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Visualizing the Dynamical Behavior of Wonderland
November-December 1997 (vol. 17 no. 6)
pp. 71-79
The analysis of complex dynamical systems produces large amounts of data that have to be interpreted efficiently. Visualizing the phase space of such systems illustrates geometrically the behavior of the underlying dynamics. This work investigates the visualization of Wonderland, a four-dimensional econometric model that describes interactions between population growth, economic activity, and environmental implications. Wonderland belongs to a class of interesting dynamical systems with pronounced slow-fast dynamics-some variables change much faster than others. The behavior of the Wonderland model can be characterized by equilibrium surfaces which are not streamsurfaces, that is, the flow does not stay within these surfaces. This article discusses the application and adaptation of various visualization techniques to analytically specified dynamical systems with these special properties.

1. A.A. Tsonis, Chaos: From Theory to Applications, Plenum Press, New York, 1992.
2. E. Gröller et al., "The Geometry of Wonderland," Chaos, Solitons, and Fractals, Vol. 7, No. 12, 1996, pp. 1,989-2,006.
3. N. Fenichel, "Geometric Singular Perturbation Theory," J. Differential Equations, Vol. 31, 1979, pp. 53-98.
4. E.F. Mishchenko and N.K. Roszow, Differential Equations with Small Parameters and Relaxation Oscillations, Plenum Press, New York, 1980.
5. C. Hansen, "Visualization of Vector Fields (2D and 3D)," Siggraph 93 Course Notes No. 2, Introduction to Scientific Visualization, Tools and Techniques, ACM Press, New York, 1993, pp. 4-1-4-9.
6. F.H. Post and T. van Walsum, “Fluid Flow Visualization,” Focus on Scientific Visualization, H. Hagen, H. Müller, G.M. Nielson, eds., Springer-Verlag, Berlin, 1993.
7. W.C. deLeeuw and J.J. van Wijk, "A Probe for Local Flow Field Visualization," Proc. Visualization '93, pp. 39-45, IEEE CS Press, Oct. 1993.
8. J.J. van Wijk, “Spot Noise-Texture Synthesis for Data Visualization,” Computer Graphics (SIGGRAPH '91 Proc.), T.W. Sederberg, ed., vol. 25, pp. 309-318, July 1991.
9. B. Cabral and L.C. Leedom, "Imaging Vector Fields Using Line Integral Convolution," Computer Graphics (SIGGRAPH '93 Proc.), pp. 263-272, 1993.
10. N. Max, R. Crawfis, and C. Grant, “Visualizing 3D Velocity Fields Near Contour Surface,” Proc. IEEE Visualization '94, pp. 248-254, 1994.
11. D.C. Banks and B.A. Singer, A Predictor-Corrector Technique for Visualizing Unsteady Flow IEEE Trans. Visualization and Computer Graphics, vol. 1, no. 2, pp. 151-163, June 1995.

Index Terms:
vector field visualization, complex dynamical systems
Rainer Wegenkittl, Eduard Gröller, Werner Purgathofer, "Visualizing the Dynamical Behavior of Wonderland," IEEE Computer Graphics and Applications, vol. 17, no. 6, pp. 71-79, Nov.-Dec. 1997, doi:10.1109/38.626972
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