Trying to Characterize Robust and Generic Dynamics

If we consider that the mathematical formulation of natural phenomena always involves simplifications of the physical laws, real significance of a model may be accorded only to those properties that are robust under perturbations. In loose terms, robustness means that some main features of a dynamical system are shared by all nearby systems. In this short article, we will explain the structure related to the presence of robust transitivity and the universal mechanisms that lead to lack of robustness. Providing a conceptual framework, the goal is to show how this approach helps to describe ‘generic’ dynamics in the space of all dynamical systems.

[1]  An extension of the Artin-Mazur theorem , 1999, math/9909196.

[2]  S. Newhouse,et al.  Diffeomorphisms with infinitely many sinks , 1974 .

[3]  Jun Yan,et al.  Existence of Diffusion Orbits in a priori Unstable Hamiltonian Systems , 2004 .

[4]  Christian Bonatti,et al.  Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective , 2004 .

[5]  J. Bochi.  Genericity of zero Lyapunov exponents , 2002, Ergodic Theory and Dynamical Systems.

[6]  Robert S. MacKay,et al.  Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor , 2003 .

[7]  D. V. Anosov,et al.  Geodesic flows on closed Riemann manifolds with negative curvature , 1969 .

[8]  Maria José Pacifico,et al.  Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers , 2004 .

[9]  José F. Alves,et al.  SRB measures for partially hyperbolic systems whose central direction is mostly expanding , 2000, 1403.2937.

[10]  R. Douady.  Stabilité ou instabilité des points fixes elliptiques , 1988 .

[11]  G. Gallavotti.  New Methods in Nonequilibrium Gases and Fluids , 1996, chao-dyn/9610018.

[12]  Generic robustness of spectral decompositions , 2003 .

[13]  R. Mañé,et al.  An Ergodic Closing Lemma , 1982 .

[14]  J. Palis,et al.  Homoclinic tangencies and fractal invariants in arbitrary dimension , 2001 .

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

[16]  W. G. Hoover.  molecular dynamics , 1986, Catalysis from A to Z.

[17]  R. Mañé.  Contributions to the stability conjecture , 1978 .

[18]  SRB States and Nonequilibrium Statistical Mechanics Close to Equilibrium , 1996, chao-dyn/9612002.

[19]  J. Bochi,et al.  The Lyapunov exponents of generic volume-preserving and symplectic maps , 2005 .

[20]  S. D. Chatterji.  Proceedings of the International Congress of Mathematicians , 1995 .

[21]  R. Mañé,et al.  Hyperbolicity, sinks and measure in one dimensional dynamics , 1985 .

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

[23]  Christian Bonatti,et al.  Connexions hétéroclines et généricité d'une infinité de puits et de sources , 1999 .

[24]  Enrique R. Pujals,et al.  On the dynamics of dominated splitting , 2009 .

[25]  Lorenzo J. Díaz,et al.  Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations , 1995 .

[26]  E. Valdinoci,et al.  Instability of resonant totally elliptic points of symplectic maps in dimension 4 , 2004 .

[27]  C. Bonatti,et al.  SRB measures for partially hyperbolic systems whose central direction is mostly contracting , 2000 .

[28]  Enrique R. Pujals,et al.  Dynamical properties of singular-hyperbolic attractors , 2007 .

[29]  T. M. Seara,et al.  A Geometric Mechanism for Diffusion in Hamiltonian Systems Overcoming the Large Gap Problem: Heuristics And Rigorous Verification on a Model , 2005 .

[30]  Enrique R. Pujals,et al.  A C^1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources , 2003 .

[31]  D. Ruelle.  Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics , 1998, chao-dyn/9812032.

[32]  R. F. Williams,et al.  Structural stability of Lorenz attractors , 1979 .

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

[34]  L. Díaz.  Robust nonhyperbolic dynamics and heterodimensional cycles , 1995, Ergodic Theory and Dynamical Systems.

[35]  Homoclinic tangencies and hyperbolicity for surface diffeomorphisms , 2000, math/0005303.

[36]  Weyl manifolds and Gaussian thermostats , 2003, math/0304461.

[37]  Zhihong Xia.  Arnold diffusion: a variational construction. , 1998 .

[38]  A. Arbieto,et al.  A pasting lemma and some applications for conservative systems , 2006, Ergodic Theory and Dynamical Systems.

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

[40]  A remark on conservative diffeomorphisms , 2004, math/0408344.

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

[42]  L. Díaz,et al.  Partial hyperbolicity and robust transitivity , 1999 .

[43]  V. Arnold.  SMALL DENOMINATORS AND PROBLEMS OF STABILITY OF MOTION IN CLASSICAL AND CELESTIAL MECHANICS , 1963 .