cheng, Z., Global dynamics near the resonance in the Sine-Gordon equation, J. Shanghai Univ., 2, 1998, 259–261. · Zbl 0927.34038 · doi:10.1007/s11741-998-0036-6 [5] Feckan, M., Bifurcation of multi-bump homoclinics in systems with normal and slow variables, J. Differential Equations, 41, 2000, 1–17. [6] Kovacic, G., Singular perturbation theory for homoclinic orbits in a class of near-integrable dissipative system, SIAM J. Math. Anal., 26, 1995, 1611–1643. · Zbl 0835.34049 · doi:10.1137/S0036141093245422 [7] Zhu, D. M., Problems in homoclinic bifurcations with higher dimensions, Acta Math. Sinica, New Ser., 14, 1998, 341–352. · Zbl 0932.37032 · doi:10.1007/BF02580437 [8] Zhu, D. M. and Xia, Z. H., Bifurcations of heteroclinic loops, Sci. China Ser. A, 41, 1998, 837–848. · Zbl 0993.34040 · doi:10.1007/BF02871667 [9] Zhu, D. M. and Han, M. A., Bifurcation of homoclinic orbits in fast variable space (in chinese), Chin. Ann. Math, 23A(4), 2002, 438–449. [10] Deng, B., Homoclinic bifurcations with nonhyperbolic equilibria, SIAM J. Math. Anal., 3, 1990, 693–720. · Zbl 0698.34037 · doi:10.1137/0521037