Qualitative Techniques for Bifurcation Analysis of Complex Systems*

Here x is an element of a finite-dimensional vector space (say R") or of a suitable Banach space of functions. In the latter case, (1) represents a partial differential equation (PDE). The control parameter p E R" is supposed to vary slowly in comparison with the evolution rate of a typical solution x ( t ) of (1). Thus we treat (1) as an m-parameter family of ODE'S. We are primarily interested in studying the qualitative changes that occur in the vector field or (semi) flow defined by (1) as p varies. The techniques used in the study of (1) draw on several fields, notably those of functional analysis and differentiable topology. In this brief paper we are only able to sketch general ideas and must therefore refer the reader to texts such as Chillingworth,' and Marsden and McCracken" for background information and further details. Both texts contain a wealth of additional references. The general problem of bifurcation of vector fields-the qualitative study of equations, such as (1)-contains as an important subproblem, the study of bifurcations of equilibria, or stationary solutions. Much of the work done so far in bifurcation theory has been addressed specifically to the latter problem. The usual definitions of a bifurcation point are couched with this in mind. Since we wish to study a more general class of problems, and, in particular, to consider the case of global bifurcations, we propose an alternative definition, which is a slight modification of the definitions due to Smale and Thorn. First we review the usual definition. Consider a map F:X x A Y, where X , A and Y are Banach spaces, and A is the parameter space. Set F(x ,X) = 0 and seek the solutions. Let x(X) be