Bifurcation Theory

Abstract

At points where the manifolds of bifurcating solutions intersect, flows may switch from one manifold to another, and sometimes take on new qualitative behavior by exchanging stability between manifolds. This signifies the occurrence of a bifurcation. Bifurcation or branching occurs in a system when the state of the system depends on some parameter and as that parameter varies the state branches to another state at some critical value of the parameter with usually a concomitant change of stability. The nature of the bifurcation is therefore determined by the dynamics on the center manifold at the bifurcation point. Further, the normal-forms reduction on the central manifold proves to be convenient for a discussion of local bifurcation because, as we saw in Chapter 1, this causes a reduction in dimensionality and, therefore, proves especially helpful in the bifurcation analysis of high-dimensional systems, (Crawford, (1991)). On the other hand, when there is a multiplicity of solutions, the solution sought out by the system is determined by stability considerations. The bifurcation theory is a study of non-uniqueness, and specifically a study of how the multiplicity of solutions varies with the parameter and the stability properties of the bifurcating solutions. Here, we will consider local bifurcation theory that addresses phenomena near a single point, (see Wiggins (1988) for a discussion of global bifurcations which often involve homoclinic and heteroclinic connections (Chapter 5)).