王媛, 王慧男, 盛正大, 王芳. 色噪声激励下含分数阶时滞项的广义Van der Pol系统的随机分岔[J]. 内江师范学院学报, 2023, 38(12): 48-54. DOI:10.13603/j.cnki.51-1621/z.2023.12.009
引用本文: 王媛, 王慧男, 盛正大, 王芳. 色噪声激励下含分数阶时滞项的广义Van der Pol系统的随机分岔[J]. 内江师范学院学报, 2023, 38(12): 48-54.DOI:10.13603/j.cnki.51-1621/z.2023.12.009
WANG Yuan, WANG Huinan, SHENG Zhengda, WANG Fang. Stochastic bifurcation of generalized Van der Pol systems with fractional order delay under colored noise excitation[J]. Journal of Neijiang Normal University, 2023, 38(12): 48-54. DOI:10.13603/j.cnki.51-1621/z.2023.12.009
Citation: WANG Yuan, WANG Huinan, SHENG Zhengda, WANG Fang. Stochastic bifurcation of generalized Van der Pol systems with fractional order delay under colored noise excitation[J].Journal of Neijiang Normal University, 2023, 38(12): 48-54.DOI:10.13603/j.cnki.51-1621/z.2023.12.009

色噪声激励下含分数阶时滞项的广义Van der Pol系统的随机分岔

Stochastic bifurcation of generalized Van der Pol systems with fractional order delay under colored noise excitation

  • 摘要:研究了在色噪声激励下含分数阶时滞项的广义Van der Pol系统的随机分岔问题.首先将其中的分数阶导数项转换为一种回复力和阻尼力的线性组合,利用均方误差最小原则,通过计算与原系统的误差,可将原分数阶系统转换为整数阶系统.其次引入广义Van der Pol变换,根据随机平均法求得系统的伊藤随机微分方程和Fokker-Planck-Kolmogorov(FPK)方程.最后通过系统的稳态概率密度函数图验证了分数阶导数阶数 p、非线性阻尼系数 μ等参数的改变可以诱发系统的随机 P-分岔.

    Abstract:The stochastic bifurcation problem of a generalized Van der Pol system with fractional order delay excited by colored noise is studied. Firstly, the fractional derivative term is transformed into a linear combination of restoring force and damping force, using the principle of minimum mean square error, by calculating the error with the original system, the original fractional order system can be transformed into integer order system. Secondly, the generalized Van der Pol transformation is introduced, based on the stochastic averaging method for system of Itô stochastic differential equations and Fokker-Planck-Kolmogorov(FPK) equations. Finally, through the steady-state probability density function diagram of the system, it is verified that the change of fractional derivative order pand nonlinear damping coefficient μcan induce stochastic P-bifurcation of the system.

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