Paths Connecting Elementary Critical Points of Dynamical Systems.

摘要

This report is concerned with the occurrence of solution paths connecting elementary critical points of systems of autonomous ordinary differential equations. Since such paths can be thought of as being formed by the intersection of two manifolds, part of the problem is very much a geometric one. Section I deals with this aspect. The major result is Theorem 2, which gives conditions insuring that the connecting paths are persistent under perturbation. A simple equation, relating the dimension of the manifolds to the dimension of the intersection, is also derived. Sections II and III are primarily concerned with defining situations where this equation is satisfied. Section IV contains one of the major results, Theorem 8. This theorem insures the existence of a stable path connecting two elementary critical points. Section V has as its primary goal the application of Theorem 8 to practical physical problems. Various papers that have appeared in the literature are discussed. Also, new work on the Shock Structure problem is presented. Under suitable assumptions the existence of a locally unique, stable solution is exhibited. (Author)