Chaotic dynamics in quasi-static systems: theory and applications

Abstract

By a dynamical system, we mean a system of N ∈ N equations that depend on x t−μ , …, x t+ ν ∈ R N , where μ, ν ∈ N . We define a static system as a dynamical system that depends only on x t . We define a quasi-static system as a dynamical system that is in a certain sense relatively close to a static system. We show that under additional conditions, a quasi-static system is chaotic in a generalized sense of Li and Yorke. This result provides easy-to-verify sufficient conditions for chaos for general multidimensional dynamical systems, including maps. We show that these conditions are stable under small C 1 perturbations. We apply these results to two types of growth models with externalities. We show that the models display chaotic dynamics for certain parameter values. We also construct a numerical example in which utility is logarithmic and the dynamics are chaotic (and the discount rate is small). Our conditions for chaos are particularly useful in analyzing dynamic versions of static models with multiple equilibria, as well as dynamic models with multiple steady states.