Bifurcation control of bounded noise excited Duffing oscillator by a weakly fractional-order feedback controller

An innovative bifurcation control method by using a weakly fractional-order feedback controller is proposed to eliminate the stochastic jump in the forced response for a bounded noise excited Duffing oscillator. The averaged It equations of amplitude modulation and phase difference are derived via stochastic averaging method, from which the reduced Fokker-Planck-Kolmogorov equation is established and solved numerically to obtain the stationary probability density of amplitude. An efficient scheme with high accuracy for simulating the fractional integral and fractional derivative is then explored. By examining the stationary probability density of amplitude of the uncontrolled and controlled systems, the fractional-order feedback controller has been demonstrated capable of eliminating the stochastic jump and alleviating the amplitude peak of primary resonance effectively, particularly for the case that the integer-order PID controller fails to perform and even leads to the unstable response. Moreover, analytical results show generally agreement with the results obtained by the proposed simulation scheme.