Computing omega-limit sets in linear dynamical systems

Abstract

Dynamical systems allow to modelize various phenomena or processes by only describing their way of evolution. It is an important matter to study the global and the limit behaviour of such systems. A possible description of this limit behaviour is via the omega-limit set: the set of points that can be limit of subtrajectories. The omega-limit set is in general uncomputable. It can be a set highly difficult to apprehend. Some systems have for example a fractal omega-limit set. However, in some specific cases, this set can be computed. This problem is important to verify properties of dynamical systems, in particular to predict its collapse or its infinite expansion. We prove in this paper that for linear continuous time dynamical systems, it is in fact computable. More, we also prove that the ω-limit set is a semi-algebraic set. The algorithm to compute this set can easily be derived from this proof. © Springer-Verlag Berlin Heidelberg 2008.