On a homoclinic origin of Hénon-like maps

Regular and Chaotic Dynamics

Volumn 15, Issue 4, 2010, Pages 462-481

  Gonchenko, S V ^a   Gonchenko, V S ^a   Shilnikov, L P ^a  

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

Author keywords

Henon like maps; homoclinic tangency; saddle focus fixed point; wild hyperbolic attractor

Indexed keywords

EID: 85027957638     PISSN: 15603547     EISSN: 14684845     Source Type: Journal    
DOI: 10.1134/S1560354710040052     Document Type: Article

References (50)

1. Hénon, M., A Two-dimensional Mapping with a Strange Attractor, Comm. Math. Phys., 1976, vol. 50, pp. 69-77.
2. Benedidicks, M. and Carleson, L., The Dynamics of the Hénon Map, Ann. Math., 1991, vol. 133, pp. 73-169.
3. Lai, Y.-C., Grebogi, C., Yorke, J. and Kan, I., How Often are Chaotic Saddles Nonhyperbolic? Nonlinearity, 1993, vol. 6, pp. 779-797.
4. Newhouse, S. E., The Abundance of Wild Hyperbolic Sets and Non-smooth Stable Sets for Diffeomorphisms, Publ. Math. IHES, 1979, vol. 50, pp. 101-151.
5. Gavrilov, N. K. and Shilnikov, L. P., On Three-dimensional Dynamical Systems Close to Systems with a Structurally Unstable Homoclinic Curve: Part I, Mat. Sb., 1972, vol. 88(130), no. 4(8), pp. 475-492 [Mathematics of the USSR-Sbornik, 1972, vol. 17, no. 4, pp. 467-485]; Part II, Mat. Sb., 1973, vol. 90(132), no. 1, pp. 139-156 [Mathematics of the USSR-Sbornik, 1973, vol. 19, no. 1, pp. 139-156].
6. Tedeschini-Lalli, L. and Yorke, J. A., How Often do Simple Dynamical Processes Have Infinitely Many Coexisting Sinks? Commun. Math. Phys., 1986, vol. 106, pp. 635-657.
7. Gonchenko, S. V., Shilnikov, L. P. and Turaev, D. V., Dynamical Phenomena in Systems with Structurally Unstable Poincaré Homoclinic Orbits, Dokl. Akad. Nauk, 1993, vol. 330, no. 2, pp. 44-147 [Acad. Sci. Dokl. Math., vol. 47, no. 3, pp. 410-415].
8. Gonchenko, S. V., Shilnikov, L. P. and Turaev, D. V., Dynamical Phenomena in Systems with Structurally Unstable Poincaré Homoclinic Orbits, Chaos, 1996, vol. 6, no. 1, pp. 15-31.
9. Gonchenko, S. V. and Gonchenko, V. S., On Andronov-Hopf Bifurcations of Two-Dimensional Diffeomorphisms with Homoclinic Tangencies, Preprint of Weierstrass Inst. for Applied Analysis and Stochastics, Berlin, 2000, no. 556; http://www. wias-berlin. de/main/publications/wias-publ/index. cgi. en.
10. Gonchenko, S. V. and Gonchenko, V. S., On Bifurcations of the Birth of Closed Invariant Curves in the Case of Two-Dimensional Diffeomorphisms with Homoclinic Tangencies, Tr. Mat. Inst. Steklova, 2004, vol. 244, pp. 87-114 [Proc. Steklov Inst. Math., 2004, vol. 244, no. 1, pp. 80-105].
11. Gonchenko, S. V., Meiss, J. D., and Ovsyannikov, I. I., Chaotic Dynamics of Three-Dimensional Hénon Maps That Originate from a Homoclinic Bifurcation, Regul. Chaotic Dyn., 2006, vol. 11, no. 2, pp. 191-212.
12. Gonchenko, S. V., Shilnikov, L. P., and Turaev, D. V., On the Dynamical Properties of Diffeomorphisms with Homoclinic Tangencies, Contemporary Mathematics and its Applications, 2003, vol. 7, pp. 92-118 (Russian) [English version: J. Math. Sci. (N. Y.), 2005, vol. 126, pp. 1317-1343].
13. Gonchenko, S. V., Sten'kin, O. V., and Shilnikov, L. P., On the Existence of Infinitely Stable and Unstable Invariant Tori for Systems from Newhouse Regions with Heteroclinic Tangencies, Rus. J. Nonlin. Dyn., 2006, vol. 2, no. 1, pp. 3-25 (Russian).
14. Gonchenko, S. V., Gonchenko, V. S., and Tatjer, J. C., Bifurcations of Three-dimensional Diffeomorphisms with Non-simple Quadratic Homoclinic Tangencies and Generalized Hénon Maps, Regul. Chaotic Dyn., 2007, vol. 12, no. 3, pp. 233-266.
15. Gonchenko, S. V., Shilnikov, L. P., and Turaev, D. V., On Dynamical Properties of Multidimensional Diffeomorphisms from Newhouse Regions: P. 1, Nonlinearity, 2008, vol. 21, no. 5, pp. 923-972.
16. Gonchenko, S. V., Shilnikov, L. P., and Turaev, D. V., On Global Bifurcations in Three-Dimensional Diffeomorphisms Leading to Wild Lorenz-Like Attractors, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 137-147.
17. Biragov, V. S., On Bifurcations in a Two-parameter Family of Conservative Maps Close to the Hénon Map, Methods of the Qualitative Theory of Differential Equations, Gorky State Univ., 1987, pp. 10-23 [English version: Selecta Math. Sovietica, 1990, vol. 9].
18. Gonchenko S. V. and Shilnikov L. P., On Two-dimensional Analytic Area-preserving Diffeomorphisms with Infinitely Many Stable Elliptic Periodic Points, Regul. Chaotic Dyn., 1997, vol. 2, pp. 106-123.
19. Gonchenko S. V. and Shilnikov L. P., On Two-dimensional Area-Preserving Maps with Homoclinic Tangencies, Dokl. Akad. Nauk, 2001, vol. 378, no. 6, pp. 727-732 (Russian).
20. Gonchenko, S. V., Shilnikov, L. P., and Turaev, D. V., Elliptic Periodic Orbits Near a Homoclinic Tangency in Four-dimensional Symplectic Maps and Hamiltonian Systems with Three Degrees of Freedom, Regul. Chaotic Dyn., 1998, vol. 3, No. 4, pp. 3-26.
21. Turaev, D. V., On Dimension of Nonlocal Bifurcational Problems, Int. J. of Bifurcation and Chaos, 1996, vol. 6, no. 5, pp. 919-948.
22. Gonchenko, S. V. and Shilnikov, L. P., Eds., Homoclinic Tangencies, Moscow-Izhevsk: RCD, 2007.
23. Gonchenko, S. V., Turaev, D. V., and Shilnikov, L. P., On the Existence of Newhouse Domains in a Neighbourhood of Systems With a Structurally Unstable Poincaré Homoclinic Curve (the Higherdimensional case), Dokl. Akad. Nauk, 1993, vol. 329, no. 4, pp. 404-407 [Russian Acad. Sci. Dokl. Math., 1993, vol. 47, no. 2, pp. 268-273].
24. Turaev, D., Polynomial Approximations of Symplectic Dynamics and Richness of Chaos in Nonhyperbolic Area-preserving Maps, Nonlinearity, 2003, vol. 16, pp. 123-135.
25. Turaev D. and Rom-Kedar, V., Elliptic Islands Appearing in Near-ergodic Flows, Nonlinearity, 1998, vol. 11, pp. 575-600.
26. Palis, J. and Takens, F., Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge: Cambridge University Press, 1993.
27. Palis, J. and Viana, M., High Dimension Diffeomorphisms Displaying Infinitely Many Periodic Attractors, Ann. Math., 1994, vol. 140, pp. 207-250.
28. Romero, N., Persistence of Homoclinic Tangencies in Higher Dimensions, Ergod. Th. and Dyn. Sys., 1995, vol. 15, pp. 735-757.
29. Lomelí, H. E. and Meiss, J. D., Quadratic Volume Preserving Maps, Nonlinearity, 1998, vol. 11, no. 3, pp. 557-574.
30. Gonchenko, S. V. and Shilnikov, L. P., Invariants of Ω-conjugacy of Diffeomorphisms with a Nontransversal Homoclinic Orbit, Ukr. Math. J., 1990, vol. 42, no. 2, pp. 134-140.
31. Gonchenko, S. V. and Shilnikov, L. P., Om Moduli of Systems with a Nontransversal Poincaré Homoclinic Orbit, Izv. Ross. Akad. Nauk Ser. Mat., 1992, vol. 56, no. 6, pp. 1165-1197 [Russian Acad. Sci. Izv. Math., 1993, vol. 41, no. 3, pp. 417-445].
32. Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V., and Chua, L., Methods of Qualitative Theory in Nonlinear Dynamics, Part I; Part II, River Edge, NJ: World Sci. Publ., 1998, 2002.
33. Gonchenko, S. V., On Stable Periodic Motions in Systems Close to a System with a Nontransversal Homoclinic Curve, Mat. Zametki, 1983, vol. 33, no. 5., pp. 745-755 [Russian Math. Notes, 1983, vol. 33, no. 5., pp. 384-389].
34. Newhouse, S. E., Diffeomorphisms with Infinitely Many Sinks, Topology, 1974, vol. 13, pp. 9-18.
35. Gonchenko, S. V., Ovsyannikov, I. I., Simó, C. and Turaev, D. V., Three-Dimensional Hénon-like Maps and Wild Lorenz-Like Attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2005, vol. 15, no. 11, pp. 3493-3508.
36. Leontovich, E. A., On Birth of Limit Cycles From Separatrixes, Math. Dokl. USSR, 1951, vol. 78, no. 4, pp. 641-644 (Russian).
37. Gonchenko, S. V. and Shilnikov, L. P., On Two-dimensional Area-preserving Diffeomorphisms with Infinitely Many Elliptic Islands, J. Stat. Phys., 2000, vol. 101, pp. 321-356.
38. Gonchenko, V. S., Kuznetsov, Yu. A., and Meijer, H. G. E., Generalized Hénon Map and Bifurcations of Homoclinic Tangencies, SIAM J. Appl. Dyn. Syst., 2005, vol. 4, no. 2, pp. 407-436.
39. Gonchenko, S. V. and Shilnikov, L. P., On Two-dimensional Area-preserving Maps with Homoclinic Tangencies That Have Infinitely Many Generic Elliptic Periodic Points, Notes of S.-Petersburg Math. Steklov Inst., 2003, vol. 300, pp. 155-166.
40. Battelli, F. and Lazzari, C., Perturbing Two-dimensional Maps Having Critical Homoclinic Orbits, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1999, vol. 9, no. 6, pp. 1189-1195.
41. Gonchenko, S. V., Dynamics and Moduli of ω-conjugacy of 4D-diffeomorphisms with a Structurally Unstable Homoclinic Orbit to a Saddle-focus Fixed Point, Amer. Math. Soc. Transl. Ser. 2, vol. 200, 2000, pp. 107-134.
42. Gonchenko, S. V., Gonchenko, V. S., and Tatjer, J. C., Three-dimensional Dissipative Diffeomorphisms with Codimension Two Homoclinic tangencies and generalized Hénon maps, Proc. of Int. Conf. dedicated to 100th Anniv. of A. A. Andronov, Vol. 1, Nizhny Novgorod, 2002, pp. 80-102.
43. Tatjer, J. C., Three-dimensional Dissipative Diffeomorphisms with Homoclinic Tangencies, Ergod. Th. and Dynam. Sys., 2001, vol. 21, no. 1, pp. 249-302.
44. Gonchenko, V. S. and Ovsyannikov, I. I., On Bifurcations of Three-dimensional Diffeomorphisms with a Homoclinic Tangency to a "Neutral" Saddle Fixed Point, Notes of S. Petersburg Math. Inst., 2003, vol. 300, pp. 167-172.
45. Gonchenko, V. S. and Shilnikov, L. P., On Bifurcations of a Homoclinic Loop of a Saddle-Focus with the Index 1/2, Dokl. Akad. Nauk, 2007, vol. 417, no. 6, pp. 727-731.
46. Gonchenko, S. V. and Gonchenko, M. S., On Cascades of Elliptic Periodic Points in Two-dimensional Symplectic Maps with Homoclinic Tangencies, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 116-136.
47. Shilnikov, A. L., Shilnikov, L. P., and Turaev, D. V., Normal Forms and Lorenz Attractors, IInternat. J. Bifur. Chaos Appl. Sci. Engrg., 1993, vol. 3, pp. 1123-1139.
48. Turaev, D. V. and Shilnikov, L. P., An Example of a Wild Strange Attractor, Mat. Sb., 1998, vol. 189, no. 2, pp. 137-160 [Sb. Math., 1998, vol. 189, nos. 1-2, pp. 291-314].
49. Turaev, D. V. and Shilnikov, L. P., Pseudohyperbolisity and the Problem on Periodic Perturbations of Lorenz-Type Attractors, Dokl. Akad. Nauk, 2008, vol. 418, no. 1, pp. 23-27.
50. Sataev, E. A., Non-existence of Stable Trajectories in Non-autonomous Perturbations of Systems of Lorenz Type, Mat. Sb., 2005, vol. 196, no. 4, pp. 99-134 [Sb. Math., 2005, vol. 196, nos. 3-4, pp. 561-594.