Nonlinear Dynamics

, Volume 83, Issue 4, pp 2183–2211 | Cite as

Synchronization of uncertain constrained hyperchaotic systems and chaos-based secure communications via a novel decomposed nonlinear stochastic estimator

  • Mohamed F. Hassan
Original Paper


In this paper, a secure chaotic scheme for communications in noisy public channel is proposed. This scheme is based on the concept of carrier encryption in addition to the typical data encryption techniques. At the transmitter end, the unified chaotic system with adaptive parameter and a hyperchaotic Rössler system with uncertain parameters are coupled, constrained and used as a new hyperchaotic system which generates waveforms that are different from those of any known chaotic oscillator. After modulating one of the outputs of the system with the encrypted data signal, the outputs of the system are encrypted using a set of pre-defined encryption rules and transmitted to the receiving end through a noisy public communications channel. At the receiving end, the received outputs are decrypted and the transmitted data are retrieved by reconstructing the constrained hyperchaotic signals using the novel discrete-time iterative decomposed uncertain constrained extended Kalman filter (IDUCEKF). The proposed state estimator, besides being used to handle the estimation problem of uncertain constrained nonlinear dynamical systems, reduces the required processing time and gives good numerical performance. Simulation results are firstly presented to illustrate the applicability of the IDUCEKF in synchronizing the states of the constrained hyperchaotic system. Then, the proposed secure communication scheme is applied to transmit images, and the quality of the transmission process is assessed. The obtained results show the effectiveness of the proposed approach.


Chaos Chaotic synchronization Extended Kalman filter Secure communication State estimation 



This research was supported by Kuwait University under research Grant No. EE 02/14.


  1. 1.
    Satish, K., Jayakar, T., Tobin, C., Kadhavi, K., Murali, K.: Chaos based spread spectrum image steganography. IEEE Trans. Consum. Electron. 50, 587–590 (2004)CrossRefGoogle Scholar
  2. 2.
    Fallahi, K., Raoufi, R., Khoshhin, H.: An application of Chen system for secure chaotic communication based on extended Kalman filter and multi-shift cipher algorithm. Nonlinear Sci. Numer. Simul. 13, 763–781 (2008)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Heidari Bateni, G., Mc Gillem, C.D.: A chaotic direct sequence spread spectrum communication system. IEEE Trans. Commun. 42, 1524–1527 (1994)CrossRefGoogle Scholar
  4. 4.
    Kolumban, G., Kennedy, M.P., Chua, L.O.: The role of synchronization in digital communications using chaos—part I: fundamentals of digital communications. IEEE Trans. Circuits Syst. I 44, 927–936 (1998)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Kolumban, G., Kennedy, M.P., Chua, L.O.: The role of synchronization in digital communications using chaos—part II: chaotic modulation and chaotic synchronization. IEEE Trans. Circuits Syst. I 45, 1129–1140 (1998)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Kolumban, G., Kennedy, M.P., Chua, L.O.: The role of synchronization in digital communications using chaos—part III: performance bounds for correlation receivers. IEEE Trans. Circuits Syst. I 47, 1673–1683 (2000)CrossRefMATHGoogle Scholar
  7. 7.
    Pecora, L.M., Carroll, T.L., Johnson, G.A., Mar, D.J.: Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos 7, 520–542 (1997)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Murali, K.: Digital signal transmission with cascaded heterogeneous chaotic systems. Phys. Rev. E 63, 016217–23 (2001)CrossRefGoogle Scholar
  9. 9.
    Calitoiu, D., Oommen, B.J., Nussbaum, D.: Desynchronizing a chaotic pattern recognition neural network to model inaccurate perception. IEEE Trans. Syst. Man Cybern. B 37, 692–704 (2007)CrossRefGoogle Scholar
  10. 10.
    Ruan, H., Zhai, T., Yaz, E.E.: A chaotic secure chaotic communication scheme with extended Kalman filter based parameter estimation. Proc. IEEE Conf. Control Appl. 1, 404–408 (2003)Google Scholar
  11. 11.
    Leung, H., Zhu, Z., Ding, Z.: An aperiodic phenomenon of the extended Kalman filter in filtering noisy chaotic signals. IEEE Trans. Signal Process. 48, 1807–1810 (2000)CrossRefGoogle Scholar
  12. 12.
    Li, S.Y., Ge, Z.M.: Fuzzy modelling and synchronization of two totally different chaotic systems via novel fuzzy mode. IEEE Trans. Syst. Man Cybern. B 41, 1015–1026 (2011)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Zhang, H., Ma, T., Huang, G.-B., Wang, Z.: Robust global exponential synchronization of uncertain chaotic delayed neural network via dual-stage impulsive control. IEEE Trans. Syst. Man Cybern. B 40, 831–844 (2010)CrossRefGoogle Scholar
  14. 14.
    Ahn, C.K.: Takagi–Sugeno fuzzy receding horizon H\(\infty \) chaotic synchronization and its application to the Lorenz system. Nonlinear Anal. Hybrid Syst. 9, 1–8 (2013)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Shi, X., Wang, Z.: Robust chaos synchronization of four-dimensional energy resource system via adaptive feedback control. Nonlinear Dyn. 60, 631–637 (2010)CrossRefMATHGoogle Scholar
  16. 16.
    Lu, J.A., Wu, X.Q., Lu, J.H.: Synchronization of a unified chaotic system and the application in secure communication. Phys. Lett. A 305, 365–370 (2002)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Chen, H.H., Sheu, G.J., Lin, Y.L., Chen, C.S.: Chaos synchronization between two different chaotic systems via nonlinear feedback control. Nonlinear Anal. Theory Methods Appl. 70, 4393–4401 (2009)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Wang, X., Song, J.: Synchronization of the unified chaotic system. Nonlinear Anal. Theory Methods Appl. 69, 3409–3416 (2008)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Lin, J.S., Yan, J.J.: Adaptive synchronization for two identical generalized Lorenz chaotic systems via a single controller. Nonlinear Anal. Real World Appl. 10, 1151–1159 (2009)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Njah, A.N.: Tracking control and synchronization of the new hyperchaotic Liu system via backstepping techniques. Nonlinear Dyn. 61(1–2), 1–9 (2010)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Xiang-Jun, W., Jing-Sen, L., Guan-Rong, C.: Chaos synchronization of Rikitake chaotic attractor using the passive control technique. Nonlinear Dyn. 53(1–2), 45–53 (2008)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Ablay, G.: Sliding mode control of uncertain unified chaotic systems. Nonlinear Anal. Hybrid Syst. 3, 531–535 (2009)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Smaoui, N., Karouma, A., Zribi, M.: Secure communication based on the synchronization of the hyperchaotic Chen and the unified chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 16, 3279–3293 (2011)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Mata-Machuca, J.L., Martínez-Guerra, R., Aguilar-López, R., Aguilar-Ibañez, C.: A chaotic system in synchronization and secure communications. Commun. Nonlinear. Sci. Numer. Simul. 17, 1706–1713 (2012)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Wang, H., Han, Z.Z., Zhang, W., Xie, Q.Y.: Synchronization of unified chaotic systems with uncertain parameters based on the CLF. Nonlinear Anal. Real World Appl. 10, 715–722 (2009)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Wang, H., Han, Z.Z., Xie, Q.Y., Zhang, W.: Finite-time synchronization of uncertain chaotic systems based on CLF. Nonlinear Anal. Real World Appl. 10, 2842–2849 (2009)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Nosrati, K., Azemi, A., Pariz, N., Shokouhi-R, A.: Chaotic synchronization of Lorenz system using Unscented Kalman Filter. Proc. Chin. Control Decis. Conf. 1, 848–853 (2011)Google Scholar
  28. 28.
    Chen, S.H., Yang, Q., Wang, C.P.: Impulsive control and synchronization of unified chaotic system. Chaos Solitons Fract. 20, 153–160 (2004)MathSciNetMATHGoogle Scholar
  29. 29.
    Yang, X., Yang, Z., Nie, X.: Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication. Commun. Nonlinear Sci. Numer. Simul. 19, 1529–1543 (2014)Google Scholar
  30. 30.
    Zhu, Z., Leung, H.: Adaptive blind equalization for chaotic communication systems using extended Kalman filter. IEEE Trans. Circuits Syst. I(48), 979–987 (2001)Google Scholar
  31. 31.
    Zhu, F.: Observer based synchronization of uncertain chaotic system and its application to secure communications. Chaos Solitons Fract. 40, 2384–2391 (2009)CrossRefMATHGoogle Scholar
  32. 32.
    Yang, J., Zhu, F.: Synchronization for chaotic systems and chaos-based secure communications via both reduced-order and step-by-step sliding mode observers. Commun. Nonlinear Sci. Numer. Simul. 18, 926–937 (2013)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Chen, M., Zhou, D., Shang, Y.: A sliding mode observer based secure communication scheme. Chaos Solitons Fract. 25, 573–578 (2005)CrossRefMathSciNetMATHGoogle Scholar
  34. 34.
    Short, K.M.: Unmasking a modulated chaotic communication scheme. Int. J. Bifurc. Chaos 6, 367–375 (1996)CrossRefMATHGoogle Scholar
  35. 35.
    Perez, G., Cerdeira, H.A.: Extracting messages masked by chaos. Phys. Rev. Lett. 74, 1970–1973 (1995)CrossRefGoogle Scholar
  36. 36.
    Tlelo-Cuautle, E., Carbajal-Gomez, V.H., Obeso-Rodelo, P.J., Rangel-Magdaleno, J.J., Núñez-Pérez, J.C.: FPGA realization of a chaotic communication system applied to image processing. Nonlinear Dyn. (2015). doi: 10.1007/s11071-015-2284-x MathSciNetGoogle Scholar
  37. 37.
    Tlelo-Cuautle, E., Rangel-Magdaleno, J.J., Pano-Azucena, A.D., Obeso-Rodelo, P.J., Nunez-Perez, J.C.: FPGA realization of multi-scroll chaotic oscillators. Commun. Nonlinear Sci. Numer. Simul. 27(1), 66–80 (2015)CrossRefMathSciNetGoogle Scholar
  38. 38.
    García-Martínez, M., Campos-Cantón, E.: Pseudo-random bit generator based on multi-modal maps. Nonlinear Dyn. (2015). doi: 10.1007/s11071-015-2303-y MathSciNetGoogle Scholar
  39. 39.
    De la Fraga, L.G., Tlelo-Cuautle, E.: Optimizing the maximum Lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators. Nonlinear Dyn. 76(2), 1503–1515 (2014)CrossRefGoogle Scholar
  40. 40.
    Hassan, M.F.: Optimal Kalman filter for large scale systems using the partitioning approach. IEEE Trans. Syst. Man Cyber. 6, 714–720 (1976)CrossRefMATHGoogle Scholar
  41. 41.
    Hassan, M.F., Salut, G., Singh, M.G., Titli, A.: A decentralized computational algorithm for the global Kalman filter. IEEE Trans. Automat. Control 23, 262–268 (1978)CrossRefMathSciNetMATHGoogle Scholar
  42. 42.
    Mahmoud, M.S., Hassan, M.F., Darwish, M.G.: Large-Scale Control Systems: Theories and Techniques. Marcel Dekker Inc., New York (1985)MATHGoogle Scholar
  43. 43.
    Hassan, M.F.: A Decomposed Estimator for Constrained Uncertain Stochastic Nonlinear Systems (submitted for publication) (2015)Google Scholar
  44. 44.
    Zhou, Q., Shi, P., Liu, H., Xu, S.: Neural-network-based decentralized adaptive output-feedback control for large-scale stochastic nonlinear systems. IEEE Trans. Syst. Man Cyber. B 42, 1608–1619 (2012)CrossRefGoogle Scholar
  45. 45.
    Liu, S., Zhang, J., Jiang, Z.: Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems. Automatica 43, 238–251 (2007)CrossRefMathSciNetMATHGoogle Scholar
  46. 46.
    Mahmoud, M.S.: Decentralized Control and Filtering in Interconnected Dynamical Systems. CRC Press, Boca Raton (2011)MATHGoogle Scholar
  47. 47.
    Wang, X.Y., Wu, X.J.: Tracking control and synchronization of four-dimensional hyperchaotic Rössler system. Chaos 16, 033121 (2006)CrossRefMATHGoogle Scholar
  48. 48.
    Meditch, J.S.: Stochastic Optimal Linear Estimation and Control. McGraw-Hill, New York (1965)MATHGoogle Scholar
  49. 49.
    Reif, K., Gunther, S., Yaz, E., Unbehauen, R.: Stochastic stability of the discrete-time extended Kalman filter. IEEE Trans. Automat. Control 44, 714–728 (1999)CrossRefMathSciNetMATHGoogle Scholar
  50. 50.
    Hassan, M.F., Alrifai, M.T., Soliman, H.M., Kourah, M.A.: Observer-based controller for constrained uncertain stochastic nonlinear discrete-time systems. Int. J. Robust Nonlinear Control (2015). doi: 10.1002/rnc.3396
  51. 51.
    Luenberger, D.: Optimization by Vector Space Methods. Wiley, New York (1969)Google Scholar
  52. 52.
    Hassan, M.F.: Iterated constrained state estimator for nonlinear discrete-time systems with uncertain parameters. Int. J. Innov. Comput. Inf. Control 8, 6141–6160 (2012)Google Scholar
  53. 53.
    Sandri, M.: Numerical calculation of Lyapunov exponents. Math. J. 6, 78–84 (1996)Google Scholar
  54. 54.
    Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)CrossRefMathSciNetMATHGoogle Scholar
  55. 55.
    Holte, J.M.: Discrete Gronwall lemma and applications, MAA-NCS Meeting at the University of North Dakota 24 (2009).

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Electrical Engineering Department, College of EngineeringKuwait UniversitySafatKuwait

Personalised recommendations