Global Dynamics of an Elliptically Excited Pendulum Model

Abstract

Using both analytical and numerical methods on the global dynamics, including the existence and uniqueness of solutions, subharmonic bifurcations and dynamic responses, of an elliptically excited pendulum model are investigated in this paper. The heteroclinic orbits, as well as periodic orbits with [Formula: see text] and [Formula: see text] types of unperturbed systems are obtained analytically. Chaotic vibrations arising from heteroclinic intersections are studied by means of the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are plotted for different system parameters. The chaotic feature on the system parameter [Formula: see text], named the ratio between the horizontal and the vertical diameter of the upright ellipse traced out by the pivot during each period, is discussed in detail. The conditions for subharmonic bifurcations with the [Formula: see text] type or the [Formula: see text] type are also presented with the subharmonic Melnikov method. It is proved rigorously that the system can undergo chaotic motions through finite subharmonic bifurcations with the [Formula: see text] type. In addition, chaotic motions can occur through infinite subharmonic bifurcations with the [Formula: see text] type. An interesting dynamical phenomenon, i.e. “controllable frequency”, which decreases monotonically with the system parameter [Formula: see text], is presented. A number of related numerical simulations are given to confirm the analytical results.