Henon equations
The Henon equations, named after the Mathematician Michel Henon, are
a special system of two coupled equations for two independent variables
x and y. By an iteration procedure consecutive values for x_{n}
and y_{n} are calculated and plotted.
The Java applet below implements (for now) two different sets of equations:
Quadratic Henon equations
x_{n+1} = x_{n} · cos(a)  (y_{n}
x_{n}^{2}) · sin(a)
y_{n+1} = x_{n} · sin(a)  (y_{n}
x_{n}^{2}) · cos(a)
The parameter a determines the strength of the coupling of the two
equations.
Transcendent Henon equations
x_{n+1} = sin(a · y_{n}) 
cos(b · x_{n})
y_{n+1} = sin(c · x_{n}) 
cos(d · y_{n})
Here we've got four free parameters a, b, c and d.
The following table shows some value sets for these parameters which
produce 'nice' plots of the Henon equations:
Set  a
 b  c  d


1  +2.01
 2.53  +1.61
 0.33

2  2.70
 0.09  0.86
 2.20

3  2.24
 +0.43  0.65
 2.43

4  +2.30
 2.80  +1.20
 0.50

5  +2.30
 2.75  1.22
 0.52

6  +3.00
 1.80  +3.00
 0.60

Using the applet
You may change value in the applet's input fields and click the 'draw'
button to plot the new graph. The following parameters may be changed:
 equation: 1=quadratic Henon equations (see above); 2=transcendent
Henon equations (see above)
 param set: number of the parameter shown in the table above (only
relevant for the transcendent equations)
 #iterations: number of iterations for each starting value of x and
y
 a, b, c, d: parameters may be plugged in by hand; for the quadratic
equations the values of b, c and d are ignored
Starting values for x and y are from 0.0 to 0.48 with a step size of
0.08. Each new pair of starting values is plotted in a different color
(10 colors are used in all, so after each ten value pairs the color
cycle starts again).
Have fun with it ! If you happen to find an interesting value set for
the parameters, please drop me an
email !
Peter Uelkes
Last modified: Mon Jan 1 15:59:13 CET 2001