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 xn
and yn are calculated and plotted.
The Java applet below implements (for now) two different sets of equations:
Quadratic Henon equations
xn+1 = xn · cos(a) - (yn-
xn2) · sin(a)
yn+1 = xn · sin(a) - (yn-
xn2) · cos(a)
The parameter a determines the strength of the coupling of the two
equations.
Transcendent Henon equations
xn+1 = sin(a · yn) -
cos(b · xn)
yn+1 = sin(c · xn) -
cos(d · yn)
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