Iterations of the complex Henon map.

This page introduces a Java applet which allows one to investigate orbits of certain complex Henon maps. A preprint "Reversible complex Henon maps" by C.R. Jordan, D.A. Jordan and J.H. Jordan giving more detail is available as a pdf file.

The complex Henon map is a map H from two-dimensional complex space to itself involving two parameters a and b:

The case in which we are interested is when b has modulus 1 and a=rb is a real multiple of b. In this case H can be written as the composition of two involutions.

The applet allows you to choose the parameters r and q (the argument of b) either graphically or numerically.

Choosing the parameters graphically

• The image which appears on the left side of the applet shows (in black) the set B of values of a, with r positive, for which the orbit of the origin is bounded.
• Click with the mouse on this image. The values of r and q (expressed as a multiple of p) will appear in the appropriate boxes.
• Click on the button "Plot Orbit" and the first 20,000 points of the orbit, starting from (0,0), will be plotted on the right of the applet, unless the orbit is found to be unbounded. Plotting stops as soon as the orbit is known to be unbounded.

Choosing the parameters numerically.

• Enter the desired value of r into the r box.
• Enter the desired multiple of p for q into the theta box.
• Click on the button "Plot Orbit" as before.

Changing the starting point of the orbit.

• Plot an orbit starting from (0,0).
• Click in the right hand area of the window. The message "Ready to start plotting at ¼" will appear.
• Click on "Plot Orbit".

Colouring

• When b is a root of unity the orbit decomposes into "coset orbits". These can be coloured using the "No. of cosets" box. For example, with q=0.3333333 and 6 in this box, bounded orbits should decompose into 6 differently coloured curves.
• In general, if b is a primitive mth root of 1 then m should be entered in the "No. of cosets" box, though on "islands" of B other numbers can be of interest.
• If you enter a number, other than 0, in the "Coset to plot" box then only one coset orbit will be plotted.

Scale etc.

If you have any comments, such as problems running the applet or the discovery of amazing orbits, please let me know by email.

Another version of the applet is an appendix to the preprint and contains instructions similar to those above but with references to the preprint.