Available online at www.sciencedirect.com Journal of Mathematical Economics 44 (2008) 423–444 The inverse limits approach to chaos Judy Kennedy a,b, David R. Stockman c,∗, James A. Yorke d a Departmentof Mathematical Sciences, University of Delaware, Newark, DE 19716, United States b Department of Mathematics, Lamar University, Beaumont, TX 77710, United States c Department of Economics, University of Delaware, Newark, DE 19716, United States d Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, United States Received 27 March 2006; received in revised form 31 October 2007; accepted 3 November 2007 Available online 22 November 2007 Abstract When analyzing a dynamic economic model, one fundamental question is are the dynamics simple or chaotic? Inverse limits,as an area of topology, has its origins in the 1920s and since the 1950s has been very useful as a means of constructing “pathological” continua. However, since the 1980s, there is a growing literature linking dynamical systems with inverse limits. In this paper, we review some results from this literature and apply them to the cash-in-advance model of money. In particular, we analyze the inverse limit of the cash-in-advance model of money and illustrate how information about the inverse limit is useful for detecting or ruling out complicated dynamics. © 2007 Elsevier B.V. All rights reserved. JEL Classification: C6; E3; E4 Keywords: Chaos; Inverse limits; Continuum theory; Topological entropy; Cash-in-advance 1. Introduction An equilibrium of a dynamic economic model can often be characterized as a solution to a dynamical system. One fundamental question when analyzing such a model is are the dynamics simple or chaotic? There are numerous definitions and measurements for chaos. Two of the more popular definitions are those of Devaney (2003) and Li and Yorke (1975). Given a definition of chaos, one still needs a means for establishing or ruling out chaos. For example, Li and Yorke (1975) show that for a continuous map on an interval, a three cycle is sufficient for establishing chaos. Other measures of complicated dynamics include topological entropy, measure-theoretic entropy, zeta function, Liapunov exponents and box dimension. In this paper, we focus on inverse limits as a method of detecting or ruling out chaotic behavior. As an area in topology, inverse limits has its origins in the 1920s and since the 1950s has been very useful as a means of constructing “pathological” continua. However, since the 1980s there is a growing literature that looks at the connection between inverse limits and topological dynamics. In this paper, we review some results from this literature and apply them to the cash-in-advance model of money. In particular, we analyze the inverse limit of the ∗ Corresponding author. E-mail address: [email protected] (D.R. Stockman). 0304-4068/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2007.11.001 424 J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 cash-in-advance model of money and illustrate how information about the inverse limit is useful for detecting or ruling out complicated dynamics. Inverse limits is a relatively new tool to economic theory that has recently been used to understand models with backward dynamics. Many nonlinear dynamical systems are single-valued moving forward, but multi-valued going backward. However, in economics we sometimes have just the opposite, namely dynamics that are multi-valued going forward, but are single-valued going backward. In such instances, we say the model (or dynamical system) has backward dynamics. Two economic models that may have backward dynamics include the overlapping generations (OLG) model and the cash-in-advance (CIA) model.1Medio and Raines (2007, 2006) use inverse limits to analyze the long-run behavior of an OLG model with backward dynamics. Even though the forward dynamics are multi-valued, they show that long-run behavior of equilibria in the model corresponds to an “attractor” of the shift map on the inverse limit space. Kennedy and Stockman (2007) use the inverse limit space to show that a model with backward dynamics is chaotic going forward in time if and only if it is chaotic going backward in time. Though inverse limits have been used to analyze models with backward dynamics, the results in this paper apply to models with forward dynamics as well. Heuristically, the inverse limit of a dynamical system is a subset of an infinite dimensional space (e.g. the Hilbert cube) where each point in the inverse limit corresponds to a backward solution (backward orbit) of the dynamical system. When considering the inverse limit space of a dynamic economic model, two questions naturally arise: (1) what does the structure of the inverse limit tell us about the underlying dynamics, and (2) what does the underlying dynamics imply for the inverse limit? Ingram and Mahavier (2004) explore the connections between these two aspects of dynamical systems for maps on the unit interval. Working within a family of one-dimensional maps, they illustrate that restrictions on parameters that lead to simple/complicated dynamics also lead to simple/complicated inverse limits. Consequently, by analyzing the inverse limit one can detect chaotic/complicated dynamics or rule out such behavior. The inverse limit approach offers an alternative method of exploring chaotic behavior that does not work directly with establishing a 3-cycle (or some other cycle and using Sarkovskii’s ordering on the integers) or calculating some measure of entropy. Moreover, when the topology of the inverse limit space changes there is a qualitative change in the dynamics. Thus the inverse limit space can be used to detect qualitative changes in chaotic behavior. The paper is organized as follows. We provide the necessary background material from dynamical systems and inverse limits in Section 2. In Section 3, we illustrate how inverse limits can be applied to the CIA model of money to establish or rule out complicated dynamics. We conclude in Section 4. 2. Dynamics, continuum theory, and inverse limits The applicability of inverse limits to detecting or ruling out chaotic behavior in a dynamic economics model like the CIA model depends on the relationship between equilibria in the model, the inverse limit, continuum theory and dynamics. In this section we discuss the necessary background from dynamics, inverse limits and continuum theory for our application to the CIA model. We do not express definitions or theorems in their most general form, but at a level sufficient for our purposes. 2.1. Dynamics In this subsection, we discuss some definitions and facts from dynamics that we will use later. Throughout this paper, we will suppose X is a compact metric space and f : X → X is continuous. We are interested in using the inverse limit space to establish or rule out complicated dynamics. What does one mean by complicated? One can say that the dynamics are complicated if the dynamical system is chaotic. There are several definitions of chaotic in the dynamical systems literature. Two properties that most definitions include are topological transitivity and sensitive dependence on initial conditions. Definition 1. A map f : X → X is topologically transitive if whenever U and V are nonempty open subsets of X, there is some integer n such that f n(U) ∩ V =∅. 1 See Grandmont (1985) for the OLG model and Michener and Ravikumar (1998) for the CIA model. J. Kennedy et al. / Journal of Mathematical Economics 44 (2008) 423–444 425 Definition 2. A map f : X → X, where X is a metric space has sensitive dependence on initial conditions on an invariant closed subset H of X if there is some number r>0 such that for each point x in H and for each >0, there is a point y in H with d(x, y) <and an integer k ≥ 0 such that d(f k(x),fk(y)) ≥ r. Two of the more commonly used definitions of chaotic are those of Li and Yorke (1975) and Devaney (2003). Definition 3. The map f : X → X is chaotic in the sense of Li and Yorke (1975) if f has sensitive dependence on initial conditions on X. The map f is chaotic in the sense of Devaney (2003) if (1) f is topologically transitive, (2) the set of periodic points in X is dense in X, and (3) f has sensitive dependence on initial conditions. Three more concepts used to indicate that a system has complicated dynamics are a homoclinic fixed point,a horseshoe and topological entropy. Definition 4. Suppose that x is a fixed point of f. We say that y is homoclinic to the fixed point x if x = y such that f n(y) → x and that there is a choice of inverse images f −1(y), f −2(y),...with f −n(y) → x.Ifx is a periodic point of period n under f,wesayy is homoclinic to x if y is homoclinic to x under f n. Definition 5. Let f : I → I be continuous where I := [a, b]. We say that f has a (one-dimensional) horseshoe if there n n are disjoint subintervals I0 and I1 of I and n0 ∈ N such that I0 ∪ I1 ⊂ f 0 (I0) and I0 ∪ I1 ⊂ f 0 (I1). The notion of topological entropy involves the concept of an (n, )-separated set. Let f : X → X be a continuous map on a compact metric space X with metric d. Let A, E ⊂ X. We say that E is (d, , A)-spanning if E is finite and f for every y ∈ A, there exists an x ∈ E such that d(x, y) <.Givenf, for n ∈ N we define a new metric dn on X given f i i f by dn (x, y):= max0≤i≤n−1d(f (x),f (y)).