From attractor to chaotic saddle: a tale of transverse instability

Suppose that a dynamical system possesses an invariant submanifold, and the restriction of the system to this submanifold has a chaotic attractor A. Under which conditions is A an attractor for the original system, and in what sense? We characterize the transverse dynamics near A in terms of the normal Liapunov spectrum of A. In particular, we emphasize the role of invariant measures on A. Our results identify the points at which A: (1) ceases to be asymptotically stable, possibly developing a locally riddled basin; (2) ceases to be an attractor; (3) becomes a transversely repelling chaotic saddle. We show, in the context of what we call `normal parameters' how these transitions can be viewed as being robust. Finally, we discuss some numerical examples displaying these transitions.

[1]  C. Caramanis.  What is ergodic theory , 1963 .

[2]  V. I. Oseledec.  A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[3]  Karl Sigmund,et al.  Generic properties of invariant measures for AxiomA-diffeomorphisms , 1970 .

[4]  J. Yorke,et al.  On the existence of invariant measures for piecewise monotonic transformations , 1973 .

[5]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[6]  David Ruelle,et al.  A MEASURE ASSOCIATED WITH AXIOM-A ATTRACTORS. , 1976 .

[7]  P. Walters.  Introduction to Ergodic Theory , 1977 .

[8]  Y. Pesin.  CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY , 1977 .

[9]  D. Ruelle,et al.  Analycity properties of the characteristic exponents of random matrix products , 1979 .

[10]  M. Jakobson.  Absolutely continuous invariant measures for one-parameter families of one-dimensional maps , 1981 .

[11]  Michał Misiurewicz,et al.  Absolutely continuous measures for certain maps of an interval , 1981 .

[12]  F. Ledrappier,et al.  A proof of the estimation from below in Pesin's entropy formula , 1982, Ergodic Theory and Dynamical Systems.

[13]  J. Palis,et al.  Geometric theory of dynamical systems : an introduction , 1984 .

[14]  John Milnor,et al.  On the concept of attractor: Correction and remarks , 1985 .

[15]  Monopole charges for arbitrary compact gauge groups and Higgs fields in any representation , 1985 .

[16]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[17]  L. Young.  Bowen-Ruelle measures for certain piecewise hyperbolic maps , 1985 .

[18]  J. Milnor.  On the concept of attractor , 1985 .

[19]  M. Shub,et al.  Expanding endomorphisms of the circle revisited , 1985, Ergodic Theory and Dynamical Systems.

[20]  Lennart Carleson,et al.  On Iterations of 1 - ax 2 on (- 1,1) , 1985 .

[21]  R. Mañé,et al.  Ergodic Theory and Differentiable Dynamics , 1986 .

[22]  Martin Golubitsky,et al.  Symmetry-increasing bifurcation of chaotic attractors , 1988 .

[23]  D. Ruelle.  Chaotic evolution and strange attractors , 1989 .

[24]  P. Grassberger,et al.  Symmetry breaking bifurcation for coupled chaotic attractors , 1991 .

[25]  James A. Yorke,et al.  Analysis of a procedure for finding numerical trajectories close to chaotic saddle hyperbolic sets , 1991, Ergodic Theory and Dynamical Systems.

[26]  Leopold Flatto,et al.  Geodesic flows, interval maps, and symbolic dynamics , 1991 .

[27]  Ian Melbourne.  An example of a non-asymptotically stable attractor , 1991 .

[28]  Ian Melbourne,et al.  An example of a nonasymptotically stable attractor , 1991 .

[29]  S. Ciliberto,et al.  Thermodynamic aspects of the transition to spatiotemporal chaos , 1992 .

[30]  Leon O. Chua,et al.  TRANSITIONS IN DYNAMICAL REGIMES BY DRIVING: A UNIFIED METHOD OF CONTROL AND SYNCHRONIZATION OF CHAOS , 1993 .

[31]  Lai-Sang Young,et al.  Sinai-Bowen-Ruelle measures for certain Hénon maps , 1993 .

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

[33]  Spiegel,et al.  On-off intermittency: A mechanism for bursting. , 1993, Physical review letters.

[34]  W. D. Melo,et al.  ONE-DIMENSIONAL DYNAMICS , 2013 .

[35]  M. Golubitsky,et al.  The structure of symmetric attractors , 1993 .

[36]  Ott,et al.  Scaling behavior of chaotic systems with riddled basins. , 1993, Physical review letters.

[37]  Michael Dellnitz,et al.  Symmetry of attractors and the Karhunen-Loegve decomposition , 1994 .

[38]  P. Ashwin,et al.  Symmetry groups of attractors , 1994 .

[39]  J. Yorke,et al.  The transition to chaotic attractors with riddled basins , 1994 .

[40]  I. Kan.  Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin , 1994, math/9407218.

[41]  Lee,et al.  Experimental observation of on-off intermittency. , 1994, Physical review letters.

[42]  Platt,et al.  Effects of additive noise on on-off intermittency. , 1994, Physical review letters.

[43]  H. B. Wilson,et al.  Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics. , 1994, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[44]  I. Stewart,et al.  Bubbling of attractors and synchronisation of chaotic oscillators , 1994 .

[45]  E. Ott,et al.  Blowout bifurcations: the occurrence of riddled basins and on-off intermittency , 1994 .

[46]  I. Stewart,et al.  Liapunov stability and adding machines , 1995, Ergodic Theory and Dynamical Systems.

[47]  Michael Dellnitz,et al.  SYMMETRY BREAKING BIFURCATIONS OF CHAOTIC ATTRACTORS , 1995 .

[48]  Peter Ashwin,et al.  Attractors stuck on to invariant subspaces , 1995 .

[49]  Lai-Sang Young,et al.  Ergodic Theory of Differentiable Dynamical Systems , 1995 .