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

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 13, Issue 9, 2003, Pages 2603-2622

  Klaus, J ^a   Knobloch, J ^a  

^a ILMENAU UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Bifurcation; Homoclinic orbit; Nonhyperbolic fixed point; Reversible system

Indexed keywords

ASYMPTOTIC STABILITY; CHAOS THEORY; EIGENVALUES AND EIGENFUNCTIONS; HAMILTONIANS; PERTURBATION TECHNIQUES; VECTORS;

HOMOCLINIC ORBITS; REVERSIBLE SYSTEMS;

BIFURCATION (MATHEMATICS);

EID: 0242425111     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127403008119     Document Type: Article

References (19)

1. Champneys, A. R. & Härterich, J. [2000] "Cascades of homoclinic orbits to a saddle-centre for reversible and perturbed Hamiltonian systems," Dyn. Syst. Appl. 15, 231-252.
2. Coppel, W. A. [1978] Dichotomies in Stability Theory, LNM, Vol. 629 (Springer-Verlag).
3. Devaney, R. [1976] "Reversible diffeomorphisms and flows," Trans. Amer. Math. Soc. 218, 89-113.
4. Hale, J. & Lin, X. B. [1986] "Heteroclinic orbits for retarded functional differential equations," J. Differ. Eqs. 65, 175-202.
5. Koltsova, O. Yu. & Lerman, L. M. [1995] "Periodic and homoclinic orbits in a two-parameter unfolding of a Hamiltonian system with a homoclinic orbit to a saddle-center," Int. J. Bifurcation and Chaos 5, 397-408.
6. Koltsova, O. Yu. & Lerman, L. M. [1996] "Families of transverse Poincaré homoclinic orbits in 2n-dimensional Hamiltonian systems close to the system with a loop to a saddle-center," Int. J. Bifurcation and Chaos 6, 991-1006.
7. Knobloch, J. [1997] "Bifurcations of degenerate homoclinics in reversible and conservative systems," J. Dyn. Differ. Eqs. 9, 427-444.
8. Knobloch, J. [1999] "Jump estimates for Lin's method," Preprint TU Ilmenau No M31/99.
9. Knobloch, J. [2000] "Lin's method for discrete dynamical systems," J. Diff. Eqs. Appl. 6, 577-623.
10. Lerman, L. M. [1991] "Hamiltonian systems with loops of a separatrix of a saddle-center," Sci. Math. Sov. 10, 297-306.
11. Lin, X.-B. [1990] "Using Melnikov's method to solve Shilnikov's problems," Proc. Roy. Soc. Edinburgh A116, 295-325.
12. Lin, X.-B. [1996] "Homoclinic bifurcations with weakly expanding center manifolds," in Dynamics Reported Vol. 5 (New Series), eds. Jones, C. K. R. T., Kirchgraber, U. & Walther, H. O. (Springer-Verlag, Berlin), pp. 99-189.
13. Mielke, A., Holmes, P. & O'Reilly, O. [1992] "Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center," J. Dyn. Differ. Eqs. 4, 95-126.
14. Sandstede, B. [1993] "Verzweigungstheorie homokliner Verdopplungen," PhD-Thesis Universität Stuttgart.
15. Schalk, U. [1995] "Homokline Punkte in periodisch gestörten Systemen gewöhnlicher Differentialgleichungen," PhD-Thesis TU Ilmenau.
16. Shilnikov, L. P., Shilnikov, A. L., Turaev, D. & Chua, L. O. [1998] Methods of Qualitative Theory in Nonlinear Dynamics, Part I (World Scientific, Singapore).
17. Vanderbauwhede, A. [1989] "Centre manifolds, normal forms and elementary bifurcations," in Dynamics Reported Vol. 2, eds. Kirchgraber, U. & Walther, H. O. (Teubner, Stuttgart), pp. 89-169.
18. Vanderbauwhede, A. [1992] "Bifurcation of degenerate homoclinics," Results in Math. 21, 211-223.
19. Vanderbauwhede, A. & Fiedler, B. [1992] "Homoclinic period blow-up in reversible and conservative systems," ZAMP 43, 292-318.