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1. Exploring Nonlinear Dynamics with a Spreadsheet: A Graphical View of Chaos for Beginners
- University of Michigan Press
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CHAPTER 1 Exploring Nonlinear Dynamics with a Spreadsheet: A Graphical View of Chaos for Beginners L. Douglas Kiel and Euel Elliott The mathematical foundation of chaos theory and the unique vernacular of this new science can deter some researchers from exploring the dynamics of nonlinear systems. Terms such as periodicity, sensitive dependence on initial conditions, and attractors are not the usual vernacular of the social sciences. However, the modem microcomputer and electronic spreadsheet software provide means for the novice to chaos research to explore the mathematics of chaos. The graphics capabilities of spreadsheet software also provide a visual means for exploring chaotic dynamics. This is particularly important considering the reliance of chaos researchers on graphical analysis. The intractability of nonlinear mathematics, while often defying solution, is now explored via visual analysis. This chapter should thus bring to light the amazing behavior and visual imagery of nonlinear dynamical systems and their relevance to social science. Fortunately, the dynamics of time-based nonlinear systems can readily be explored by researchers new to the study of chaos theory. This exploration is accomplished here via the use of a simple algebraic formula and the computational and graphical powers of an electronic spreadsheet. The electronic spreadsheet readily allows the researcher to generate nonlinear time series and then examine these series graphically. Only a minimal knowledge of spreadsheets is necessary for the reader to follow the examples in this chapter. Readers are also urged to examine in greater detail the mathematical formulation and its various dynamics presented here. By examining the chaotic and, more generally, nonlinear behavior in this chapter, the reader will understand that a simple deterministic equation can generate very complex behavior over time. This has considerable value for social scientists as we learn that systems evolve from the simple to the complex (Prigogine and Stengers 1984). This chapter also reveals the importance of history to social systems. The initial starting point of a social system has much to do with its eventual structure and behavior. 19 20 Chaos Theory in the Social Sciences As noted in the introduction to this volume, nonlinear systems can take on a wide array of behaviors over time. Scientists have, however, classified these behaviors into three distinct types of time-based regimes. These behavioral regimes are (I) convergence to an equilibrium or steady state; (2) periodic behavior or a stable oscillation; and (3) chaos. The most widely used mathematical formula for exploring these three behavioral regimes is a firstorder nonlinear difference equation, labeled the logistic map. This mapping takes on the form The variable to be examined is the value x. The parameter, or boundary value, of the formula is a constant, k. Remember, chaotic behavior occurs within defined parameters. The subscript t represents time and is the current value of the variable x. The subscript t + I represents one time period of the variable x following the previous Xt. Mapping this formula also requires an initial starting value. The starting point, usually called the initial condition, is represented by the first value of Xt' Xo. Once the first value of Xt and the parameter value are determined, a simple "copy" command with the spreadsheet can be used to generate the time series. The copy command serves the purpose of recursion or feedback by using the previous value to generate the current value of Xt. A couple of rules must be followed when using the logistic map. First, the initial condition must be a fractional value such that 0 ~ 0.4 0 0.3 .. 0.2 -- .. 0.1 - --- ---- --------.--- - ----- - -- -- ---- __ O -TrorT"T'TT'T"T'1r"T'TT"TTT'T"'T'TT"TT'T"T~r"T'TT"T"T'T"'I""1"T"T"'~""f""r'T'1r"T'T"1"'~r-T'T'T'"~~r"T'T"I"'''''''''"''''''~'I'"''I'''T''I'~rT'T'''I''T''T'T''I'''T'TT''.,..,.,..,.~ 25 50 75 Iterations 1-xo =0.97 k = 2.827 Fig. 1.1. Stable equilibrium Periodic Behavior A second type of nonlinear behavior that can occur over time is periodic behavior. Periodic behavior is cyclical or oscillatory behavior that repeats an identifiable pattern. Such periodic behavior starts to occur when k > 3. This regime initiates instability into the equation as the data start to oscillate. Such a change in the qualitative behavior of the time series is referred to as a bifurcation, or a branching to a new regime of behavior. This can be seen in column one of table 1.2. This first...