In this paper, we consider a family of dynamical systems on the same compact metric space. We then consider the dynamics given when the given flow shifts between these different flows at regular time intervals. We further require that shifts be allowed by a given directed graph. We then define a type of set, called a chain set, that exhibits many similar properties to chain transitive sets of flows. By considering the dynamics as a skew product flow, we are able to demonstrate that chain sets can be lifted to a chain transitive set if the given graph is complete.
Chain recurrence Skew product flows Chain controllability Hybrid systems Chain transitivity Dynamical systems on directed graphs
This is a preview of subscription content, log in to check access.
Ackerman J., Ayers K., Beltran E.J., Bonet J., Lu D., Rudelius T. A behavioral characterization of discrete time dynamical systems over directed graphs. Qualitative Theory of Dynamical Systems 2014;13:161–180.MathSciNetCrossRefzbMATHGoogle Scholar
Alongi J., Nelson G. Recurrence and Topology, Graduate studies in mathematics. Providence: American Mathematical Society; 2007.zbMATHGoogle Scholar
Ayers K., Garcia X., Kunze J., Rudelius T., Sanchez A., Shao S., Speranza E. Limit and morse sets for deterministic hybrid systems. Qualitative Theory of Dynamical Systems 2013;12:335–360.MathSciNetCrossRefzbMATHGoogle Scholar
Colonius F., Kliemann W. The Dynamics of Control, Systems & Control: Foundations & Applications. Boston: Birkhäuser; 2000.zbMATHGoogle Scholar
Colonius F., Kliemann W. Dynamical Systems and Linear Algebra: Graduate Studies in Mathematics. Providence: American Mathematical Society; 2014.CrossRefzbMATHGoogle Scholar