Generation of stochastic mixed-mode oscillations in a pair of VDP oscillators with direct-indirect coupling

Xiaojun Huang; Zigen Song; Xiaojun Huang; Zigen Song

doi:10.3934/mbe.2024032

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 1: 765-777. doi: 10.3934/mbe.2024032

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Research article

Generation of stochastic mixed-mode oscillations in a pair of VDP oscillators with direct-indirect coupling

Xiaojun Huang ,
Zigen Song ^,

School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China

Academic Editor: Yang Kuang

Received: 06 September 2023 Revised: 26 November 2023 Accepted: 07 December 2023 Published: 19 December 2023

Environmental noise can lead to complex stochastic dynamical behavior in nonlinear systems. In this paper, we studied the phenomenon of a pair of Van der Pol (VDP) oscillators with direct-indirect coupling affected by Gaussian white noise. That is to say, a noise-induced equilibrium transition oscillation was observed in three types of different parameter regions, where the deterministic system had two kinds of stable equilibrium points. Meanwhile, with the noise intensity increasing, we found that the stochastic system will constantly switch between two stable equilibrium points. To analyze the stochastic behavior, we used the stochastic sensitivity equation and confidence ellipse method. When the confidence ellipsoid crossed the boundary of the attraction basin of the equilibrium point, the system entered into the state of stochastic mixed-mode oscillations, which was consistent with the simulation results.
- VDP oscillation,
- direct-indirect coupling,
- stochastic dynamic,
- mixed-mode oscillation,
- confidence ellipse method,
- basins of attraction
Citation: Xiaojun Huang, Zigen Song. Generation of stochastic mixed-mode oscillations in a pair of VDP oscillators with direct-indirect coupling[J]. Mathematical Biosciences and Engineering, 2024, 21(1): 765-777. doi: 10.3934/mbe.2024032

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Abstract

Environmental noise can lead to complex stochastic dynamical behavior in nonlinear systems. In this paper, we studied the phenomenon of a pair of Van der Pol (VDP) oscillators with direct-indirect coupling affected by Gaussian white noise. That is to say, a noise-induced equilibrium transition oscillation was observed in three types of different parameter regions, where the deterministic system had two kinds of stable equilibrium points. Meanwhile, with the noise intensity increasing, we found that the stochastic system will constantly switch between two stable equilibrium points. To analyze the stochastic behavior, we used the stochastic sensitivity equation and confidence ellipse method. When the confidence ellipsoid crossed the boundary of the attraction basin of the equilibrium point, the system entered into the state of stochastic mixed-mode oscillations, which was consistent with the simulation results.

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