Research ArticleOpen AccessGoogle Scholar indexed
Exponential Dichotomies and Homoclinic Orbits from Heteroclinic Cycles
Yiyang Medical College Hunan Pro of China, Yiyang, China
Yiyang Medical College Hunan Pro of China, Yiyang, China
Yiyang Medical College Hunan Pro of China, Yiyang, China
- 1 Yiyang Medical College Hunan Pro of China, Yiyang, China
- 2 Yiyang Medical College Hunan Pro of China, Yiyang, China
- 3 Yiyang Medical College Hunan Pro of China, Yiyang, China
American Journal of Computational Mathematics·Volume 02 (2012)·Pages 106–113·Published 21 June 2012·DOI10.4236/ajcm.2012.22014
Copy link · social · email
Abstract
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.
KeywordsExponential DichotomiesHomoclinic OrbitsHeteroclinic CycleMelnikov Function
- S. Wiggins, “Golbal Bifurcations and Chaos,” Springer-Verlag, New York, 1988.
- 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
- 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
- S. Campbell and P. Holmes, “Bifurcation from O(2)sy Mmetric Heterclinic Cycles with Three Interacing Mofes,” Nonlinearity, Vol. 4, 1991, pp. 697-726.
- 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.
- 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
- 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
- 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.
- 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
- 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
- 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
- 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
- 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