Dynamic response and bifurcation for Rayleigh-Liénard oscillator under multiplicative colored noise

The generalized mixed Rayleigh-Liénard (R-L) oscillator with both periodic and parametric excitations subjected to multiplicative colored noise is investigated in this paper. First, the deterministic global properties are calculated by the iterative compatible cell mapping method, showing the coexistence of chaotic attractor and periodic attractor. Then, under different damping control parameters, the evolutionary processes of transient and steady-state probability density functions (PDFs) of the response under multiplicative colored noise are discussed by the generalized cell mapping based on the short-time Gaussian approximation (GCM/STGA). To solve the storage problem and improve the computational efficiency of one-step transition probability density matrix, we propose to introduce the block matrix procedure into the GCM/STGA method. The results show that the evolutionary direction of PDFs is in accordance with the shape of the unstable manifold. We also observe that it takes longer to achieve the steady state for the case with the coexistence of chaotic attractor and periodic attractor than the case with only periodic attractor. Besides, the stochastic P-bifurcation occurs with changing the intensity and the correlation time for the colored noise. The GCM/STGA scheme shows good agreement with Monte Carlo simulations, which proves the effectiveness of the proposed strategy for systems with chaotic attractor and multiplicative colored noise.