Robust Synchronization of Chaotic Systems: A Proportional Integral Approach

  • Ricardo Femat
  • Gualberto Solis-Perales
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 378)


In previous chapter the chaos suppression was discussed. However, there is one more interesting problem in chaos control: the synchronization. Synchronize means to share the same time and signifies that two or more events occurs at same time. In nonlinear science diverse synchronization phenomena have been found in chaotic systems. Thus, such a problem results in very interesting dynamical phenomena and has technological applications, as in communication [1], and scientific impact as, for example, in animal gait [2],[3] or cells of human organs [4]. A continuation path for synchronization is in spatially extended systems [5] where synchronization phenomena are already being studied. Other interesting issues on synchronization is, on the one hand, the cost of synchronizing chaotic systems [6]; that is, to measure the energy required to achieve chaotic synchronization. Here, the control theory can be exploited to include cost function at design of synchronization command by computing optimal, sub-optimal and/or robust controllers [7]. On the other, the geometrical properties of synchronization are also a raising theme [8], [9]. Here, geometrical control theory can be used to compute the invariant manifolds [10]. This Chapter is related to the robust synchronization, and is centred on the robust analysis and some interpretations about robustness in synchronization. To this end we exploit the simpler controller in Chapter 2: the Proportional-Integral feedback and some approaches.


Chaotic System Linear Feedback Synchronization Error Message Signal Slave System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Wu, C.W., Chua, L.O.: A simple way to synchronize chaotic systems with application to secure communication systems. Int. J. of Bifur. and Chaos 3, 1619 (1993)zbMATHCrossRefGoogle Scholar
  2. 2.
    Solís-Perales, G.: Synchronization in polipode gait, Ms. Sc. Thesis, Universidad Autónoma de San Luis Potosí, México (1999) (in Spanish)Google Scholar
  3. 3.
    Collins, J.J., Stewart, I.N.: Hexapodal gaits and coupled nonlinear oscillator model. Biol. Cybernetics 68, 287 (1993)zbMATHCrossRefGoogle Scholar
  4. 4.
    Holstein-Rathlou, N.-H., Yip, K.-P., Sosnovtseva, O.V., Mosekilde, E.: Synchronization phenomena in nephron-nephron interaction. Chaos 11, 417 (2001)CrossRefGoogle Scholar
  5. 5.
    Bragard, J., Boccaletti, S.: Integral behavior for localized synchronization in nonidentical extended systems. Phys. Rev. E 62, 6346 (2000)CrossRefGoogle Scholar
  6. 6.
    Sarasola, C., Torrealdea, F.J., d́Anjou, A., Graña, M.: Cost of synchronizing different chaotic systems. Math. Comp. In Simulation 58, 309 (2002)zbMATHCrossRefGoogle Scholar
  7. 7.
    Zhou, K., Doyle, J.C., Glover, K.: Robust and optimal control. Prentice-Hall, USA (1996)zbMATHGoogle Scholar
  8. 8.
    Martens, M., Pécou, E., Tresser, C., Workfolk, P.: On the geometry of master-slave synchronization. Chaos 12, 316 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Josić, K.: Synchronizaiton of chaotic systems and invariant manifolds. Nonlinearity 13, 1321 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Nijmeijer, H., van der Schaft, A.: Nonlinear dynamical control systems. Springer, USA (1990)zbMATHGoogle Scholar
  11. 11.
    Perez, G., Cerdeirea, H.A.: Extracting messages masked by chaos. Phys. Rev. Lett. 74, 1970 (1995)CrossRefGoogle Scholar
  12. 12.
    Pyragas, K.: Transmission of signals via synchronization of chaotic time delay systems. Int. Jour. of Bifur. and Chaos 8, 1839 (1997)CrossRefGoogle Scholar
  13. 13.
    Rabinovich, M.I., Abarbanel, H.: The role of chaos in neural systems. Neuroscience 87, 5 (1998)CrossRefGoogle Scholar
  14. 14.
    Kapitaniak, T., Skeita, M., Ogorzalek, M.: Monotone synchronization of chaos. Int. Jour. of Bifur. and Chaos 6, 211 (1996)zbMATHCrossRefGoogle Scholar
  15. 15.
    Mosayebi, F., Qammar, H.K., Hertley, T.T.: Adaptive estimation and synchronization of chaotic systems. Phys. Lett. A 161, 255 (1991)CrossRefGoogle Scholar
  16. 16.
    Cazelles, B., Boudjema, G., Chau, N.P.: Adaptive control of systems in a noisy environment. Phys. Lett. A 196, 326 (1995)CrossRefGoogle Scholar
  17. 17.
    Zhuo, K., Doyle, J.C.: Essential of robust control. Prentice-Hall, USA (1998)Google Scholar
  18. 18.
    Kocarev, L., Parlitz, U.: General approach for chaotic synchronization with application to communication. Phys. Rev. Lett. 74, 5028 (1995)CrossRefGoogle Scholar
  19. 19.
    Femat, R., Alvarez-Ramirez, J.: Synchronization of a class of strictly different chaotic oscillators. Phys. Lett. A 236, 307 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Femat, R., Alvarez-Ramírez, J., Fernandez Anaya, G.: Adaptive synchronization of high order chaotic systems: A feedback with low parameterization. Physica D 139, 231 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Femat, R., Capistrán-Tobías, D., Solís-Perales, G.: Laplace domain controllers for chaos control. Phys. Lett. A 252, 27 (1999)CrossRefGoogle Scholar
  22. 22.
    Koslov, A.K., Shalfeev, V.D., Chua, O.L.: Exact synchronization of mismatched chaotic systems. Int. Jour. of Bifur. and Chaos 6, 569 (1996)CrossRefGoogle Scholar
  23. 23.
    Campos-Delgado, D.U., Femat, R., Martínez-López, F.J.: Laplace domain controllers for chaos control: sub-optimal approaches. IEEE Trans. Circ. and Syst. I (submitted, 2003)Google Scholar
  24. 24.
    Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, N.Y (1990)Google Scholar
  25. 25.
    Isidori, A.: Nonlinear Control Systems. Springer, Berlin (1989)zbMATHGoogle Scholar
  26. 26.
    Alvarez-Ramirez, J., Femat, R., Barreiro, A.: A PI controller with disturbance estimation. Ind. Eng. Chem. Res. 36, 3668 (1997)CrossRefGoogle Scholar
  27. 27.
    Haken, H.: Synergetic: an introduction. Springer, Berlin (1983)Google Scholar
  28. 28.
    Campos-Delgado, D.U., Femat, R., Ruiz-Velazquez, E.: Design of reduced- order controlers via H ∞ and parametric optimisation: Comparison for an active suspension system. In: Landau, I.D., Karimi, A., Hjalmarson, H. (eds.) Special isue on Design and Optmisation of restricted complexity controlers. Eur. J. Control, vol. 9, pp. 48–60 (2003)Google Scholar
  29. 29.
    Femat, R., Solís-Perales, G.: On the chaos synchronization phenomena. Phys. Let. A 262, 50 (1999)zbMATHCrossRefGoogle Scholar
  30. 30.
    Martínez-López, F.J.: Suppresion of chaos in third-order dynamical systems and robustnes analysis, Mc. Sc. Thesis, Universidad Autónoma de San Luis Potosí, S.L.P., México (2003) (in Spanish)Google Scholar
  31. 31.
    Schuster, H.G.: Deterministic chaos, an introduction, Germany (1989)Google Scholar
  32. 32.
    Brown, R., Chua, L.O.: Clarifying chaos: examples and counterexamples. Int. Jour. of Bifur. and Chaos 6, 219 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Xiaofeng, G., Lai, C.H.: On the synchronization of different chaotic oscillators. Chaos Solitons and Fractals 11, 1231 (2000)zbMATHCrossRefGoogle Scholar
  34. 34.
    Femat, R., Alvarez-Ramírez, J., González, J.: A strategy to control chaos in nonlinear driven oscillators with least prior knowledge. Phys. Lett. A 224, 271 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Pecora, L.M., Carrol, T.L.: Synchronization in Chaotic Systems. Phys. Rev. Lett. 64, 821 (1990)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: Synchronization in a population of globaly coupled oscillators. Europhys. Letts. 34, 165 (1996)CrossRefGoogle Scholar
  37. 37.
    van Vreswijk, C.: Partial synchronization in population of pulse-cupled oscillators. Phys. Rev. E 54, 5522 (1996)CrossRefGoogle Scholar
  38. 38.
    Brown, R., Kocarev, L.: A unifying definition of synchronization for dynamical systems. Chaos 10, 344 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Colonius, F., Kliemann, W.: The Dynamics of Control. Birkhäuser, Basel (2000)Google Scholar
  40. 40.
    Femat, R.: Chaos in a clas of reacting systems induced by robust asymptotic feedback. Physica D 136, 193 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Pérez, M., Albertos, P.: Regular and chaotic behavior of a PI-controled CSTR. In: Proc. of the V NOLCOS 2001, St. Petersburg, Rusia, p. 1386 (2001)Google Scholar
  42. 42.
    Alvarez-Ramírez, J.: Nonlinear feedback for controling the Lorenz equation. Phys. Rev. E 53, 2339 (1994)CrossRefGoogle Scholar
  43. 43.
    Aguirre, L.A., Billings, S.A.: Closed-loop suppresion of chaos in nonlinear driven oscillators. Nonlinear Sci. 5, 189 (1995)zbMATHCrossRefGoogle Scholar
  44. 44.
    Morari, M., Zafiriou, E.: Robust Proces Control. Prentice-Hall, USA (1989)Google Scholar
  45. 45.
    Zhou, C., Lai, C.-H.: Extracting messages masked by chaotic signals of time delay systems. Phys. Rev. E 60, 320 (1999)CrossRefGoogle Scholar
  46. 46.
    Rulkov, N.F., Suschik, M.M.: Robustnes of synchronized chaotic systems. Int. Jour. of Bifur. and Chaos 7, 625 (1997)zbMATHCrossRefGoogle Scholar
  47. 47.
    Abbott, L.F., van Vresweijk, C.: Asynchronous states in networks of pulse- coupled oscillators. Phys. Rev. E 48, 1483 (1993)CrossRefGoogle Scholar
  48. 48.
    Traub, R.D., Wong, R.S.W.: Celular mechanisms of neuronal synchronization in epilepsy. Science 216, 745 (1982)CrossRefGoogle Scholar
  49. 49.
    Short, K.M.: Steps toward unmasking secure communication. Int. J. of Bifur. and Chaos 4, 959 (1994)zbMATHCrossRefGoogle Scholar
  50. 50.
    Cuomo, K.M., Oppenheim, A.V.: Circuit implementation of synchronization chaos with applications to communications. Phys. Rev. Lett. 71, 65 (1993)CrossRefGoogle Scholar
  51. 51.
    Cuomo, K.M., Oppenheim, A.V., Strogatz, S.H.: Synchronization of Lorenz- based chaotic with application to communications. IEEE Trans. on Circuits and Sistems II 40, 626 (1993)CrossRefGoogle Scholar
  52. 52.
    Zhu, Z., Leung, H.: Adaptive identification of nonlinear systems with application to chaotic communications. IEEE Trans. Circ. and Syst. I 47, 1072 (2000)zbMATHCrossRefGoogle Scholar
  53. 53.
    Femat, R., Alvarez-Ramírez, J., Castilo Toledo, B., González, J.: On robust chaos suppresion in a clas of nondriven oscillators: Application to Chua’s circuit. IEEE Trans. Circuits and Systems I 46, 1150 (1999)zbMATHCrossRefGoogle Scholar
  54. 54.
    Teel, A., Praly, L.: Tools for semiglobal stabilization by partial state and output feedback. SIAM J. Contr. Opt. 33, 1443 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Arena, P., Baglio, S., Fortuna, L., Manganaro, G.: Experimental signal transmission using synchronised state controlled celular neural networks. Electronic Lett. 32, 362 (1996)CrossRefGoogle Scholar
  56. 56.
    Torres, L.A.B., Aguirre, L.A.: Inductorles Chuaś circuit. Electronic Letts. 36, 1915 (2000)CrossRefGoogle Scholar
  57. 57.
    Chua, L.O., Yang, T., Zhoung, G.-Q., Wu, C.W.: Adaptive synchronization of Chuaś oscillators. Int. J. of Bifur. and Chaos 6, 189 (1996)CrossRefGoogle Scholar
  58. 58.
    Nijmeijer, H., Mareels, I.M.Y.: An observer looks at synchronization. IEEE Trans. on Circuits and Systems I 44, 882 (1997)CrossRefMathSciNetGoogle Scholar
  59. 59.
    Kocarev, L., Halle, K.S., Eckert, K., Chua, L.O., Parlitz, U.: Experimental demostration of secure communication via chaotic synchronization. Int. J. of Bifur. and Chaos 2, 709 (1992)zbMATHCrossRefGoogle Scholar
  60. 60.
    Suykens, J.A., Curan, P.F., Vandewalle, J., Chua, L.O.: Nonlinear H ∞ synchronization of chaotic Luré systems. Int. Jour. of Bifur. and Chaos 7, 1323 (1997)zbMATHCrossRefGoogle Scholar
  61. 61.
    Suykens, J.A.K., Curran, P.F., Chua, L.O.: Robust nonlinear H ∞ synchronization of the chaotic Luré systems. Trans. on Circuits and Systems I 44, 891 (1997)CrossRefMathSciNetGoogle Scholar
  62. 62.
    van der Schaft, A.: L 2-gain and pasivity techniques in nonlinear control, 2nd edn. Springer, London (2000)Google Scholar

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Authors and Affiliations

  • Ricardo Femat
    • Gualberto Solis-Perales

      There are no affiliations available

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