Vector Difference Equations

Abstract

For maps of Euclidean spaces of dimension higher than 1, a theory of stability that is as satisfactory as the one in Section 2.1 does not yet exist. Nevertheless, there are some general results in the literature that suitably fit within the context of this book. This section presents a general theory of stability for invariant sets of maps of any Euclidean space. Such invariant sets include (but are not limited to) equilibria and limit cycles; in particular, they include “strange attractors.” The differential-equations version of the theory in question dates back to A.M. Liapunov’s memoir of 1892, which became the subject of much attention as the “Liapunov’s second method” (or “direct method,” so called because explicit solutions are not needed in the qualitative analysis).