Periodic Motions with Impact Chatters in an Impact Double-well Duffing Oscillator

In this paper, the dynamics of a double-well Duffing oscillator impacting with single displacement boundary is discussed. The flow switchability theory is used for developing the analytical conditions for the impact, grazing and stuck motions of the impact double-well Duffing oscillator at the displacement boundary. Through the discretization of a nonlinear dynamical system, the implicit mapping is developed for the solution of a double-well Duffing oscillator. The mapping structures for period-{n} motions with impact chatters are introduced for impact periodic motions of the double-well Duffing oscillator. The bifurcation trees of impact chatter periodic motions are developed. The corresponding saddle-node and period-doubling bifurcations of periodic motions are determined through the eigenvalue analysis, and the periodic motion switching are determined through the grazing bifurcations. The impact chattered periodic motions from period-1 to period-8 motions with/without grazing at the impact boundary is presented numerically and analytically for illustrations of motion complexity in the impact double-well Duffing oscillator. Such a study can help one understand impact dynamics and motion complexity of the double-well Duffing systems with a displacement boundary. The method presented in this paper can be applied to other discontinuous nonlinear dynamical systems.