Controlling Chaos - Instructions

This applet demonstrates the OGY control scheme.

Parameters

a,b,c	Parameters of the map functions
dp	Change in checked parameter for derivative and control
x_min, y_min, x_max, y_max	Range of variables plotted
eps1, eps2	Tolerances for finding periodic orbits. For a period p orbit the pth iterate must be within a ditance eps1 and intermediate points must be outside a distance eps2
points	Number of points used to fit the periodic points
period	Period of orbit to find or control

Method

There are 4 steps to achieve control.

Find the periodic orbit of period period and the eigenvectors and eigenvalues of a linearization about the periodic orbit. This is done by finding the fixed point of F^(period) and then the eigenvalues and eigenvectors linearizing F^(period) about this point. For the sake of accuracy the eigenvalues and eigenvectors are calculated separately for each of the period points in the periodic orbit.
Find the periodic points for a small increment of the parameter to be used for control. The periodic points and eigenvalues are written out, and the periodic points and the stable and unstbale directions are displayed on the screen.
Find the periodic points for a small decrement of the parameter to be used for control, and use with (2) to calculate the derivate of the periodic point positions with respect to the control parameter.
Use the information calculated in (1)-(3) to control. The derivative of the fixed point position with respect to the control parameter, and the ratio of the control parameter shift needed at the next iteration to the deviation along the unstable direction of the current iteration from the fixed point are written out, and the line along which the period point moves as the chosen control parameter is changed is plotted in red.

Obviously the parameters of the map and the period of the orbit should not be changed between these four processes, although the region of the plane plotted, the tolerances eps1, eps2 and the number points of fit points may be changed. Also it is important that the same periodic orbit is found in the first three steps: this must be visually checked, and the run stopped and Reset if the "wrong" orbit is found in steps (2) or (3).

Remarks

Points that will be used to find the periodic orbit (i.e. satisfying the criteria within the tolerances eps1,eps2) are plotted in red.
After the fit to a periodic orbit, the directions of the eigenvectors at the fitted point are shown on the plot as blue lines, and the next period iterations from the point are shown as black squares. If there are not period black squares with nearby red points, the fit is bad, and should be repeated by using Reset.
The slider changes the speed of the iteration. Setting the speed too high may make the applet unresponsive or crash, depending on the Java implementation. However you will usually want to increase the speed from the default. During the control phase the speed is automatically reduced after locking on to a periodic point to show the approach to the point.
A new initial condition may be chosen by clicking within the running plot. In stages (1)-(3) clicking near the likely location of the periodic points may speed the accumulation of the points needed to perform the fit. In stage (4), this may be used after locking on to a periodic orbit to restart to repeat the control process.
If you change a text entry, you must hit "Enter" for the change to take effect.
Dragging the mouse on a stopped plot will reset the scale to the outlined region, allowing finer detail to be seen. Only control to the points displayed will be performed. A single click outside the plot box will restore the default range.

[Equations]

Last modified Sunday, February 27, 2000
Michael Cross