Chaotic sets and Euler equation branching

Some macroeconomic models may exhibit a type of indeterminacy known as Euler equation branching (e.g., the one-sector growth model with a production externality). The dynamics in such models are governed by a differential inclusion x ˙ ∈ H ( x ) , where H is a set-valued function. In this paper, we introduce the concept of a chaotic set and explore its implications for Devaney chaos, Li–Yorke chaos and distributional chaos (adapted to dynamical systems generated by a differential inclusion). We show that a chaotic set will imply Devaney and Li–Yorke chaos and that a chaotic set with Euler equation branching will imply distributional chaos. We show that the existence of a steady state for a differential inclusion on the plane will generate a chaotic set and hence Devaney and Li–Yorke chaos. As an application, we show how these results can be applied to a one-sector growth model with a production externality – extending the results of Christiano and Harrison (1999). We show that chaotic (Devaney, Li–Yorke and distributional) and cyclic equilibria are possible and that this behavior is not dependent on the steady state being “locally” a saddle, sink or source.