2.43 Dynamic System
(type=dynamic, dynamic2)
These fractals are based on a cyclic system of differential equations:
x'(t) = -f(y(t))
y'(t) = f(x(t))
These equations are approximated by using a small time step dt, forming
a time-discrete dynamic system:
x(n+1) = x(n) - dt*f(y(n))
y(n+1) = y(n) + dt*f(x(n))
The initial values x(0) and y(0) are set to various points in the plane,
the dynamic system is iterated, and the resulting orbit points are
plotted.
In fractint, the function f is restricted to: f(k) = sin(k + a*fn1(b*k))
The parameters are the spacing of the initial points, the time step dt,
and the parameters (a,b,fn1) that affect the function f. Normally the
orbit points are plotted individually, but for a negative spacing the
points are connected.
This fractal is similar to the Pickover Popcorn (p. 56).
A variant is the implicit Euler approximation:
y(n+1) = y(n) + dt*f(x(n))
x(n+1) = x(n) - dt*f(y(n+1))
This variant results in complex orbits. The implicit Euler
approximation is selected by entering dt<0.
There are two options that have unusual effects on these fractals. The
Orbit Delay value controls how many initial points are computed before
the orbits are displayed on the screen. This allows the orbit to settle
down. The outside=summ option causes each pixel to increment color
every time an orbit touches it; the resulting display is a 2-d
histogram.
These fractals are discussed in Chapter 14 of Pickover's "Computers,
Pattern, Chaos, and Beauty".