Higher Order Bifurcation near Equilibrium

Abstract

Most of the discussion in the previous chapters on dynamics have centered around the case where the unperturbed autonomous equation is a bifurcation point of degree one. In particular, for any equilibrium point, this implies that the linear variational equation must either have only zero as a simple eigenvalue and no other eigenvalue on the imaginary axis or have only a pair of simple complex eigenvalues on the imaginary axis. Generically, the first alternative corresponds to a saddle-node bifurcation and the second alternative to the generic Hopf bifurcation. When the bifurcation point has degree greater than one, there can be more eigenvalues on the imaginary axis. In this chapter, we discuss situations where this occurs. As we shall see, the behavior of solutions of the perturbed equation near such a bifurcation point depends upon more global concepts; for example, homoclinic orbits can occur as well as all of the bifurcations encountered in the previous chapters. The method of investigation is to scale variables in such a way as to “blow up” the neighborhood of the equilibrium point in order to see the fine structure of the flow.KeywordsPeriodic OrbitEquilibrium PointHopf BifurcationBifurcation PointHomoclinic OrbitThese 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.