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Bifurcation of degenerate homoclinic orbits to saddle-center in reversible systems. (English) Zbl 1197.34067

This paper deals with bifurcations from a homoclinic orbit Γ to a saddle-center in a two parameter family of reversible systems in 2n+2 . The saddle-center has a two-dimensional center manifold filled with periodic orbits and n-dimensional stable and unstable manifolds in each case. Further it is assumed that the homoclinic orbit Γ is degenerate, meaning that along Γ the stable and unstable manifolds (of the equilibrium) have an additional common tangent (linearly independent into the direction of the vector field).

Using a Liapunov-Schmidt reduction, the nearby 1-homoclinic orbits to the equilibrium are studied.

34C37Homoclinic and heteroclinic solutions of ODE
37G25Bifurcations connected with nontransversal intersection
34C23Bifurcation (ODE)
34C14Symmetries, invariants (ODE)
[1]Champneys, A. R., Malomed, B. A., Yang, J. and Kaup, D. J., Embedded solitons: solitary waves in resonance with the linear spectrum, Phys. D, 152, 2001, 340–354. · Zbl 0976.35087 · doi:10.1016/S0167-2789(01)00178-6
[2]Champneys, A. R., Homoclinic orbits in reversible system and their applications in mechanics, fluids and optics, Phys. D, 112, 1998, 158–186. · Zbl 1194.37154 · doi:10.1016/S0167-2789(97)00209-1
[3]Wagenknecht, T. and Champneys, A. R., When gap solitons become embedded solitons: an unfolding via Lin’s method, Phys. D, 177, 2003, 50–70. · Zbl 1011.37037 · doi:10.1016/S0167-2789(02)00773-X
[4]Vanderbauwhede, A. and Fiedler, B., Homoclinic period blow-up in reversible and conservative systems, Z. Angew. Math. Phys., 43, 1992, 292–318. · Zbl 0762.34023 · doi:10.1007/BF00946632
[5]Shilnikov, L. P., Shilnikov, A. L., Turaev, D. and Chua, L. O., Methods of Qualitative Theory in Nonlinear Dynamics, Part I, World Scientific, Singapore, 1998.
[6]Yagasaki, K. and Wagenknecht, T., Detection of symmetric homoclinic orbits to saddle-center in reversible systems, Phys. D, 214, 2006, 169–181. · Zbl 1101.37017 · doi:10.1016/j.physd.2006.01.009
[7]Lin, X. B., Using Melnikov’s method to solve Shilnikov’s problem, Proc. Roy. Soc., Edinburgh, 116A, 1990, 295–325.
[8]Klaus, J. and Knobloch, J., Bifurcation of homoclinic orbits to a saddle-center in reversible systems, Int. J. of Bifur. and Chaos, 13(9), 2003, 2603–2622. · Zbl 1052.37043 · doi:10.1142/S0218127403008119
[9]Wiggins, S., Global Bifurcation and Chaos, Springer-Verlag, New York, 1988.