Transient responses of nonlinear dynamical systems under colored noise

Abstract

The influence of colored noise on the transient response of nonlinear dynamical systems is investigated by the generalized cell mapping (GCM) based on the short-time Gaussian approximation (STGA) scheme. The block matrix procedure is introduced into the GCM/STGA method to solve the storage problem caused by the dimensionality of the system. In addition, a parallel calculation strategy can be implemented due to the independence of the storage and the computation of the block matrixes. Taking the well-known Mathieu-Duffing oscillator as an example, the deterministic global properties are first computed with the digraph cell mapping method, in which the coexistence of multiple attractors is observed. Then, the evolutionary processes of transient probability density functions (PDFs) of the response under colored noise from different initial distributions are revealed. Due to the attractive characteristics of the attractor and disturbance generated by colored noise, it can be seen that it takes longer for transient responses from the attractor to achieve the steady-state than the case of the initial distribution which locates around the saddle. The evolutionary processes of transient responses are quite disparate from different initial distributions with the influence of the colored noise, although they both concentrate around the attractor after enough time, which is consistent with the global structure without noise. Furthermore, the effects of the intensity and the correlation time of the colored noise on the mulistability are discussed. The stochastic P-bifurcation occurs with the increase of these two parameters, respectively. The evolutionary directions of the colored-noise–induced bifurcations are opposite in the two cases. Monte Carlo (MC) simulations are in good agreement with the results obtained by the GCM/STGA method based on the block matrix procedure.