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Using small perturbations to control chaos

Nature volume 363, pages 411–417 (1993)Cite this article

Abstract

The extreme sensitivity of chaotic systems to tiny perturbations (the ‘butterfly effect’) can be used both to stabilize regular dynamic behaviours and to direct chaotic trajectories rapidly to a desired state. Incorporating chaos deliberately into practical systems therefore offers the possibility of achieving greater flexibility in their performance.

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References

  1. Dyson, F. Infinite in All Directions, 183–184 (Harper and Row, New York, 1988).

    Google Scholar 

  2. Ott, E., Grebogi, C. & Yorke J. A. Phys. Rev. Lett. 64, 1196–1199 (1990).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  3. Ott, E., Grebogi, C. & Yorke, J. A. in Chaos: Soviet-American Perspectives on Nonlinear Science (ed. Campbell, D. K.) 153–172 (Am. Inst. Phys., New York, 1990).

    Google Scholar 

  4. Ott, E., Romeiras, F. J., Grebogi, C. & Dayawansa, W. P. Physica D 58, 165–192 (1992).

    MathSciNet  Google Scholar 

  5. Auerbach, D., Grebogi, C., Ott, E. & Yorke, J. A. Phys. Rev. Lett. 69, 3479–3482 (1992).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  6. Fowler, T. B. IEEE Trans. Auto. Control 34, 201–205 (1989).

    Article  Google Scholar 

  7. Dressler, U. & Nitsche, G. Phys. Rev. Lett 68, 1–4 (1992).

    Article  ADS  CAS  Google Scholar 

  8. Pyragas, K. Phys. Lett. A170, 421–428 (1992).

    Article  Google Scholar 

  9. Ditto, W. L., Rauseo, S. N. & Spano, M. L. Phys. Rev. Lett. 65, 3211–3214 (1990).

    Article  ADS  CAS  Google Scholar 

  10. Roy, R., Murphy, T. W., Maier, T. D., Gills, A. & Hunt, E. R. Phys. Rev. Lett. 68, 1259–1262 (1992).

    Article  ADS  CAS  Google Scholar 

  11. Gills, Z., Iwata, C., Roy, R., Schwartz, I. B. & Triandaf, I. Phys. Rev. Lett. 69, 3169–3172 (1992).

    Article  ADS  CAS  Google Scholar 

  12. Bielawski, S., Bouazaoui, M., Derozier, D. & Glorieux, P. Proc. Nonlinear Dynam. Optical Syst. Topical Mtg, Alpbach, Austria, June 22–26 (Optical Society of America, Washington, DC 1992).

    Google Scholar 

  13. Bielawski, S., Bouazaoui, M., Derozier, D. & Glorieux, P. Stabilization and Characterization of Unstable Steady States in a Laser (preprint, 1992).

    Google Scholar 

  14. Reyl, C., Flepp, L., Badii, R. & Brun, E. Control of NMR-laser Chaos in High-dimensional Embedding Space (preprint, 1992).

    Google Scholar 

  15. Hunt, E. R. Phys. Rev. Lett. 67, 1953–1955 (1991).

    Article  ADS  CAS  Google Scholar 

  16. Singer, J., Wang, Y. Z. & Bau, H. H. Phys. Rev. Lett. 66, 1123–1125 (1991).

    Article  ADS  CAS  Google Scholar 

  17. Garfinkel, A., Spano, M. L., Ditto, W. L. & Weiss, J. N. Science 257, 1230–1235 (1992).

    Article  ADS  CAS  Google Scholar 

  18. Abed, E. H., Wang, H. O. & Lee, H.-C. Proc. 1992 Am. Control Conf., 2236–2237 (Chicago, 1992).

    Google Scholar 

  19. Wang, H. O. & Abed, E. H. Proc. 2nd IFAC Workshop Syst. Struct. Control. 494–497 (Prague, 1992).

    Google Scholar 

  20. Wang, H. O. & Abed, E. H. Proc. 2nd IFAC Nonlinear Control Sys. Design Symp., 57–62 (Bordeaux, France, June 1992).

    Google Scholar 

  21. Schwartz, I. B. & Triandaf, I. Phys. Rev. A46, 7439–7444 (1992).

    Article  ADS  CAS  Google Scholar 

  22. Carroll, T., Triandaf, I., Schwartz, I. B. & Pecora, L. Phys. Rev. A46, 6189–6192 (1992).

    Article  ADS  CAS  Google Scholar 

  23. Chen, G. & Dong, X. J. Circ. Syst. Comput. 3 (in the press).

  24. Genesio, R. & Tesi, A. Automatica 28, 531–548 (1992).

    Article  Google Scholar 

  25. Tél, T. J. Phys. A24, L1359–L1368 (1992).

    ADS  Google Scholar 

  26. Bloch, A. M. & Marsden, J. E. Theor. comput. Fluid Dynam. 1, 179–190 (1989).

    Article  ADS  Google Scholar 

  27. Ding, M. & Kelso, J. A. S. in Measuring Chaos in the Human Brain (eds Duke, D. & Pritchard, W.) 17–31 (World Scientific, Singapore, 1991).

    Google Scholar 

  28. Petrov, V., Gáspár, V., Masere, J. & Showalter, K. Nature 361, 240–243 (1993).

    Article  ADS  CAS  Google Scholar 

  29. Hübinger, B., Doerner, R. & Martienssen, W. Controlling Chaotic Motion in Noisy Systems (preprint, Phys. Inst. der J. W. Goethe-Universität, Frankfurt-am-Main, 1992).

    MATH  Google Scholar 

  30. Vassiliadis, D. Physica D (in the press).

  31. Huberman, B. A. & Lumer, E. IEEE Trans. Circ. Syst. 37, 547–550 (1990).

    Article  Google Scholar 

  32. Sinha, S. & Ramaswamy, R. Physica D43, 118–128 (1990).

    MathSciNet  Google Scholar 

  33. Shinbrot, T., Ott, E., Grebogi, C. & Yorke, J. A. Phys. Rev. Lett. 65, 3215–3218 (1990).

    Article  ADS  CAS  Google Scholar 

  34. Shinbrot, T., Ott, E., Grebogi, C. & Torke, J. A. Phys. Rev. A45, 4165–4168 (1992).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  35. Shinbrot, T., Ott, E., Grebogi, C. & Yorke, J. A. Phys. Rev. Lett. 68, 2863–2866 (1992).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  36. Shinbrot, T., Grebogi, C., Ott, E. & Yorke, J. A. Phys. Lett A169, 349–354 (1992).

    Article  MathSciNet  Google Scholar 

  37. Kostelich, E., Grebogi, C., Ott, E. & Yorke, J. A. Phys. Rev. E47, 305–310 (1993).

    ADS  MathSciNet  CAS  Google Scholar 

  38. Bradley, E. in Lecture Notes Control and Information Sciences. No. 165 (eds Jacob, G. & Lamnabhi-Lagarrigue, F.) 307–325 (Springer, Berlin, 1991).

    Google Scholar 

  39. Farquhar, R., Muhonen, D. & Church, L. C. J. astronaut. Sci. 33, 235–254 (1985).

    ADS  Google Scholar 

  40. Muhonen, D., Davis, S. & Dunham, D. J. astronaut. Sci. 33, 255–273 (1985).

    ADS  Google Scholar 

  41. Dunham, D. W. & Davis, S. A. J. astronaut. Sci. 33, 275–288 (1985).

    ADS  Google Scholar 

  42. Muhonen, D. & Folta, D. J. astronaut. Sci. 33, 289–300 (1985).

    ADS  Google Scholar 

  43. Efron, L., Yeomans, D. K. & Schanzle, A. F. J. astronaut. Sci. 33, 301–323 (1985).

    ADS  Google Scholar 

  44. So, P., Ott, E. & Dayawansa, W. P. Phys. Lett. A. (in the press).

  45. Chen, G. & Dong, X. From Chaos to Order—Perspectives and Methodologies in Controlling Chaotic Dynamical Systems Tech. Rep. 92-07 (Univ. Houston, Texas, 1992).

    Google Scholar 

  46. Vincent, T. L. & Yu, J. Dynam. Control 1, 35–52 (1991).

    Article  MathSciNet  Google Scholar 

  47. Grebogi, C., Ott, E. & Yorke, J. A. Phys. Rev. A37, 1711 (1988).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  48. Auerbach, D., Cvitanović, P., Eckmann, J.-P., Gunaratne, G. & Procaccia, I. Phys. Rev. Lett. 58, 2387 (1987).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  49. Hénon, M. Commun. Math. Phys. 50, 69–77 (1976).

    Article  ADS  Google Scholar 

  50. Ogata, K. Modern Control Engineering, 2nd Ed, 347–889 (Prentice-Hall, Englewood Cliffs, 1990).

    MATH  Google Scholar 

  51. Packard, N. H., Crutchfield, J. P., Farmer, J. D. & Shaw, R. S. Phys. Rev. Lett. 45, 712–716 (1980).

    Article  ADS  Google Scholar 

  52. Eckmann, J.-P. & Ruelle, D. Rev. mod. Phys. 57, 617–656 (1985).

    Article  ADS  CAS  Google Scholar 

  53. Whitney, H. Ann. Math. 37, 645–671 (1936).

    Article  MathSciNet  Google Scholar 

  54. Takens, F. in Lecture Notes in Mathematics 898 (eds Rand, D. A. & Young, L. S.) 366–381 (Springer, Berlin, 1981).

    Google Scholar 

  55. Mehta, N. J. & Henderson, R. M. Phys. Rev. A44, 4861–4865 (1991).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  56. Schuster, H. G. Deterministic Chaos, 2nd ed., 37–38 (VCH, Weinheim, 1989).

    Google Scholar 

  57. Lorenz, E. J. atmos. Sci. 20, 130–141 (1963).

    Article  ADS  Google Scholar 

  58. Yorke, J. A., Yorke, E. D. & Mallet-Paret, J. Physica D24, 279–291 (1987).

    MathSciNet  Google Scholar 

  59. Hayes, S., Grebogi, C. & Ott, E. Phys. Rev. Lett. 70, 3031–3034 (1993).

    Article  ADS  CAS  Google Scholar 

  60. Chua, L. O., Komura, M. & Matsumoto, T. IEEE Trans. Circ. Syst 11, 1073–1118 (1986).

    ADS  Google Scholar 

  61. Hübler, A. & Lüscher, E. Naturwissenschaften 76, 67–69 (1989).

    Article  ADS  Google Scholar 

  62. Hübler, A. W. Helv. phys. Acta 62, 343–346 (1989).

    Google Scholar 

  63. Plapp, B. B. & Hübler, A. W. Phys. Rev. Lett. 65, 2302–2305 (1990).

    Article  ADS  CAS  Google Scholar 

  64. Jackson, E. A. & Hübler, A. W. Physica D44, 407–420 (1990).

    MathSciNet  Google Scholar 

  65. Jackson, E. A. Phys. Lett. A151, 478–484 (1990).

    Article  MathSciNet  Google Scholar 

  66. Jackson, E. A. Phys. Rev. A44, 4839–4853 (1991).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  67. Lüscher, E. & Hübler, A. W. Helv. phys. Acta 62, 544–551 (1989).

    Google Scholar 

  68. Georgii, R., Eberl, W. & Lüscher, E. Helv. phys. Acta 62, 290–293 (1989).

    Google Scholar 

  69. Alekseev, V. V. & Loskutov, A. Y. Sov. Phys. Dokl. 32, 1346–1348 (1987).

    Google Scholar 

  70. Pettini, M. in Dynamics and Stochastic Processes (ed. Lima, R., Streit, L. & Vilela Mendes, R. 242–250 (Springer, Berlin, 1990).

    Google Scholar 

  71. Lima, R. & Pettini, M. Phys. Rev. A41, 726–733 (1990).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  72. Braiman, Y. & Goldhirsch, I. Phys. Rev. Lett 66, 2545–2548 (1991).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  73. Azevedo, A. & Rezende, S. M. Phys. Rev. Lett. 66, 1342–1345 (1991).

    Article  ADS  CAS  Google Scholar 

  74. Herzel, H. A. Angew. Math. Mech. 68, 11–12 (1988).

    Article  Google Scholar 

  75. Fahy, S. & Hamann, D. R. Phys. Rev. Lett. 69, 761–764 (1992).

    Article  ADS  CAS  Google Scholar 

  76. Ditto, W. L. et al. Phys. Rev. Lett 63, 923–966 (1989).

    Article  ADS  CAS  Google Scholar 

  77. Lai, Y.-C., Ding, M.-Z. & Grebogi, C. Phys. Rev. E47, 86–92 (1992).

    Google Scholar 

  78. Abed, E. H., Wang, H. O. & Chen, R. Physica D (in the press).

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Author information

Author notes

  1. Troy Shinbrot

    Present address: Laboratory for Fluid Mechanics, Chaos and Mixing, Northwestern University, Evanston, Illinois, 60208, USA

Authors and Affiliations

  1. Department of Physics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland, 20742, USA

    Troy Shinbrot

  2. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland, 20742, USA

    Celso Grebogi & James A. Yorke

  3. Department of Electrical Engineering, the Department of Physics and the Institute for Systems Research, University of Maryland, College Park, Maryland, 20742, USA

    Edward Ott

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  3. James A. Yorke
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  4. Edward Ott
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Shinbrot, T., Grebogi, C., Yorke, J. et al. Using small perturbations to control chaos. Nature 363, 411–417 (1993). https://doi.org/10.1038/363411a0

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