Nonlinear Systems

Abstract

If a nonlinear system has an equilibrium, then the behavior of the orbits near that point is often mirrored by a linear system obtained by discarding the small nonlinear terms. We already know from Chapter 6 how to analyze linear systems; their behavior is determined by the eigenvalues of the associated matrix for the system. Therefore the general idea is to approximate the nonlinear system by a linear system in a neighborhood of the equilibrium and use the properties of the linear system to deduce the properties of the nonlinear system. This analysis, which is standard fare in differential equations, is called local stability analysis.