Continuous unimodal maps in economic dynamics: On easily verifiable conditions for topological chaos

Abstract

In this paper, we offer necessary and sufficient conditions for the presence of odd-period cycles and turbulence in a continuous unimodal interval map. The characterizations we present are original both to the economic and the mathematical literature, and go beyond existential assertions to easy verifiability. We apply our two theorems to six different canonical models in the literature on economic dynamics, all being grounded in the fact that their policy functions are given by continuous unimodal maps. An unintended outcome of the work presented here is to alert the economics profession to a richer conception of erratic and chaotic dynamics, one that goes considerably beyond the 1964-1975 Sharkovsky-Li-Yorke emphasis on three-period cycles and on uncountable scrambled sets.