Examples of lorenz-like attractors in hénon-like maps

Mathematical Modelling of Natural Phenomena

Volumn 8, Issue 5, 2013, Pages 48-70

  Gonchenko, S V ^a   Gonchenko, A S ^a   Ovsyannikov, I I ^a,b   Turaev, D V ^b  

^a Institute for Applied Mathematics and Cybernetics   (Russian Federation)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Bifurcations; Homoclinic orbit; Homoclinic tangency; Normal form; Pseudohyperbolicity; Strange attractor; Three dimensional maps; Volume hyperbolicity; Wild set

Indexed keywords

HOMOCLINIC ORBITS; HOMOCLINIC TANGENCIES; NORMAL FORM; PSEUDOHYPERBOLICITY; STRANGE ATTRACTORS; THREE-DIMENSIONAL MAPS; VOLUME HYPERBOLICITY; WILD SET;

BIFURCATION (MATHEMATICS); THREE DIMENSIONAL;

DYNAMICAL SYSTEMS;

EID: 84885099877     PISSN: 09735348     EISSN: 17606101     Source Type: Journal    
DOI: 10.1051/mmnp/20138504     Document Type: Article

References (146)

1. S.V. Gonchenko, I.I. Ovsyannikov, C. Simó, D. Turaev. Three-dimensional Hénon-like maps and wild Lorenz-like attractors. Bifurcation and Chaos, 15 (2005), 3493-3508. (Pubitemid 43201369)
2. V.S. Afraimovich, V.V. Bykov, L.P. Shilnikov. The origin and structure of the Lorenz attractor. Sov. Phys. Dokl., 22 (1977), 253-255.
3. V.S. Afraimovich, V.V. Bykov, L.P. Shilnikov. On attracting structurally unstable limit sets of lorenz attractor type. Trans. Mosc. Math. Soc., 44 (1982), 153-216.
4. J. Guckenheimer. A strange, strange attractor. In The Hopf Bifurcation Theorem and its Applications, eds. J. Marsden and M. McCracken, Springer-Verlag, 1976, 368-381.
5. J. Guckenheimer, R.F. Williams. Structural stability of Lorenz attractors. IHES Publ. Math., 50 (1979), 59-72.
6. R.F. Williams. The structure of Lorenz attractors. IHES Publ. Math., 50 (1979), 73-79.
7. L.A. Bunimovich, Ya.G. Sinai. Stochasticity of the attractor in the Lorenz model. In Nonlinear Waves (Proc. Winter School, Moscow), Nauka, 1980, 212-226.
8. C. Robinson. Differentiability of the stable foliation for the model Lorenz equations. Lect. Notes Math., 898 (1981), 302-315.
9. V.S. Aframovich, L.P. Shilnikov. Strange attractors and quasiattractors. in Nonlinear Dynamics and Turbulence, eds. G.I.Barenblatt, G.Iooss, D.D.Joseph, Boston, Pitmen, 1983.
10. 2. Proc. A.M.S. Symp. in Pure Math., 1 (1970), 191-202.
11. S.E. Newhouse. Diffeomorphisms with infinitely many sinks. Topology, 13 (1974), 9-18.
12. S.E. Newhouse. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. IHES Publ. Math., 50 (1979), 101-151.
13. M. Benedicks, L. Carleson. The dynamics of the Henon map. Ann. Math., 133 (1991), 73-169.
14. L. Mora, M. Viana. Abundance of strange attractors. Acta Math., 171 (1993), 1-71.
15. Q.D. Wang, L.-S. Young. Strange attractors with one direction of instability. Commun. Math. Phys., 218 (2001), 1-97. (Pubitemid 33381244)
16. R. Ures. On the Approximation of Hénon-like Attractors by Homoclinic Tangencies. Ergod. Th. Dyn. Sys. 15 (1995), 1223-1229.
17. A.S. Gonchenko, S.V. Gonchenko. On existence of Lorenz-like attractors in a nonholonomic model of celtic stones. Rus. J. Nonlin. Dyn., 9 (2013), 77-89.
18. D.V. Turaev, L.P. Shilnikov. An example of a wild strange attractor. Sb. Math., 189 (1998), 291-314.
19. D.V. Turaev, L.P. Shilnikov. Pseudo-hyperbolisity and the problem on periodic perturbations of Lorenz-like attractors. Russian Dokl. Math., 467 (2008), 23-27.
20. T. Shimizu, N. Morioka. On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model. Phys. Lett. A, 76 (1980), 201-204.
21. A.L. Shilnikov. Bifurcation and chaos in the Morioka-Shimizu system. Methods of qualitative theory of differential equations, Gorky, 1986, 180-193 [English translation in Selecta Math. Soviet., 10 (1991) 105-117]; II. Methods of Qualitative Theory and Theory of Bifurcations, Gorky, 1989, 130-138.
22. A.L. Shilnikov. On bifurcations of the Lorenz attractor in the Shimuizu-Morioka model. Physica D, 62 (1993), 338-346.
23. A.L. Shilnikov, L.P. Shilnikov, D.V. Turaev. Normal forms and Lorenz attractors. Bifurcation and Chaos, 3 (1993), 1123-1139.
24. H.W.E. Jung. Uber ganze birationale Transformationen der Ebene. J. Reine Angew. Math., 184 (1942), 161-174.
25. S. Friedland, J. Milnor. Dynamical properties of plane polynomial automorphisms. Ergod. Th. Dyn. Sys., 9 (1989), 67-99.
26. A.J. Dragt, D.T. Abell. Symplectic maps and computation of orbits in particle accelerators. In Integration algorithms and classical mechanics (Toronto, ON, 1993), Amer. Math. Soc., Providence, RI, 1996, 59-85.
27. H.E. Lomeli, J.D. Meiss. Quadratic volume preserving maps. Nonlinearity, 11 (1998), 557-574.
28. S.V. Gonchenko, V.S. Gonchenko, L.P. Shilnikov. On a homoclinic origin of Henon-like maps. Regul. Chaotic Dyn., 15 (2010), 462-481.
29. L. Tedeschini-Lalli, J.A. Yorke. How often do simple dynamical processes have infinitely many coexisting sinks? Commun. Math. Phys., 106 (1986), 635-657.
30. M. Henon. A two-dimensional mapping with a strange attractor. Commun. Math. Phys., 50 (1976), 69-77.
31. N.K. Gavrilov, L.P. Shilnikov. On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. I. Math. USSR Sbornik, 17 (1972), 467-485; II. Math. USSR Sbornik, 19 (1973), 139-156.
32. S.V. Gonchenko, D.V. Turaev, L.P. Shilnikov. On models with a structurally unstable Poincare homoclinic curve. Sov. Math., Dokl., 44 (1992), 422-426.
33. S.V. Gonchenko, L.P. Shilnikov, D.V. Turaev. On models with non-rough Poincare homoclinic curves. Physica D, 62 (1993), 1-14.
34. S.V. Gonchenko, D.V. Turaev, L.P. Shilnikov. Dynamical phenomena in multi-dimensional systems with a non-rough Poincare homoclinic curve. Russ. Acad. Sci. Dokl. Math., 47 (1993), 410-415.
35. S.V. Gonchenko, D.V. Turaev, L.P. Shilnikov. On the existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case). Russian Acad. Sci.Dokl.Math. 47 (1993), 268-283.
36. J. Palis, M. Viana. High dimension diffeomorphisms displaying infinitely many sinks. Ann. Math., 140 (1994), 91-136.
37. N. Romero. Persistence of homoclinic tangencies in higher dimensions. Ergod. Th. Dyn. Sys., 15 (1995), 735-757.
38. E. Colli. Infinitely many coexisting strange attractors. Ann. IHP, Anal. Non Lineaire, 15 (1998), 539-579. (Pubitemid 128433733)
39. J.C. Tatjer. Three dimensional dissipative diffeomorphisms with homoclinic tangencies. Ergod. Th. Dyn. Systems, 21 (2001), 249-302.
40. A.R. Champneys, J. Haerterich, B. Sandstede. A non-transverse homoclinic orbit to a saddle-node equilibrium. Ergod. Th. Dyn. Sys., 16 (1996), 431-450.
41. D.V. Turaev D.V. On dimension of non-local bifurcational problems. Bifurcation and Chaos, 6 (1996), 919-948.
42. S.V. Gonchenko, L.P. Shilnikov, D.V. Turaev. Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits. Chaos, 6 (1996), 15-31.
43. S.V. Gonchenko, L.P. Shilnikov, D.V. Turaev. On Newhouse regions of two-dimensional diffeomorphisms close to a diffeomorphism with a nontransversal heteroclinic cycle. Proc. Steklov Inst. Math., 216 (1997), 70-118.
44. S.V. Gonchenko, D.V. Turaev, L.P. Shilnikov Quasiattractors and homoclinic tangencies. Comput. Math. Appl., 34 (1997), 195-227. (Pubitemid 127400515)
45. S.V. Gonchenko, D.V. Turaev, L.P. Shilnikov. Homoclinic tangencies of an arbitrary order in Newhouse domains. In Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., 67 (1999), 69-128. [English translation in J. Math. Sci., 105 (2001), 1738-1778].
46. S.V. Gonchenko, L.P. Shilnikov, D.V. Turaev. On dynamical properties of diffeomorphisms with homoclinic tangencies. Contemprary Mathematics and Its Applications, 7 (2003), 92-118. [English translation in J. Math. Sci., 126 (2005), 1317-1343].
47. S. Gonchenko, L. Shilnikov, D. Turaev. Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps. Nonlinearity, 20 (2007), 241-275.
48. S.V. Gonchenko, L.P. Shilnikov, D.V. Turaev. On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. I. Nonlinearity, 21 (2008), 923-972.
49. S.V. Gonchenko, L.P. Shilnikov, D.V. Turaev. On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. II. submitted to Nonlinearity (2013).
50. S.V. Gonchenko, I.I. Ovsyannikov, D. Turaev. On the effect of invisibility of stable periodic orbits at homoclinic bifurcations. Physica D, 241 (2012), 1115-1122.
51. S.V.Gonchenko, V.S.Gonchenko. On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies. preprint WIAS No.556, Berlin, 2000.
52. S.V. Gonchenko, V.S. Gonchenko. On bifurcations of the birth of closed invariant curves in the case of two-dimensional diffeomorphisms with homoclinic tangencies. Proc. Steklov Inst. Math., 244 (2004), 80-105.
53. V.S. Gonchenko, Yu.A. Kuznetsov, H.G.E. Meijer. Generalized Henon map and bifurcations of homoclinic tangencies. SIAM J. Appl. Dyn. Sys., 4 (2005), 407-436 (Pubitemid 43385182)
54. S.V. Gonchenko, O.V. Sten'kin, L.P. Shilnikov. On the existence of infinitely many stable and unstable invariant tori for systems from Newhouse regions with heteroclinic tangencies. Rus. J. Nonlinear Dynamics, 2 (2006), 3-25
55. S.V. Gonchenko, V.S. Gonchenko, J.C. Tatjer. Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps. Regul. Chaotic Dyn., 12 (2007), 233-266.
56. V.S. Gonchenko, L.P. Shilnikov. On bifurcations of systems with homoclinic loops to a saddle-focus with saddle index 1/2. Russian Dokl. Math., 76 (2007), 929-933.
57. A. Gonchenko, S. Gonchenko. Towards a classification of linear and nonlinear Smale horseshoes. Rus. Nonlinear Dynamics, 3 (2007), 423-433.
58. S. Gonchenko, M.-C. Li, M. Malkin. Generalized Henon maps and Smale horseshoes of new types. Bifurcation and Chaos, 18 (2008), 3029-3052.
59. A. Gonchenko, S. Gonchenko, M. Malkin. On classification of classical and half-orientable horseshoes in terms of border points. Rus. Nonlinear Dynamics, 6 (2010), 549-566.
60. V.S. Biragov. Bifurcations in two-parameter family of conservative mappings that are close to the Hénon map. Methods of Qualitative Theory and Theory of Bifurcations, Gorky, 1987, 10-24 [English translation in Selecta Math. Sov., 9 (1990), 273-282].
61. V. Biragov, L. Shilnikov. On the bifurcation of a saddle-focus separatrix Loop in a three-dimensional conservative dynamical system. Methods of Qualitative Theory and Theory of Bifurcations, Gorky, 1989, 25-34 [English translation in Selecta Math. Soviet., 11 (1992), 333-340].
62. L. Mora, N. Romero. Moser's invariant curves and homoclinic bifurcations. Dyn. Sys. Appl., 6 (1997), 29-42. (Pubitemid 127039659)
63. P. Duarte. Abundance of elliptic isles at conservative bifurcations. Dynam. Stability Systems, 14 (1999), 339-356.
64. P. Duarte. Persistent Homoclinic Tangencies for Conservative Maps near the Identity. Ergod. Th. Dyn. Sys., 20 (2000), 393-438.
65. P. Duarte. Elliptic Isles in Families of Area-Preserving Maps. Ergod. Th. Dyn. Sys., 28 (2008), 1781-1813.
66. V. Rom-Kedar, D. Turaev. Big islands in dispersing billiard-like potentials. Physica D, 130 (1999), 187-210. (Pubitemid 129587669)
67. D. Turaev, V. Rom-Kedar. Soft billiards with corners. J. Stat. Phys., 112 (2003), 765-813.
68. S.V. Gonchenko, L.P. Shilnikov. On two-dimensional analytic area-preserving diffeomorphisms with infinitely many stable elliptic periodic points. Regul. Chaotic Dyn., 2 (1997), 106-123
69. S.V. Gonchenko, L.P. Shilnikov. On two-dimensional area-preserving diffeomorphisms with infinitely many elliptic islands. J. Stat. Phys., 101 (2000), 321-356
70. S.V. Gonchenko, L.P. Shilnikov. On two-dimensional area-preserving maps with homoclinic tangencies that have infinitely many generic elliptic periodic points. Notes Sci. seminars PDMI, 300 (2003). [English translation in J. Math. Sci. 128 (2), 2767-2773].
71. M.S. Gonchenko. On the structure of 1:4 resonances in Hénon maps. Bifurcation and Chaos, 15 (2005), 3653-3660. (Pubitemid 43201381)
72. M.S. Gonchenko, S.V. Gonchenko. On cascades of elliptic periodic points in two-dimensional symplectic maps with homoclinic tangencies. Regul. Chaotic Dyn., 14 (2009), 116-136
73. A. Gorodetski, V. Kaloshin. Conservative homoclinic bifurcations and some applications. Proc. Steklov Inst. Math., 267 (2009), 76-90.
74. J.S.W. Lamb, O.V. Sten'kin. Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits. Nonlinearity, 17 (2004), 1217-1244. (Pubitemid 38919691)
75. S.V. Gonchenko, L.P. Shilnikov, D. Turaev. Elliptic periodic orbits near a homoclinic tangency in four-dimensional symplectic maps and Hamiltonian systems with three degrees of freedom. Regul. Chaotic Dyn., 3 (1998), 3-26.
76. S.V. Gonchenko, D.V. Turaev, L.P. Shilnikov. Infinitely many elliptic periodic orbits in four-dimensional symplectic diffeomorphisms with a homoclinic tangency. Proc. Steklov Inst. Math., 244 (2004), 106-131.
77. S.V. Gonchenko, J.D. Meiss, I.I. Ovsyannikov. Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation. Regul. Chaotic Dyn., 11 (2006), 191-212. (Pubitemid 43978372)
78. S.V. Gonchenko, L. Shilnikov, D. Turaev. On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors. Regul. Chaotic Dyn., 14 (2009), 137-147.
79. S.V. Gonchenko, I.I. Ovsyannikov. On bifurcations of three-dimensional diffeomorphisms with a non-transversal heteroclinic cycle containing saddle-foci. Rus. J. Nonlinear Dynamics, 6 (2010), 61-77
80. D. Turaev. Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps. Nonlinearity, 16 (2003), 123-135. (Pubitemid 37342383)
81. D.Turaev. Maps close to identity and universal maps in the Newhouse domain. preprint arXiv:1009.0858, 2010.
82. A. Delshams, S.V. Gonchenko, V.S. Gonchenko, J.T. Lazaro, O. Sten'kin. Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps. Nonlinearity, 26 (2013), 1-33.
83. Homoclinic Tangencies, eds. S.V.Gonchenko and L.P.Shilnikov, Regular and Chaotic Dynamics, Moscow-Izhevsk, 2007.
84. L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev, L.O. Chua. Methods of qualitative theory in nonlinear dynamics. Part I, World Scientific, 1998; Part II, World Scientific, 2001.
85. V.S. Afraimovich, L.P. Shilnikov. On small periodic perturbations of autonomous systems. Soviet Math., Dokl., 15 (1974), 206-211.
86. V.S. Afraimovich, L.P. Shilnikov. On some global bifurcations connected with the disappearance of saddle-node fixed point. Soviet. Math., Dokl.. 15 (1974), 1761-1765.
87. V.S. Afraimovich, L.P. Shilnikov. The ring principle in problems of interaction between two self-oscillating systems. J. Appl. Math. Mech., 41 (1977), 632-641
88. 2. In The Structure of Attractors in Dynamical Systems, eds. J.C. Martin, N.G. Markley, W. Perrizo, Lecture Notes Math., 668, Springer, Berlin, 1978, 48-66.
89. D.G. Aronson, M.A. Chory, R.P. Mcgehee, G.R. Hall. Bifurcation from an invariant circle for two-parameter families of maps of the plane. Commun. Math. Phys., 83 (1982), 303-354.
90. V.S. Afraimovich, L.P. Shilnikov. On invariant two-dimensional tori, their breakdown and stochasticity. Methods of the Qualitative Theory of Differential Equations, Gorky, 1983, 3-26 [English translation in: Amer. Math. Soc. Transl., 149 (1991), 201-212].
91. A. Shilnikov, L. Shilnikov, D. Turaev. On some mathematical topics in classical synchronization. Bifurcation and Chaos, 14 (2004), 2143-2160. (Pubitemid 40242096)
92. A.S. Gonchenko, S.V. Gonchenko, L.P. Shilnikov. Towards scenarios of chaos appearance in three-dimensional maps. Rus. J. Nonlinear Dynamics, 8 (2012), 3-28.
93. A.V. Borisov, A.Yu. Jalnine, S.P. Kuznetsov, I.R. Sataev, J.V. Sedova. Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback. Regul. Chaotic Dyn., 17 (2012), 512-532.
94. 1-generic dynamics: tame and wild behaviour. Proceedings of the ICM, Beijing 2002, V. 3, 279-294.
95. 1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources. Ann. Math., 158 (2003), 355-418. (Pubitemid 38033620)
96. C. Bonatti, L. Diaz, M. Viana. Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective. Encyclopaedia of Mathematical Sciences, V. 102, Springer, 2005.
97. C. Morales, E.R. Pujals. Singular strange attractors on the boundary of Morse-Smale systems. Ann. Sci. Ecole Norm. Sup., 30 (1997), 693-717. (Pubitemid 127162105)
98. V. Araujo, M.J. Pacifico. Three-Dimensional Flows. Springer, 2010.
99. S. Luzzatto, I. Melbourne, F. Paccaut. The Lorenz attractor is mixing. Comm. Math. Phys., 260 (2005), 393-401. (Pubitemid 41407110)
100. Ya. B. Pesin. Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergod. Th. Dyn. Sys., 12 (1992), 123-151.
101. E.A. Sataev. Invariant measures for hyperbolic maps with singularities. Russ. Math. Surv., 47 (1992), 191-251.
102. V.S. Afraimovich, N.I. Chernov, E.A. Sataev. Statistical properties of 2-D generalized hyperbolic attractors. Chaos 5 (1995), 238-252.
103. C. Bonatti, A. Pumarino, M. Viana. Lorenz attractors with arbitrary expanding dimension. C.R. Acad. Sci. Paris, 325-I (1997), 883-888. (Pubitemid 127172193)
104. J. Auslander, P. Seibert. Prolongations and stability in dynamical systems. Ann. Inst. Forier., Genoble, 14 (1964), 237-268.
105. J. Auslander, N.P. Bhatia, P. Seibert. Attractors in dynamical systems. Bol. Soc. Mat. Mex., 9 (1964), 55-66.
106. J. Milnor. On the concept of attractor. Comm. Math. Phys., 99 (1985), 177-195; Correction and remarks. Comm. Math. Phys., 102 (1985), 517-519.
107. A. Gorodetski, Yu. Ilyashenko. Minimal and strange attractors. Bifurcation and Chaos, 6 (1996), 1177-1183.
108. C. Robinson. What is a Chaotic Attractor? Qualitative Theory Dyn. Sys., 7 (2008), 227-236.
109. C. Conley. Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series Math., 38 (1978), Am. Math. Soc., Providence RI.
110. D. Ruelle. Small random perturbations of dynamical systems and the definition of attractors. Comm. Math. Phys., 82 (1981), 137-151.
111. M. Hurley. Attractors: persistence and density of their basins. Trans. Amer. Math. Soc., 269 (1982), 247-271.
112. V.A. Dobrynsky, A.N. Sharkovsky. Genericity of the dynamical systems, almost all trajectories of which are stable underconstantly acting perturbations. Soviet. Math. Dokl., 211 (1973).
113. E.A. Sataev. Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type. Sb. Math., 196 (2005), 561-594. (Pubitemid 41017395)
114. R. Bamon, J. Kiwi, J. Rivera. Wild Lorenz like attractors. Preprint (2005), arXiv:math/0508045.
115. L.J. Diaz, J. Rocha. Nonconnected heteroclinic cycles: bifurcation and stability. Nonlinearity, 5 (1992), 1315-1341.
116. L.J. Diaz. Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations. Nonlinearity, 8 (1995), 693-713.
117. Ch. Bonatti, L.J. Diaz. Persistent nonhyperbolic transitive diffeomorphisms. Ann. Math., 143 (1996), 367-396.
118. C. Bonatti, L.J. Diaz, E. Pujals, J. Rocha. Robust transitivity and heterodimensional cycles. Asterisque, 286 (2003), 187-222.
119. 1-generic dynamics. J. Inst. of Math. Jussieu, 7 (2008), 469-525.
120. L.J. Diaz, A. Gorodetski. Non-hyperbolic invariant measures for non-hyprerbolic homoclinic classes. Ergod. Th. Dyn. Sys., 29 (2009), 479-513.
121. L.P. Silnikov. Structure of the neighborhood of a homoclinic tube of an invariant torus. Sov. Math. Dokl., 9 (1968), 624-628.
122. V.S. Afrajmovich, L.P. Shil'nikov. On a bifurcation of codimension 1 leading to the appearance of a countable set of tori. Sov. Math. Dokl., 25 (1982), 101-105.
123. A. Gorodetski, Yu. Ilyashenko. Some new robust properties of invariant sets and attractors of dynamical systems. Functional Analysis and Applications, 33 (1999), 16-30.
124. A. Gorodetski, Yu. Ilyashenko. Some properties of skew products over the horseshoe and solenoid. Proc. Steklov Inst. Math., 231 (2000), 96-118.
125. A. Gorodetski, Yu. Ilyashenko, V. Kleptsyn, M. Nalskij. Non-removable zero Lyapunov exponent. Functional Analysis and Applications, 39 (2005), 27-38.
126. V.N. Pisarevsky, A. Shilnikov, D. Turaev. Asymptotic normal forms for equilibria with a triplet of zero characteristic exponents in systems with symmetry. Regul. Chaotic Dyn., 2 (1998), 19-27.
127. L.P. Shilnikov. The theory of bifurcations and quasiattractors. Russ. Math. Survey, 36 [4] (1981), 240-241.
128. C. Robinson. Homoclinic bifurcation to a transitive attractor of the Lorenz type. Nonlinearity, 2 (1989), 495-518; II. SIAM J. Math. Anal., 23 (1992), 1255-1268.
129. M. Rychlic. Lorenz attractors through Shilnikov type bifurcation. Ergod. Th. Dyn. Sys., 10 (1990), 793-821.
130. G. Tigan, D. Turaev. Analytical search for homoclinic bifurcations in Morioka-Shimizu model. Physica D, 240 (2011), 985-989.
131. V.V. Bykov. On the structure of a neighborhood of a separatrix contour with a saddle-focus. Methods of the Qualitative Theory of Differencial Equations, Gorky, 1978, 3-32.
132. V.V. Bykov. On bifurcations of dynamical systems close to systems with a separatrix contour containing a saddle-focus. Methods of the Qualitative Theory of Differencial Equations, Gorky, 1980, 44-72.
133. V.V. Bykov. The bifurcations of separatrix contours and chaos. Physica D, 62 (1993), 290-299.
134. N.V. Petrovskaya, V.I. Yudovich. Homoclinic loops of the Saltzman-Lorenz system. Methods of qualitative theory of differential equations, Gorky, 1980, 73-83.
135. V.V. Bykov, A.L. Shilnikov. On the boundaries of the domain of existence of the Lorenz attractor. Methods of qualitative theory and bifurcation theory, Gorky, 1989, 151-159 [English translation in Selecta Math. Soviet., 11 (1992), 375-382].
136. V.S. Afraimovich, V.V. Bykov, L.P. Shilnikov. On the existence of stable periodic orbits in the Lorenz model. Russ. Math. Survey, 35 [4] (1980), 164-165.
137. A. Chenciner. Bifurcations de points fixes elliptiques. I. Courbes invariantes. IHES Publ. Math., 61 (1985), 67-127; II. Orbites periodiques et ensembles de Cantor invariants. Invent. Math., 80 (1985), 81-106; III. Orbites periodiques de "petites" periodes et elimination resonante des couples de courbes invariantes. IHES Publ. Math., 66 (1987), 5-91.
138. 3 and doubling bifurcation. Nonlinearity, 2 (1989), 149-174.
139. H.W. Broer, G.B. Huitema, F. Takens. Unfoldings of quasi-periodic tori. Mem. AMS, 83 (1990), 1-82.
140. B.L.J. Braaksma, H.W. Broer, G.B. Huitema. Toward a quasi-periodic bifurcation theory. Mem. AMS, 83 (1990), 83-175.
141. A. Arneodo, P.H. Coullet, E.A. Spiegel. Cascade of period doublings of tori. Phys. Lett. A, 94 (1983), 1-5.
142. V. Franceschini. Bifurcations of tori and phase locking in a dissipative system of differential equations. Physica D, 6 (1983), 285-304. (Pubitemid 14545461)
143. V.S. Anishchenko, V.V. Astakhov, T.E. Letchford, M.A. Safonova. Structure of the quasihyperbolic stochasticity in an inertial self-excited oscillator. Radiophysics and Quantum Electronics, 26 (1983), 619-628. (Pubitemid 14571773)
144. V.S. Anishenko. Complex oscillations in simple systems. Nauka, Moscow, 1990.
145. K. Kaneko. Collapse of Tori and Genesis of Chaos in Dissipative Systems. World Scientific, Singapore, 1986.
146. C. Bonatti, Y. Shi, Robustly transitive attractor derived from Lorenz, in preparation.