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On a homoclinic origin of Hénon-like maps. (English) Zbl 1203.37035
Summary: We review bifurcations of homoclinic tangencies leading to Hénon-like maps of various kinds.

##### MSC:
 37C29 Homoclinic and heteroclinic orbits for dynamical systems 37G25 Bifurcations connected with nontransversal intersection in dynamical systems 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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##### References:
 [1] Hénon, M., A Two-dimensional Mapping with a Strange Attractor, Comm. Math. Phys., 1976, vol. 50, pp. 69–77. · Zbl 0576.58018 [2] Benedidicks, M. and Carleson, L., The Dynamics of the Hénon Map, Ann. Math., 1991, vol. 133, pp. 73–169. · Zbl 0724.58042 [3] Lai, Y.-C., Grebogi, C., Yorke, J. and Kan, I., How Often are Chaotic Saddles Nonhyperbolic? Nonlinearity, 1993, vol. 6, pp. 779–797. · Zbl 0785.58035 [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. · Zbl 0445.58022 [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. · Zbl 0602.58036 [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. · Zbl 1055.37578 [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. · Zbl 1079.37046 [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]. · Zbl 1079.37046 [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. · Zbl 1164.37306 [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. · Zbl 1229.37040 [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. · Zbl 1160.37019 [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. · Zbl 1229.37041 [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). · Zbl 1041.37033 [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. · Zbl 0956.37048 [21] Turaev, D.V., On Dimension of Nonlocal Bifurcational Problems, Int. J. of Bifurcation and Chaos, 1996, vol. 6, no. 5, pp. 919–948. · Zbl 0879.58058 [22] Gonchenko, S.V. and Shilnikov, L.P., Eds., Homoclinic Tangencies, Moscow-Izhevsk: RCD, 2007. · Zbl 1132.37023 [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. · Zbl 1022.37037 [25] Turaev D. and Rom-Kedar, V., Elliptic Islands Appearing in Near-ergodic Flows, Nonlinearity, 1998, vol. 11, pp. 575–600. · Zbl 0902.58017 [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. · Zbl 0790.58014 [27] Palis, J. and Viana, M., High Dimension Diffeomorphisms Displaying Infinitely Many Periodic Attractors, Ann. Math., 1994, vol. 140, pp. 207–250. · Zbl 0817.58004 [28] Romero, N., Persistence of Homoclinic Tangencies in Higher Dimensions, Ergod. Th. and Dyn.Sys., 1995, vol. 15, pp. 735–757. · Zbl 0833.58020 [29] Lomelí, H.E. and Meiss, J. D., Quadratic Volume Preserving Maps, Nonlinearity, 1998, vol. 11, no. 3, pp. 557–574. · Zbl 0902.58010 [30] Gonchenko, S.V. and Shilnikov, L.P., Invariants of {$$\Theta$$}-conjugacy of Diffeomorphisms with a Nontransversal Homoclinic Orbit, Ukr. Math. J., 1990, vol. 42, no. 2, pp. 134–140. · Zbl 0705.58044 [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. · Zbl 0941.34001 [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]. · Zbl 0519.58029 [34] Newhouse, S.E., Diffeomorphisms with Infinitely Many Sinks, Topology, 1974, vol. 13, pp. 9–18. · Zbl 0275.58016 [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. · Zbl 1097.37023 [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. · Zbl 0987.37062 [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. · Zbl 1090.37036 [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. · Zbl 1089.37530 [41] Gonchenko, S.V., Dynamics and Moduli of {$$\omega$$}-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. · Zbl 0990.37012 [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. · Zbl 1229.37040 [43] Tatjer, J.C., Three-dimensional Dissipative Diffeomorphisms with Homoclinic Tangencies, Ergod.Th. and Dynam.Sys., 2001, vol. 21, no. 1, pp. 249–302. · Zbl 0972.37013 [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. · Zbl 1229.37054 [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. · Zbl 0885.58080 [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]. · Zbl 0927.37017 [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.
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