Stochastic averaging of quasi-integrable Hamiltonian systems with delayed feedback control

A stochastic averaging method for quasi-integrable Hamiltonian systems with time-delayed feedback control is proposed. First, a quasi-integrable Hamiltonian system with delayed feedback control subjected to Gaussian white noise excitations is formulated and then transformed into Itô stochastic differential equations without time delay. Then, the averaged Itô stochastic differential equations for the system are derived and the stationary solution of the averaged Fokker–Planck–Kolmogorov (FPK) equation associated with the averaged Itô equations is obtained for both non-resonant and resonant cases. Finally, three examples are worked out in detail to illustrate the application and effectiveness of the proposed method and the effect of time delayed feedback control on the response of the systems.