Stationary random response evaluation of nonlinear oscillators with Preisach hysteresis discretized by RNN model

This manuscript is devoted to evaluating random stationary responses of nonlinear oscillators including Preisach hysteresis, which is modeled by a powerful tool, viz. Recurrent Neural Networks (RNN). First, a generalized harmonic transformation is introduced by adopting a geometric viewpoint of RNN-described hysteresis and intentionally separating the hysteretic force into conservative and dissipative components, to circumvent direct mathematical calculations of amounts of difference equations. The hysteretic force could be equivalently mimicked by a force that only depends on the present system states. Then, the system dimension is reduced by stochastic averaging, which is derived from the corresponding low-dimensional Fokker–Planck–Kolmogorov (FPK) equation associated with the probability density of slow-varying process. Additionally, solving the FPK equation yields the stationary probability density as well as various-order response statistics. Finally, two numerical examples, i.e., Duffing oscillator and van der Pol oscillator, are investigated to verify the efficacy of the proposed procedure.