Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits

Nonlinearity

Volumn 17, Issue 4, 2004, Pages 1217-1244

  Lamb, Jeroen S W ^a   Stenkin, Oleg V ^a,b  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b Res Inst Appl Math and Cybernet   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 3142692473     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/17/4/005     Document Type: Article

References (32)

1. Biragov V S 1990 Bifurcations in a two-parameter family of conservative mappings that are close to the Henon mapping (Russian) Transl. Selecta Math. Sov. 9 273-82 Biragov V S 1987 Methods of the qualitative theory of differential equations (Russian) Gor'kov. Gos. Univ. Gorki 11 10-24
2. Biragov V S 1990 Bifurcations in a two-parameter family of conservative mappings that are close to the Henon mapping (Russian) Transl. Selecta Math. Sov. 9 273-82 Biragov V S 1987 Methods of the qualitative theory of differential equations (Russian) Gor'kov. Gos. Univ. Gorki 11 10-24
3. Champneys A R 1998 Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics Physica D 112 158-86
4. Duarte P 1999 Abundance of elliptic isles at conservative bifurcations Dyn. Stab. Syst. 14 339-56
5. Duarte P 2000 Persistent homoclinic tangencies for conservative maps near the identity Ergod. Theory Dyn. Syst. 20 393-438
6. Gavrilov N K and Shilnikov L P 1972 Three-dimensional dynamical systems that are close to systems with a structurally unstable homoclinic curve: I Mat. Sb. 88 475-92 (in Russian) Gavrilov N K 1973 and Shilnikov L P Three-dimensional dynamical systems that are close to systems with a structurally unstable homoclinic curve: II Mat. Sb. 90 139-56 (in Russian)
7. Gavrilov N K and Shilnikov L P 1972 Three-dimensional dynamical systems that are close to systems with a structurally unstable homoclinic curve: I Mat. Sb. 88 475-92 (in Russian) Gavrilov N K 1973 and Shilnikov L P Three-dimensional dynamical systems that are close to systems with a structurally unstable homoclinic curve: II Mat. Sb. 90 139-56 (in Russian)
8. Gonchenko V S 2003 About existence of attractors and repellers in reversible systems with additional symmetry Proc. Conf., NNGU (Nyzhny Novgorod) pp 98-100 (in Russian)
9. Gonchenko S V 1996 Moduli of Ω-conjugacy of two-dimensional diffeomorphisms with a structurally unstable heteroclinic contour Sbornik: Math. 187 1261-81
10. Gonchenko S V and Shilnikov L P 1990 Invariants of Ω-conjugacy of diffeomorphisms with a structurally unstable homoclinic trajectory Ukrainian Math. J. 42 134-40
11. Gonchenko S V and Shilnikov L P 1993 On the moduli of systems with a non-rough Poincaré homoclinic curve Russian Acad. Sci. Izv. Math. 41 417-45
12. Gonchenko S V and Shilnikov L P 1997 On two-dimensional analytical area-preserving diffeomorphisms with infinitely many stable elliptic periodic points Regular Chaotic Dyn. 2 106-23 (in Russian)
13. Gonchenko S V and Shilnikov L P 2000 On two-dimensional area-preserving diffeomorphisms with infinitely many elliptic islands J. Stat. Phys. 101 321-56
14. Gonchenko S V, Shilnikov L P and Stenkin O V 2002 On Newhouse regions with infinite many stable and unstable invariant tori Progress in Nonlinear Science: Proc. Int. Conf. Dedicated to A A Andronov, Vol I: Mathematical Problems of Nonlinear Dynamics University of Nizhny Novgorod, pp 80-102
15. Gonchenko S V, Shilnikov L P and Turaev D V 1993 On models with non-Rough Poincaré homoclinic curves Physica D 62 1-14
16. Gonchenko S V, Shilnikov L P and Turaev D V 1998 Elliptic periodic orbits near a homoclinic tangency in four-dimensional symplectic maps and Hamiltonian systems with three degrees of freedom Regular Chaotic Dyn. 3 3-26
17. Gonchenko S V, Shilnikov L P and Turaev D V 2002 Infinitely many elliptic periodic orbits in four-dimensional symplectic maps with a homoclinic tangency Proc. Steklov Inst. Math. WIAS-Preprint no 791, Berlin submitted
18. Gonchenko S V, Turaev D V and Shilnikov L P 1992 On models with a structurally unstable homoclinic curve Sov. Math. Dokl. 44 422-6
19. Gonchenko S V, Turaev D V and Shilnikov L P 1993 On the existence of Newhouse regions near systems with a structurally unstable homoclinic Poincaré curve (multidimensional case) Russian Acad. Sci. Dokl. Math. 47 268-73
20. Gonchenko S V, Turaev D V and Shilnikov L P 1993 Dynamical phenomena in multi-dimensional systems with a structurally unstable homoclinic Poincaré curve Russian Acad. Sci. Dokl. Math. 47 410-15
21. Gonchenko S V, Turaev D V and Shilnikov L P 2001 Homoclinic tangencies of arbitrary high orders in the Newhouse regions J. Math. Sci. 105 1738-778
22. Gonchenko S V, Turaev D V and Shilnikov L P 1997 On Newhouse domains of two-dimensional diffeomorphisms which are close to a diffeomorphism with a structurally unstable heteroclinic cycle Proc. Steklov Inst. Math. 216 7-118
23. Iooss G 1998 Travelling water-waves, as a paradigm for bifurcations in reversible infinite-dimensional 'dynamical' systems Proc. Int. Congr. Mathematicians (Berlin, 1998) vol III Doc. Math. extra vol III, pp 611-22
24. Lamb J S W and Roberts JAG 1998 Time-reversal symmetry in dynamical systems: a survey Physica D 112 1-39
25. Lamb J S W, Teixeira, M A and Webster K N Heteroclinic cycle bifurcations near Hopf-zero bifurcation in 3D reversible vector fields Preprint Dynamic 2003-10
26. Michelson D 1986 Steady solutions of the Kuramoto-Sivashinsky equation Physica D 19 89-111
27. Newhouse S E 1974 Diffeomorphisms with infinitely many sinks Topology 13 9-18
28. Newhouse S E 1979 The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms Publ. Math. IHES. 50 101-51
29. 2 Proc. AMS Symp. Pure Math. 14 191-202
30. Palis J and Viana M 1994 High dimension diffeomorphisms displaying infinitely many sinks Ann. Math. 140 207-50
31. Topaj D and Pikovsky A 2002 Reversibility vs synchronization in oscillator lattices Physica D 170 118-30
32. Romero N 1995 Persistence of homoclinic tangencies in higher dimensions Ergod. Theory Dyn. Syst. 15 735-57