Elements of Contemporary Theory of Dynamical Chaos: A Tutorial. Part I. Pseudohyperbolic Attractors

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth...

[1]  I. I. Ovsyannikov,et al.  CHAOTIC DYNAMICS OF THREE-DIMENSIONAL H ENON MAPS THAT ORIGINATE FROM A HOMOCLINIC BIFURCATION , 2005, nlin/0510061.

[2]  C. Conley Isolated Invariant Sets and the Morse Index , 1978 .

[3]  L. Shilnikov,et al.  NORMAL FORMS AND LORENZ ATTRACTORS , 1993 .

[4]  S.V.Gonchenko,et al.  Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points , 2014, 1412.0738.

[5]  L. Shilnikov,et al.  On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: I , 2008 .

[6]  S. Gonchenko,et al.  On Global Bifurcations of Three-dimensional Diffeomorphisms Leading to Lorenz-like Attractors , 2013 .

[7]  L. Chua,et al.  The double scroll family , 1986 .

[8]  S. Kuznetsov,et al.  Example of a physical system with a hyperbolic attractor of the Smale-Williams type. , 2005, Physical review letters.

[9]  I. R. Sataev,et al.  The reversal and chaotic attractor in the nonholonomic model of Chaplygin’s top , 2014 .

[10]  L. Shilnikov,et al.  Pseudohyperbolicity and the problem on periodic perturbations of Lorenz-type attractors , 2008 .

[11]  Sergey P. Kuznetsov,et al.  Autonomous coupled oscillators with hyperbolic strange attractors , 2007 .

[12]  I. R. Sataev,et al.  Spiral chaos in the nonholonomic model of a Chaplygin top , 2016 .

[13]  I. I. Ovsyannikov,et al.  Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model , 2015 .

[14]  Dmitry Turaev,et al.  An example of a wild strange attractor , 1998 .

[15]  C. Simó,et al.  Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight , 2013 .

[16]  L. Shilnikov,et al.  Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps , 2007 .

[17]  S. Gonchenko,et al.  Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps , 2015, 1510.02252.

[18]  A. Kazakov,et al.  Richness of chaotic dynamics in nonholonomic models of a celtic stone , 2013, Regular and Chaotic Dynamics.

[19]  Dmitry Turaev,et al.  Three-Dimensional HÉnon-like Maps and Wild Lorenz-like attractors , 2005, Int. J. Bifurc. Chaos.

[20]  Dmitry Turaev,et al.  On models with non-rough Poincare´ homoclinic curves , 1993 .

[21]  M. Hénon,et al.  A two-dimensional mapping with a strange attractor , 1976 .

[22]  Alain Arneodo,et al.  Oscillators with chaotic behavior: An illustration of a theorem by Shil'nikov , 1982 .

[23]  S. Newhouse,et al.  The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms , 1979 .

[24]  O. Rössler An equation for continuous chaos , 1976 .

[25]  Sergey P. Kuznetsov,et al.  A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system , 2006 .

[26]  V. S. Gonchenko,et al.  Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps , 2007 .

[27]  Armin Schmidt,et al.  Zur Umsetzung von Trichloracetimidsäuremethylester mit Antimon(V)-chlorid. , 1976 .

[28]  D. Turaev,et al.  On three types of dynamics and the notion of attractor , 2017, Proceedings of the Steklov Institute of Mathematics.

[29]  L. Shilnikov CHUA’S CIRCUIT: RIGOROUS RESULTS AND FUTURE PROBLEMS , 1994 .

[30]  N. Romero Persistence of homoclinic tangencies in higher dimensions , 1995, Ergodic Theory and Dynamical Systems.

[31]  Dmitry Turaev,et al.  Quasiattractors and Homoclinic Tangencies , 1997 .

[32]  Alain Arneodo,et al.  Possible new strange attractors with spiral structure , 1981 .

[33]  L. P. Šil'nikov,et al.  ON THREE-DIMENSIONAL DYNAMICAL SYSTEMS CLOSE TO SYSTEMS WITH A STRUCTURALLY UNSTABLE HOMOCLINIC CURVE. II , 1972 .

[34]  Leonid P Shilnikov Mathematical Problems of Nonlinear Dynamics: A Tutorial , 1997 .

[35]  R. Macarthur,et al.  Graphical Representation and Stability Conditions of Predator-Prey Interactions , 1963, The American Naturalist.

[36]  Ale Jan Homburg,et al.  Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbits to saddle-focus equilibria , 2002 .

[37]  D. Turaev,et al.  Examples of Lorenz-like Attractors in Hénon-like Maps , 2013 .

[38]  Dmitry Turaev,et al.  Simple Scenarios of Onset of Chaos in Three-Dimensional Maps , 2014, Int. J. Bifurc. Chaos.

[39]  Andrey Shilnikov,et al.  On bifurcations of the Lorenz attractor in the Shimizu-Morioka model , 1993 .

[40]  W. Tucker The Lorenz attractor exists , 1999 .

[41]  Marcelo Viana,et al.  Abundance of strange attractors , 1993 .

[42]  L. Shilnikov,et al.  Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits. , 1996, Chaos.

[43]  D. Turaev,et al.  Homoclinic Tangencies of an Arbitrary Order in Newhouse Domains , 2001 .

[44]  S. V. Gonchenko Stable periodic motions in systems close to a structurally unstable homoclinic curve , 1983 .

[45]  D. Ruelle Small random perturbations of dynamical systems and the definition of attractors , 1981 .

[46]  Dmitry Turaev,et al.  On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors , 2009 .

[47]  S. Gonchenko,et al.  Criteria on existence of horseshoes near homoclinic tangencies of arbitrary orders , 2018 .

[48]  Yuri A. Kuznetsov,et al.  Belyakov Homoclinic Bifurcations in a Tritrophic Food Chain Model , 2001, SIAM J. Appl. Math..

[49]  J. C. Tatjer Three-dimensional dissipative diffeomorphisms with homoclinic tangencies , 2001, Ergodic Theory and Dynamical Systems.

[50]  L. Shilnikov,et al.  Simple Bifurcations Leading to Hyperbolic Attractors , 1997 .

[51]  Alain Arneodo,et al.  Occurence of strange attractors in three-dimensional Volterra equations , 1980 .

[52]  Eduardo Colli Infinitely many coexisting strange attractors , 1998 .

[53]  J. Palis,et al.  High dimension diffeomorphisms displaying infinitely many periodic attractors , 1994 .

[54]  Lennart Carleson,et al.  The Dynamics of the Henon Map , 1991 .