加性和乘性三值噪声激励下周期势系统的动力学分析

DYNAMICAL ANALYSIS OF A PERIODIC POTENTIAL DRIVEN BY ADDITIVE AND MULTIPLICATIVE TRICHOTOMOUS NOISES

周期势系统是一类在机械工程、物理、化学、神经生物等领域应用十分广泛的系统,其随机动力学特性的研究是非线性科学的一个热点和难点问题.三值噪声是真实噪声的典型模型,不仅包含二值噪声和高斯白噪声情形,而且能更好地描述自然界中随机环境扰动的多样性,本文研究了由加性和乘性三值噪声驱动的周期势系统中概率密度的演化和随机共振.通过计算系统的平均稳态联合概率密度函数和瞬态联合概率密度函数,发现随着外周期力振幅的增大,单自由度系统在多个稳态之间跃迁,其平均稳态联合概率密度具有多峰结构.此外,利用随机能量法揭示了系统的随机共振,发现存在最优的噪声强度和外周期力振幅使得平均输入能量曲线存在一个极大值,即出现随机共振现象.对于仅考虑加性噪声或乘性噪声激励的情况,平均输入能量曲线随噪声转迁率是否出现共振现象依赖于外周期激励振幅的大小.特别是仅考虑加性噪声的情形,对于较小的外周期激励振幅,加性噪声转迁率诱导产生抑制共振现象,而对于较大的外周期激励振幅,加性噪声转迁率诱导产生随机共振现象.

The periodic potentials have been widely applied in the fields of engineering, physics, chemistry and neurobiology, whose stochastic dynamics has become the focus of nonlinear science. Trichotomous noise is a kind of three-level Markovian noise and can converge to dichotomous noise or Gaussian white noise under some limits. The trichotomous noise is a better representation of real noise than the widely used Gaussian white noise to display the diversity of environmental excitation. This paper studies the evolutions of probability density function(PDF) and stochastic resonance(SR) in a periodic potential driven by additive and multiplicative trichotomous noise. The average stationary joint PDF and transient joint PDF are obtained by numerical simulation. It is found that the shape of the stationary PDF has the multi-modal structure as the amplitude of the periodic force increases, which indicates the noise-induced hopping among the potential wells. Furthermore, the stochastic energy method is used to explore the SR phenomenon. The obtained results show that the average input energy curve attains a maximum at an optimal noise intensity and amplitude of periodic force. That is, the SR happens. Moreover, the SR happens at an optimal transition rate of noise and the proper amplitude of periodic force under the excitation of only multiplicative or additive noise. Especially, for the case of additive noise,transition rate of additive noise can induce the suppression of SR for a small amplitude of periodic force. While the SR appears at a proper transition rate of additive noise for a large amplitude of periodic force.