Chaotic equilibria in models with backward dynamics

Some economic models like the cash-in-advance model of money or overlapping generations model have the property that the dynamical system characterizing equilibria in the model are multi-valued going forward in time, but single-valued going backward in time, i.e., the model or dynamical system has backward dynamics. In such instances, what does it mean for a dynamical system (set-valued) to be chaotic? Furthermore, under what conditions are such dynamical systems chaotic? In this paper, we provide a definition of chaos that is in the spirit of Devaney for a dynamical system with backward dynamics. We utilize the theory of inverse limits to provide necessary and sufficient conditions for such a dynamical system to be chaotic.