Abstract
The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.
Highlights
Fractional calculus is a generalization of integer-order calculus, which has a history of more than 300 years
In order to reveal the influences with the variations of parameters p and D2 on the stochastic P-bifurcation of the system, the characteristics of the Probability Density Function (PDF) for p(a) and p(x, x) are analyzed for a point (p, D2) in each sub-area in Fig. 11, the analytical solutions are compared with the Monte Carlo simulation results for original system (3) based on the numerical method for fractional derivative [43, 55], and the corresponding results are displayed in Fig. 12 and Fig. 13
In order to reveal the influences with the variations of parameters τ and D2 on the stochastic P-bifurcation of the system, we analyze the characteristics of PDFs for p(a) and p(x, x) for a point (τ, D2) in each sub-area in Fig. 14, the analytical solutions are compared with the Monte Carlo simulation results for original system (3) based on the numerical method for fractional derivative [43, 55], and the corresponding results are illustrated in Fig. 15 and Fig. 16
Summary
Fractional calculus is a generalization of integer-order calculus, which has a history of more than 300 years. Chen et al studied the primary resonance response of a Van der Pol system under fractional-order delayed negative feedback and forced excitation, obtained the approximate analytical solution for the system based on the averaging method [39]. Chen et al proposed a stochastic averaging technique to analyze the dynamic behavior of a randomly excited strongly nonlinear system with a delayed feedback fractional-order proportional-derivative controller and obtained the stationary probability density function of the system [41]. Due to the complexity of fractional derivatives, analyzing them is difficult, and the parametric vibration characteristics can only be analyzed qualitatively, while the critical conditions of parametric influences cannot be found These problems affect the design and analysis of such systems, in part because the stochastic P-bifurcation of bi-stability for fractional-order time-delay coupled system has not been reported. Substituting Eq (15) into Eq (18), the explicit expression of stationary PDF of the system amplitude a can be obtained:
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