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
twodimensional complex space to itself involving two parameters
a and b:
H 
æ ç
è


 
ö ÷
ø

= 
æ ç
è


 
ö ÷
ø

. 

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.
The other buttons enable you to zoom in or out from an orbit, or move about
the screen. The new orbit is replotted from scratch each time. "Clear"
removes all orbits and "Reset" clears the orbits and puts the screen parameters
back to their initial values.
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.
File translated from T_{E}X by T_{T}H, version 1.67.