Detection of symmetric homoclinic orbits to saddle-centres in reversible systems

Physica D: Nonlinear Phenomena

Volumn 214, Issue 2, 2006, Pages 169-181

  Yagasaki, Kazuyuki ^a   Wagenknecht, Thomas ^b  

^a GIFU UNIVERSITY   (Japan)

^b UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Embedded soliton; Homoclinic orbit; Perturbation technique; Reversible system; Saddle centre

Indexed keywords

BIFURCATION (MATHEMATICS); COMPUTATION THEORY; DIFFERENTIAL EQUATIONS; NONLINEAR OPTICS; SOLITONS; WATER WAVES;

EMBEDDED SOLITONS (ES); HOMOCLINIC ORBITS; REVERSIBLE SYSTEMS; SADDLE-CENTRE;

PERTURBATION TECHNIQUES;

EID: 32844464987     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2006.01.009     Document Type: Article

References (32)

1. M. Abramowitz I.A. Stegun Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables 1965 Dover New York
2. A.R. Champneys, and J. Härterich Cascades of homoclinic orbits to a saddle-centre for reversible and perturbed Hamiltonian systems Dyn. Stab. Sys. 15 2000 231 252
3. A.R. Champneys, and B.A. Malomed Embedded solitons in a three-wave system Phys. Rev. E 61 1999 463466
4. A.R. Champneys, B.A. Malomed, J. Yang, and D.J. Kaup Embedded solitons: solitary waves in resonance with the linear spectrum Physica D 152 2001 340 354
5. E.A. Coddington, and N. Levinson Theory of Ordinary Differential Equations 1955 McGraw-Hill New York
6. R.L. Devaney Reversible diffeomorphisms and flows Trans. Amer. Math. Soc. 218 1976 89 113
7. E. Doedel, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B. Sandstede, X. Wang, AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), Concordia University, Montreal, 1997
8. A. Delshams, and J.T. Lázaro Pseudo-normal form near saddle-center or saddle-focus equilibria J. Differential Equations 208 2 2005 312 343
9. A. Erdélyi Higher Transcendental Functions vol. I 1953 McGraw-Hill New York
10. C. Grotta-Ragazzo Nonintegrability of some Hamiltonian systems, scattering and analytic continuation Comm. Math. Phys. 166 1994 255 277
11. C. Grotta-Ragazzo Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers Comm. Pure Appl. Math. 50 1997 105 147
12. J. Guckenheimer, and P.J. Holmes Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 1983 Springer New York
13. L.M. Lerman Hamiltonian systems with loops of a separatrix of a saddle-center Selecta Math. Sov. 10 1991 297 306
14. B. Malomed, T. Wagenknecht, A.R. Champneys, and M.J. Pearce Accumulation of embedded solitons in systems with quadratic nonlinearity Chaos 15 2005 037116
15. K.R. Meyer, and G.R. Hall Introduction to Hamiltonian Dynamical Systems and the N-Body Problem 1992 Springer New York
16. A. Mielke, P.J. Holmes, and O. O'Reilly Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center J. Dynam Differential Equations 4 1992 95 126
17. J.J. Morales-Ruiz Differential Galois Theory and Non-Integrability of Hamiltonian systems 1999 Birkhäuser Basel
18. J.J. Morales-Ruiz, and J.M. Peris On a Galoisian approach to the splitting of separatrices Ann. Fac. Sci. Toulouse Math. 8 1999 125 141
19. J. Moser On the generalization of a theorem of A. Liapunoff Commun. Pure Appl. Math. 11 1958 257 271
20. H. Rüssmann Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung Math. Ann. 154 1964 285 300
21. I. Towers, A.V. Buryak, R.A. Sammut, and B.A. Malomed Stable localized vortex solitons Phys. Rev. E 63 2001 055601
22. A. Vanderbauwhede, and B. Fiedler Homoclinic period blow-up in reversible and conservative systems Z. Angew. Math. Phys. 43 1992 292 318
23. M. Van der Put, and M.F. Singer Galois Theory of Linear Differential Equations 2003 Springer New York
24. S. Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos 1990 Springer New York
25. S. Wolfram The Mathematica Book 5th ed. 2003 Wolfram Media Champaign, IL
26. K. Yagasaki The method of Melnikov for perturbations of multi-degree-of-freedom Hamiltonian systems Nonlinearity 12 1999 799 822
27. K. Yagasaki Horseshoes in two-degree-of-freedom Hamiltonian systems with saddle-centers Arch. Ration. Mech. Anal. 154 2000 275 296
28. K. Yagasaki Galoisian obstructions to integrability and Melnikov criteria for chaos in two-degree-of-freedom Hamiltonian systems with saddle-centers Nonlinearity 16 2003 2003 2012
29. K. Yagasaki Detection of homoclinic and heteroclinic behavior in Hamiltonian systems with saddle-centers F. Dumortier H. Broer J. Mawhin A. Vanderbauwhede S.V. Lunel Equadiff 2003: International Conference on Differential Equations, Hasselt 2003 2005 World Scientific New Jersey 984 989
30. K. Yagasaki, Chaos and diffusion in four-dimensional, non-conservative, reversible systems with saddle-centers, in: Proceedings of Fifth EUROMECH Nonlinear Dynamics Conference, 7-12 August, Eindhoven, The Netherlands, 2005, Paper No. 13-312
31. J. Yang Dynamics of embedded solitons in the extended KdV equations Stud. Appl. Math. 106 2001 337 365
32. J. Yang, B.A. Malomed, D.J. Kaup, and A.R. Champneys Embedded solitons: A new type of solitary wave Math. Comput Simulation 56 2001 585 600