Bifurcation Dynamics of Periodic Motions in an Impact Duffing Oscillator with Two Boundaries

In this paper, the bifurcation dynamics of a Duffing oscillator impacting with two symmetric boundaries is studied to obtain the switching of complex periodic motions with grazing and period-doubling bifurcations. The analytical conditions for the grazing and stuck motions at the two symmetric displacement boundaries are presented. The periodic motions in the impact Duffing oscillator are obtained through the implicit mapping method. The bifurcation trees of switching velocities and phases varying with excitation frequency are obtained semi-analytically for a periodically forced Duffing oscillator, and the corresponding stability analysis of impact periodic motions is carried out. The saddle-node and period-doubling bifurcations of periodic motions are determined through the eigenvalue analysis, and the periodic motion switching is determined through the grazing bifurcations. Numerical illustrations of periodic motions with impacts are given, and periodic motions with grazing are also demonstrated. This study can help one further understand complex impact behaviors in nonlinear dynamical systems. The method presented herein can characterize the semi-analytical, stable and unstable impact periodic motions in discontinuous nonlinear dynamical systems.