Document Type : Research Article

Authors

1 Department of Mathematics, Payame Noor University of Iran

2 Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

Abstract

In this study, we propose a secure communication scheme based on the synchronization of two identical fractional-order chaotic systems. The fractional-order derivative is in Caputo sense, and for synchronization, we use a robust sliding-mode control scheme. The designed sliding surface is taken simply due to using special technic for fractional-order systems. Also, unlike most manuscripts, the fractional-order derivatives of state variables can be chosen differently. The stability of the error system is proved using the Lyapunov stability of fractional-order systems. Numerical simulations illustrate the ability and effectiveness of the proposed method. Moreover, synchronization results are applied to secure communication using the masking method. The security analysis demonstrates that the introduced algorithm has a large keyspace, high sensitivity to encryption keys, higher security, and the acceptable performance speed.

Keywords

 [1] Rui-Guo Li and Huai-Ning Wu. Adaptive synchronization control based on "qpso" algorithm with interval estimation for fractional-order chaotic systems and its application in secret communicatio. Nonlinear Dynamics, 92(3):935–959, 2018.
[2] Maamar Bettayeb, Ubaid Muhsen Al-Sagga, and Said Djennoune. Single channel secure communication scheme based on synchronization of fractional-order chaotic chua’s systems. Transactions of the Institute of Measurement and Control, 40(13):3651–3664, 2017.
[3] Jialin Hou, Rui Xi, Ping Liu, and Tianliang Liu. The switching fractional order chaotic system and its application to image encryption. IEEE/CAA Journal of Automatica Sinica, 4(2), 2017.
[4] Bagley R. L. and Calico R. A. Fractional-order state equations for the control of viscoelastically damped structures. Journal of Guidance, Control, and Dynamics, pages 304–311.
[5] Jenson V. G. and Jeffreys G. V. Mathematical methods in chemical engineering. In Elsevier, page 559, 1977.
[6] Vijay K. Yadav, Subir Das, Beer Singh Bhadauria, Ashok K. Singh, and Mayank Srivastava. Stability analysis, chaos control of a fractional order chaotic chemical reactor system and its function projective synchronization with parametric uncertainties. Chinese Journal of Physics, 55(3):594–605, 2017.
[7] Yong Xu, Hua Wang Yongge Li, and Bin Pei. Image encryption based on synchronization of fractional chaotic systems. Communications in Nonlinear Science and Numerical Simulation, 19:3735–3744, 2014.
[8] Ying Luo, YangQuan Chen, and Youguo Pi. Experimental study of fractional-order proportional derivative controller synthesis for fractionalorder systems. Mechatronics, 21:204–214, 2011.
[9] Sabatier J., Cugnet M., Laruelle S., Grugeon S., Sahut B., Oustaloup A., and Tarascon J. M. A fractional-order model for lead-acid battery crankability estimation. Communications in Nonlinear Science and Numerical Simulation, 15:1308–1317, 2010.
[10] Vedat Suat Erturk, Zaid M. Odibat, and Shaher Momani. Anapproximatesolutionofafractionalorder differential equation model of human t-cell lymphotropic virus i (htlv-i) infection of cd4+ t-cells. Computers and Mathematics with Applications, 62:996–1002, 2011.
[11] Aida Mojaver and Hossein Kheiri. Mathematical analysis of a class of hiv infection models of cd4+ t-cells with combined antiretroviral therapy. Applied Mathematics and Computation, 259:258– 270, 2015.
[12] Nick Laskin. Fractional market dynamics. Physica A, 287:482–492, 2000.
[13] Yongguang Yu, Han-Xiong Li, Sha Wang, and Junzhi Yud. Dynamic analysis of a fractionalorder lorenz chaotic system. Chaos, Solitons and Fractals, 42:1181–1189, 2009.
[14] Chunguang Li and Guanrong Chen. Chaos and hyperchaos in the fractional-order rösslere quations. Physica A: Statistical Mechanics and its Applications, 341:55–61, 2004.
[15] Mohammad Mostafa Asheghan, Mohammad Taghi Hamidi, Beheshti, and Mohammad Saleh Tavazoei. Robust synchronization of perturbed chen’s fractional-order chaotic systems. Communications in Nonlinear Science and Numerical Simulation, 16(2):1044–1051, 2011.
[16] Jun Guo Lu. Chaotic dynamics of the fractionalorder lü system and its synchronization. Physics Letters A, 354(4):305–311, 2006.
[17] Xiaojun Liu, Ling Hong, and Lixin Yang. Fractional-order complex t system: bifurcations, chaos control and synchronization. Nonlinear Dynamics, 75(3):586–602, 2014.
[18] Runzi Luo, Haipeng Su, and Yanhui Zeng. Synchronizationofuncertainfractional-orderchaotic systems via a novel adaptive controller. Chinese Journal of Physics, 55(2):342–349, 2017.
[19] Louis M. Pecora and Thomas L. Carroll. Synchronization in chaotic systems. PHYSICAL REVIEW LETTERS, 64:821–824, 1990.
[20] Bashir Naderi and Hossein Kheiri. Exponential synchronization of chaotic system and application in securecommunication. OptikInternational Journal for Light and Electron Optics, 127(5):2407–2412, 2016.
[21] Hossein Kheiri and Bashir Naderi. Dynamical behavior and synchronization of chaotic chemical reactors model. Iranian Journal of Mathematical Chemistry, 6(1):81–92, 2015.
[22] Hossein Kheiri and Bashir Naderi. Dynamical behavior and synchronization of hyperchaotic complex t-system. Journal of Mathematical Modelling, 3(1):15–32, 2015.
[23] Chao-JungCheng. Robustsynchronizationofuncertain unified chaotic systems subject to noise and its application to secure communication. Applied Mathematics and Computation, 219:2698– 2712, 2012.
[24] Fei Yu and Chunhua Wang. Secure communication based on a four-wing chaotic system subjectto disturbance inputs. Optik, 125:5920–5925, 2014.
[25] N.Smaoui, A.Karouma, and M.Zribi. Secure communications based on the synchronization of the hyperchaotic chen and the unified chaotic systems. Communications in Nonlinear Science and Numerical Simulation, 16:3279–3293, 2011.
[26] Amin Kajbaf, Mohammad Ali Akhaee, and Mansour Sheikhana. Fast synchronization of nonidentical chaotic modulation-based secure systems using a modified sliding mode controller. Chaos Solitons And Fractals, 84:49–57, 2016.
[27] M.Mossa Al-sawalha and M.S.M.Noorani. On anti-synchronization of chaotic systems via nonlinear control. Chaos Solitons and Fractals, 2009:170–179, 2009.
[28] Zuolei Wang. Anti-synchronization in two non-identical hyperchaotic systems with known or unknown parameters. Communications in Nonlinear Science and Numerical Simulation, 14(5):2366–2372, 2009.
[29] E.M.E labbasy and M.M.El-Dessoky. Adaptive anti-synchronization of different chaotic dynamical systems. Chaos Solitons Fractals, 42:2174– 2180, 2009.
[30] Synchronization and anti-synchronization of a hyperchaotic chen system.
[31] Sachin Bhalekar Varsha and Daftardar-Gejji. Synchronization of different fractional-order chaotic systems using active control. Communications in Nonlinear Science and Numerical Simulation, 15:3536–3546, 2010.
[32] RazminiaA. Fullstatehybridprojectivesynchronization of a novel incommensurate fractionalorder hyperchaotic system using adaptive mechanism. Indian Journal of Physics, 87(2):161–167, 2013.
[33] Agrawal S. K. and Das S. A modified adaptive control method for synchronization of some fractional chaotic systems with unknown parameters. Nonlinear Dynamics, 73:907–919, 2013.
[34] Abolhassan Razminia and Dumitru Baleanu. Complete synchronization of commensurate fractional-order chaotic systems using sliding mode control. Mechatronics, 23:873–879, 2013. [35] Kalidass Mathiyalagan, Ju H. Park, and Rathinasamy Sakthivel. Exponential synchronization for fractional-order chaotic systems with mixed uncertainties. Complexity, 21(1):114–125, 2015.
[36] P. Muthukumar and P. Balasubramaniam. Feedback synchronization of the fractional-order reverse butterfly-shaped chaotic system and its application to digital cryptography. Nonblinear Dynamics, 74:1169–1181, 2013.
[37] Mohammad Pourmahmood Aghababa. Finitetime chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique. Nonlinear Dynamic, 69:247–261, 2012.
[38] T.T. Hartley, C.F. Lorenzo, and H. Killory Qammer. Chaos in a fractional-order chua’s system. circuits and systems i: Fundamental theory and applications. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42(8):485–490, 1995.
[39] J. Tenreiro Machado, Virgini aKiryakova, and Francesco Mainardi. Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation,16(3):1140–1153, 2011.
[40] Mohammad Pourmahmood Aghababa. A novel terminalslidingmodecontrollerforaclassofnonautonomous fractional-order systems. Nonlinear Dynamics, 73(1):679–688, 2o13.
[41] Leon Glass. Synchronization and rhythmic processes in physiology. Nature, 410(6825):277–284, 2001.
[42] W.H. Deng and C.P. Li. deng. Physica A: Statistical Mechanics and its Applications, 353:61–72, 2005.
[43] Weihua Deng and Changpin Li. Synchronization of chaotic fractional chen system. Journal of the Physical Society of Japan, 74(6):1645–1648, 2005.
[44] Ping Zhou and Wei Zhu. Function projective synchronization for fractional-order chaotic systems. Nonlinear Analysis: Real World Applications, 12(2):811–816, 2012.
[45] Lj Kocarev, K. S. Halle, K. Eckert, L. O. Chua, and U. Parlitz. Experimental demonstration of secure communication via chaotic synchronization. International Journal of Bifurcation and Chaos, 2:709–713, 1992.
[46] M. Boutayeb, Mohamed Darouach, Mohamed Darouach, Hugues Rafaralahy, and Hugues Rafaralahy. Generalized state observers for chaotic synchronization and secure communication. IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications, 49(3):345–349, 2002.
[47] U. Parlitz, L.O. Chua, Lj. Kocarev, K.S. Halle, and A. Shang. Transmission of digital signals by chaotic synchronization. International Journal of Bifurcation and Chaos, 2:937–977, 1992. [48] K.M. Cuomo, A.V. Oppenheim, and S.H. Strogatz. Synchronization of lorenz-based chaotic circuitswithapplicationstocommunications. IEEE Transactions on Circuits and Systems, 40:626– 633, 1993.
[49] H. Dedieu, M.P. Kennedy, and M. Hasler. Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing chua’s circuits. IEEE Transactions on Circuits and Systems II, 40:634–642, 1993.
[50] Tao Yang and L.O. Chua. Secure communication via chaotic parameter modulation. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 43:817–819, 1996.
[51] Rafael Martinez-Guerra, Juan J.Montesinos Garcia, and Sergio M.Delfín Prieto. Secure communications via synchronization of liouvillian chaotic systems. Journal of the Franklin Institute, 353:4384–4399, 2016.
[52] Bashir Naderi, Hossein Kheiri, Aghileh Heydari, and Reza Mahini. Optimal synchronization of complex chaotic t-systems and its application in secure communication. Journal of Control, Automation and Electrical Systems, 27(4):379– 390, 2016.
[53] Chih-Chiang, Cheng Yan-SiLin, and Shiue-Wei Wu. Design of adaptive sliding mode tracking controllers for chaotic synchronization and application to secure communications. Journal of the Franklin Institute, 349:2626–2649, 2012.
[54] Chaohai Tao and Xuefei Liu. Feedback and adaptive control and synchronization of a set of chaotic and hyperchaotic systems. Chaos, Solitons and Fractals, 32(4):1572–1581, 2007. [55] Wang Xingyuan and Wang Mingjun. Adaptive synchronization for a class of high-dimensional autonomous uncertain chaotic systems. International Journal of Modern Physics C, 18(3):399– 406, 2007.
[56] Wu Xiangjun, Li Yang, and Kurths Jurgen. A new color image encryption scheme using cml and a fractional-order chaotic system. PLOS ONE, 10(3):1–28, 2015.
[57] Francis Austin, Wen Sun, and Xiaoqing Lu. Estimation of unknown parameters and adaptive synchronization of hyperchaotic systems. Communications in Nonlinear Science and Numerical Simulation, 14(12):4264–4272, 2009.
[58] Mohamad F. Haroun and T. Aaron Gulliver. A new 3d chaotic cipher for encrypting two data streams simultaneously. Nonlinear Dynamics, 81:1053–1066, 2015.
[59] Mohammad PourmahmoodAghababa. Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller. Communications in Nonlinear Science and Numerical Simulation, 17:2670–2681, 2012.
[60] Norelys Aguila-Camacho, Manuel A. DuarteMermoud, and Javier A. Gallegos. Lyapunov functions for fractional-order systems. Communications in Nonlinear Science and Numerical Simulation, 19(9):2951–2957, 2014.
[61] A.A. Kilbas, H. M. Srivastava, and J.J. Trujillo. Theory and applications of fractional differential equations. In Amsterdam: Elsevier, page 540, 2006.
[62] Hassan K. Khalil. Nonlinear systems. In Nonlinear systems, page 748, 1950.
[63] Jean-Jacques Slotine and Weiping Li. Applied nonlinear control. In New York, Prentice-Hall. [64] Mohammad Pourmahmood Aghababa and HasanPourmahmoodAghababa. Adaptivefinitetime synchronization of non-autonomous chaotic systems with uncertainty. Journal of Computational and Nonlinear Dynamics, 8(3):1–11, 2013.
[65] Douglas Robert Stinson. Cryptography: Theory and practice. page 580. CRC Press, 2005. [66] Narendra K. Pareek. Design and analysis of a novel digital image encryption schem. International Journal of Network Security and Its Applications, 4(2):95–108, 2012.
[67] Lilian Huang, Donghai Shi, and Jie Gao. The design and its application in secure communication and image encryption of a new lorenz-like system with varying parameter. Mathematical Problems in Engineering, (8973583):1–11, 2016.
[68] Abdul Hanan Abdullah, Rasul Enayatifar, and Malrey Lee. A hybrid genetic algorithm and chaotic function model for image encryption. AEU - International Journal of Electronics and Communications, 66:806–816, 2012.
[69] Ch. K. Volos, I. M. Kyprianidis, and I.N.Stouboulos. Image encryption process based on chaotic synchronization phenomena. Signal Processing, 93:1328–1340, 2013.
[70] Shannon C. E. Communication theory of secrecy systems. The Bell System Technical Journal, 28(4):656–715, 1949.