Stochastic response and bifurcation of periodically driven nonlinear oscillators by the generalized cell mapping method

• Extended short-time Gaussian approximation to periodically driven systems.
• Introduced construction of the transition probability matrix over one period.
• Studied stochastic response by the generalized cell mapping method.
• Studied transient and steady-state responses of SD oscillator.
• Found stochastic P-bifurcation of SD oscillator.

The stochastic response of nonlinear oscillators under periodic and Gaussian white noise excitations is studied with the generalized cell mapping based on short-time Gaussian approximation (GCM/STGA) method. The solutions of the transition probability density functions over a small fraction of the period are constructed by the STGA scheme in order to construct the GCM over one complete period. Both the transient and steady-state probability density functions (PDFs) of a smooth and discontinuous (SD) oscillator are computed to illustrate the application of the method. The accuracy of the results is verified by direct Monte Carlo simulations. The transient responses show the evolution of the PDFs from being Gaussian to non-Gaussian. The effect of a chaotic saddle on the stochastic response is also studied. The stochastic P-bifurcation in terms of the steady-state PDFs occurs with the decrease of the smoothness parameter, which corresponds to the deterministic pitchfork bifurcation.