Summary

Long description

How to do it

• run program Mandel. Now you are on parameter plane with cursor in the point c=0. The function is complex quadratic polynomial
• go to the bifurcation point on the boundary of period 1 component ( main cardioid) with combinatorial rotation number = 13/34 ( use main menu/Points/Bifurcate or key c) See function int mndlbrot::bifurcate from mndlbrot.cpp. Now you are in the root point c :

 c = -0.392571548476155  +0.585781365897037 i    period = 10000

Period = 10 000 means that period was not detected ( see function ???? )

• zoom in into parameter plane around above root point c. After zooming in for a few times, you will see smooth bands instead of fractal trees. Then increase the maximal number of iterations with the key n. ( maximal Nmax=65 000) When you see large rectangular pixels, the limit of possible magnifications is reached.
• draw the wake for combinatorial rotation number = 13/34 ( using key w or main Manu/Rays/Angles of the wake ) See function ???

The 13/34-wake of the main cardioid is
bounded by the parameter rays with the angles
4985538889/17179869183  or  p0100101001001010010100100101001001  and
4985538890/17179869183  or  p0100101001001010010100100101001010 .
Do you want to draw the rays and to shift c
to the center of the satellite component?

Copy angle of first ray , here 4985538889/17179869183.

Now you are in the center of period 34 component  :

  c = -0.392958628058490  +0.586546408285511 i    period = 34

• go to the dynamic plane ( F2 key, or click on right window )
• increase the maximal number of iterations with the key n. ( maximal Nmax=65 000). Redrawind now take some time so please wait !
• draw external rays landing on alfa fixed point = orbit portrait ( key O or Main Many/Rays/Orbit portrait, Paste here angle of first ray that land on the root point )
• go to the alfa fixed point ( key x or Main Manu/Points/ find end enter 1 ) See function int mndlbrot::find from mndlbrot.cpp

Theory

See demo 4 page 8 from program Mandel. Try a simple example like 3/7 by hand.

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