This article aims at studying the stochastic P-bifurcation of tri-stable van der Pol-Duffing oscillator subjected to multiplicative colored noise. First, the stationary probability density of amplitude is derived by using the stochastic averaging method. Then the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory. And the different types of stationary probability densities of amplitude are also obtained, which are in good agreement with the results from Monte-Carlo numerical simulation. Based on these results, the effects of the noise correlation time, noise intensity and linear damping coefficient on the P-bifurcation and the stable response behavior of the system are studied.
The differential equations of motion are established for a three-degree-freedom wing dynamic model subjected to unsteady aerodynamic loads and random perturbations. The system is dimensionally reduced by the improved average method to obtain the standard equations. Flutter problems of the deterministic wing system with high-order structural nonlinearity are studied using Hopf bifurcation theory and numerical simulation, the critical flutter speed is obtained and the effectiveness of the improved average method in the process of dimensionality reduction is verified. The stochastic P-bifurcation behaviors of the system are analyzed considering the effects of random perturbations of the longitudinal airflow by examining the qualitative variations of the probability density function curves. The results show that the deterministic wing system has a secondary bifurcation, a bistable phenomenon in which the equilibrium and the limit cycle oscillations coexist. The random disturbances significantly increases the critical flutter speed of the wing system, and the amplitude of limit cycle oscillations increases after including random perturbations for the same incoming flow speed.
The stochastic P-bifurcation behavior of tri-stability in a fractional-order
van der Pol system with time-delay feedback under additive Gaussian white
noise excitation is investigated. Firstly, according to the equivalent
principle, the fractional derivative and the time-delay term can be
equivalent to a linear combination of damping and restoring forces, so the
original system can be simplified into an equivalent integer-order system.
Secondly, the stationary probability density function of the system
amplitude is obtained by the stochastic averaging, and based on the
singularity theory, the critical parameters for stochastic P-bifurcation of
the system are found. Finally, the properties of stationary probability
density function curves of the system amplitude are qualitatively analyzed
by choosing corresponding parameters in each sub-region divided by the
transition set curves. The consistence between numerical results obtained by
Monte-Carlo simulation and analytical solutions has verified the accuracy of
the theoretical analysis. The method used in this paper has a direct
guidance in the design of fractional-order controller to adjust the dynamic
behavior of the system.
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