Period-1 to Period-2 Motions in a Discontinuous Oscillator

Abstract

In this paper, grazing bifurcations on bifurcation trees in a discontinuous dynamical oscillator are discussed. Once the grazing bifurcation occurs, periodic motions switches from the old motion to a new one. Thus, grazing bifurcations on a bifurcation tree of period-1 to period-2 motions varying spring stiffness are presented in a discontinuous oscillator with three domains divided by circular boundaries. The stability and bifurcations of period-1 and period-2 motions are discussed. From analytical predictions, periodic motions are simulated numerically. Stiffness effects on the periodic motions are discussed. Such studies will help one understand parameter effects in discontinuous dynamical systems, which can be applied for system design and control.