Controlling Chaos - Demo 2
Controlling to the Period 2 Orbit in the Henon Map
This demonstration controls to the period 2 orbit of the Henon map using the map parameter a as the control variable.
The properties of the periodic orbit are first established for a=a0=1.4 and then for a=a0+dp and a=a0-dp with dp=0.02.
In the control phase, control is implemented whenever the iterated point falls within eps of one of the fitted periodic points, and the change in a needed to implement control (see text) is less than the value set by dp. When this occurs the change in a used at each step is shown in the text area.
After control is achieved you can start a new iteration by clicking somewhere in the running plot (which restarts from the point clicked on), or hitting Stop and then Start (which restarts with a new random initial condition).
- Test the dependence on the maximum variation of the control variable allowed dp.
- Investigate how the success of control depends on the number points of fit points used for the periodic orbits, and the values of tolerances eps1,eps2 and the variation dp used in calculation the derivative of the position of the periodic point.
- Enlarge the region around one of the periodic points by dragging the mouse on the stopped plot and watch how the points approach the periodic point. (This is easiest for a relatively large value allowed for the variation of a in the control phase.)
- Try control using the parameter b.
For some initial values iterations run away to infinity rather than converge to the Henon attractor. Simply Stop and then Start to restart from another initial condition. The range of initial values leading to divergence depends on the parameters a,b and so when these are varied you might have more trouble finding "good" initial points.
Last modified Wednesday, December 15, 1999