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 bcd
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:

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 !

A Java applet appears here.

Peter Uelkes
Last modified: Mon Jan 1 15:59:13 CET 2001