Equation-free bifurcation analysis of a stochastically excited Duffing oscillator

In this paper, an extensive analysis of a stochastically excited one-degree-of-freedom mechanical system with cubic nonlinearity is presented. This is motivated by the need for realistic bifurcation analyses of stochastic dynamical systems, given that many physical applications contain significant time-varying uncertainty that can lead to drastically different solutions from the deterministic case. The proposed methodology is based on pseudo arc-length continuation combined with the moment-map method, which allows the investigation of dynamical systems with stochastic behaviour. It is shown that the introduction of noise in the excitation leads to the destabilisation of stable periodic orbits over time for the underlying system. For better interpretation and lower computational costs, instead of the combined stochastic continuation, the deterministic bifurcation diagram is modified with the detection and approximation of the mean first passage time of the stochastic system by introducing three different methods. Two semi-analytical approaches with Markov models, together with a completely numerical detection method for the mean first passage time, are compared to each other with similar results. The parameter sensitivity analysis and the comparison of the different methods support the proposed methodology for structural dynamic cases. •Stable oscillation of the stochastically forced mechanical system is highly time-dependent.•Improved calculation costs of the stochastic bifurcation analysis method.•Transient vibrations vary the boundary of stability loss, which leads to the jump phenomenon.•Identification of jump phenomenon in time domain simulations.•A novel, semi-analytical approximation of the first passage time (jump phenomenon).