Bifurcation of degenerate homoclinic orbits to saddle-center in reversible systems

Abstract

The authors study the bifurcation of homoclinic orbits from a degenerate homoclinic orbit in reversible system. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in two-dimensional manifolds. A perturbation technique for the detection of symmetric and nonsymmetric homoclinic orbits near the primary homoclinic orbits is developed. Some known results are extended.

Keywords

Reversible system / Homoclinic orbits / Saddle-center / Bifurcation

References

[1] Champneys A. R., Malomed B. A., Yang J., Kaup D. J.. Embedded solitons: solitary waves in resonance with the linear spectrum. Phys. D, 2001, 152: 340-354

[2] Champneys A. R.. Homoclinic orbits in reversible system and their applications in mechanics, fluids and optics. Phys. D, 1998, 112: 158-186

[3] Wagenknecht T., Champneys A. R.. When gap solitons become embedded solitons: an unfolding via Lin’s method. Phys. D, 2003, 177: 50-70

[4] Vanderbauwhede A., Fiedler B.. Homoclinic period blow-up in reversible and conservative systems. Z. Angew. Math. Phys., 1992, 43: 292-318

[5] Shilnikov L. P., Shilnikov A. L., Turaev D., Chua L. O.. Methods of Qualitative Theory in Nonlinear Dynamics, Part I, 1998, Singapore: World Scientific

[6] Yagasaki K., Wagenknecht T.. Detection of symmetric homoclinic orbits to saddle-center in reversible systems. Phys. D, 2006, 214: 169-181

[7] Lin X. B.. Using Melnikov’s method to solve Shilnikov’s problem. Proc. Roy. Soc., Edinburgh, 1990, 116A: 295-325

[8] Klaus J., Knobloch J.. Bifurcation of homoclinic orbits to a saddle-center in reversible systems. Int. J. of Bifur. and Chaos, 2003, 13(9): 2603-2622

[9] Wiggins S.. Global Bifurcation and Chaos, 1988, New York: Springer-Verlag