Stochastic nonlinear dynamics analysis of double-well duffing systems under random parametric excitations

Based on the orthogonal polynomial approximation theory, the stochastic dynamical behaviors of double-well Duffing systems under random parametric excitations were investigated. Firstly, the complex dynamical behaviors of deterministic Duffing systems were studied by means of the Poincaré sections. Then, the Duffing systems with random stiffnesses and damping parameters were reduced to equivalent deterministic expanded-order systems, and the effectiveness of this approximation method was proved. Thus, the ensemble-mean responses of the equivalent systems were applied to reveal the stochastic dynamical properties and the effects of the random variable intensity on the double-well Duffing systems. The numerical simulation results indicate that, in the case of coexistent attractors, the double-well Duffing system with random stiffness parameters has the similar stable dynamical behaviors to those in deterministic cases. However, for the Duffing system with random damping parameters, during the increase of the random variable intensity, the bifurcation phenomena occur to some coexistent attractors.