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On the number of limit cycles in double homoclinic bifurcations. (English) Zbl 1013.34026
The authors consider perturbations of Hamiltonian systems in ${ℝ}^{2}$. They assume that the unperturbed system possesses a double homoclinic loop (figure-eight-configuration) at a hyperbolic fixed point. The main results concern the maximal number of limit cycles near the given homoclinic orbits. The condition permitting the existence of the limit cycles are formulated in terms of Melnikov functions.
##### MSC:
 34C05 Location of integral curves, singular points, limit cycles (ODE)
##### Keywords:
homoclinic orbits; limit cycles; bifurcation
##### References:
 [1] Roussarir, R., On the number of limit cycles which appear by perturbation of separatnx loop of plannar fields, Bol. Soc. Brasil Mat., 1986, 17: 67. · Zbl 0628.34032 · doi:10.1007/BF02584827 [2] Han Maoan, Ye Yanqian, On the coefficients appearing in the expansion of Melnikov functions in homoclinic bifurcations, Ann. of Diff. Equs., 1998, 14(2): 156. [3] Han Maoan, Cyclicity of plannar homoclinic loops and quadratic integratable systems, Science in China, Ser. A, 1997, 40(12): 1247. · Zbl 0930.37035 · doi:10.1007/BF02876370 [4] Joyal, P., Generalized Hopf bifurcation and its dual generalized homoclinic bifurcation, SIAM J. Math., 1988, 48: 481. · Zbl 0642.34041 · doi:10.1137/0148027 [5] Han Maoan, Luo Dingjun, Zhu Deming, The uniqueness of limit cycles bifurrated fiwm a separatrix cycle (II), Acta Math. Sinica (in Chinese), 1992, 4: 541. [6] Han Maoan, Bifurcations of limit cycles from a heteroclinic cycle of Hamiltonian systems, Chin. Ann. of Math., 1998, 19B(2): 189. [7] Han Maoan, Zhu Deming, Bifurcation Theory of Differential Equations (in Chinese), Beijing: Coal Industry Publishing House, 1994. [8] Chow, S. N., Hale, J. K., Methods of Bifurcation Theory, New York: Springer-Verlag, 1982.