This directory contains programs for Macintosh ( Intel version only ). These programs are for experts in complex dynamical systems.

Some snap shots of Second Julia set for regular polynomial maps

- Skew product family whose base dynamics has an interval as Julia set.
- Another Skew product family with interval of base Julia set.
- Symmetric polynomial family from square to tricorn.
- Symmetric product family from torus to Moebius band.
- Symmetric polynomial family from tricorn to fractal tree.
- Skew product with interval base and circle fiber.
- Skew product with St Marco type base Julia set.
- General regular polynomials of degrees 2.
- Another General regular polynomials of degrees 2.

Skew product family

Pictures for skew product maps ( c = 0 in the following family):

X = x^2 + a*x + b*y

Y = y^2 + c*x + d*y

In this family, a=-2, c=0, d=-2 are fixed. Parameter b is varied from 0 to two thirds. For b=0, the base dynamics, Y=y^2-2*y, and the fiber dynamics, X=x^2-2*x, have interval from -1 to 3 as their Julia sets. So, the second Julia set of the product dynamics is a square in the real plane.

A square second Julia set for product map for parameters a=-2, b=0, c=0,d=-2.

A square second Julia set is broken by varying parameter b slightly.
For non zero parameter b, The dynamical system is a skew product.

Parameter b is varied furhter.

Parameters b varied still further.

Parameters b varied still further.

Parameters b varied still further.

Parameters b varied still further.

Parameters b varied still further. Side view of the second Julia set.

A snap shot of second Julia set for skew product map for parameters a=-2,b=0.4667,c=0,d=-2.

A snap shot of second Julia set for skew product map for parameters a=-2,b=0.6667,c=0,d=-2.

Put this directory(1021MB Heavy!) in your directory "~/RP2Pts".

Some further deformation of the second Julia set.

Put this directory(1021MB Heavy!) in your directory "~/RP2Pts".

This family begins with a square of second Julia set.

By a slight change of parameters, fences and branches emanate from the square.

This second Julia set looks like the Sierpinskii Gasket.
Most of the points have four preimages.
This dynamics does not satisfy the so-called "open set condition".

Data of this family is in this directory(1021MB Heavy!). Try to make the data for yourself, instead of downloading.

Data of this family is in this directory(1021MB Heavy!). Try to make the data for yourself, instead of downloading.

Data of this family is in this directory(1021MB Heavy!).
Try to make the data for yourself, instead of downloading.

Data of this family is in this directory(1300MB Heavy!).
Try to make the data for yourself, instead of downloading.

Data of this family is in this directory(1020MB Heavy!).
Try to make the data for yourself, instead of downloading.

Data of this family is in this directory(1020MB Heavy!).
Try to make the data for yourself, instead of downloading.

Data of this family is in this directory(220MB Heavy!). Try to make the data for yourself, instead of downloading.

In this family, our regular polynomial maps is postcomposed with a linear map.
The parameters for this linear maps are not indicted in the title line of the windows of StereoViewer.

Data of this family is in this directory(220MB Heavy!). Try to make the data for yourself, instead of downloading.