Near Equilibria

Abstract

In this chapter, we investigate the stability and instability properties of equilibrium points of planar differential equations. It is evident from our foregoing discussions that the stability type of an equilibrium point of a linear system is determined by the eigenvalues of its coefficient matrix. Analogous to the results in Section 1.3, we prove several theorems to show that, under certain conditions, the stability type of an equilibrium point of a nonlinear planar differential equation is determined by the linear approximation of the vector field in a sufficiently small neighborhood of the equilibrium point. In order to determine how large these “small” neighborhoods can be, we present another, somewhat more geometric, technique—the direct method of Liapunov—for investigating the stability of an equilibrium of a nonlinear system. We continue our presentation with an analysis of some of the finer geometric details of the flows of nonlinear systems in a neighborhood of an equilibrium point of saddle type. Next, we include a discussion of deciding the local equivalence of flows of nonlinear systems from that of their linear approximations. We conclude the chapter with an example illustrating the global dynamics of saddle points—saddle connections.