Instructions for the Henon map time series program
This program plots the (2 dimensional) orbits of the
Henon map. Enter the values for the parameters a
and b that define the Henon map, initial point
(x_o, y_o), the x,y coordinates of the bottom
left point and top right point of the square that you
wish to observe (xbl,ybl and xtr,ytr
respectively), and the number N of points in
the orbit. Then run the program by clicking the 2D
Time Series button at the top.
Note that unless you choose the parameter values a
and b properly, the orbits will typically run
off to infinity and the plot of the orbit will not show
much more than the first few points of the orbit. What you
are looking for are 'attractors'; regions that attract nearby
orbits. These attractors can be simple (such as periodic
orbits) or extremely complicated ( like fractals).
For example, if a is between 0.4 and 1.05, with
b=0.3, there is a period orbit that attracts nearby
orbits. If a is between 1.1 and 1.42, a much
more comlicated attractor is seen; this is a 'strange attractor'
because it is not composed of points, but rather is a
complicated, fractal-like curve (see Figure 12.12 in the text).
The applet
Henon movie shows the motion of a point on the attractor
as it moves under iteration by the Henon map.
Clicking the button 1D Time Series will plot the time series
of the x and y coordinates of points in the orbit.
This is useful for showing the periodic nature of an orbit.
After plotting the 1 dimensional time series of an orbit, you can
also plot the histograms of these 1 dimensional time series
by clicking the Histogram button on the top of the
1 dimensional time series window. If you plot the histograms of
several orbits as a increases from 0.4 to 1.05, you
will observe a series of period doubling bifurcations that
lead to ergodic orbits, much like that for the logistic
equation (see Figure 12.15 in the text).