CHAOS AND CRISES IN MORE THAN TWO DIMENSIONS

Instituto de Astronomi´ayFi´sica del Espacio (CONICET), Casilla de Correo 67, Socursal 28,1428 Buenos Aires, Argentina~Received 23 December 1996!Noisy chaotic trajectories, with finite-time Lyapunov exponents that fluctuate about zero, are basicallyunshadowable @S. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett 73, 1927 ~1994!#. This canoccur when periodic orbits, with different numbers of unstable directions, coexist inside the attractor. Thepresence of a He´non-type chaotic saddle~i.e., a nonattracting chaotic set with a structure similar to that of theHe´non attractor! guarantees such coexistence in a persistent manner @S. P. Dawson, Phys. Rev. Lett. 76, 4348~1996!#. In this paper, we describe how these sets appear naturally in maps of more than two dimensions, howthey can be found, and what crises they produce. @S1063-651X~97!13005-7#PACS number~s!: 05.45.1b, 05.40.1jI. INTRODUCTION

[1]  Grebogi,et al.  Obstructions to shadowing when a Lyapunov exponent fluctuates about zero. , 1994, Physical review letters.

[2]  J. Yorke,et al.  A numerical procedure for finding accessible trajectories on basin boundaries , 1991 .

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

[4]  Ott,et al.  Fractal dimensions of chaotic saddles of dynamical systems. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  R. Devaney.  An Introduction to Chaotic Dynamical Systems , 1990 .

[6]  Grebogi,et al.  Unstable periodic orbits and the dimension of chaotic attractors. , 1987, Physical review. A, General physics.

[7]  Gabriel B. Mindlin,et al.  Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction , 1991 .

[8]  Grebogi,et al.  Critical exponents for crisis-induced intermittency. , 1987, Physical review. A, General physics.

[9]  James A. Yorke,et al.  A procedure for finding numerical trajectories on chaotic saddles , 1989 .

[10]  Tél,et al.  Exploring transient chaos in an NMR-laser experiment. , 1994, Physical review letters.

[11]  Celso Grebogi,et al.  Multi-dimensioned intertwined basin boundaries: Basin structure of the kicked double rotor , 1987 .

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

[13]  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.

[14]  Ditto,et al.  Experimental control of chaos. , 1990, Physical review letters.

[15]  Intrinsic fluctuations in chaotic dynamics , 1992 .

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

[17]  E. Kostelich,et al.  Characterization of an experimental strange attractor by periodic orbits. , 1989, Physical review. A, General physics.

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

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

[20]  Grebogi,et al.  Metamorphoses of basin boundaries in nonlinear dynamical systems. , 1986, Physical review letters.

[21]  Tél,et al.  Time-series analysis of transient chaos. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  S. Smale.  Differentiable dynamical systems , 1967 .

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

[24]  Wu,et al.  Internal fluctuations and deterministic chemical chaos. , 1993, Physical review letters.

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

[26]  Dawson.  Strange nonattracting chaotic sets, crises, and fluctuating Lyapunov exponents. , 1996, Physical review letters.

[27]  A. Avez,et al.  Ergodic theory of dynamical systems , 1966 .

[28]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[29]  T. Tél,et al.  Transient chaos as the backbone of dynamics on strange attractors beyond crisis , 1994 .