Conservative and Gradient Systems

Abstract

In this chapter, we investigate the dynamics of two classes of vector fields with special characteristics—conservative and gradient. Both types of vector fields have the common property that they are defined in terms of functions; however, their flows are completely different. While periodic and homoclinic orbits may be omnipresent in conservative systems, the limit sets of orbits of gradient systems are necessarily part of the set of equilibria. We first uncover certain basic relations between the phase portraits of these systems and the geometry of underlying functions. Then we identify subsets of desirable “generic” functions. The vector fields of generic functions are structurally stable in the restricted sense that they are insensitive to small perturbations of the underlying functions. Analysis in the generic situations is made possible by the fact that the flows of both types of vector fields are essentially determined by the unstable manifolds of the saddle points. We also illustrate typical one-parameter bifurcations of conservative and gradient systems in nongeneric cases. Of course, the setting for the bifurcation theory of these systems has the important restriction that change of parameters preserve the conservative or gradient character of vector fields.