Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

Nonlinearity

Volumn 26, Issue 1, 2013, Pages 1-33

  Delshams, A ^a   Gonchenko, S V ^b   Gonchenko, V S ^b   Lazaro J T ^a   Sten'Kin, O ^b  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^b LOBACHEVSKY STATE UNIVERSITY OF NIZHNY NOVGOROD   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84870399480     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/26/1/1     Document Type: Article

References (37)

1. Afraimovič V S and Šil'nikov L P 1973 The singular sets of Morse-Smale systems Trudy Moskov. Mat. Obšč. 28 181-214 (in Russian)
2. Bochner S 1945 Compact groups of differentiable transformations Ann. Math. 46 372-81 (second Series)
3. Biragov V S 1987 Methods of the qualitative theory of differential equations Gor'kov. Gos. Univ. Gorki 1987 10-24 (in Russian)
4. Biragov V S 1990 Bifurcations in a two-parameter family of conservative mappings that are close to the Hénon mapping Selecta Math. Sov. 9 273-82 (Engl. transl.)
5. Delshams A and Lázaro J T 2005 Pseudo-normal form near saddle-center or saddle-focus equilibria J. Diff. Eqns. 208 312-43 (Pubitemid 40030555)
6. Duarte P 2000 Persistent homoclinic tangencies for conservative maps near the identity Ergod. Theory Dyn. Syst. 20 393-438
7. Gavrilov N K and Shilnikov L P 1972 On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve I: Math. USSR Sb. 17 467-85
8. Gavrilov N K and Shilnikov L P 1973 On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve: II Math. USSR Sb. 19 139-56
9. Gonchenko S V and Shilnikov L P 1990 Invariants of -conjugacy of diffeomorphisms with a structurally unstable homoclinic trajectory Ukr. Math. J. 42 134-40
10. Gonchenko S V and Shil'nikov L P 1992 Moduli of systems with a structurally unstable homoclinic Poincaré curve Izv. Ross. Akad. Nauk. Ses. Mat. 56 1165-97 (in Russian)
11. Gonchenko S V and Shil'nikov L P 1993 Moduli of systems with a structurally unstable homoclinic Poincaré curve Russian Acad. Sci. Izv. Math. 41 417-45 (Engl. transl.)
12. Gonchenko S V, Turaev D V and Shil'nikov L P 1993 Dynamical phenomena in multidimensional systems with a structurally unstable homoclinic Poincaré curve Dokl. Akad. Nauk. 330 144-7 (in Russian)
13. Gonchenko S V, Turaev D V and Shil'nikov L P 1993 Dynamical phenomena in multidimensional systems with a structurally unstable homoclinic Poincaré curve Russ. Acad. Sci. Dokl. Math. 47 410-5 (Engl. transl.)
14. Gonchenko S V, Turaev D V and Shil'nikov L P 1997 On Newhouse domains of two-dimensional diffeomorphisms that are close to a diffeomorphism with a structurally unstable heteroclinic contour Tr. Mat. Inst. Steklova 216 76-125 (in Russian)
15. Gonchenko S V, Turaev D V and Shil'nikov L P 1997 On Newhouse domains of two-dimensional diffeomorphisms that are close to a diffeomorphism with a structurally unstable heteroclinic contour Proc. Steklov Inst. Math. 216 70-118 (Engl. transl.)
16. 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
17. Gonchenko S V and Gonchenko V S 2004 On bifurcations of birth of closed invariant curves in the case of two-dimensional diffeomorphisms with homoclinic tangencies Proc. Steklov Inst. 244 80-105
18. Gonchenko S V, Shilnikov L P and Turaev D 2007 Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps Nonlinearity 20 241-75
19. Gonchenko S V, Shilnikov L P and Turaev D 2008 On dynamical properties of multidimensional diffeomorphisms from Newhouse regions Nonlinearity 21 923-72
20. Lamb J S W and Stenkin O V 2004 Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits Nonlinearity 17 1217-44 (Pubitemid 38919691)
21. Montgomery D and Zippin L 1955 Topological Transformation Groups (New York: Interscience)
22. 2 Proc. Am. Math. Soc. Symp. Pure Math. 14 191-202
23. Newhouse S 1974 Diffeomorphisms with infinitely many sinks Topology 13 9-18
24. Newhouse S E 1979 The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms Publ. Math. Inst. Hautes Etudes Sci. 50 101-51
25. Palis J and Viana M 1994 High dimension diffeomorphisms displaying infinitely many periodic attractors Ann. Math. 140 207-50
26. Pikovsky A and Topaj D 2002 Reversibility versus synchronization in oscillator latties Physica D 170 118-30 (Pubitemid 34813924)
27. Roberts J A G 1997 Some characterisations of low-dimensional dynamical systems with time-reversal symmetry Control and Chaos (Mathematical Modelling vol 8) (Boston, MA: Birkhäuser) pp 106-33
28. Roberts J A G and Quispel G R W 1992 Chaos and time-reversal symmetry: order and chaos in reversible systems Phys. Rep. 216 63-177
29. Romero N 1995 Persistence of homoclinic tangencies in higher dimensions Ergod. Theory Dyn. Syst. 15 735-57
30. Shilnikov L P 1967 On a Poincaré-Birkhoff problem Math. USSR Sb. 3 91-102
31. Shilnikov L P 1977 The bifurcation theory of dynamical systems with Poincaré homoclinic curves VII Internationale Konferenzüber nichtlineare Schwingungen, Band 1, 2 (Berlin: Academie) 279-93
32. Shilnikov L P, Shilnikov A L, Turaev D V and Chua L O 1998 Methods of Qualitative Theory in Nonlinear Dynamics: I (Singapore: World Scientific)
33. Siegel C C L and Moser J K 1995 Lectures on Celestial Mechanics (Berlin: Springer)
34. Tedeschini-Lalli L and Yorke J A 1986 How often do simple dynamical processes have infinitely many coexisting sinks? Commun. Math. Phys. 106 635-57
35. Turaev D 1996 On dimension of nonlocal bifurcational problems Int. J. Bifurcation Chaos 6 919-48
36. Turaev D 2003 Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps Nonlinearity 16 123-35
37. Yorke J A and Alligood K T 1983 Cascades of period-doubling bifurcations: a prerequisite for horseshoes Bull. Am. Math. Soc. 9 319-22