Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Systems

Abstract

In the previous two chapters we studied bifurcations of equilibria and fixed points in generic one-parameter systems having the minimum possible phase dimensions. Indeed, the systems we analyzed were either one- or two-dimensional. This chapter shows that the corresponding bifurcations occur in “essentially” the same way for generic n-dimensional systems. As we shall see, there are certain parameter-dependent one- or two-dimensional invariant manifolds on which the system exhibits the corresponding bifurcations, while the behavior off the manifolds is somehow “trivial,” for example, the manifolds may be exponentially attractive. Moreover, such manifolds (called center manifolds) exist for many dissipative infinite-dimensional dynamical systems. Below we derive explicit formulas for the approximation of center manifolds in finite dimensions and for systems restricted to them at bifurcation parameter values. In Appendix 1 we consider a reaction-diffusion system on an interval to illustrate the necessary modifications of the technique to handle infinite-dimensional systems.