Switching Dynamics and Periodic Motions in a Switching Duffing Oscillator

In this paper, switching dynamics and periodic motions in a periodically forced Duffing oscillator with switching in different vector fields are studied semi-analytically. Between two switches, specific nonlinear dynamical subsystems in the switching system are discretized to obtain discrete implicit mappings. From discrete implicit mappings, specific mapping structures are developed for periodic motions in such a switching system. With periodicity conditions, periodic motions in the switching dynamical systems are determined and the corresponding stability and bifurcations are carried out through eigenvalue analysis. The analytical bifurcation trees of stable and unstable period-1 motions to chaos are obtained for the switching dynamical system. From the analytical solutions, initial conditions are chosen for numerical simulations. The numerical and analytical solutions of stable periodic motions are compared, and the results match very well. The method presented in this paper can be applied to other switched nonlinear systems, nonlinear system controls and MEMS, etc.