Poincaré recurrence and output reversibility in linear dynamical systems

Reversibility of dynamical processes arises in many physical dynamical systems. For example, lossless Newtonian and Hamiltonian mechanical systems exhibit trajectories that can be obtained by time going forward and backward, providing an example of time symmetry that arises in natural sciences. Another example of such time symmetry is the phenomenon known as Poincaré recurrence wherein the dynamical system exhibits trajectories that return infinitely often to neighborhoods of their initial conditions. In this paper, we study output reversibility in linear dynamical systems, that is, the backward recoverability of the system output while time is going forward. Specifically, we provide necessary and sufficient conditions for output reversibility in terms of the spectrum of the system dynamics. In addition, we provide sufficient conditions for the absence of output reversibility. Furthermore, we establish that no system trajectory can retrace its time history backwards with time going forward which is also natural in light of the uniqueness of solutions to linear dynamical systems. Finally, we draw connections between output reversibility and Poincaré recurrence.