On the topological entropy, nonwandering set and chaos of monotone and competitive dynamical systems

Abstract

In this paper, we prove that for strongly monotone and competitive dynamical systems, the topological entropy is equal to the supremum of the topological entropy on invariant and unordered submanifolds with codimension one, and the nonwandering set of a strongly monotone system is the union of the nonwandering sets on invariant and unordered submanifolds with codimension one. Using these conclusions, we prove that a two-dimensional strongly monotone or competitive system has positive topological entropy iff the system has a periodic orbit whose period is not a power of two, and that the depth of centre of two-dimensional monotone systems is not more than two. Finally, we generalize many well-known results about one-dimensional systems to two-dimensional strongly monotone systems.