# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Bifurcation of degenerate homoclinic orbits to saddle-center in reversible systems. (English) Zbl 1197.34067

This paper deals with bifurcations from a homoclinic orbit ${\Gamma }$ to a saddle-center in a two parameter family of reversible systems in ${ℝ}^{2n+2}$. 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 ${\Gamma }$ is degenerate, meaning that along ${\Gamma }$ 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.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions of ODE 37G25 Bifurcations connected with nontransversal intersection 34C23 Bifurcation (ODE) 34C14 Symmetries, invariants (ODE)
##### References:
 [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.