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Chaotic Synchronization and Its Applications in Secure Communications

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Applications of Chaos and Nonlinear Dynamics in Engineering - Vol. 1

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

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.

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Notes

  1. 1.

    ℝ⟨u, y⟩ denotes the differential field generated by the field ℝ, the input u, the measurable output y, and the time derivatives of u and y.

  2. 2.

    This is a realistic assumption taking into account that in the physical circuit these variables are related to currents and voltages, which can be measurable.

  3. 3.

    Let us consider the matrix \(\mathcal{A} ={ \left [{a}_{ij}\right ]}_{1\leq i,j\leq n}\), then (see Chap. 5 in [60])

    $${ \left \|\mathcal{A}\right \|}_{\infty } := {n\max }_{1\leq i,j\leq n}\vert {a}_{ij}\vert $$

References

  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. 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. Aguilar-López, R., Martínez-Guerra, R.: Synchronization of a class of chaotic signals via robust observer design. Chaos, Solitons Fractals 37, 581–587 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anulova, S.V.: Random disturbances of the operation of control systems in the sliding mode. Autom. Rem. Contr. 47, 474–479 (1986)

    MATH  Google Scholar 

  5. Ayati, M., Khaloozadeh, H.: Stable chaos synchronisation scheme for nonlinear uncertain systems. IET Contr. Theor. Appl. 4, 437–447 (2010)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Chen, G., Dong, X.: On feedback control of chaotic continuous-time systems. IEEE Trans. Circ. Syst. 40, 591–601 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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. De Carlo, R., Zak, S., Drakunov, S.: Variable structure and sliding mode control. Control Handbook, Electrical Engineering Handbook Series (1996)

    Google Scholar 

  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. 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. 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. Feki, M.: Observer-based exact synchronization of ideal and mismatched chaotic systems. Phys. Lett. A 309, 53–60 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Feki, M.: An adaptive chaos synchronization scheme applied to secure communication. Chaos, Solitons Fractals 18, 141–148 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fotsin, H.B., Daafouz, J.: Adaptive synchronization of uncertain chaotic Colpitts oscillator based on parameter identification. Phys. Lett. A 339, 304–315 (2005)

    Article  MATH  Google Scholar 

  19. Fradkov, A.L.: Cybernetical physics: from control of chaos to quantum control. Springer, Berlin (2007)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  21. Fradkov, A.L., Nijmeijer, H., Markov, A.: Adaptive observer-based synchronisation for communications. Int. J. Bifurc. Chaos 10, 2807–2814 (2000)

    Article  MATH  Google Scholar 

  22. Garfinkel, S., Spafford, G.: Practical unix and internet security. O’ Reilly & Associates Inc., Sebastopol, CA (1996)

    Google Scholar 

  23. Gauthier, J., Hammouri, H., Othman, S.: A simple observer for nonlinear systems: aplications to bioreactors. IEEE Trans. Autom. Contr. 37, 875–880 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gavalas, G.R.: Non-linear Differential Equations of Chemically Reacting Systems. Springer, New York (1968)

    Book  Google Scholar 

  25. Ghosh, D., Banerjee, S., Chowdhury, A.: Synchronization between variable time-delayed systems and cryptography. Europhys. Lett. 80(30006), 1–6 (2007)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gray, P., Scott, S.K.: Chemical Oscillations and Instabilities. Clarendon Press, Oxford (1990)

    Google Scholar 

  28. Guo-Hui, L.: Synchronization and anti-synchronization of Colpitts oscillators using active control. Chaos, Solitons Fractals 26, 87–93 (2005)

    Article  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  31. Hua, C., Guan, X.: Synchronization of chaotic systems based on PI observer design. Phys. Lett. A 334, 382–389 (2005)

    Article  MATH  Google Scholar 

  32. Huijberts, H., Nijmeijer, H., Willems, R.: System identification in communication with chaotic systems. IEEE Trans. Circ. Syst. I 47, 800–808 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Isidori, A.: Nonlinear Control Theory. Springer, New York (1995)

    Google Scholar 

  34. Jorgensen, D.V., Aris, R.: On the dynamics of a stirred tank with consecutive reactions. Chem. Eng. Sci. 38, 45–53 (1983)

    Article  Google Scholar 

  35. Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. Series D 82, 35–45 (1960)

    Article  Google Scholar 

  36. Keller, H.: Non-linear observer design by transformation into a generalized observer canonical form. Int. J. Contr. 46, 1915–1930 (1987)

    Article  MATH  Google Scholar 

  37. Kennedy, M.P.: Chaos in Colpitts oscillator. IEEE Trans. Circ. Syst. I 41, 771–774 (1994)

    Article  Google Scholar 

  38. Khalil, H.: Nonlinear Systems, 3rd edn. Englewood Cliffs, NJ: Prentice–Hall (2002)

    Google Scholar 

  39. Krener, A.J., Isidori, A.: Linearization by output injection and nonlinear observers. Syst. Contr. Lett. 3, 47–54 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  40. Levant, A.: Universal SISO sliding mode controllers with finite tine convergence. IEEE Trans. Autom. Contr. 46, 1447–1451 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  44. Luenberger, D.: An introduction to observers. IEEE Trans. Autom. Contr. 16, 592–602 (1971)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  48. Martínez-Guerra, R., Diop, S.: Diagnosis of nonlinear systems: An algebraic and differential approach. IEE Proc. Contr. Theor. Appl. 151, 130–135 (2004)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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. Martínez-Guerra, R., Yu, W.: Chaotic communication and secure communication via sliding-mode observer. Int. J. Bifur. Chaos 18, 235–243 (2008)

    Article  MATH  Google Scholar 

  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. 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. Morgül, O., Feki, M.: A chaotic masking scheme by using synchronized chaotic systems. Phys. Lett. A 251, 169–176 (1999)

    Article  Google Scholar 

  56. Morgül, O., Solak, E.: Observed based synchronization of chaotic systems. Phys. Rev. E 54, 4803–4811 (1996)

    Article  Google Scholar 

  57. Nijmeijer, H., Mareels, I.M.Y.: An observer looks at synchronization. IEEE Trans. Circ. Syst. I 44, 882–890 (1997)

    Article  MathSciNet  Google Scholar 

  58. Pecora, L.M., Carrol, T.L.: Synchronization in chaotic systems. Phys. Rev. A 64, 821–824 (1990)

    Google Scholar 

  59. Pérez, G., Cerdeira, H.A.: Extracting messages masked by chaos. Phys. Rev. Lett. 74, 1970–1973 (1995)

    Article  Google Scholar 

  60. Poznyak, A.S.: Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, vol. 1, pp. 77–212. Elsevier (2008)

    Google Scholar 

  61. Raghavan, S., Hedrick, J.: Observer design for a class of nonlinear systems. Int. J. Contr. 59, 515–528 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  62. Röbenack, K., Lynch, A.F.: High-gain nonlinear observer design using the observer canonical form. IET Contr. Theor. Appl. 1, 1574–1579 (2007)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  64. Slotine, J., Hedricks, J., Misawa, E.: On sliding observers for nonlinear systems. J. Dyn. Meas. Contr. 109, 245–252 (1987)

    Article  MATH  Google Scholar 

  65. Soroush, M., Tyner, D., Grady, M.: Adaptive temperature control of multiproduct jacketed reactors. Ind. Eng. Chem. Res. 38, 4337–4344 (1999)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  68. Tao, Y.: Chaotic secure communication systems history and new results. Telecommun. Rev. 9, 597–634 (1999)

    Google Scholar 

  69. Ushio, Y.: Synthesis of synchronized chaotic systems based on observers. Int. J. Bifurc. Chaos 9, 541–546 (1999)

    Article  MATH  Google Scholar 

  70. Wang, C., Ge, S.: Adaptive backstepping control of uncertain Lorenz system. Int. J. Bifurc. Chaos 11, 1115–1119 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  71. Wang, F., Liu, C.: A new criterion for chaos and hyperchaos synchronization using linear feedback control. Phys. Lett. A 360, 274–278 (2006)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Zhu, F.: Observer-based synchronization of uncertain chaotic systems and its application to secure communications. Chaos, Solitons Fractals 40, 2384–2391 (2009)

    Article  MATH  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Departamento de Control Automático, CINVESTAV-IPN, Av. IPN 2508, San Pedro Zacatenco, 07360, Distrito Federal, México, México

    Rafael Martínez-Guerra, Juan L. Mata-Machuca & Andrés Rodríguez-Bollain

  2. Departamento de Biotecnolog y Bioingeniería, CINVESTAV-IPN, Av. IPN 2508, San Pedro Zacatenco, 07360, Distrito Federal, México, México

    Ricardo Aguilar-López

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  1. Rafael Martínez-Guerra
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  2. Juan L. Mata-Machuca
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  3. Ricardo Aguilar-López
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  4. Andrés Rodríguez-Bollain
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Correspondence to Rafael Martínez-Guerra .

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Editors and Affiliations

  1. Politechnico di Torino, Department of Mathematics, Corso Duca degli Abruzzi 24, Torino, 10129, Italy

    Santo Banerjee

  2. and Technology, Department of Physics, Camellia School of Engineering, Nadibag PO Kajipara Barasat, Kolkata, 700124, India

    Mala Mitra

  3. , Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, Torino, 10129, Italy

    Lamberto Rondoni

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Martínez-Guerra, R., Mata-Machuca, J.L., Aguilar-López, R., Rodríguez-Bollain, A. (2011). Chaotic Synchronization and Its Applications in Secure Communications. In: Banerjee, S., Mitra, M., Rondoni, L. (eds) Applications of Chaos and Nonlinear Dynamics in Engineering - Vol. 1. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21922-1_8

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