### 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 |

Number of pages | 17 |

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

Volume | 67 |

Issue number | 2 |

DOIs | |

Publication status | Published - 19 Feb 2003 |

### Keywords

- on-off intermittency
- generalized dimensions
- coherence resonance
- strange attractors
- stochastic chaos
- fractal measures
- synchronization
- brain
- sets
- oscillators

### Cite this

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

*67*(2), [026210]. 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, Z H ; Billings, L ; Schwartz, I B .

Research output: Contribution to journal › Article

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

}

TY - JOUR

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

AU - Lai, Ying-Cheng

AU - Liu, Z H

AU - Billings, L

AU - Schwartz, I B

PY - 2003/2/19

Y1 - 2003/2/19

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.

KW - on-off intermittency

KW - generalized dimensions

KW - coherence resonance

KW - strange attractors

KW - stochastic chaos

KW - fractal measures

KW - synchronization

KW - brain

KW - sets

KW - oscillators

U2 - 10.1103/PhysRevE.67.026210

DO - 10.1103/PhysRevE.67.026210

M3 - Article

VL - 67

JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

SN - 1539-3755

IS - 2

M1 - 026210

ER -