# 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=a*_{0}=1.4 and then for *a=a*_{0}+dp and *a=a*_{0}-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).

## Other Runs

- 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*.

## Notes

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.

[Previous Demonstration]
[Next Demonstration]
[Introduction]

Last modified Wednesday, December 15, 1999

Michael Cross