
dynamic

    Time-discrete dynamic system.

      x(0) = y(0) = start position.
      y(n+1) = y(n) + f( x(n) )
      x(n+1) = x(n) - f( y(n) )
      f(k) = sin(k + a*fn1(b*k))

    For implicit Euler approximation: x(n+1) = x(n) - f( y(n+1) )

    Five parameters: start position step, dt, a, b, and the function fn1.


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 xmfract, 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 and Popcorn types.

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.

