Reversibility and PoincarÉ Recurrence in Linear Dynamical Systems

Abstract

In this paper, we study the Poincare recurrence phenomenon for linear dynamical systems, that is, linear systems whose trajectories return infinitely often to neighborhoods of their initial condition. Specifically, we provide several equivalent notions of Poincare recurrence and review sufficient conditions for nonlinear dynamical systems that ensure that the system exhibits Poincare recurrence. Furthermore, we establish necessary and sufficient conditions for Poincare recurrence in linear dynamical systems. In addition, we show that in the case of linear systems the absence of volume-preservation is equivalent to the absence of Poincare recurrence implying irreversibility of a dynamical system. Finally, we introduce the notion of output reversibility and show that in the case of linear systems, Poincare recurrence is a sufficient condition for output reversibility.