Fractal escape-time

FIGURE 23.1. Escape-time fractal for parameters EHPLWTDJRCAP. (See text for explanation of parameter code. Also see color plates 3-6 and 10-12 for additional examples.)... 

Another way you can display the solution is to solve the equations with many different starting values and count the number of iterations required for the solution to wander outside some region in the xy-plane. You can use the final number of iterations to determine the color for that point in the plane of initial xy-values. Plots produced in this way are called escape-time fractals because the color contours indicate the time required for the orbit to escape from the region. Some initial values may have orbits that never escape, and so you need to have a bailout condition beyond which your program stops iterating and moves on to test the next initial condition. 

Some of the most artistic examples of escape-time fractals explored in the past have been Julia sets, fractal representations describing the behavior of the map mew = -I- c, where z and c are complex numbers." In terms of x and y this map is... 

This program, ESCAPE.BAS, automatically produces an unlimited number of escape-time fractals similar to those shown in Chapter 23. 

Escape-time fractal for parameters KUONOVSVFLAR. See Escape from Fractalia" (chapter 23) for more information. 

Figure 2.13 Escape times as a function of initial location, showing singularities on a fractal set for an ensemble of particles released on a line segment in the vortex-sink flow (from Karolyi and Tel (1997)).

The chaotic saddle and its manifolds are also sets of zero measure with fractal structure. The set of points, seen in Fig. 2.13 corresponding to inflow coordinates with very large, singular, escape times, typically form also a fractal set determined by the intersection of the saddle s stable manifold and the line containing the initial conditions. There is a connection between the dimension of the chaotic saddle and the dimensions of its manifolds. The trajectories on the chaotic saddle have a set of Lyapunov exponents whose number is equal to the dimension of the full space, d. The sum of the Lyapunov exponents is zero due to incompressibility and chaotic dynamics implies... 

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