Complex Crises

We have seen (Real crises), that near bottom of every window of periodicity of real maps iterations are changed dramatically and critical orbits are spread from narrow band to a very large region. Let us look for a complex crisis near tiny copy of the Mandelbrot set with period-4. Here each view is zoomed by 50 times.
The last two pictures are self similar again and symmetry of the filaments is changed "after" the bifurcation point.

You see below the superstable periodic critical orbit with period-4 and narrow band of the period-32 orbit "before" crisis on the dynamical plane. The period-39 critical orbit goes away from the band "after" the crisis on the fourth picture.
On the last two pictures only 1,5,9,13... points of orbit are ploted. On the third picture these points are located in the center near the z = 0 point. This fact is used in the applet below to detect components of the period-4 M-set copy.

Many Crises make a "Catastrophe"

Any period-n M-set copy on the complex plane is "separated" from the main Mandelbrot set by an infinite (countable) set of crisis (preperiodic) points. We can use critical orbit of the map fc on to detect points of the tiny M-set. Below all points of the Mandelbrot set for which critical orbit of fc o4 is confined in the |z| < ε region (where ε ~ 0.2) are painted in white.
Therefore you can determine components of the period-4 M-set copy by changes in filaments color or symmetry and by the Mandelbrot set color. In a similar way one can detect any period-n tiny M-set.

It is amazing, that a part of the Mandelbrot set is "equal" to the whole set (as like as the set of even integers are equal to the whole integers set).


Contents   Previous: interior crisis points   Next: Transition to chaos through intermittency
updated 26 June 2002