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Noise-induced chaotic-attractor escape route

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Abstract

In the present article, a combination of numerical and experimental studies is undertaken to comprehend the influence of noise on the responses of continuous-time dynamical systems. In particular, the influence of white Gaussian noise on the chaotic and periodic responses of bistable, Duffing oscillators is the focus of this work. The noteworthy result of the conducted studies concerns the presence of a pair of attractors, one being periodic and the other being chaotic: the chaotic attractor response can be controlled and terminated with an appropriate noise level. For trajectories in the basin of the chaotic attractor, white Gaussian noise is added at a barely sufficient level to allow trajectories to eventually leave (within some specified time). The authors report that trajectories leave via a special escape route: the unstable manifold of a fixed point saddle on the basin boundary between the two basins of attraction. Striking similarities and differences between experimental and numerical investigation are discussed in the work.

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Abbreviations

x :

Nondimensional oscillator displacement

\(x_{1}\) :

Nondimensional oscillator position in state space

\(x_{2}\) :

Nondimensional oscillator velocity in state space

\(\eta \) :

Nondimensional damping factor

\(\alpha \) :

Nondimensional linear stiffness

\(\sigma _E\) :

Amplitude of noise for experimental studies

\(\varOmega \) :

Nondimensional forcing frequency

\(F_0\) :

Nondimensional forcing amplitude

\({\dot{W}}(t)\) :

White Gaussian noise

\(\sigma _N\) :

Amplitude of noise for numerical simulations

\(\beta \) :

Nondimensional nonlinear stiffness

\(\omega _n\) :

Linear natural frequency

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Acknowledgements

Support received for this work through NSF Grant Nos. CMMI1436141 and CMMI 1760366 are gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Maryland, College Park, MD, 20742, USA

    Vipin Agarwal & Balakumar Balachandran

  2. Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    James A. Yorke

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  1. Vipin Agarwal
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  2. James A. Yorke
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  3. Balakumar Balachandran
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Correspondence to Balakumar Balachandran.

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Agarwal, V., Yorke, J.A. & Balachandran, B. Noise-induced chaotic-attractor escape route. Nonlinear Dyn 102, 863–876 (2020). https://doi.org/10.1007/s11071-020-05873-3

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