As we have seen earlier, chaotic systems, like the logistic map, depict simple periodic and complex chaotic behavior. For certain practical applications it might be preferable to be able to convert a system which is in a chaotic state, into its periodic behavior. The trivial way for this would be to turn the system parameter back to his original value where the periodic solution is stable. However, another way would be to leave the parameter at its present value, and add only small changes to it, in such a way that the unstable periodic state becomes stabilized. Well, that indeed is typically meant when we say controlling chaos - that is stabilizing the unstable periodic orbit (which is embedded in the chaotic attractor). A simple java applet is provided that implements both OPF (described below) and nonlinear control strategy.

For example imagine the logistic map would describe the temperature of an engine, where the parameter b describes the fuel throughput. A simple period one behavior of the system is probably the one most desirable. Now, using chaos control, we want to be able to increase the throughput given by the system parameter b, but avoid getting into the regime of chaotic temperature variations. In case of the logistic map, this can be archived by using the OPF chaos control method.

OCCASIONAL PROPORTIONAL FEEDBACK CONTROL

The Occasional Proportional Feedback control method (OPF), is commonly used in control of chaotic systems such as electronic circuits and lasers. To stabilize a period one fixed point x_fp, we apply small control changes dp to a system parameter b, proportional to the distance of the system state x_i from the fixed point. This is

dpi = adxi (1)

where constant a is the OPF control gain, and dx_i = (x_i-x_fp). The control is only applied within a small neighborhood of the fixed point x_fp. The control is implemented by adding dp_i to the setpoint of the system parameter b in every iteration. The new logistic map including control becomes

xi+1 = (b+dpi) xi(1-xi) (2)

For the logistic map we can derive the optimal value of the OPF control gain a for the fixed point x_fp to be stablized. Expressing the logistic map F(x,b) in linear order,

dxi+1 = dF dx dxi + dF delta p dpi (3)

For the fixed point to be stable dx_i+1 = 0. After substitution the differentials of F evaluated at x_fp and solving for dp_i we get

dpi = adxi (4)

with

a = - (1-2 xfp)b xfp-xfp2 (5)

NONLINEAR CONTROL OF LOGISTIC MAP

For the logistic map we can derive the optimal value of the Nonlinear control by a process similar to derived above. We proceed by doing the Taylor series expansion about the fixed point that we intend to control but this time even the nonlinear terms are retained. We retain terms upto second order and all other terms are ignored. Thus,

dxi+1 = ¶F ¶x dxi + ¶F ¶p dpi + ¶2 F 2 ¶x2 (dxi)2 + ¶2 F ¶x¶p dxi dpi (6)

where F is the logistic map F(x,b). For the fixed point to be stable we set dx_i+1 = 0 and derive the control law by solving for dp_i. This gives the control perturbations to be applied to control the unstable fixed point as

dpi = - ¶F ¶x dxi + ¶2 F (dxi)2 2 ¶x2 ¶F ¶p + ¶2 F ¶x¶p dxi (7)

where all partial derivatives of the map F is evaluated at x_i = x_f and p_i = p₀. Note, the above expression reduces to the OPF control law if the second terms from the numerator and the denominator are set to zero. The java applet demonstration shown below implements the above nonlinear control law to stabilize the unstable periodic orbit.

JAVA APPLET

The java applet below is designed to demonstrate OPF control on the logistic map. Current iterates of x_i are displayed in red on two viewgraphs. The fixed point x_fp is shown in green. The left graph displays the logistic mapping function x_i = F(x_i-1). Plotting successive iterates in this way is commonly called a return map. On the right hand side values of x_i are plotted versus the parameter value b. Clicking on the drift up bottom increases the value of b. Further increasing of b will yield to period doubling of x_i, building up the bifurcation diagram for the map.

If b increases beyond 4, x_i leave its valid range. After drifting b down below 4, clicking the reset button will assing a random value to x between 0 and 1 and effectively restart the system.

OPF control can be implemented by clicking on the control button. Notice that control will also work in the presence of adding random noise to b, or when simultaneously changing b. Drifting the controlled system beyond 4, does in effect extend the useful working range of the system.

CONCLUSION

Controlling the logistic map using the OPF method serves as a basic example to demonstrate important features of chaos control. In practice however, the mapping function of the nonlinear system might not be known and information of the system dynamics must then be extracted from time series data.

Check other implementations of different control algorithms on various other systems. Click here to download an X-Window based implementation of High Dimensional RPF control algorithm on Henon map (it was used as a test ground), low dimensional RPF impact systems and the real full fledged implementation of high dimensional control algorithm on Hyper-chaotic Rossler system (Borland 5.0 C++). You can download the paper from here. The above methods are implemented concurrently with the adaptive learning algorithm which enables tracking of the controlled state through the parameter space. The implementation assumes no knowledge of the model equations and implements control as though the data is coming from an experimental setup. Thus, control is implemented in the embedded delay time space.

Here is a generic help file for each of the XWindow based control programs. Generally speaking the control, learning, tracking (drift) and noise can be switched on or off as a toggle. The default setting are control (ON), learning (ON), noise (ON), drift (OFF). The key C/c, L/l, N/n, and D/d can be used to toggle there values. Each tar-gzipped file is provided with the make file and you may need to modify the Makefiles to compile the program (No automatic configurations, please!). The program has been tested on Sun systems (and the various mutants) and Linux (both on alpha and intel machines).

Further information about the research activities at Ohio University can be found at http://www.phy.ohiou.edu/research/chaos.html. For paper on high dimensional control, presented at the St. Louis APS march meeting 1996 look here