Dynamics of a stochastic system driven by cross-correlated sine-Wiener bounded noises


The sine-Wiener noise, as one new type of bounded noise and a natural tool to model fluctuations in dynamical systems, has been applied to problems in a variety of areas, especially in biomolecular networks and neural models. In this paper, by virtue of the Novikov theorem, Fox’s approach, and the ansatz of Hanggi, an approximate Fokker–Planck equation is derived for an one-dimensional Langevin-type equation with cross-correlated sine-Wiener noise. Meanwhile, the dynamical characters of a bistable system driven by cross-correlated sine-Wiener noise are investigated by applying the approximate theoretical method. For the bistable system, the cross-correlation intensity \(\lambda \) can induce the reentrance-like phase transition, while the other noise intensities and the self-correlation time, except for the self-correlation time of additive bounded noise, can induce the first-order-like phase transition. The transition from the stable state to another one can be accelerated by \(\alpha \) (additive bounded noise intensity), \(\tau _1\) (the self-correlation time of the multiplicative bounded noise), and \(\tau _2\) (the self-correlation time of the additive bounded noise) and can be restrained with \(\lambda \) and \(\tau _3\) (self-correlation time of the cross-correlation bounded noise). It is interesting that the noise-enhanced stability phenomenon is observed with D (multiplicative bounded noise intensity) varying for the positive correlation (\(\lambda >0\)) and is enhanced as \(\lambda \) increases. The numerical results are in basic agreement with the theoretical predictions.

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This work is supported by the National Natural Science Foundation of China ( Grant No. 11205006), the Natural Science New Star of Science and Technologies Research Plan in Shaanxi Province of China (Grant No. 2014KJXX-77), the Science Foundation of the Education Bureau of Shaanxi Province, China (Grant No. 15JK1045), and the Natural Science Foundation of Shaanxi Province, China (Grant No. 2018JM1034).

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Correspondence to Can-Jun Wang.

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Wang, C., Lin, Q., Yao, Y. et al. Dynamics of a stochastic system driven by cross-correlated sine-Wiener bounded noises. Nonlinear Dyn 95, 1941–1956 (2019). https://doi.org/10.1007/s11071-018-4669-0

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  • Cross-correlated sine-Wiener bounded noises
  • Fokker–Planck equation
  • Bistable dynamical system