Codimension Two Bifurcation and Its Computational Algorithm

Abstract

In this paper we discuss a method for calculating the bifurcation value of parameters of periodic solution in nonlinear ordinary differential equations. The generic bifurcations of the periodic solution are known as codimension one bifurcations: tangent bifurcation, period doubling bifurcation and the Hopf bifurcation. At the parameters for which bifurcation occurs, if a periodic solution satisfies two bifurcation conditions, then the bifurcation is referred to as a codimension two bifurcation. Our method enables us to obtain directly both codimension one and two bifurcation values from the original equations without special coordinate transformation. Hence we can easily trace out various bifurcation sets in an appropriate parameter plane, which correspond to nonlinear phenomena such as jump and hysteresis phenomena, frequency entrainment, etc. Some electrical circuit examples are analyzed and shown to illustrate the validity of our method.KeywordsPeriodic SolutionHopf BifurcationBifurcation DiagramPeriodic PointChaotic AttractorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.