Fractals/Iterations in the complex plane/av velocity

=Average velocity by Chris King =

"Discrete Velocity of non-attracting Basins and Petals: Compute, for the points that don't escape, the average discrete velocity on the orbit:"

$$\left\vert z_{n+1} -z_n  \right\vert $$

Algorithm
On dynamical plane one can see :
 * Exterior of filled Julia set (blue) colored by level set method,
 * Interior of Julia set showing irrational flow (green) coloured by the sine of the velocity

For the points that don’t escape compute the average discrete velocity of orbit  :

$$ \upsilon = \frac{S_n} {n} $$

where :

$$S_n = \sum_{n=1}^{n_{max}} d_n =\sum_{n=1}^{n_{max}} |Z_{n+1} - Z_n |$$

In Octave it looks : d=0; iter = 0;
 * 1) octave code

while (iter < maxiter) && (abs(z)<ER) h=z; # previous point = z_(n) z=z*z+c; # next point = z_(n+1) iter = iter+1; d=d+abs(z-h); # sum of distances along orbit end

if iter < maxiter # exterior measure = iter; myflag=3; # escaping to infinity else # iter==maxiter ( inside filled julia set ) measure=20*d/iter; # average distance (d/iter) = 0.5

In Chris King Maple code this discrete velocity is measured only by sum of distances between points

$$S_n = \sum_{n=1}^{n_{max}} d_n =\sum_{n=1}^{n_{max}} |Z_{n+1} - Z_n |$$

Because : so distance is a good measure into which Siegel orbit point fall
 * all forward orbit from interior of Julia set fall into SIegel disc
 * inside Siegel disc points turn around its center ( indifferent periodic point )

Using periodic function ( sin, cos) creates bands showing dynamics inside Julia set ( siegel disc and its preimages ).

Octave src code
=Images with src code ( see commons page)=

=References=