Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T01:12:17.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Examples of Lorenz-like Attractors in Hénon-likeMaps

Published online by Cambridge University Press:  17 September 2013

S.V. Gonchenko ,
A.S. Gonchenko ,
I.I. Ovsyannikov  and
D.V. Turaev
S.V. Gonchenko*
Affiliation:
Research Institute of Applied Mathematics and Cybernetics, 10, Ulyanova Str. 603005 Nizhny Novgorod, Russia
A.S. Gonchenko
Affiliation:
Research Institute of Applied Mathematics and Cybernetics, 10, Ulyanova Str. 603005 Nizhny Novgorod, Russia
I.I. Ovsyannikov
Affiliation:
Research Institute of Applied Mathematics and Cybernetics, 10, Ulyanova Str. 603005 Nizhny Novgorod, Russia Imperial College, SW7 2 AZ London, UK
D.V. Turaev
Affiliation:
Imperial College, SW7 2 AZ London, UK
*
⋆ Corresponding author. E-mail: gonchenko@pochta.ru
Get access

Abstract

We display a gallery of Lorenz-like attractors that emerge in a class ofthree-dimensional maps. We review the theory of Lorenz-like attractors for diffeomorphisms(as opposed to flows), define various types of such attractors, and find sufficientconditions for three-dimensional Henon-like maps to possess pseudohyperbolic Lorenz-likeattractors. The numerically obtained scenarios of the creation and destruction of theseattractors are also presented.

Keywords

Strange attractorthree-dimensional mapspseudohyperbolicityvolume hyperbolicitybifurcationshomoclinic orbithomoclinic tangencywild setnormal form
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 48 - 70
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Gonchenko, S.V., Ovsyannikov, I.I., Simó, C., Turaev, D.. Three-dimensional Hénon-like maps and wild Lorenz-like attractors. Bifurcation and Chaos, 15 (2005), 3493–3508. CrossRefGoogle Scholar
Afraimovich, V.S., Bykov, V.V., Shilnikov, L.P.. The origin and structure of the Lorenz attractor. Sov. Phys. Dokl., 22 (1977), 253–255. Google Scholar
Afraimovich, V.S., Bykov, V.V., Shilnikov, L.P.. On attracting structurally unstable limit sets of lorenz attractor type. Trans. Mosc. Math. Soc., 44 (1982), 153–216. Google Scholar
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.
Guckenheimer, J., Williams, R.F.. Structural stability of Lorenz attractors. IHES Publ. Math., 50 (1979), 59–72. CrossRefGoogle Scholar
Williams, R.F.. The structure of Lorenz attractors. IHES Publ. Math., 50 (1979), 73–79. CrossRefGoogle Scholar
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.
Robinson, C.. Differentiability of the stable foliation for the model Lorenz equations. Lect. Notes Math., 898 (1981), 302–315. CrossRefGoogle Scholar
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.
Newhouse, S.E.. Non-density of Axiom A(a) on S2. Proc. A.M.S. Symp. in Pure Math., 1 (1970), 191–202. CrossRefGoogle Scholar
Newhouse, S.E.. Diffeomorphisms with infinitely many sinks. Topology, 13 (1974), 9–18. CrossRefGoogle Scholar
Newhouse, S.E.. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. IHES Publ. Math., 50 (1979), 101–151. CrossRefGoogle Scholar
Benedicks, M., Carleson, L.. The dynamics of the Henon map. Ann. Math., 133 (1991), 73–169. CrossRefGoogle Scholar
Mora, L., Viana, M.. Abundance of strange attractors. Acta Math., 171 (1993), 1–71. CrossRefGoogle Scholar
Wang, Q.D., Young, L.-S.. Strange attractors with one direction of instability. Commun. Math. Phys., 218 (2001), 1–97. CrossRefGoogle Scholar
Ures, R.. On the Approximation of Hénon-like Attractors by Homoclinic Tangencies. Ergod. Th. Dyn. Sys. 15 (1995), 1223–1229. CrossRefGoogle Scholar
Gonchenko, A.S., Gonchenko, S.V.. On existence of Lorenz-like attractors in a nonholonomic model of celtic stones. Rus. J. Nonlin. Dyn., 9 (2013), 77-89. Google Scholar
Turaev, D.V., Shilnikov, L.P.. An example of a wild strange attractor. Sb. Math., 189 (1998), 291–314. CrossRefGoogle Scholar
Turaev, D.V., Shilnikov, L.P.. Pseudo-hyperbolisity and the problem on periodic perturbations of Lorenz-like attractors. Russian Dokl. Math., 467 (2008), 23–27. Google Scholar
Shimizu, T., Morioka, N.. On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model. Phys. Lett. A, 76 (1980), 201–204. CrossRefGoogle Scholar
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.
Shilnikov, A.L.. On bifurcations of the Lorenz attractor in the Shimuizu-Morioka model. Physica D, 62 (1993), 338–346. CrossRefGoogle Scholar
Shilnikov, A.L., Shilnikov, L.P., Turaev, D.V.. Normal forms and Lorenz attractors. Bifurcation and Chaos, 3 (1993), 1123–1139. CrossRefGoogle Scholar
Jung, H.W.E.. Uber ganze birationale Transformationen der Ebene. J. Reine Angew. Math., 184 (1942), 161–174. Google Scholar
Friedland, S., Milnor, J.. Dynamical properties of plane polynomial automorphisms. Ergod. Th. Dyn. Sys., 9 (1989), 67–99. Google Scholar
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.
Lomeli, H.E., Meiss, J.D.. Quadratic volume preserving maps. Nonlinearity, 11 (1998), 557–574. CrossRefGoogle Scholar
Gonchenko, S.V., Gonchenko, V.S., Shilnikov, L.P.. On a homoclinic origin of Henon-like maps. Regul. Chaotic Dyn., 15 (2010), 462–481. CrossRefGoogle Scholar
Tedeschini-Lalli, L., Yorke, J.A.. How often do simple dynamical processes have infinitely many coexisting sinks? Commun. Math. Phys., 106 (1986), 635–657. CrossRefGoogle Scholar
Henon, M.. A two-dimensional mapping with a strange attractor. Commun. Math. Phys., 50 (1976), 69–77. CrossRefGoogle Scholar
Gavrilov, N.K., Shilnikov, L.P.. 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. CrossRefGoogle Scholar
Gonchenko, S.V., Turaev, D.V., Shilnikov, L.P.. On models with a structurally unstable Poincare homoclinic curve. Sov. Math., Dokl., 44 (1992), 422–426. Google Scholar
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.. On models with non-rough Poincare homoclinic curves. Physica D, 62 (1993), 1–14. CrossRefGoogle Scholar
Gonchenko, S.V., Turaev, D.V., Shilnikov, L.P.. Dynamical phenomena in multi-dimensional systems with a non-rough Poincare homoclinic curve. Russ. Acad. Sci. Dokl. Math., 47 (1993), 410–415. Google Scholar
Gonchenko, S.V., Turaev, D.V., Shilnikov, L.P.. 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. Google Scholar
Palis, J., Viana, M.. High dimension diffeomorphisms displaying infinitely many sinks. Ann. Math., 140 (1994), 91–136. CrossRefGoogle Scholar
Romero, N.. Persistence of homoclinic tangencies in higher dimensions. Ergod. Th. Dyn. Sys., 15 (1995), 735–757. Google Scholar
Colli, E.. Infinitely many coexisting strange attractors. Ann. IHP, Anal. Non Lineaire, 15 (1998), 539–579. CrossRefGoogle Scholar
Tatjer, J.C.. Three dimensional dissipative diffeomorphisms with homoclinic tangencies. Ergod. Th. Dyn. Systems, 21 (2001), 249–302. Google Scholar
Champneys, A.R., Haerterich, J., Sandstede, B.. A non-transverse homoclinic orbit to a saddle-node equilibrium. Ergod. Th. Dyn. Sys., 16 (1996), 431–450. Google Scholar
Turaev D.V., D.V. On dimension of non-local bifurcational problems. Bifurcation and Chaos, 6 (1996), 919–948. CrossRefGoogle Scholar
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.. Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits. Chaos, 6 (1996), 15–31. CrossRefGoogle ScholarPubMed
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.. On Newhouse regions of two-dimensional diffeomorphisms close to a diffeomorphism with a nontransversal heteroclinic cycle. Proc. Steklov Inst. Math., 216 (1997), 70–118. Google Scholar
Gonchenko, S.V., Turaev, D.V., Shilnikov, L.P. Quasiattractors and homoclinic tangencies. Comput. Math. Appl., 34 (1997), 195–227. CrossRefGoogle Scholar
Gonchenko, S.V., Turaev, D.V., Shilnikov, L.P.. 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].
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.. 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]. Google Scholar
Gonchenko, S., Shilnikov, L., Turaev, D.. Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps. Nonlinearity, 20 (2007), 241–275. CrossRefGoogle Scholar
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.. On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. I. Nonlinearity, 21 (2008), 923–972. CrossRefGoogle Scholar
S.V. Gonchenko, L.P. Shilnikov, D.V. Turaev. On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. II. submitted to Nonlinearity (2013).
Gonchenko, S.V., Ovsyannikov, I.I., Turaev, D.. On the effect of invisibility of stable periodic orbits at homoclinic bifurcations. Physica D, 241 (2012), 1115–1122. CrossRefGoogle Scholar
S.V.Gonchenko, V.S.Gonchenko. On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies. preprint WIAS No.556, Berlin, 2000.
Gonchenko, S.V., Gonchenko, V.S.. 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. Google Scholar
Gonchenko, V.S., Kuznetsov, Yu.A., Meijer, H.G.E.. Generalized Henon map and bifurcations of homoclinic tangencies. SIAM J. Appl. Dyn. Sys., 4 (2005), 407–436 CrossRefGoogle Scholar
Gonchenko, S.V., Sten’kin, O.V., Shilnikov, L.P.. 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 Google Scholar
Gonchenko, S.V., Gonchenko, V.S., Tatjer, J.C.. Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps. Regul. Chaotic Dyn., 12 (2007), 233–266. CrossRefGoogle Scholar
Gonchenko, V.S., Shilnikov, L.P.. On bifurcations of systems with homoclinic loops to a saddle-focus with saddle index 1/2. Russian Dokl. Math., 76 (2007), 929–933. CrossRef
Gonchenko, A., Gonchenko, S.. Towards a classification of linear and nonlinear Smale horseshoes. Rus. Nonlinear Dynamics, 3 (2007), 423–433. Google Scholar
Gonchenko, S., Li, M.-C., Malkin, M.. Generalized Henon maps and Smale horseshoes of new types. Bifurcation and Chaos, 18 (2008), 3029–3052. CrossRefGoogle Scholar
Gonchenko, A., Gonchenko, S., Malkin, M.. On classification of classical and half-orientable horseshoes in terms of border points. Rus. Nonlinear Dynamics, 6 (2010), 549–566. Google Scholar
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].
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].
Mora, L., Romero, N.. Moser’s invariant curves and homoclinic bifurcations. Dyn. Sys. Appl., 6 (1997), 29–42. Google Scholar
Duarte, P.. Abundance of elliptic isles at conservative bifurcations. Dynam. Stability Systems, 14 (1999), 339–356. CrossRefGoogle Scholar
Duarte, P.. Persistent Homoclinic Tangencies for Conservative Maps near the Identity. Ergod. Th. Dyn. Sys., 20 (2000), 393–438. CrossRefGoogle Scholar
Duarte, P.. Elliptic Isles in Families of Area-Preserving Maps. Ergod. Th. Dyn. Sys., 28 (2008), 1781–1813. CrossRefGoogle Scholar
Rom-Kedar, V., Turaev, D.. Big islands in dispersing billiard-like potentials. Physica D, 130 (1999), 187–210. CrossRefGoogle Scholar
Turaev, D., Rom-Kedar, V.. Soft billiards with corners. J. Stat. Phys., 112 (2003), 765–813. CrossRefGoogle Scholar
Gonchenko, S.V., Shilnikov, L.P.. On two-dimensional analytic area-preserving diffeomorphisms with infinitely many stable elliptic periodic points. Regul. Chaotic Dyn., 2 (1997), 106–123 Google Scholar
Gonchenko, S.V., Shilnikov, L.P.. On two-dimensional area-preserving diffeomorphisms with infinitely many elliptic islands. J. Stat. Phys., 101 (2000), 321–356 CrossRefGoogle Scholar
Gonchenko, S.V., Shilnikov, L.P.. 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]. Google Scholar
Gonchenko, M.S.. On the structure of 1:4 resonances in Hénon maps. Bifurcation and Chaos, 15 (2005), 3653–3660. CrossRefGoogle Scholar
Gonchenko, M.S., Gonchenko, S.V.. On cascades of elliptic periodic points in two-dimensional symplectic maps with homoclinic tangencies. Regul. Chaotic Dyn., 14 (2009), 116–136 CrossRefGoogle Scholar
Gorodetski, A., Kaloshin, V.. Conservative homoclinic bifurcations and some applications. Proc. Steklov Inst. Math., 267 (2009), 76–90. CrossRefGoogle Scholar
Lamb, J.S.W., Sten’kin, O.V.. Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits. Nonlinearity, 17 (2004), 1217–1244. CrossRefGoogle Scholar
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.. 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. CrossRefGoogle Scholar
Gonchenko, S.V., Turaev, D.V., Shilnikov, L.P.. Infinitely many elliptic periodic orbits in four-dimensional symplectic diffeomorphisms with a homoclinic tangency. Proc. Steklov Inst. Math., 244 (2004), 106–131. Google Scholar
Gonchenko, S.V., Meiss, J.D., Ovsyannikov, I.I.. Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation. Regul. Chaotic Dyn., 11 (2006), 191–212. CrossRefGoogle Scholar
Gonchenko, S.V., Shilnikov, L., Turaev, D.. On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors. Regul. Chaotic Dyn., 14 (2009), 137–147. CrossRefGoogle Scholar
Gonchenko, S.V., Ovsyannikov, I.I.. On bifurcations of three-dimensional diffeomorphisms with a non-transversal heteroclinic cycle containing saddle-foci. Rus. J. Nonlinear Dynamics, 6 (2010), 61–77 Google Scholar
Turaev, D.. Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps. Nonlinearity, 16 (2003), 123–135. CrossRefGoogle Scholar
D.Turaev. Maps close to identity and universal maps in the Newhouse domain. preprint arXiv:1009.0858, 2010.
Delshams, A., Gonchenko, S.V., Gonchenko, V.S., Lazaro, J.T., Sten’kin, O.. Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps. Nonlinearity, 26 (2013), 1–33. CrossRefGoogle Scholar
Homoclinic Tangencies, eds. S.V.Gonchenko and L.P.Shilnikov, Regular and Chaotic Dynamics, Moscow-Izhevsk, 2007.
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.
Afraimovich, V.S., Shilnikov, L.P.. On small periodic perturbations of autonomous systems. Soviet Math., Dokl., 15 (1974), 206–211. Google Scholar
Afraimovich, V.S., Shilnikov, L.P.. On some global bifurcations connected with the disappearance of saddle-node fixed point. Soviet. Math., Dokl.. 15 (1974), 1761–1765. Google Scholar
Afraimovich, V.S., Shilnikov, L.P.. The ring principle in problems of interaction between two self-oscillating systems. J. Appl. Math. Mech., 41 (1977), 632–641 CrossRefGoogle Scholar
J.H. Curry, J.A. Yorke. A transition from Hopf bifurcation to chaos: computer experiments with maps in R 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.
Aronson, D.G., Chory, M.A., Mcgehee, R.P., Hall, G.R.. Bifurcation from an invariant circle for two-parameter families of maps of the plane. Commun. Math. Phys., 83 (1982), 303–354. CrossRefGoogle Scholar
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].
Shilnikov, A., Shilnikov, L., Turaev, D.. On some mathematical topics in classical synchronization. Bifurcation and Chaos, 14 (2004), 2143–2160. CrossRefGoogle Scholar
Gonchenko, A.S., Gonchenko, S.V., Shilnikov, L.P.. Towards scenarios of chaos appearance in three-dimensional maps. Rus. J. Nonlinear Dynamics, 8 (2012), 3–28. Google Scholar
Borisov, A.V., Jalnine, A.Yu., Kuznetsov, S.P., Sataev, I.R., Sedova, J.V.. Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback. Regul. Chaotic Dyn., 17 (2012), 512–532. CrossRefGoogle Scholar
Bonatti, C.. C1-generic dynamics: tame and wild behaviour. Proceedings of the ICM, Beijing 2002, V. 3, 279–294. Google Scholar
Bonatti, C., Diaz, L., Pujals, E.R.. A C1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources. Ann. Math., 158 (2003), 355–418. CrossRefGoogle Scholar
C. Bonatti, L. Diaz, M. Viana. Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective. Encyclopaedia of Mathematical Sciences, V. 102, Springer, 2005.
Morales, C., Pujals, E.R.. Singular strange attractors on the boundary of Morse–Smale systems. Ann. Sci. Ecole Norm. Sup., 30 (1997), 693–717. CrossRefGoogle Scholar
V. Araujo, M.J. Pacifico. Three-Dimensional Flows. Springer, 2010.
Luzzatto, S., Melbourne, I., Paccaut, F.. The Lorenz attractor is mixing. Comm. Math. Phys., 260 (2005), 393–401. CrossRefGoogle Scholar
Pesin, Ya. B.. Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergod. Th. Dyn. Sys., 12 (1992), 123–151. Google Scholar
Sataev, E.A.. Invariant measures for hyperbolic maps with singularities. Russ. Math. Surv., 47 (1992), 191–251. CrossRefGoogle Scholar
Afraimovich, V.S., Chernov, N.I., Sataev, E.A.. Statistical properties of 2-D generalized hyperbolic attractors. Chaos 5 (1995), 238–252. CrossRefGoogle Scholar
Bonatti, C., Pumarino, A., Viana, M.. Lorenz attractors with arbitrary expanding dimension. C.R. Acad. Sci. Paris, 325-I (1997), 883–888. CrossRefGoogle Scholar
Auslander, J., Seibert, P.. Prolongations and stability in dynamical systems. Ann. Inst. Forier., Genoble, 14 (1964), 237–268. CrossRefGoogle Scholar
Auslander, J., Bhatia, N.P., Seibert, P.. Attractors in dynamical systems. Bol. Soc. Mat. Mex., 9 (1964), 55–66. Google Scholar
Milnor, J.. On the concept of attractor. Comm. Math. Phys., 99 (1985), 177–195; Correction and remarks. Comm. Math. Phys., 102 (1985), 517–519. CrossRefGoogle Scholar
Gorodetski, A., Ilyashenko, Yu.. Minimal and strange attractors. Bifurcation and Chaos, 6 (1996), 1177–1183. CrossRefGoogle Scholar
Robinson, C.. What is a Chaotic Attractor? Qualitative Theory Dyn. Sys., 7 (2008), 227–236. CrossRefGoogle Scholar
C. Conley.Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series Math., 38 (1978), Am. Math. Soc., Providence RI.
Ruelle, D.. Small random perturbations of dynamical systems and the definition of attractors. Comm. Math. Phys., 82 (1981), 137–151. CrossRefGoogle Scholar
Hurley, M.. Attractors: persistence and density of their basins. Trans. Amer. Math. Soc., 269 (1982), 247–271. CrossRefGoogle Scholar
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).
Sataev, E.A.. Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type. Sb. Math., 196 (2005), 561–594. CrossRefGoogle Scholar
R. Bamon, J. Kiwi, J. Rivera. Wild Lorenz like attractors. Preprint (2005), arXiv:math/0508045.
Diaz, L.J., Rocha, J.. Nonconnected heteroclinic cycles: bifurcation and stability. Nonlinearity, 5 (1992), 1315–1341. CrossRefGoogle Scholar
Diaz, L.J.. Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations. Nonlinearity, 8 (1995), 693–713. CrossRefGoogle Scholar
Bonatti, Ch., Diaz, L.J.. Persistent nonhyperbolic transitive diffeomorphisms. Ann. Math., 143 (1996), 367–396. CrossRefGoogle Scholar
Bonatti, C., Diaz, L.J., Pujals, E., Rocha, J.. Robust transitivity and heterodimensional cycles. Asterisque, 286 (2003), 187–222. Google Scholar
Bonatti, Ch., Diaz, L.J.. Robust heterodimensional cycles and C1-generic dynamics. J. Inst. of Math. Jussieu, 7 (2008), 469–525. CrossRefGoogle Scholar
Diaz, L.J., Gorodetski, A.. Non-hyperbolic invariant measures for non-hyprerbolic homoclinic classes. Ergod. Th. Dyn. Sys., 29 (2009), 479–513. Google Scholar
Silnikov, L.P.. Structure of the neighborhood of a homoclinic tube of an invariant torus. Sov. Math. Dokl., 9 (1968), 624–628. Google Scholar
Afrajmovich, V.S., Shil’nikov, L.P.. On a bifurcation of codimension 1 leading to the appearance of a countable set of tori. Sov. Math. Dokl., 25 (1982), 101–105. Google Scholar
Gorodetski, A., Ilyashenko, Yu.. Some new robust properties of invariant sets and attractors of dynamical systems. Functional Analysis and Applications, 33 (1999), 16–30. Google Scholar
Gorodetski, A., Ilyashenko, Yu.. Some properties of skew products over the horseshoe and solenoid. Proc. Steklov Inst. Math., 231 (2000), 96–118. Google Scholar
Gorodetski, A., Ilyashenko, Yu., Kleptsyn, V., Nalskij, M.. Non-removable zero Lyapunov exponent. Functional Analysis and Applications, 39 (2005), 27–38. Google Scholar
Pisarevsky, V.N., Shilnikov, A., Turaev, D.. Asymptotic normal forms for equilibria with a triplet of zero characteristic exponents in systems with symmetry. Regul. Chaotic Dyn., 2 (1998), 19–27. CrossRefGoogle Scholar
Shilnikov, L.P.. The theory of bifurcations and quasiattractors. Russ. Math. Survey, 36[4] (1981), 240–241. Google Scholar
Robinson, C.. Homoclinic bifurcation to a transitive attractor of the Lorenz type. Nonlinearity, 2 (1989), 495–518; II. SIAM J. Math. Anal., 23 (1992), 1255–1268. CrossRefGoogle Scholar
Rychlic, M.. Lorenz attractors through Shilnikov type bifurcation. Ergod. Th. Dyn. Sys., 10 (1990), 793–821. Google Scholar
Tigan, G., Turaev, D.. Analytical search for homoclinic bifurcations in Morioka-Shimizu model. Physica D, 240 (2011), 985–989. CrossRefGoogle Scholar
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.
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.
Bykov, V.V.. The bifurcations of separatrix contours and chaos. Physica D, 62(1993), 290–299. CrossRefGoogle Scholar
N.V. Petrovskaya, V.I. Yudovich. Homoclinic loops of the Saltzman-Lorenz system. Methods of qualitative theory of differential equations, Gorky, 1980, 73–83.
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].
Afraimovich, V.S., Bykov, V.V., Shilnikov, L.P.. On the existence of stable periodic orbits in the Lorenz model. Russ. Math. Survey, 35[4] (1980), 164–165. Google Scholar
Chenciner, A.. 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. CrossRefGoogle Scholar
Los, J.E.. Nonnormally hyperbolic invariant curves for maps in R3 and doubling bifurcation. Nonlinearity, 2 (1989), 149–174. CrossRefGoogle Scholar
Broer, H.W., Huitema, G.B., Takens, F.. Unfoldings of quasi-periodic tori. Mem. AMS, 83 (1990), 1–82. Google Scholar
Braaksma, B.L.J., Broer, H.W., Huitema, G.B.. Toward a quasi-periodic bifurcation theory. Mem. AMS, 83 (1990), 83-175. Google Scholar
Arneodo, A., Coullet, P.H., Spiegel, E.A.. Cascade of period doublings of tori. Phys. Lett. A, 94 (1983), 1–5. CrossRefGoogle Scholar
Franceschini, V.. Bifurcations of tori and phase locking in a dissipative system of differential equations. Physica D, 6 (1983), 285–304. CrossRefGoogle Scholar
Anishchenko, V.S., Astakhov, V.V., Letchford, T.E., Safonova, M.A.. Structure of the quasihyperbolic stochasticity in an inertial self-excited oscillator. Radiophysics and Quantum Electronics, 26 (1983), 619–628. CrossRefGoogle Scholar
V.S. Anishenko. Complex oscillations in simple systems. Nauka, Moscow, 1990.
K. Kaneko. Collapse of Tori and Genesis of Chaos in Dissipative Systems. World Scientific, Singapore, 1986.
C. Bonatti, Y. Shi, Robustly transitive attractor derived from Lorenz, in preparation.