Synchronization of different dimensions fractional-order chaotic systems with uncertain parameters and secure communication
Resumo
In this paper, an adaptive modified function projective synchronization (AMFPS) scheme of different dimensions fractional-order chaotic systems with fully unknown parameters is presented. On the basis of fractional Lyapunov stability theory and adaptive control law, a new fractional-order controller and suitable update rules for unknown parameters are designed to realize the AMFPS of different fractional-order chaotic systems with non-identical orders and different dimensions. Theoretical analysis and numerical simulations are given to verify the validity of the proposed method. Additionally, synchronization results are applied to secure communication via modified masking method. Due to the unpredictability of the scale function matrix and using of fractional-order systems with different dimensions and unequal orders, the proposed scheme has higher security. The security analysis demonstrate that the proposed algorithm has a large key space and high sensitivity to encryption keys and it is resistance to all kind of attacks.
Downloads
Referências
M. A. Ansari, D. Arora and S. P. Ansari, Chaos control and synchronization of fractional order delay-varying computer virus propagation model, Math. Meth. Appl. Sci., 39, 1197-1205, (2016). DOI: 10.1002/mma.3565.
D. Wei, X. Wang, J. Hou and P. Liu, Hybrid projective synchronization of complex Duffing-Holmes oscillators with application to image encryption, Math. Meth. Appl. Sci., 40(12), 4259-4271, (2017). DOI: 10.1002/mma.4302.
X. Wu, Y. Li and J. Kurths, A new color image encryption scheme using CML and a fractional-order chaotic system, PLOS ONE, 10(3), 1-28, (2015). DOI:10.1371/journal.pone.0119660
S. K. Agrawal and S. Das, Projective synchronization between different fractional-order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique, Math. Meth. Appl. Sci., 37(14), 2164-2176, (2014). DOI: 10.1002/mma.2963.
H. Liang, Z. Wang, Z. Yue and R. Lu, Generalized synchronization and control for incommensurate fractional unified chaotic system and applications in secure communication, Kybernetika, 48(2), 190-205, (2012).
C. Feng, X. Lei and L. Chun-Guang, Wavelet Phase Synchronization of fractional- order Chaotic Systems, CHIN. PHYS. LETT., 29(7), 070501, (2012). DOI:10.1088/0256-307X/29/7/070501
Y. Xu, H. Wang, D. Liu and H. Huang, Sliding mode control of a class of fractional chaotic systems in the presence of parameter perturbations, J. Vib. Control, 21(3), 435-448, (2015). DOI:10.1177/1077546313486283.
R. Mainieri and J. Rehacek, Projective synchronization in three dimensional chaotic systems, Phys. Rev. Lett., 82(15), 3042-3045, (1999).
G. H. Li, Modified projective synchronization of chaotic system, Chaos Solitons Fractals, 32(5), 1786-1790, (2007). DOI:10.1016/j.chaos.2005.12.009
Y. Chen and X. Li, Function projective synchronization between two identical chaotic systems, Internat. J. Modern Phys. C, 18(5), 883-888, (2007). DOI:10.1142/S0129183107010607
H. Y. Du, Q. S. Zeng and C.H. Wang, Modified function projective synchronization of chaotic system, Chaos Solitons Fractals, 42(4), 2399-2404, (2009). DOI:10.1016/j.chaos.2009.03.120
H. T. Yau, Y. C. Pu and S. C. Li, Application of a Chaotic Synchronization System to Secure Communication, Inf. Technol. control, 41(3), 274-282, (2012). DOI:10.5755/j01.itc.41.3.1137
M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys J. R. Astron. Soc., 13(5), 529-539, (1967). DOI:10.1111/j.1365-246X.1967.tb02303.x.
Y. Li, Y. Chen and I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag Leffler stability Comput Math Appl., 59(5), 1810-1821, (2010). DOI:10.1016/j.camwa.2009.08.019.
M. A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos and R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 22(1), 650-659, (2015). DOI:10.1016/j.cnsns.2014.10.008
J. J. Slotine and W. Li, Applied nonlinear control, Prentice Hall, (1991).
K. Diethelm, N. Ford and A. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29(1), 3-22, (2002). DOI:10.1023/A:1016592219341
C. G. Li and G. R. Chen, Chaos in the fractional order Chen system and its control, Chaos Solitons Fractals, 22(3), 549-554, (2004). DOI:10.1016/j.chaos.2004.02.035
X. Wu and Y. Lu, Generalized projective synchronization of the fractional-order Chen hyperchaotic system, Nonlinear Dynam, 57(1), 25-35, (2009). DOI:10.1007/s11071-008-9416-5
D. R. Stinson, Cryptography: Theory and Practice, Boca Raton, CRC Press, (2005).
C. E. Shannon, Communication Theory of Secrecy Systems, Bell Syst. Tech. J., 28(4), 656-715, (1949). DOI:10.1002/j.1538-7305.1949.tb00928.x
N. K. Pareek, Design and analysis of a novel digital image encryption schem, Internat. J. Netw. Secur. Appl., 4(2), 95-108, (2012). DOI: 10.5121/ijnsa.2012.4207
Copyright (c) 2020 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
