Monotonicity and mutability

Abstract

This is a study of systems which possess several steady states and whose solutions approach the set of the steady states as time tends to infinity. One is especially interested in the property of mutability, which implies the ability of the system to execute transitions from one steady state to another. It will be shown that this type of behavior is secured if the system possesses some very simple properties of monotonicity. The general concept of mutability includes many cases studied before. Closely related to this paper are especially J. Moser's work on nonoscillating networks [1] and G. A. Leonov's paper [2], concerned with global asymptotic stability. Mutable systems have been often encountered, more or less explicitly, in investigating the limits of the methods of the theory of stability (see, for instance, J. A. Nohel and D. F. Shea [3]). In this paper one studies mutable systems of special structures, aiming at identifying systems with high mobility, whose transitions from state to state are sharp, fast and easily controllable. The method of study is based on a comparison approach (section 4) which establishes similarities between the investigated system and a model. A simple example is the system