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WHAT IS IT?
This is an agent-based model of the log map, a well-known formula for the demonstration of chaotic dynamics in nonlinear systems. It illusrates the "butterfly effect" or how miniscule differences in initial conditions can create large divergences in the path evolution of a nonlinear system. (See Lorenz 1963.)
Nonlinear dynamics and the butterfly effect have profound implications for the social sciences for two reasons. One is our traditional use of statistical methods works only for linear and log-linear systems; they are not useful for nonlinear systems. Second, the log map illustrates that even a simple deterministic system may produce dynamics and values that appear "random."
The model adapts Kiel and Elliott's (1997) demonstration of nonlinear dynamics using the log map and a spreadsheet.
HOW IT WORKS
The model consists of two agents. At each step in the simulation, each agent calculates a value for the log map using the function:
x1 = k * x0 * (1 -x0)
The agents differ only in their initial value of x0. By changing the difference in initial values of the two systems using the "initial-difference" slider, one can explore the sensitivity of the log map to its initial conditions.
The plot records the values of X for both agents (blue and red lines). This allows one to observe how the respective "systems" for each agent evolve over time. The plot also includes a black line to record the differences between the red agent's value of X and that of the blue agent. The black line thus visually presents the convergent or divergence of the two systems; the user can turn this feature on by toggling the "plot-difference?" on-off switch.
The model also visually presents the differences between the two systems by having the red and blue agents run a "race" across the top of the screen. Each moves forward by five times its value of X.
HOW TO USE IT
The user selects the initial conditions for each "system"--that is, for each agent's calculation of X. The log map has two parameters: its initial value (x0) and k. To test the sensitivity of the log map to initial conditions, the user can set a difference in the initial value x0 for each agent. By moving the "initial-difference" slider the user can vary the starting points for the two system by as much as 0.3 or as little as 0.00001.
Once you have configured the parameters, click "setup" and "go."
THINGS TO NOTICE
Click on the "converge" button. Notice that the two systems differ in their initial values by 0.5; both systems however have a k value of 2.95. Now click "go" and watch the plot of the red and blue values. What happens to the two systems?
Now click on the "chaos" button. The two systems have identical values of k = 3.9. The two differ in their initial values by only 0.00001, analogous the butterfly flapping its wings. How does this initial difference affect the two systems.
In the first scenario, despite a relatively large initial difference the two systems converged over time. In the second scenario, a very small initial difference led to chaotic differences over time between the two systems. Why?
THINGS TO TRY
Maintain the same initial value and the same initial difference for the two systems. Change the k-value using the slider to 2.5. What happens to the red and blue systems? Now change the k-value to 3.25. How does this change the behavior of the respective systems?
At what value of k does the system switch from convergence to periodic cycles, and from cycles to chaos?
Do the red and blue systems become "co-cyclical"? Are there conditions under which the systems share the same periodicity and amplitude, but their periods do not overlap?
HINT: set the initial-value slider to 0.2, the initial-difference to 0.5, and the k-value to 3.33. Run the model. Then tick the k-value up one notch to 3.34 and run the model again.
CREDITS AND REFERENCES
Model developed by:
David C. Earnest, Ph.D.
Assitant Professor of Political Science & International Studies
Old Dominion University
BAL 700
Norfolk, VA 23529
dearnest@odu.edu
References:
Kiel, L. Douglas and Euel Elliott. 1997. "Exploring Nonlinear Dynamics with a Spreadsheet: A Graphical View of Chaos for Beginners." L. Douglas Kiel and Euel Elliott, eds. Chaos Theory in the Social Sciences. Ann Arbor: University of Michigan Press.
Lorenz, Edward. 1963. "Deterministic nonperiodic flow."Journal of Atmospheric Sciences. Vol.20 : 130—141.
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