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Projective Synchronization of The Modified Fractional-Order Hyperchaotic R\"{o}ssler System and Its Application in Secure Communication

Year 2021, Volume: 4 Issue: 2, 50 - 58, 30.06.2021
https://doi.org/10.32323/ujma.739649

Abstract

In this paper, we propose a new approach to investigate the chaos projective synchronization of the modified fractional-order hyperchaotic Rossler system and its application in secure communication. The proposed communication system consists of four main elements including: modulation, master system, slave system and demodulation. The main idea of this approach is to inject the bounded or unbounded message into one of the parameters of the proposed system using the exponential function. However, the way of injecting the message in the modulation parameter must not remove the hyperchaotic character of the signal sent to the slave system. The slave system adaptively synchronizes with the master system, and the information signal can be recovered. Based on the Lyapunov stability theory, an adaptation laws and adaptive control are designed to achieve projection synchronization of the modified system. Numerical simulations are performed to show the feasibility of the proposed secure communication scheme.

Supporting Institution

This research was supported by the Algerian General Directorate for Scientific Research and Technological Development (DG-RSDT).

References

  • [1] T. L. Carroll, L. M. Pecora, Synchronizing chaotic circuits, IEEE Trans. Circuits Syst., 38 (4) (1991), 453-456.
  • [2] I. S. Jesus, J. T. Machado, Fractional control of heat diffusion systems, Nonlinear Dynamics, 54 (3) (2008), 263-282.
  • [3] F. Tlacuahuac, L. T. Biegler, Optimization of fractional order dynamic chemical processing systems, Industrial and Engineering Chemistry Research, 53 (13) (2014), 5110-5127.
  • [4] R. Darling, J. Newman, On the short behaviour of porous interaction electrodes, J. of the Electrochemical Society, 144 (1997), 3057-3063.
  • [5] R. T. Hernandez, V. R. Ramirez, G. A. Iglesias-Silva, M. U. Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems, Part I: Fractional models for biological reactions, Chemical Engineering Science, 117 (2014), 217-228.
  • [6] R. L. Bagley, R. A. Calico, Fractional order state equations for the control of viscoelastically damped structures, Journal of Guid Control Dyn., 14 (2) (1991), 304-311.
  • [7] I. M. Olga, A. K. Alexey, R. H. Alexander, Generalized synchronization of chaos for secure communication: remarkable stability to noise, Physics Letters A, 374 (29) (2010), 2925-2931.
  • [8] M. S. Abdelouahab, N. Hamri, Fractional-order Hybrid Optical System and its Chaos Control Synchronization, Electronic Journal of Theoretical Physics, 11 (30) (2014), 49-62.
  • [9] E. I. Gonzalez, C. Hernandez, Double hyperchaotic encryption for security in biometric systems, Nonlinear Dynamics and Systems Theory, 13 (1) (2013), 55-68.
  • [10] T. Menacer, N. Hamri, Synchronization of different chaotic fractional-order systems via approached auxiliary system the modified Chua oscillator and the modified Van der Pol-Dufing oscillator, Electronic Journal of Theoretical Physics, 28 (25) (2011), 253-266.
  • [11] H. E. Guitian, L. U. O. Maokang, Dynamic behavior of fractional order Dufing chaotic system and its synchronization via singly active control, Appl. Math. Mech.-Engl. Ed., 33 (5) (2012), 567-582.
  • [12] Q. Gan, Y. Yang, S. Fan, Y. Wang, Synchronization of stochastic Fuzzy cellular neural networks with leakage delay based on adaptive control, Differ. Equ. Dyn. Syst., 22 (2014), 319-332.
  • [13] A. Bouzeriba, A. Boulkroune, T. Bouden, Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control, Neural Comput. Applic., (2016), 1349-1360.
  • [14] T. L. Carroll, L. M. Pecora, Synchronizing chaotic circuits, IEEE Trans. Circuits Syst., 38 (4) (1991), 453-456.
  • [15] M. Rehan, Synchronization and anti-synchronization of chaotic oscillators under input saturation, Appl. Math. Model., 37 (2013), 6829-6837.
  • [16] S. Kaouache, M. S. Abdelouahab, Generalized synchronization between two chaotic fractional non-commensurate order systems with different dimensions, Nonlinear Dynamics and Systems Theory, 18 (3) (2018), 273-284.
  • [17] R. Manieri, J. Rehacek, Projective synchronization in three-dimensional chaotic systems, Phys. Rev. Lett., 82 (15) (1999), 3042-3045.
  • [18] G. H. Li, Modified projective synchronization of chaotic system, Chaos Solitons Fractals, 32 (5) (2007), 1786-1790.
  • [19] S. Liu, F. Zhang, Complex function projective synchronization of complex chaotic system and its applications in secure communication, Nonlinear Dyn., 76 (2014), 1087-1097.
  • [20] X. Wu, H. Wang, H. Lu, Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application in secure communication, Nonlinear Anal. RWA, 13 (2012), 1441-1450.
  • [21] C. J. Cheng, Robust synchronization of uncertain unified chaotic systems subject to noise and its application to secure communication, Appl. Math. Comput., 219 (2012), 2698-712.
  • [22] W. Xiangjun, F. Zhengye, K. J¨urgen, A secure communication scheme based generalized function projective synchronization of a new 5D hyperchaotic system, Phys. Scr., 90 (4) (2015), Article ID 045210, 12 pages, doi:10.1088/0031-8949/90/4/045210 .
  • [23] S. Kaouache, M. S. Abdelouahab, Modified Projective Synchronization between Integer Or der and Fractional Order Hyperchaotic Systems, Jour. of Adv. Research in Dynamical and Control Systems, 10 (5) (2018), 96-104
Year 2021, Volume: 4 Issue: 2, 50 - 58, 30.06.2021
https://doi.org/10.32323/ujma.739649

Abstract

References

  • [1] T. L. Carroll, L. M. Pecora, Synchronizing chaotic circuits, IEEE Trans. Circuits Syst., 38 (4) (1991), 453-456.
  • [2] I. S. Jesus, J. T. Machado, Fractional control of heat diffusion systems, Nonlinear Dynamics, 54 (3) (2008), 263-282.
  • [3] F. Tlacuahuac, L. T. Biegler, Optimization of fractional order dynamic chemical processing systems, Industrial and Engineering Chemistry Research, 53 (13) (2014), 5110-5127.
  • [4] R. Darling, J. Newman, On the short behaviour of porous interaction electrodes, J. of the Electrochemical Society, 144 (1997), 3057-3063.
  • [5] R. T. Hernandez, V. R. Ramirez, G. A. Iglesias-Silva, M. U. Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems, Part I: Fractional models for biological reactions, Chemical Engineering Science, 117 (2014), 217-228.
  • [6] R. L. Bagley, R. A. Calico, Fractional order state equations for the control of viscoelastically damped structures, Journal of Guid Control Dyn., 14 (2) (1991), 304-311.
  • [7] I. M. Olga, A. K. Alexey, R. H. Alexander, Generalized synchronization of chaos for secure communication: remarkable stability to noise, Physics Letters A, 374 (29) (2010), 2925-2931.
  • [8] M. S. Abdelouahab, N. Hamri, Fractional-order Hybrid Optical System and its Chaos Control Synchronization, Electronic Journal of Theoretical Physics, 11 (30) (2014), 49-62.
  • [9] E. I. Gonzalez, C. Hernandez, Double hyperchaotic encryption for security in biometric systems, Nonlinear Dynamics and Systems Theory, 13 (1) (2013), 55-68.
  • [10] T. Menacer, N. Hamri, Synchronization of different chaotic fractional-order systems via approached auxiliary system the modified Chua oscillator and the modified Van der Pol-Dufing oscillator, Electronic Journal of Theoretical Physics, 28 (25) (2011), 253-266.
  • [11] H. E. Guitian, L. U. O. Maokang, Dynamic behavior of fractional order Dufing chaotic system and its synchronization via singly active control, Appl. Math. Mech.-Engl. Ed., 33 (5) (2012), 567-582.
  • [12] Q. Gan, Y. Yang, S. Fan, Y. Wang, Synchronization of stochastic Fuzzy cellular neural networks with leakage delay based on adaptive control, Differ. Equ. Dyn. Syst., 22 (2014), 319-332.
  • [13] A. Bouzeriba, A. Boulkroune, T. Bouden, Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control, Neural Comput. Applic., (2016), 1349-1360.
  • [14] T. L. Carroll, L. M. Pecora, Synchronizing chaotic circuits, IEEE Trans. Circuits Syst., 38 (4) (1991), 453-456.
  • [15] M. Rehan, Synchronization and anti-synchronization of chaotic oscillators under input saturation, Appl. Math. Model., 37 (2013), 6829-6837.
  • [16] S. Kaouache, M. S. Abdelouahab, Generalized synchronization between two chaotic fractional non-commensurate order systems with different dimensions, Nonlinear Dynamics and Systems Theory, 18 (3) (2018), 273-284.
  • [17] R. Manieri, J. Rehacek, Projective synchronization in three-dimensional chaotic systems, Phys. Rev. Lett., 82 (15) (1999), 3042-3045.
  • [18] G. H. Li, Modified projective synchronization of chaotic system, Chaos Solitons Fractals, 32 (5) (2007), 1786-1790.
  • [19] S. Liu, F. Zhang, Complex function projective synchronization of complex chaotic system and its applications in secure communication, Nonlinear Dyn., 76 (2014), 1087-1097.
  • [20] X. Wu, H. Wang, H. Lu, Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application in secure communication, Nonlinear Anal. RWA, 13 (2012), 1441-1450.
  • [21] C. J. Cheng, Robust synchronization of uncertain unified chaotic systems subject to noise and its application to secure communication, Appl. Math. Comput., 219 (2012), 2698-712.
  • [22] W. Xiangjun, F. Zhengye, K. J¨urgen, A secure communication scheme based generalized function projective synchronization of a new 5D hyperchaotic system, Phys. Scr., 90 (4) (2015), Article ID 045210, 12 pages, doi:10.1088/0031-8949/90/4/045210 .
  • [23] S. Kaouache, M. S. Abdelouahab, Modified Projective Synchronization between Integer Or der and Fractional Order Hyperchaotic Systems, Jour. of Adv. Research in Dynamical and Control Systems, 10 (5) (2018), 96-104
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Smail Kaouache 0000-0001-7231-9422

Publication Date June 30, 2021
Submission Date May 19, 2020
Acceptance Date June 18, 2021
Published in Issue Year 2021 Volume: 4 Issue: 2

Cite

APA Kaouache, S. (2021). Projective Synchronization of The Modified Fractional-Order Hyperchaotic R\"{o}ssler System and Its Application in Secure Communication. Universal Journal of Mathematics and Applications, 4(2), 50-58. https://doi.org/10.32323/ujma.739649
AMA Kaouache S. Projective Synchronization of The Modified Fractional-Order Hyperchaotic R\"{o}ssler System and Its Application in Secure Communication. Univ. J. Math. Appl. June 2021;4(2):50-58. doi:10.32323/ujma.739649
Chicago Kaouache, Smail. “Projective Synchronization of The Modified Fractional-Order Hyperchaotic R\"{o}ssler System and Its Application in Secure Communication”. Universal Journal of Mathematics and Applications 4, no. 2 (June 2021): 50-58. https://doi.org/10.32323/ujma.739649.
EndNote Kaouache S (June 1, 2021) Projective Synchronization of The Modified Fractional-Order Hyperchaotic R\"{o}ssler System and Its Application in Secure Communication. Universal Journal of Mathematics and Applications 4 2 50–58.
IEEE S. Kaouache, “Projective Synchronization of The Modified Fractional-Order Hyperchaotic R\"{o}ssler System and Its Application in Secure Communication”, Univ. J. Math. Appl., vol. 4, no. 2, pp. 50–58, 2021, doi: 10.32323/ujma.739649.
ISNAD Kaouache, Smail. “Projective Synchronization of The Modified Fractional-Order Hyperchaotic R\"{o}ssler System and Its Application in Secure Communication”. Universal Journal of Mathematics and Applications 4/2 (June 2021), 50-58. https://doi.org/10.32323/ujma.739649.
JAMA Kaouache S. Projective Synchronization of The Modified Fractional-Order Hyperchaotic R\"{o}ssler System and Its Application in Secure Communication. Univ. J. Math. Appl. 2021;4:50–58.
MLA Kaouache, Smail. “Projective Synchronization of The Modified Fractional-Order Hyperchaotic R\"{o}ssler System and Its Application in Secure Communication”. Universal Journal of Mathematics and Applications, vol. 4, no. 2, 2021, pp. 50-58, doi:10.32323/ujma.739649.
Vancouver Kaouache S. Projective Synchronization of The Modified Fractional-Order Hyperchaotic R\"{o}ssler System and Its Application in Secure Communication. Univ. J. Math. Appl. 2021;4(2):50-8.

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