Two fundamental interdependent characterizations of chaos are
- exponential sensitivity to small perturbations (also known as the Butterfly Effect), and
- complex orbit structure (see Symbolic Dynamics).
Typically, attribute (1) is quantified by the largest Lyapunov exponent, while attribute (2) is most often quantified by an entropy (e.g., the metric entropy or the topological entropy or other definitions of entropy of ergodic systems). Both of these fundamental attributes of chaotic dynamical systems can be taken advantage of to devise chaos control methods.
Here by control we mean feedback control; i.e., measurements of the state of the system are regularly taken, and, on the basis of these measurements, some controllable parameter (or set of parameters) is adjusted so as to achieve some goal. The desirable goals may vary, and different types of goals lead to qualitatively different control problems. Important control goals include:
- Given a steadily running chaotic system, how can one improve its time averaged performance? Here performance is defined with reference to the specific function the system is meant to carry out.
- Given a chaotic system in a given state at some specific time, how can one nudge the subsequent orbit to evolve rapidly from its current state to a different target location in state space?
- How can the symbolic dynamics sequence of a chaotic orbit be controlled? (This goal is relevant to certain schemes for using chaotic signals for communication.)
For the first and third goals, the complex orbit structure is most relevant; for the second goal, exponential sensitivity is most relevant. In all cases, the fundamental attributes of chaos imply that the control goals can be potentially achieved by use of only small controlling perturbations. Thus one consequence of chaos is that control can be accomplished with low-energy/low-force controllers.
Controlling a Steadily Running Chaotic Process
The performance of a system, in general, depends on the state of the system and its history. Let \(x(t)\) denote the system state as a function of time \(t\ .\) Assume that the performance \(P\) of some steadily running process can be given as the time average of some quantity \(f\) that is a function of the system state, \(P=\langle f(x(t))\rangle\ ,\) where the angle brackets denote a time average.
For example, consider Figure 1 showing a schematic of a chemical reactor. Several pipes feed chemicals into the reactor. The inputs are stirred, and react within the reactor vessel. The reaction products flow out of the reactor in the pipe on the right of the figure. It has been shown that the chemical rate equations can lead to chaotic dynamics. That is, the state \(x(t)\) within the tank can vary chaotically in time. Here the components of the vector \(x\) are the amounts of different chemical species within the tank. Even though the inflow is presumed steady, and the stirring assumed fast enough that the chemicals are essentially uniformly distributed throughout the volume of the tank, the amounts of each chemical in the reactor can vary chaotically with time. As a consequence, the flux of different chemical species out of the tank also varies with time. A natural measure of performance is to take \(P\) to be the time averaged output flow rate of some desired reaction product. If the process is uncontrolled, the performance \(P\) will be the average of \(f(x)\) over the chaotic motion \(x_c(t)\) on the attractor of the system. The goal of the control strategy is to increase \(P\ ,\) and, in particular, to make it larger than its natural value, \(P_c=\langle f(x_c(t))\rangle\ ,\) associated with the uncontrolled chaotic motion. Now imagine that there is a control valve that can be used to regulate the flow in one of the input pipes in the figure. Because the dynamics within the reactor is chaotic, it will be possible that only small (but well-chosen) changes in the controlled flow can achieve a relatively large increase in \(P\ .\)
In order to see how this might be done, we first note that one aspect of the complex orbit structure of chaos is that typical chaotic attractors have embedded within them an infinite number of unstable periodic orbits (UPO's). Thus if one had the ability to perfectly place an initial condition on any chosen one of these UPO's, then an infinite number of different types of motion could be achieved. Moreover, if no noise were present in the system, such motions would ideally continue indefinitely in time. For each UPO, a different state trajectory \(x_i(t)\) results, where the subscript \(i\) labels the particular UPO. Consequently, each unstable periodic orbit \(i\) will have associated with it a performance \(P_i=\langle f(x_i(t))\rangle\ ,\) and these performance values will typically be different for each \(i\ .\) Moreover, it can be shown that, under suitable conditions, the performance \(P_c\) for the chaotic orbit is a weighted average of the performances \(P_i\) attained by the periodic orbits. The implication of this is that some of the \(P_i\) will be larger than \(P_c\ .\) If the system can be controlled to such a periodic orbit, then the system performance will be improved.
The next question is how to control the system so that it follows the chosen UPO rather than the chaotic orbit \(x_c(t)\ .\) One way of doing this is to wait until the ergodic uncontrolled chaotic orbit \(x_c(t)\) comes close to the desired UPO, and then give it a small kick to place it on or very near the UPO. Due to several factors (system noise, our inability to kick the orbit to a precise location, an imperfect knowledge of precisely where the desired orbit lies) the kicked orbit will not be exactly on the desired UPO. Since the UPO is unstable, the system orbit will begin to move away from the UPO. As soon as this is discerned, a small kick can be reapplied to reposition the system orbit closer to the desired UPO. By doing this continually, the orbit can be kept close to the desired UPO indefinitely. Note that for small noise and inaccuracy, the size of the kicks required to maintain the system orbit near the desired UPO is correspondingly small, approaching zero for the noiseless, absolutely accurate case. Thus we expect that, in many situations, the goal can be achieved with small controls. In comparison, one should note that in other proportional feedback control schemes where the targeted state might not be a part of the intrinsic dynamics, large controlling signals might be needed even if the system is noiseless.
This control strategy (Ott et al. 1990) has been implemented in a wide variety of experimental situations. These include implementations on
- mechanical systems (e.g., Ditto et al. 1990),
- lasers (e.g., Gills et al. 1992),
- cardiac tissue (e.g., Garfinkel et al. 1992), and
- chemical reactions such as in Figure 1 (e.g., Petrov et al. 1994).
Two issues that must be considered for such implementations are
- how to determine and locate UPO's embedded in a chaotic attractor, and
- how to make the small controlling kicks.
We now discuss these issues.
Determination of UPO's
If a very accurate analytical model of the system is available, then standard numerical techniques (e.g., Newton's method applied to the fixed point equation of the \(n\) times iterated map) can be applied to determine UPO's. However, in many cases of experimental interest, an analytical model of the system under study may not be available. In such cases, it is still possible to determine UPO's purely from data recording the trajectory of a free-running (i.e., uncontrolled) chaotic orbit (e.g., Ditto et al. 1992). The idea is to use state space embedding and attractor reconstruction techniques. Techniques for finding UPO's from data have been discussed by So et al. (1997) and by Pierson and Moss (1995), among others. Another issue is how many UPO's on the chaotic attractor need to be determined. It is relevant in this regard to point out that the number of UPO's with periods below some maximum period \(\tau_m\) typically increases exponentially with \(\tau_m\) as \(\exp(h_T\tau_m)\) for large \(\tau_m\ ,\) where \(h_T\) is the topological entropy. Thus it can become infeasible to obtain all UPO's below \(\tau_m\) if \(\tau_m\) is too large. However, Hunt and Ott (1996) have shown that, in a suitable sense, when considering the performances attained for all individual UPO's, maximal, or nearly maximal performance, typically occurs on low period UPO's. For this reason, it is seldom necessary to determine a very large number of UPO's.
UPO Control Algorithms
Having chosen a suitable UPO embedded within the attractor, it remains to specify how the small controls should be programmed to maintain the system orbit on the chosen UPO. Several ideas have been proposed to accomplish this:
- The technique employed in Ott et al. (1990) is to use the control to place the orbit on or near the stable manifold of the desired UPO;
- Romeiras et al. (1992) discuss use of the `pole-placement' technique, which is standard in control theory;
- Pyragas (1992) proposed stabilize UPO by using continuous time delay feedback;
- Dressler and Nitsche (1992) and So and Ott (1995) show how UPO control can be implemented using only time-series measurements of a single scalar state variable;
- Socolar et al. (1994) present a technique particularly useful for the control of very fast dynamics (e.g., lasers).
Targeting refers to the control goal of bringing an orbit to some desired location in state space. The basic idea is that since chaos is exponentially sensitive to small orbit perturbations, such orbit perturbations become large in a relatively short time. Moreover, if the perturbations are very carefully chosen, then there is the hope that the orbit can be efficiently directed to the target using only small controls. Several techniques for achieving this goal have been formulated (Shinbrot et al. 1990; Kostelich et al. 1993; Bollt and Meiss 1995; Schroer and Ott 1997).
An early example illustrating the possibility of targeting control with small controls is illustrated by the first encounter of a spacecraft with a comet, achieved by NASA in 1985. The key point is that the success of this project was dependent on the existence of chaos in the free (uncontrolled) motion of the spacecraft in the presence of the gravitational fields of the Earth and Moon.
In 1982, NASA scientists began to plan a spacecraft encounter with the Giaccobini-Zinner comet, expected to arrive in mid-1985. Because it would be expensive to launch a spacecraft from Earth for this purpose, a cheaper alternative was sought. In particular, a previously launched probe (that had been used for another mission) was still aloft and had a small amount of remaining fuel. Farquhar et al. (1985) showed that a comet encounter orbit, shown in Figure 2, could be designed within the fuel constraint. The orbit is very complicated as befits a chaotic system. In the final part of the orbit, it passes very close to the surface of the Moon, then is slung out, leaving the Earth-Moon system, and travels halfway across the solar system to make close observations of the comet.
Communicating with Chaos
Consider an electronic, microwave, or laser oscillator producing a chaotic signal. Typically, the orbit corresponding to such a chaotic signal is associated with a symbolic dynamical sequence. A simple example is a signal characterized by a binary symbol sequence, where the symbols are labeled 1 or 0, and the chaotic time series of a relevant variable displays a random-like sequence of large positive and negative peaks where a 1 is associated with a positive peak, and a 0 with a negative peak. Thus one has an information-containing bit sequence. The object is then to use small controls to cause the orbit to follow a binary sequence that encodes the information to be transmitted. The feasibility of this communication technique has been demonstrated by Hayes et al. (1993). The advantage of such a scheme is that the chaotic power stage that generates the waveform for transmission can remain simple and efficient (chaotic behavior occurs in simple systems), while the controls are at low power. Such a scheme may be advantageous when high-power signals are necessary and there is a practical constraint on the size or weight of the signal generator.
- Bollt, E.M., and Meiss, J.D. (1995) Targeting Chaotic Orbits to the Moon Through Recurrence, Phys. Lett. A 204:373.
- Ditto, W.L., Rauseo, S.N., and Spano, M.L. (1990) Experimental Control of Chaos, Phys. Rev. Lett. 65:3211.
- Dressler, U., and Nitsche, G. (1992) Controlling Chaos Using Time Delay Coordinates, Phys. Rev. Lett. 68:1.
- Farquhar, R., Muhonen, D., and Davis, S. A. (1985) Trajectories and Orbit Maneuvers for the ISEE-3/ICE Comet Mission (1985) J. Astronaut. Sci. 33:235.
- Garfinkel, A., Spano, M., Ditto, W., and Weiss, J. (1992) Controlling Cardiac Chaos, Science 257:1230.
- Gills, Z., Iwata, C., Roy, R., Schwartz, I., and Triandaf, I. (1992) Tracking Unstable Steady States: Extending the Stability Regime of a Multimode Laser System Phys. Rev. Lett. 69:3169.
- Hayes, S., Grebogi, C., and Ott, E. (1993) Communicating with Chaos, Phys. Rev. 70:3031.
- Hunt, B.R., and Ott, E. (1996) Optimal Periodic Orbits of Chaotic Systems, Phys. Rev. Lett. 76:2254.
- Kostelich, E., Grebogi, C., Ott, E., and Yorke, J.A. (1993) Higher Dimensional Targeting, Phys. Rev. E 47:305.
- Ott, E., Grebogi, C., and Yorke, J.A. (1990) Controlling Chaos, Phys. Rev. Lett. 64:1196.
- Petrov, V., Gaspar, V., Masere, J., and Showalter, K. (1994) Controlling Chaos in the Belousov-Zhabotinsky Reaction, Nature 361:240.
- Pierson, D., and Moss, F. (1995) Detecting Unstable Points in Noisy Chaotic and Limit Cycle Attractors with Application to Biology, Phys. Rev. Lett. 75:2124.
- Pyragas, K. (1992) Continuous Control of Chaos by Self-Controlling Feedback, Phys. Lett. A 170, 421-427.
- Pyragas, K., Tamasevicius, A. (1993) Experimental Control of Chaos by Delayed Self-Control Feedback, Phys. Lett. A 180:99.
- Romeiras, F. J., Ott, E., Grebogi, D., and Dayawansa, W.P. (1992) Controlling Chaotic Dynamical Systems, Physica D 58:165.
- Schroer, C., and Ott, E. (1997) Targeting in Hamiltonian Systems with Mixed KAM/Chaotic Phase Spaces, Chaos 7:512.
- Shinbrot, T., Ott, E., Grebogi, C., and Yorke, J.A. (1990) Using Chaos to Direct Trajectories to Targets, Phys. Rev. Lett. 65:3250.
- So, P., and Ott, E. (1995) Controlling Chaos Using Time Delay Coordinates via Stabilization of Unstable Periodic Orbits, Phys. Rev. E 51:2955.
- So, P., Ott, E., Sauer, T., Gluckman, B.J., Grebogi, C., and Schiff, S.J. (1997) Extracting Unstable Periodic Orbits from Chaotic Time Series, Phys. Rev. E 55:5398.
- Socolar, J.E.S., Sukow, D.W., and Gauthier, D.J. (1994) Stabilizing Unstable Periodic Orbits in Fast Dynamical Systems, Phys. Rev. E 50:3245.
- John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
- Timothy D. Sauer (2006) Attractor reconstruction. Scholarpedia, 1(10):1727.
- Jan A. Sanders (2006) Averaging. Scholarpedia, 1(11):1760.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
Some Use Reviews of Controlling Chaos
- Boccaletti, S., Grebogi, C., Lai, Y.-C., Mancini, H., and Maza, D. (2000) The Control of Chaos: Theory and Applications, Physics Reports 329:103.
- Chen, G., and Dong, X. (1993) From Chaos to Order: Perspectives and Methodologies in Controlling Chaotic Nonlinear Dynamical Systems, Int. J. Bif. and Chaos 3:1363.
- Ott, E. (2002) Chaos in Dynamical Systems, 2nd Edition (Cambridge University Press) Sections 10.1-10.3.
- Ott, E., and Spano, M. (1995) Controlling Chaos, Phys. Today 48(5):34.
- Shinbrot, T., Grebogi, C., Ott, E., and Yorke, J.A. (1993) Using Small Perturbations to Control Chaos, Nature 363:411.