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Wei-Bin Zhang (Ritsumeikan Asia Pacific University, Japan)

DOI: 10.4018/978-1-4666-5202-6.ch040

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Top*Chaos* theory has been an active discipline which has had a profound effect on a wide variety of fields in natural as well as social sciences (Prigogine & Stengers, 1984; Devaney, 1989; Peitgen, Jörgens, & Saupe, 1992; Guckenheimer & Holmes, 1986; Mainzer, 1996; Bertuglia & Vaio, 2005). Chaos theory describes the behavior of certain systems, which are highly sensitive to initial conditions. As a result of this high sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the chaotic systems behave “randomly” even though these systems are deterministic. The early applications of chaos theory include high sensitivity of weather patterns, chaotic swings in insect populations, waves in water, nonlinear behavior of electric systems, vibrations in mechanical structures, brain waves, heat beats, coupled chemical reactions, chaotic monetary economic growth, *Malthus’ population theory*, and many other applications. Chaos theory is now applied to almost every branch of sciences. In many dynamic systems, even simple non-linear interactions may contain chaotic behavior.

The first discovery of *chaos* was Henri Poincaré (Peterson, 1993). When he was examining the three-body problem which essentially consists of nine simultaneous differential equations, he found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. The main catalyst for the development of chaos theory was the electric computer. The identification of chaos involves repeated iterations of mathematical formula, which is often impractical by hand. Computers made repeated calculations practical. In particular, figures and images made it possible to visualize these systems. An early pioneer of the theory was Edward Lorenz (1963) whose research on chaos came about accidentally when he was studying weather prediction in 1961, using a simple digital computer to run his weather simulation. He found out that a tiny difference in the initial conditions can lead to quite different weather patterns, even though according to the traditional theory the resulted weather patterns should be almost the same. Chaos was observed by a number of experiments before it was recognized. For instance, Yoshisuke Ueda identified a chaotic phenomenon as such by using an analog computer on November 21, 1961 (Ueda, 2001). Nevertheless, his supervising professor did not believe in chaos and he prohibited Ueda from publishing his findings until 1970. In December 1977, the New York Academy of Science organized the first symposium on chaos. Since then, chaos theory progressively emerged as a transdisciplinary and institutional discipline, partly under the name of nonlinear dynamic theory. Alluding to Kuhn’s concept of paradigm, some chaologists claimed that this new theory was an example of such a shift.

This study shows how chaos theory has been applied to economics. Our inquiry is specially related to nonlinear studies of social and economic issues. The study is concentrated on theoretical models of economic chaos. By introducing the economic models which exhibit chaotic behavior both in discrete and continuous times in different dimensions, we demonstrate wide applications of chaos theory in different schools of economics. In particular, the paper argues that chaos theory is a basic tool for integrating various economic theories. Application 1 introduces the logistic map and examines the one-dimensional discrete growth model with population by Haavelmo and Stutzer. Application 2 identifies economic chaos in the disequilibrium inventory model by Hommes. Application 3 studies a long-run competitive two-periodic OLG model with money and capital. Application 4 discusses the Lorenz equations and its application to urban dynamics. Application 5 introduces the traditional optimal growth model with multiple capital goods, demonstrating the existence of periodic and aperiodic solutions of the traditional growth model. Finally we conclude the study.

Utility Function: u ( x ) of an individual assigns a numerical value to each element in the set X of possible alternatives, ranking the elements of X in accordance with the individual’s preferences.

Production Function: Is a function that shows the amount of output that can be produced by a firm or an entire economy by using any given quantities of inputs (such as capital and labor) for a given technology.

Malthus’ Population Theory: Holds that population tends to increase faster than food supply, with inevitably disastrous results, unless the increase in population is checked by moral restraints or by war, famine, and disease.

Logistic Map: Refers to the difference equation x ( n +1) = ax ( n )[1- x ( n )],, where a is a parameter, and x ( n ) is, for instance, a population size in generation n . The map f ( x ) = ax (1- x ) is called the logistic map.

Bifurcation: Roughly speaking, refers to the phenomenon of a system exhibiting new dynamical behavior as the parameter is varied.

Chaotic Dynamics: Roughly speaking, refers to the dynamic behavior of certain equations F which possess: (a) a non-degenerate n-period point for each n =1, and (b) an uncountable set S ? J (=[0,1]) containing no periodic points and no asymptotically periodic points.

Neoclassical Growth Theory: Asserts that the growth rate of output is determined by exogenous technological growth.

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