International Journal of Nonlinear Analysis and Applications
Application of a fractional-order financial system with disturbance in encryption and decryption
Document Type : Research Paper
Authors
Department of Mathematics, Payame Noor University, Tehran, Iran
Abstract
This paper focuses on the anti-synchronization of two identical and non-identical chaotic fractional-order financial systems with disturbance observe (FOFSDO), such that the anti-synchronization is discussed with new parameters and disturbance in the slave system by using the nonlinear active control technique. The stability of the scheme is proved by applying the Lyapunov stability method for the error system. The result of anti-synchronization with disturbance is applied in cryptography. Numerical examples and simulation analysis indicate the application and validity of the scheme and considered system.
Keywords
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[10] S. Dadras and H.R. Momeni, Control of a fractional-order economical system via sliding mode, Phys. A: Statistic. Mech. Appl. 389 (2010), no. 12, 2434–2442.
[11] M.-F. Danca, R. Garrappa, W.K.S. Tang, and G. Chen, Sustaining stable dynamics of a fractional-order chaotic financial system by parameter switching, Comput. Math. Appl. 66 (2013), no. 5, 702–716.
[12] 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 Trans. Circuits Syst. II: Analog Digital Signal Process. 40 (1993), no. 10, 634–642.
[13] P.Y. Dousseha, C. Ainamona, C.H. Miwadinoua, A.V. Monwanoua, and J.B. Chabi-Oroua, Chaos control and synchronization of a new chaotic financial system with integer and fractional order, J. Nonlinear Sci. Appl. 14 (2021), no. 6.
[14] L. Fang, T. Li, Z. Li, and R. Li, Adaptive terminal sliding mode control for anti-synchronization of uncertain chaotic systems, Nonlinear Dyn. 74 (2013), no. 4, 991–1002.
[15] A. Hajipour and H. Tavakoli, Analysis and circuit simulation of a novel nonlinear fractional incommensurate order financial system, Optik 127 (2016), no. 22, 10643–10652.
[16] M.F. Haroun and T.A. Gulliver, A new 3d chaotic cipher for encrypting two data streams simultaneously, Nonlinear Dyn. 81 (2015), no. 3, 1053–1066.
[17] T.T. Hartley and C.F. Lorenzo, Dynamics and control of initialized fractional-order systems, Nonlinear Dyn. 29 (2002), no. 1, 201–233.
[18] R. Hilfer, Applications of fractional calculus in physics, World scientific, 2000.
[19] M. Ichise, Y. Nagayanagi, and T. Kojima, An analog simulation of non-integer order transfer functions for analysis of electrode processes, J. Electroanal. Chem. Interf. Electrochem. 33 (1971), no. 2, 253–265.
[20] W. Jawaada, M.S.M. Noorani, and M.M. Al-sawalha, Robust active sliding mode anti-synchronization of hyperchaotic systems with uncertainties and external disturbances, Nonlinear Anal.: Real World Appl. 13 (2012), no. 5, 2403–2413.
[21] A. Khan and A. Tyagi, Disturbance observer-based adaptive sliding mode hybrid projective synchronisation of identical fractional-order financial systems, Pramana 90 (2018), no. 5, 1–14.
[22] L.J. Kocarev, K.S. Halle, K. Eckert, L.O. Chua, and U. Parlitz, Experimental demonstration of secure communications via chaotic synchronization, Int. J. Bifurcat. Chaos 2 (1992), no. 03, 709–713.
[23] R.C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51 (1984), no. 2, 299–307.
[24] Nick Laskin, Fractional market dynamics, Phys. A: Statistic. Mech. Appl. 287 (2000), no. 3-4, 482–492.
[25] C. Li and W. Deng, Remarks on fractional derivatives, Appl. Math. Comput. 187 (2007), no. 2, 777–784.
[26] H.-L. Li, Y.-L. Jiang, and Z.-L. Wang, Anti-synchronization and intermittent anti-synchronization of two identical hyperchaotic chua systems via impulsive control, Nonlinear Dyn. 79 (2015), no. 2, 919–925.
[27] R. Mart´ınez-Guerra, J.J.M. Garc´ıa, and S.M.D. Prieto, Secure communications via synchronization of liouvillian chaotic systems, J. Franklin Inst. 353 (2016), no. 17, 4384–4399.
[28] D. Matignon, Stability results for fractional differential equations with applications to control processing, Comput. Engin. Syst. Appl., vol. 2, Citeseer, 1996, pp. 963–968.
[29] P. Muthukumar, P. Balasubramaniam, and K. Ratnavelu, A novel cascade encryption algorithm for digital images based on anti-synchronized fractional order dynamical systems, Multimedia Tools Appl. 76 (2017), no. 22, 23517– 23538.
[30] B. Naderi and H. Kheiri, Exponential synchronization of chaotic system and application in secure communication,Optik 127 (2016), no. 5, 2407–2412.
[31] B. Naderi, H. Kheiri, A. Heydari, and R. Mahini, Optimal synchronization of complex chaotic t-systems and its
application in secure communication, J. Control Autom. Electric. Syst. 27 (2016), no. 4, 379–390.
[32] U. Parlitz, L.O. Chua, L.J. Kocarev, K.S. Halle, and A. Shang, Transmission of digital signals by chaotic synchronization, Int. J. Bifurc. Chaos 2 (1992), no. 04, 973–977.
[33] M. Srivastava, S. Agrawal, and S. Das, Reduced-order anti-synchronization of the projections of the fractional order hyperchaotic and chaotic systems, Open Phys. 11 (2013), no. 10, 1504–1513.
[34] M. Srivastava, S.P. Ansari, S.K. Agrawal, S. Das, and A.Y.T. Leung, Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method, Nonlinear Dyn. 76 (2014), no. 2, 905–914.
[35] O.I. Tacha, J.M. Munoz-Pacheco, E. Zambrano-Serrano, I.N. Stouboulos, and V.-T. Pham, Determining the chaotic behavior in a fractional-order finance system with negative parameters, Nonlinear Dyn. 94 (2018), no. 2, 1303–1317.
[36] H. Taghvafard and G.H. Erjaee, Phase and anti-phase synchronization of fractional order chaotic systems via active control, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 10, 4079–4088.
[37] C. Tao and X. Liu, Feedback and adaptive control and synchronization of a set of chaotic and hyperchaotic systems, Chaos Solitons Fractals 32 (2007), no. 4, 1572–1581.
[38] S. Wang, S. He, A. Yousefpour, H. Jahanshahi, R. Repnik, and M. Perc, Chaos and complexity in a fractional-order financial system with time delays, Chaos Solitons Fractals 131 (2020), 109521.
[39] X. Wang and M. Wang, Adaptive synchronization for a class of high-dimensional autonomous uncertain chaotic systems, Int. J. Modern Phys. C 18 (2007), no. 03, 399–406.
[40] Z Wang and X. Huang, Synchronization of a chaotic fractional order economical system with active control, Procedia Engin. 15 (2011), 516–520.
[41] Z. Wang, X. Huang, and H. Shen, Control of an uncertain fractional order economic system via adaptive sliding mode, Neurocomput. 83 (2012), 83–88.
[42] B. Xin and J. Zhang, Finite-time stabilizing a fractional-order chaotic financial system with market confidence, Nonlinear Dyn. 79 (2015), no. 2, 1399–1409.
[43] Y. Xu, H. Wang, Y. Li, and B. Pei, Image encryption based on synchronization of fractional chaotic systems, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), no. 10, 3735–3744.
[44] V.K. Yadav, S.K. Agrawal, M. Sirvestava, and S. Das, Phase and anti-phase synchronizations of fractional order hyperchaotic systems with uncertainties and external disturbances using nonlinear active control method, Int. J. Dyn. Control 5 (2017), 259–268.
[45] T. Yang and L.O. Chua, Secure communication via chaotic parameter modulation, IEEE Trans. Circuits Syst. I: Fund. Theory Appl. 43 (1996), no. 9, 817–819.
[46] F. Yu and C. Wang, Secure communication based on a four-wing chaotic system subject to disturbance inputs, Optik 125 (2014), no. 20, 5920–5925.
[47] L. Yuan, S. Zheng, and Z. Alam, Dynamics analysis and cryptographic application of fractional logistic map, Nonlinear Dyn. 96 (2019), no. 1, 615–636.
[48] Z. Zhang, J. Zhang, F. Cheng, and F. Liu, A novel stability criterion of time-varying delay fractional-order financial systems based a new functional transformation lemma, Int. J. Control Autom. Syst. 17 (2019), no. 4, 916–925.
)
Volume 14, Issue 4
April 2023Pages 125-138
- Receive Date: 14 June 2022
- Accept Date: 06 September 2022