AJCM  Vol.2 No.2 , June 2012
Exponential Dichotomies and Homoclinic Orbits from Heteroclinic Cycles
In this paper, we investigate the homoclinic bifurcations from a heteroclinic cycle by using exponential dichotomies. We give a Melnikov—type condition assuring the existence of homoclinic orbits form heteroclinic cycle. We improve some important results.

Cite this paper
T. Chen, Y. Xiang and Y. Chen, "Exponential Dichotomies and Homoclinic Orbits from Heteroclinic Cycles," American Journal of Computational Mathematics, Vol. 2 No. 2, 2012, pp. 106-113. doi: 10.4236/ajcm.2012.22014.

[1]   S. Wiggins, “Golbal Bifurcations and Chaos,” Springer-Verlag, New York, 1988.

[2]   K. J. Palmer, “Transversal Heteroclinic Orbits and Cherry’s Example of a Nonintegrable Hamiltonian System,” Journal of Differential Equations, Vol. 65, No. 3, 1986, pp. 321-360. doi:10.1016/0022-0396(86)90023-9

[3]   K. J. Palmer, “Exponential Dichotomies and Transversal Homoclinic Points,” Journal of Differential Equations, Vol. 55, No. 2, 1984, pp. 225-256. doi:10.1016/0022-0396(84)90082-2

[4]   S. Campbell and P. Holmes, “Bifurcation from O(2)sy Mmetric Heterclinic Cycles with Three Interacing Mofes,” Nonlinearity, Vol. 4, 1991, pp. 697-726.

[5]   K. R. Meyer and G. R. Sell, “Melnikov Transforms, Bernoulli Bundle and Almost Periodic Perturbations,” Transactions of the American Mathematical Society, Vol. 314, No. 1, 1989, pp. 63-105.

[6]   H. Kokubu, “Homoclinic and Heteroclinic Bifurcations of Vector Fields,” Japan Journal of Industrial and Applied Mathematics, Vol. 5, No. 3, 1988, pp. 455-501. doi:10.1007/BF03167912

[7]   S. N. Chow, B. Deng and D. Terman, “The Bifurcations of a Homoclinic and a Periodic Orbit from Two Hetero- clinic Orbits,” SIAM Journal on Mathematical Analysis, Vol. 21, No. 1, 2000, pp. 179-204. doi:10.1137/0521010

[8]   J. M. Gambaudo, P. Glendinning and C. Tresser, “Collages de Cycles et Suites de Farey,” Comptes Rendus de l'Académie des Sciences, Vol. 299, 1984, pp. 711-714.

[9]   S. N. Chow, B. Deng and D. Terman, “The Bifurcation of a Homoclinic Orbit from Two Heteroclinic Orbits—A Topological Approach,” Applicable Analysis: An International Journal, Vol. 42, No. 1-4, 1991, pp. 1057-1080. doi:10.1080/00036819108840047

[10]   X. B. Lin, “Using Melnikov’s Method to Solve Silnikov Problems,” Proceedings of the Royal Society of Edin- burgh: Section A Mathematics, Vol. 116, No. 3-4, 1990, pp. 295-325. doi:10.1017/S0308210500031528

[11]   B. Sandstede and A. Scheel, “Forced Symmetry Breaking of Homoclinic Cycles,” Nonlinearity, Vol. 8, No. 3, 2009, pp. 333-365. doi:10.1088/0951-7715/8/3/003

[12]   J. Guckenheimer and P. Holmes, “Strucarrlly Stable Pulse Heteroclinic Cycles,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 103, No. 1, 2008, pp. 189-192. doi:10.1017/S0305004100064732

[13]   M. Krupa and I. Melbourne, “Asymptotic Stability of Heteroclinic Cycles in Systems with Symmetry,” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 134, No. 6, 2004, pp. 1177-1197. doi:10.1017/S0308210500003693

[14]   M. Krupa, “Robust Heteroclinic Cycles,” Journal of Non- linear Science, Vol. 7, No. 2, 2011. doi:10.1007/BF02677976

[15]   W. A. Coppel, “Dichotomies in Stability Theory, Lecture Notes in Mathematics,” Springer-Verlag, New York, 1978.

[16]   R. J. Sacker and G. R. Sell, “A Spectral Theory for Linear Differential Systems,” Journal of Differential Equations, Vol. 27, No. 3, 1978, pp. 320-385. doi:10.1016/0022-0396(78)90057-8

[17]   P. Chossat, M. Krupa, I. Melbourne and A. Scheel, “Trans- verse Bifurcations of Homkoclinic Cycles,” Physica D: Nonlinear Phenomena, Vol. 100, No. 1-2, 2011, pp. 85-100. doi:10.1016/S0167-2789(96)00186-8

[18]   V. Naudot, “Hyperbolic Dynamice in the Unfolding of a Degenerate Homoclinic Orbit,” Preprint.

[19]   W. Y. Zeng, “Exponential Dichotomies and Transversal Homoclinic Orbits in Degenerate Cases,” Journal of Dynamics and Differential Equations, Vol. 7, No. 4, 1995, pp. 521-548. doi:10.1007/BF02218723

[20]   G. R. Sell, “Bifurcation of Higher Dimensional Tori,” Archive for Rational Mechanics and Analysi, Vol. 69, No. 3, 1979, pp. 199-230. doi:10.1007/BF00248134