Effect of noise on the neutral direction of chaotic attractor

Chaos

Volumn 14, Issue 1, 2004, Pages 189-192

  Lai, Ying Cheng ^a   Liu, Zonghua ^a  

^a ARIZONA STATE UNIVERSITY   (United States)

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EID: 1842430976     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1637735     Document Type: Article

References (22)

1. Some pioneering works in this direction are the following. The effect of noise on period-doubling transition to chaos was studied by Crutchfield et al. (Refs. 2 and 3), where a renormalization-group approach was used to analyze the scaling behavior of the Lyapunov exponent near the transition (Ref. 3). The effect of noise on type-I intermittency was investigated by Hirsch et al. (Ref. 4). The influence of noise on periodic attractors for the Lorenz system was studied by Fedchenia et al (Ref. 5). Noise-induced chaos in a system with homoclinic points was discussed by Anishchenko and Herzel (Ref. 6) and the opposite phenomenon of noise stabilization of chaotic dynamics was studied by Herzel (Ref. 7). The problem of noise-induced chaos also has similarities with the problem of noise activation of excitable systems (Ref. 8). Transition to noisy chaos for dynamical systems in periodic windows has recently been investigated (Ref. 9), which is relevant to problems in, for instance, laser physics (Ref. 10) and biology (Ref. 11).
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11. Z. Liu, Y.-C. Lai, L. Billings, and I. B. Schwartz, Phys. Rev. Lett. 88, 124101 (2002); Y.-C. Lai, Z. Liu, L. Billings, and I. B. Schwartz, Phys. Rev. E 67, 026210 (2003); B. Xu, Y.-C. Lai, L. Zhu, and Y. Do, Phys. Rev. Lett. 90, 164101 (2003).
12. Z. Liu, Y.-C. Lai, L. Billings, and I. B. Schwartz, Phys. Rev. Lett. 88, 124101 (2002); Y.-C. Lai, Z. Liu, L. Billings, and I. B. Schwartz, Phys. Rev. E 67, 026210 (2003); B. Xu, Y.-C. Lai, L. Zhu, and Y. Do, Phys. Rev. Lett. 90, 164101 (2003).
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18. The mechanism responsible for the scaling laws described here can be expected in higher dimensions as well. For instance, for chaotic attractors with multiple scrolls in phase spaces of dimensions more than three, we expect its neutral direction to be destroyed by small noise and the same scaling laws to hold. The scaling exponent α, however, will be different as it is determined by the dynamical property of an unstable steady state or a periodic orbit in the switching region.
19. A similar scaling law has been observed in coupled chaotic oscillators, which characterizes the variation of the Lyapunov exponent with the coupling strength [Z. Liu, Y.-C. Lai, and M. Matias, Phys. Rev. E 67, R045203 (2003)].
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21. For review and recent advances on phase synchronization, see, for example, S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, Phys. Rep. 366, 1 (2002); J. Kurths, S. Boccaletti, C. Grebogi, and Y.-C. Lai, Chaos 13, 126 (2003).
22. For review and recent advances on phase synchronization, see, for example, S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, Phys. Rep. 366, 1 (2002); J. Kurths, S. Boccaletti, C. Grebogi, and Y.-C. Lai, Chaos 13, 126 (2003).