David R. Stockman

Fixed points imply chaos for a class of differential inclusions that arise in economic models

We consider multi-valued dynamical systems with continuous time of the form ẋ ∈ F (x), where F (x) is a set-valued function. Such models have been studied recently in mathematical economics. We provide a definition for chaos, ω-chaos and topological entropy for these differential inclusions that is in terms of the natural R-action on the space of all solutions of the model. By considering this more complicated topological space and its R-action we show that chaos is the ‘typical’ behavior in these models by showing that near any hyperbolic fixed point there is a region where the system is chaotic, ω-chaotic, and has infinite topological entropy.

Basins of measures on inverse limit spaces for the induced homeomorphism

Let fV X! X be continuous and onto, where X is a compact metric space. Let YVD lim .X; f/ be the inverse limit and FV Y! Y the induced homeomorphism. Suppose that is an f -invariant measure, and let m be the measure induced on Y by .;; : : :/ . We show that B is a basin of if and only if 1 1 .B/ is a basin of m. From this it follows that if is an SRB measure for f on X , then the induced measure m on Y is an inverse- limit SRB measure for F. Conversely, if m is an inverse-limit SRB measure for F on Y , then the induced measure on X is an SRB measure for f .

Li-Yorke chaos in models with backward dynamics

Abstract Some economic models like the cash-in-advance model of money, the overlapping generations model and a model of credit with limited commitment may 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. Such models or dynamical systems are said to have 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, I provide a definition of chaos that is in the spirit of Li and Yorke for a dynamical system with backward dynamics. I utilize the theory of inverse limits to provide sufficient conditions for such a dynamical system to be Li-Yorke chaotic.

Uniform Measures On Inverse Limit Spaces

Motivated by problems from dynamic economic models, we consider the problem of defining a uniform measure on inverse limit spaces. Let where X is a compact metric space and f is continuous, onto and piecewise one-to-one and . Then starting with a measure μ1 on the Borel sets , we recursively construct a sequence of probability measures on satisfying for each and . This sequence of probability measures is then uniquely extended to a probability measure on the inverse limit space Y. If μ1 is a uniform measure, we argue that the measure induced on the inverse limit space by the recursively constructed sequence of measures is a uniform measure. As such, the measure has uses in economic theory for policy evaluation and in dynamical systems in providing an ambient measure (when Lebesgue measure is not available) with which to define a Sinai–Ruelle–Bowen (SRB) measure or a metric attractor for the shift map on the inverse limit space.