Bifurcations of a particular forced Van der Pol Oscillator

Abstract

The averaged equations of a forced non-linear oscillator, with both non-linear frictional and restoring forces, are considered as a two parameter system. The local and global bifurcations of the averaged two-parameter system are investigated. The local bifurcations are of Hopf and saddle-node type and are located in the parameter plane. The exceptional parameter points on the local bifurcation curves are investigated and the resulting global bifurcations are catalogued. Saddle connections, coalescence of limit cycles and saddle node creations on limit cycles are shown to occur using normal forms for vector fields and cycles without contact techniques. The behaviour shows the dramatic increase in complexity obtained by adding a nonlinear restoring force to the standard forced Van der Pol Oscillator.