On Unbounded Limit Sets of Dynamical Systems

Abstract

This paper is focused on studies of properties of unbounded \(\omega \)-limit sets of dynamical systems. It is proved that if the \(\omega \)-limit set \(\Omega \) is not connected, then each of its components is unbounded, which clarifies the well-known property confirming that if the \(\omega \)-limit set is not connected then it is unbounded. It is established that the family of connectivity components of the \( \omega \)-limit set of a planar analytic system can be finite or countable while nonanalytic systems may admit the \(\omega \)-limit set with continuum family of components. Examples of an analytic system possessing \(\omega \)-limit set with infinite number of components and a polynomial system where the \(\omega \)-limit set has exactly N connectivity components are given. Some open problems are formulated as well.