Chaotic Synchronization and Its Applications in Secure Communications

  • Rafael Martínez-GuerraEmail author
  • Juan L. Mata-Machuca
  • Ricardo Aguilar-López
  • Andrés Rodríguez-Bollain
Part of the Understanding Complex Systems book series (UCS)


This work deals with the chaotic synchronization of a class of nonlinear systems and some applications (e.g., temperature regulation of chaotic chemical systems, secure communications). We present two approaches, the first one is based on observer design theory in a master-slave configuration, thus, chaos synchronization problem can be posed as an observer design procedure, where the coupling signal is viewed as measurable output and the slave system is regarded as observer. Some results of the differential and algebraic framework are applied in order to determine if the system is observable and parameters are identifiable with the available output. As applications of this technique we show the synchronization and parameter estimation of the Colpitts oscillator considered as a Chaotic Liouvillian System (CLS) in a real-time implementation; other example is shown by using an sliding-mode uncertainty observer which allows us to achieve secure communications. The second approach is related with the feedback control design, the aim of this technique is the synthesis of a robust control law for the control of CLS, we apply this method to a class of chaotic Liouvillian chemical systems with success.


Chaotic System Slave System Chaotic Synchronization Reduce Order Observer Colpitts Oscillator 
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.



Juan L. Mata-Machuca and Andrés Rodríguez-Bollain are grateful with CONACyT (Mexico) for the corresponding postgraduate scholarships.


  1. [1].
    Aguilar-Iba\(\tilde{\text{ n}}\)ez, C., Martínez-Guerra, R., Aguilar-López, R., Mata-Machuca, J.L.: Synchronization and parameter estimations of an uncertain Rikitake system. Phys. Lett. A 374, 3625–3628 (2010)Google Scholar
  2. [2].
    Aguilar-López, R., Femat, R., Martínez-Guerra, R.: Importance of chaos synchronization on technology and science. In: Banerjee, S. (ed.) Chaos Synchronization and Cryptography for Secure Communications: Applications for Encryption, pp. 210–246, IGI Global (2010)Google Scholar
  3. [3].
    Aguilar-López, R., Martínez-Guerra, R.: Synchronization of a class of chaotic signals via robust observer design. Chaos, Solitons Fractals 37, 581–587 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4].
    Anulova, S.V.: Random disturbances of the operation of control systems in the sliding mode. Autom. Rem. Contr. 47, 474–479 (1986)zbMATHGoogle Scholar
  5. [5].
    Ayati, M., Khaloozadeh, H.: Stable chaos synchronisation scheme for nonlinear uncertain systems. IET Contr. Theor. Appl. 4, 437–447 (2010)MathSciNetCrossRefGoogle Scholar
  6. [6].
    Boutayeb, M., Darouach, M., Rafaralahy, H.: Generalized state observers for chaotic synchronization and secure communication. IEEE Trans. Circ. Syst. I 49, 345–349 (2002)MathSciNetCrossRefGoogle Scholar
  7. [7].
    Bowong, S.: Stability analysis for the synchronization of chaotic systems with different order: application to secure communications. Phys. Lett. A 326, 102–113 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8].
    Chelouah, A.: Extensions of differential flat fields and Liouvillian systems. Proceedings of the 36th IEEE Conference Decision Control, San Diego, CA, pp. 4268–4273 (1997)Google Scholar
  9. [9].
    Chen, G., Dong, X.: On feedback control of chaotic continuous-time systems. IEEE Trans. Circ. Syst. 40, 591–601 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10].
    Cherrier, E., Boutayeb, M., Ragot, J.: Observers-based synchronization and input recovery for a class of nonlinear chaotic models. IEEE Trans. Circ. Syst. I 53, 1977–1988 (2006)CrossRefGoogle Scholar
  11. [11].
    Cuomo, K.M., Oppenheim, A.V., Strogatz, S.H.: Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Trans. Circ. Syst. I 40, 626–633 (1993)Google Scholar
  12. [12].
    De Carlo, R., Zak, S., Drakunov, S.: Variable structure and sliding mode control. Control Handbook, Electrical Engineering Handbook Series (1996)Google Scholar
  13. [13].
    Drakunov, S.V., Utkin, V.: Sliding mode observers: Tutorial. In: Proceedings of the 34th IEEE Conference Decision Control (CDC), pp. 3376–3378 (1995)Google Scholar
  14. [14].
    Elabbasy, E., Agiza, H., El-Dessoky, M.: Global chaos synchronization for four scroll attractor by nonlinear control. Sci. Res. Essay 1, 65–71 (2006)Google Scholar
  15. [15].
    Emadzadeh, A., Haeri, M.: Global Synchronization of two different chaotic systems via nonlinear control. In: Proceedings of the ICCAS, Gyeonggi-Do, Korea (2005)Google Scholar
  16. [16].
    Feki, M.: Observer-based exact synchronization of ideal and mismatched chaotic systems. Phys. Lett. A 309, 53–60 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17].
    Feki, M.: An adaptive chaos synchronization scheme applied to secure communication. Chaos, Solitons Fractals 18, 141–148 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18].
    Fotsin, H.B., Daafouz, J.: Adaptive synchronization of uncertain chaotic Colpitts oscillator based on parameter identification. Phys. Lett. A 339, 304–315 (2005)zbMATHCrossRefGoogle Scholar
  19. [19].
    Fradkov, A.L.: Cybernetical physics: from control of chaos to quantum control. Springer, Berlin (2007)zbMATHGoogle Scholar
  20. [20].
    Fradkov, A.L., Andrievsky, B., Evans, R.J.: Adaptive observer-based synchronization of chaotic systems with first-order coder in the presence of information constraints. IEEE Trans. Circ. Syst. I 55, 1685–1694 (2008)MathSciNetCrossRefGoogle Scholar
  21. [21].
    Fradkov, A.L., Nijmeijer, H., Markov, A.: Adaptive observer-based synchronisation for communications. Int. J. Bifurc. Chaos 10, 2807–2814 (2000)zbMATHCrossRefGoogle Scholar
  22. [22].
    Garfinkel, S., Spafford, G.: Practical unix and internet security. O’ Reilly & Associates Inc., Sebastopol, CA (1996)Google Scholar
  23. [23].
    Gauthier, J., Hammouri, H., Othman, S.: A simple observer for nonlinear systems: aplications to bioreactors. IEEE Trans. Autom. Contr. 37, 875–880 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24].
    Gavalas, G.R.: Non-linear Differential Equations of Chemically Reacting Systems. Springer, New York (1968)CrossRefGoogle Scholar
  25. [25].
    Ghosh, D., Banerjee, S., Chowdhury, A.: Synchronization between variable time-delayed systems and cryptography. Europhys. Lett. 80(30006), 1–6 (2007)MathSciNetGoogle Scholar
  26. [26].
    Ghosh, D., Chowdhury, A., Saha, P.: On the various kinds of synchronization in delayed Duffing-Van der Pol system. Commun. Nonlinear Sci. Numer. Simulat. 13, 790–803 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27].
    Gray, P., Scott, S.K.: Chemical Oscillations and Instabilities. Clarendon Press, Oxford (1990)Google Scholar
  28. [28].
    Guo-Hui, L.: Synchronization and anti-synchronization of Colpitts oscillators using active control. Chaos, Solitons Fractals 26, 87–93 (2005)zbMATHCrossRefGoogle Scholar
  29. [29].
    Harb, A., Ahmad, W.: Chaotic systems synchronization in secure communication systems. Proc. World Congress Computer Science Computer Engineering, and Applied Computing. Las Vegas (2006)Google Scholar
  30. [30].
    He, Z., Li, K., Yuang, L., Sui, Y.: A robust digital structure communications scheme based on sporadic chaos synchronization. IEEE Trans. Circ. Syst. I 47, 397–403 (2000)CrossRefGoogle Scholar
  31. [31].
    Hua, C., Guan, X.: Synchronization of chaotic systems based on PI observer design. Phys. Lett. A 334, 382–389 (2005)zbMATHCrossRefGoogle Scholar
  32. [32].
    Huijberts, H., Nijmeijer, H., Willems, R.: System identification in communication with chaotic systems. IEEE Trans. Circ. Syst. I 47, 800–808 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33].
    Isidori, A.: Nonlinear Control Theory. Springer, New York (1995)Google Scholar
  34. [34].
    Jorgensen, D.V., Aris, R.: On the dynamics of a stirred tank with consecutive reactions. Chem. Eng. Sci. 38, 45–53 (1983)CrossRefGoogle Scholar
  35. [35].
    Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. Series D 82, 35–45 (1960)CrossRefGoogle Scholar
  36. [36].
    Keller, H.: Non-linear observer design by transformation into a generalized observer canonical form. Int. J. Contr. 46, 1915–1930 (1987)zbMATHCrossRefGoogle Scholar
  37. [37].
    Kennedy, M.P.: Chaos in Colpitts oscillator. IEEE Trans. Circ. Syst. I 41, 771–774 (1994)CrossRefGoogle Scholar
  38. [38].
    Khalil, H.: Nonlinear Systems, 3rd edn. Englewood Cliffs, NJ: Prentice–Hall (2002)Google Scholar
  39. [39].
    Krener, A.J., Isidori, A.: Linearization by output injection and nonlinear observers. Syst. Contr. Lett. 3, 47–54 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40].
    Levant, A.: Universal SISO sliding mode controllers with finite tine convergence. IEEE Trans. Autom. Contr. 46, 1447–1451 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41].
    Li, D., Lu, J., Wu, X.: Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems. Chaos, Solitons Fractals 23, 79–85 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  42. [42].
    Lian, K.Y., Chiang, T.S., Chiu, C.S., Liu, P.: Synthesis of fuzzy model-based designs to synchronization and secure communication for chaotic systems. IEEE Trans. Syst. Man Cybern. B 31, 66–83 (2001)CrossRefGoogle Scholar
  43. [43].
    Liao, T.L., Huang, N.S.: An observer-based approach for chaotic synchronization with applications to secure communication. IEEE Trans. Circ. Syst. II 46, 1144–1150 (1999)zbMATHCrossRefGoogle Scholar
  44. [44].
    Luenberger, D.: An introduction to observers. IEEE Trans. Autom. Contr. 16, 592–602 (1971)CrossRefGoogle Scholar
  45. [45].
    Maggio, G.M., De Feo, O., Kennedy, M.P.: Nonlinear analysis of the Colpitts oscillator and applications to design. IEEE Trans. Circ. Syst. I 46, 1118–1130 (1999)zbMATHCrossRefGoogle Scholar
  46. [46].
    Martínez-Guerra, R., Aguilar, R., Poznyak, A.: A new robust sliding-mode observer design for monitoring in chemical reactors. Trans. ASME J. Dyn. Syst. Meas. Contr. 126, 473–478 (2004)CrossRefGoogle Scholar
  47. [47].
    Martínez-Guerra, R., Cruz, J., Gonzalez, R., Aguilar, R.: A new reduced-order observer design for the synchronization of Lorenz systems. Chaos, Solitons Fractals 28, 511–517 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  48. [48].
    Martínez-Guerra, R., Diop, S.: Diagnosis of nonlinear systems: An algebraic and differential approach. IEE Proc. Contr. Theor. Appl. 151, 130–135 (2004)CrossRefGoogle Scholar
  49. [49].
    Martínez–Guerra, R., Mendoza–Camargo, J.: Observers for a class of Liouvillian and, non-differentially flat systems. IMA J. Math. Contr. Inf. 21, 493–509 (2004)Google Scholar
  50. [50].
    Martínez-Guerra, R., Poznyak, A., Díaz, V.: Robustness of high-gain observers for closed-loop nonlinear systems: theoretical study and robotics control application. Int. J. Syst. Sci. 31, 1519–1529 (2000)CrossRefGoogle Scholar
  51. [51].
    Martínez-Guerra, R., Rincón Pasaye, J.J.: Synchronization and anti-synchronization of chaotic systems: A differential and algebraic approach. Chaos, Solitons Fractals 28, 511–517 (2009)Google Scholar
  52. [52].
    Martínez-Guerra, R., Yu, W.: Chaotic communication and secure communication via sliding-mode observer. Int. J. Bifur. Chaos 18, 235–243 (2008)zbMATHCrossRefGoogle Scholar
  53. [53].
    Martínez-Guerra, R., Yu, W., Cisneros-Salda{n}a, E.: A new model-free sliding observer to synchronization problem. Chaos, Solitons Fractals 36, 1141–1156 (2008)Google Scholar
  54. [54].
    Min, L., Jing, J.: A new theorem to synchronization of unified chaotic systems via adaptive control. Chaos, Solitons Fractals 24, 1363–1371 (2004)Google Scholar
  55. [55].
    Morgül, O., Feki, M.: A chaotic masking scheme by using synchronized chaotic systems. Phys. Lett. A 251, 169–176 (1999)CrossRefGoogle Scholar
  56. [56].
    Morgül, O., Solak, E.: Observed based synchronization of chaotic systems. Phys. Rev. E 54, 4803–4811 (1996)CrossRefGoogle Scholar
  57. [57].
    Nijmeijer, H., Mareels, I.M.Y.: An observer looks at synchronization. IEEE Trans. Circ. Syst. I 44, 882–890 (1997)MathSciNetCrossRefGoogle Scholar
  58. [58].
    Pecora, L.M., Carrol, T.L.: Synchronization in chaotic systems. Phys. Rev. A 64, 821–824 (1990)Google Scholar
  59. [59].
    Pérez, G., Cerdeira, H.A.: Extracting messages masked by chaos. Phys. Rev. Lett. 74, 1970–1973 (1995)CrossRefGoogle Scholar
  60. [60].
    Poznyak, A.S.: Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, vol. 1, pp. 77–212. Elsevier (2008)Google Scholar
  61. [61].
    Raghavan, S., Hedrick, J.: Observer design for a class of nonlinear systems. Int. J. Contr. 59, 515–528 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  62. [62].
    Röbenack, K., Lynch, A.F.: High-gain nonlinear observer design using the observer canonical form. IET Contr. Theor. Appl. 1, 1574–1579 (2007)CrossRefGoogle Scholar
  63. [63].
    Schuler, H., Schmidt, C.: Calorimetric-state estimator for chemical reactors diagnosis and control: Review of methods and applications. Chem. Eng. Sci. 47, 899–908 (1992)CrossRefGoogle Scholar
  64. [64].
    Slotine, J., Hedricks, J., Misawa, E.: On sliding observers for nonlinear systems. J. Dyn. Meas. Contr. 109, 245–252 (1987)zbMATHCrossRefGoogle Scholar
  65. [65].
    Soroush, M., Tyner, D., Grady, M.: Adaptive temperature control of multiproduct jacketed reactors. Ind. Eng. Chem. Res. 38, 4337–4344 (1999)CrossRefGoogle Scholar
  66. [66].
    Suykens, J.A.K., Curran, P.F., Chua, L.O.: Robust synthesis for master-slave synchronization of Lures systems. IEEE Trans. Circ. Syst. I 46, 841–850 (1999)zbMATHCrossRefGoogle Scholar
  67. [67].
    Suykens, J.A.K., Curran, P.F., Vandewalle, J., Chua, L.O.: Robust nonlinear H synchronization of chaotic Lure’s systems. IEEE Trans. Circ. Syst. I 44, 891–940 (1999)MathSciNetCrossRefGoogle Scholar
  68. [68].
    Tao, Y.: Chaotic secure communication systems history and new results. Telecommun. Rev. 9, 597–634 (1999)Google Scholar
  69. [69].
    Ushio, Y.: Synthesis of synchronized chaotic systems based on observers. Int. J. Bifurc. Chaos 9, 541–546 (1999)zbMATHCrossRefGoogle Scholar
  70. [70].
    Wang, C., Ge, S.: Adaptive backstepping control of uncertain Lorenz system. Int. J. Bifurc. Chaos 11, 1115–1119 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  71. [71].
    Wang, F., Liu, C.: A new criterion for chaos and hyperchaos synchronization using linear feedback control. Phys. Lett. A 360, 274–278 (2006)zbMATHCrossRefGoogle Scholar
  72. [72].
    Yang, T., Chua, L.O.: Impulsive stabilization for control and synchronization of chaotic systems theory and application to secure communication. IEEE Trans. Circ. Syst. I 44, 976–988 (1997)MathSciNetCrossRefGoogle Scholar
  73. [73].
    Young, J., Farrel, J.: Observer based backstepping control using online approximation. Proceedings of the IEEE American Control Conference, Chicago, IL, pp. 3646–3650 (2000)Google Scholar
  74. [74].
    Zhu, F.: Observer-based synchronization of uncertain chaotic systems and its application to secure communications. Chaos, Solitons Fractals 40, 2384–2391 (2009)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Rafael Martínez-Guerra
    • 1
    Email author
  • Juan L. Mata-Machuca
    • 1
  • Ricardo Aguilar-López
    • 2
  • Andrés Rodríguez-Bollain
    • 1
  1. 1.Departamento de Control AutomáticoCINVESTAV-IPNDistrito Federal, MéxicoMéxico
  2. 2.Departamento de Biotecnolog y BioingenieríaCINVESTAV-IPNDistrito Federal, MéxicoMéxico

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