Controlling Chaos

MAY 01, 1995

The extreme sensitivity and complex behavior that characterize chaotic systems prohibit long‐range prediction of their behavior but paradoxically allow one to control them with tiny perturbations.

DOI: 10.1063/1.881461

Edward Ott

Mark Spano

Scientists in many fields are recognizing that the systems they study often exhibit a type of time evolution known as chaos. Its hallmark is wild, unpredictable behavior, a state often perplexing and unwelcome to those who encounter it. Indeed this highly structured and deterministic phenomenon was in the past frequently mistaken for noise and viewed as something to be avoided in most applications. Recently researchers have realized that chaos can actually be advantageous in many situations and that when it is unavoidably present, it can often be controlled to obtain desired results. In this article, we present some of the basic ideas behind the feedback control of chaos, review a few illustrative experimental results and assess the status and future promise of the field.

References

1. For reviews of controlling chaos, see T. Shinbrot, C. Grebogi, E. Ott, J. A. Yorke, Nature 363, 411 (1993); https://doi.org/NATUAS
W. L. Ditto, L. Pecora, Sci. Am., August 1993, p. 78;
G. Chen, X. Dong, Int. J. Bifurcations and Chaos 3, 1363 (1993);
E. Hunt, G. Johnson, IEEE Spectrum, November 1993, p. 32;
R. Roy, Z. Gills, K. S. Thornburg, Opt. Photon. News, May 1994, p. 8.
2. W. L. Ditto, S. N. Rauseo, M. L. Spano, Phys. Rev. Lett. 65, 3211 (1990).https://doi.org/PRLTAO
3. F. Takens, in Dynamical Systems and Turbulence, D. A. Rand, L. S. Young, eds., Springer‐Verlag, Berlin (1981) p. 366.
For reviews of this topic, see H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, L. S. Tsimiring, Rev. Mod. Phys. 65, 1331 (1993); https://doi.org/RMPHAT
E. Ott, T. Sauer, J. A. Yorke, eds., Coping with Chaos, Wiley, New York (1994)
E. Ott, Chaos in Dynamical Systems, Cambridge U.P., New York (1993).
4. E. Ott, C. Grebogi, J. A. Yorke, Phys. Rev. Lett. 64, 1996 (1990). https://doi.org/PRLTAO
For a discussion of control using delay coordinate embeddings, see U. Dressler, G. Nitsche, Phys. Rev. Lett. 68, 1 (1992).https://doi.org/PRLTAO
5. E. R. Hunt, Phys. Rev. Lett. 67, 1953 (1991). https://doi.org/PRLTAO
R. W. Rollins, P. Parmananda, P. Sherard, Phys. Rev. E 47, R780 (1993).https://doi.org/PLEEE8
6. R. Roy, T. W. MurphyJr, T. D. Maier, Z. Gills, Phys. Rev. Lett. 68, 1259 (1992).https://doi.org/PRLTAO
7. V. Petrov, V. Gaspar, J. Masere, K. Showalter, Nature 361, 240 (1993). https://doi.org/NATUAS
P. Parmananda, P. Sherard, R. W. Rollins, H. D. Dewald, Phys. Rev. E 47, R3003 (1993).https://doi.org/PLEEE8
8. A. Garfinkel, M. L. Spano, W. L. Ditto, J. Weiss, Science 257, 1230 (1992).https://doi.org/SCIEAS
9. F. X. Witkowski, K. M. Kavanagh, P. A. Penkoske, R. Plonsey, M. L. Spano, W. L. Ditto, D. T. Kaplan, preprint, available from Spano.
10. S. Hayes, C. Grebogi, E. Ott, A. Mark, Phys. Rev. Lett. 73, 1781 (1994). https://doi.org/PRLTAO
S. Hayes, C. Grebogi, E. Ott, Phys. Rev. Lett. 70, 3031 (1993).https://doi.org/PRLTAO
11. I. Schwartz, I. Triandaf, Phys. Rev. A 46, 7439 (1992).https://doi.org/PLRAAN
12. T. L. Carroll, I. Triandaf, I. Schwartz, L. Pecora, Phys. Rev. A 46, 6189 (1992). https://doi.org/PLRAAN
Z. Gills, C. Iwata, R. Roy, I. B. Schwartz, I. Triandaf, Phys. Rev. Lett. 69, 3169 (1992). https://doi.org/PRLTAO
V. Petrov, M. J. Crowley, K. Showalter, Phys. Rev. Lett. 72, 2955 (1994). https://doi.org/PRLTAO
V. In, W. L. Ditto, M. L. Spano, Phys. Rev. E 51, 2689 (1995).https://doi.org/PLEEE8
13. T. Shinbrot, E. Ott, C. Grebogi, J. A. Yorke, Phys. Rev. Lett. 65, 3215 (1990). https://doi.org/PRLTAO
T. Shinbrot, W. Ditto, C. Grebogi, E. Ott, M. Spano, J. A. Yorke, Phys. Rev. Lett. 68, 2863 (1992).https://doi.org/PRLTAO
14. R. Farquhar, D. Muhonen, L. C. Church, J. Astronaut. Sci. 33, 235 (1985).https://doi.org/JALSA6
15. For discussion of a nonfeedback technique for eliminating chaos, see A. Hubler, Helv. Phys. Acta 62, 343 (1989); https://doi.org/HPACAK
E. A. Jackson, Phys. Lett. A 151, 478 (1990) and references therein.
An example of another nonfeedback technique that attempts to entrain a chaotic system to an externally applied periodic perturbation is discussed in Z. Qu, G. Hu, G. Yang, G. Qin, Phys. Rev. Lett. 74, 1736 (1995) and references therein.https://doi.org/PRLTAO
16. S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano, W. L. Ditto, Nature 370, 615 (1994).https://doi.org/NATUAS
17. W. Yang, M. Ding, A. J. Mandell, E. Ott, Phys. Rev. E 51, 102 (1995).https://doi.org/PLEEE8
18. V. In, S. Mahan, W. L. Ditto, M. L. Spano, Phys. Rev. Lett. (1995), in press.

More about the Authors

Edward Ott. University of Maryland, College, Park.

Mark Spano. Naval Surface Warfare Center, Silver Spring, Maryland.