| Title:
|
Rigorous numerics for symmetric homoclinic orbits in reversible dynamical systems (English) |
| Author:
|
Hiraoka, Yasuaki |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
43 |
| Issue:
|
6 |
| Year:
|
2007 |
| Pages:
|
797-806 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We propose a new rigorous numerical technique to prove the existence of symmetric homoclinic orbits in reversible dynamical systems. The essential idea is to calculate Melnikov functions by the exponential dichotomy and the rigorous numerics. The algorithm of our method is explained in detail by dividing into four steps. An application to a two dimensional reversible system is also treated and the existence of a symmetric homoclinic orbit is rigorously verified as an example. (English) |
| Keyword:
|
rigorous numerics |
| Keyword:
|
exponential dichotomy |
| Keyword:
|
homoclinic orbits |
| MSC:
|
34C37 |
| MSC:
|
34D09 |
| MSC:
|
37C29 |
| MSC:
|
37M20 |
| MSC:
|
65G20 |
| MSC:
|
65P30 |
| idZBL:
|
Zbl 1138.65107 |
| idMR:
|
MR2388394 |
| . |
| Date available:
|
2009-09-24T20:29:37Z |
| Last updated:
|
2013-09-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135816 |
| . |
| Reference:
|
[1] Chow S.-N., Deng, B., Fiedler B.: Homoclinic bifurcation at resonant eigenvalues.J. Dyn. Differential Equations 2 (1990), 177–244 Zbl 0703.34050, MR 1050642, 10.1007/BF01057418 |
| Reference:
|
[2] Coddington E. A., Levinson L.: Theory of Ordinary Differential Equations.McGraw-Hill, New York 1955 Zbl 0064.33002, MR 0069338 |
| Reference:
|
[3] Coppel W. A.: Dichotomies in Stability Theory.(Lecture Notes in Mathematics 629.), Springer-Verlag, Berlin 1978 Zbl 0376.34001, MR 0481196 |
| Reference:
|
[4] Deng B.: The Sil’nikov problem, exponential expansion, strong $\lambda $-lemma, $C^1$-linearization, and homoclinic bifurcation.J. Differential Equations 79 (1989), 189–231 MR 1000687, 10.1016/0022-0396(89)90100-9 |
| Reference:
|
[5] Guckenheimer J., Holmes P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.Springer–Verlag, Berlin 1983 Zbl 0515.34001, MR 0709768 |
| Reference:
|
[6] Hiraoka Y.: in preparatio. |
| Reference:
|
[7] Iooss G., Pérouème M. C.: Perturbed homoclinic solutions in reversible 1:1 resonance vector fields.J. Differential Equations 102 (1993), 62–88 Zbl 0792.34044, MR 1209977, 10.1006/jdeq.1993.1022 |
| Reference:
|
[8] Kapitula T.: The Evans function and generalized Melnikov integrals.SIAM J. Math. Anal. 30 (1999), 273–297 Zbl 0921.34009, MR 1664760, 10.1137/S0036141097327963 |
| Reference:
|
[9] Kokubu H.: Homoclinic and heteroclinic bifurcations in vector fields.Japan J. Appl. Math. 5 (1988), 455–501 MR 0965875, 10.1007/BF03167912 |
| Reference:
|
[10] Kisaka M., Kokubu, H., Oka H.: Bifurcations to $N$-Homoclinic orbits and $N$-periodic orbits in vector fields.J. Dyn. Differential Equations 5 (1993), 305–357 Zbl 0784.34038, MR 1223451, 10.1007/BF01053164 |
| Reference:
|
[11] Lohner R. J.: Einschliessung der Lösung gewonhnlicher Anfangs- and Randwertaufgaben und Anwendungen.Thesis, Universität Karlsruhe (TH) 1988 |
| Reference:
|
[12] Melnikov V. K.: On the stability of center for time periodic perturbations.Trans. Moscow Math. Soc. 12 (1963) , 1–57 MR 0156048 |
| Reference:
|
[13] Oishi S.: Research Institute for Mathematical Sciences Kôkyûroku, 928 (1995), 14–1. |
| Reference:
|
[14] Vanderbauwhede A., Fiedler B.: Homoclinic period blow-up in reversible and conservative systems.Z. Angew. Math. Phys. 43 (1992), 292–318 Zbl 0762.34023, MR 1162729, 10.1007/BF00946632 |
| Reference:
|
[15] Wilczak D., Zgliczyński P.: Heteroclinic connections between periodic orbits in planar restricted circular three-body problem – a computer assisted proof.Comm. Math. Phys. 234 (2003), 37–75 Zbl 1055.70005, MR 1961956, 10.1007/s00220-002-0709-0 |
| Reference:
|
[16] Yamamoto N.: A numerical verification method for solutions of boundary value problems with local uniqueness by Banach’s fixed-point theorem.SIAM J. Numer. Anal. 35 (1998), 2004–2013 Zbl 0972.65084, MR 1639986, 10.1137/S0036142996304498 |
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