### Abstract

Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapunov exponent becomes positive from zero in a continuous manner, and its scaling with the variation of the noise amplitude is algebraic. Formulas for the scaling exponent are derived in all dimensions. Numerical support using both low- and high-dimensional systems is provided.

Original language | English |
---|---|

Article number | 026210 |

Pages (from-to) | 262101-2621017 |

Number of pages | 1 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 67 |

Issue number | 2 |

DOIs | |

State | Published - 1 Feb 2003 |

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*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*67*(2), 262101-2621017. [026210]. https://doi.org/10.1103/PhysRevE.67.026210

}

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 67, no. 2, 026210, pp. 262101-2621017. https://doi.org/10.1103/PhysRevE.67.026210

**Noise-induced unstable dimension variability and transition to chaos in random dynamical systems.** / Lai, Ying Cheng; Liu, Zonghua; Billings, Lora; Schwartz, Ira B.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Noise-induced unstable dimension variability and transition to chaos in random dynamical systems

AU - Lai, Ying Cheng

AU - Liu, Zonghua

AU - Billings, Lora

AU - Schwartz, Ira B.

PY - 2003/2/1

Y1 - 2003/2/1

N2 - Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapunov exponent becomes positive from zero in a continuous manner, and its scaling with the variation of the noise amplitude is algebraic. Formulas for the scaling exponent are derived in all dimensions. Numerical support using both low- and high-dimensional systems is provided.

AB - Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapunov exponent becomes positive from zero in a continuous manner, and its scaling with the variation of the noise amplitude is algebraic. Formulas for the scaling exponent are derived in all dimensions. Numerical support using both low- and high-dimensional systems is provided.

UR - http://www.scopus.com/inward/record.url?scp=85037239893&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.67.026210

DO - 10.1103/PhysRevE.67.026210

M3 - Article

AN - SCOPUS:0038057925

VL - 67

SP - 262101

EP - 2621017

JO - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

JF - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

SN - 1063-651X

IS - 2

M1 - 026210

ER -