Exploring hyperchaotic synchronization of a fractional-order system without equilibrium points: A sliding mode control approach

Document Type : Research Article

Authors

• R.A. Meskine
• S. Kaouache

Laboratory of mathematics and their interactions, Department of Mathematics, Institute of Mathematics and Computer Science, Abdelhafid Boussouf University Center, Mila, Algeria.

10.22067/ijnao.2025.89779.1511

Abstract

Recently, constructing hidden attractors of chaotic systems without equi-librium point has become a key discussion point in the application fields of chaos and hyperchaos science. This paper introduces a novel hyperchaotic system without equilibrium points, distinct from existing systems that rely on the Shilnikov criterion for demonstrating hyperchaos. This study inves-tigates the qualitative properties of the system, including its hyperchaotic attractors, Poincare map, Lyapunov exponents, and Kaplan-Yorke dimen-sion. To enhance the practical applicability of this system, an integral sliding mode control method for synchronization is proposed. Lyapunov theory ensures the stability and effectiveness of the synchronization scheme.The efficiency of the approach is demonstrated by numerical simulations, which validate the potential of the system for various applications.

Keywords

• Fractional derivative
• Synchronization
• Lyapunov stability theory
• Lyapunov exponents
• Sliding mode control

Main Subjects

• Modeling and Simulation

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References

[1] Al-sawalha, M.M. Synchronization of different fractional-order chaotic systems using modify adaptive sliding mode control, Adv. Diff. Equ. 1 (2020), 417.
[2] Bensimessaoud, S. and Kaouache, S. A general fractional control scheme for compound combination synchronization between different fractional-order identical chaotic systems, Aust. J. Math. Anal. Appl. 20 (2) (2023), Art. 12, 9 pp.
[3] Bensimessaoud, S., Kaouache, S. and Abdelouahab, M.-S. Chaos combi-nation anti-synchronization (CCAS) of some fractional-order uncertain chaotic systems by some random noise, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 31 (2024), 199–211.
[4] Bhalekar, S. and Daftardar-Gejji, V. Synchronization of different fractional-order chaotic systems using active control, Commun. Non-linear Sci. Numer. Simul. 15 (2010), 3536–3546.
[5] Bhat, M. A. and Shikha. Complete synchronization of non-identical frac-tional order hyperchaotic systems using active control, Int. J. Autom. Control, 13(2) (2019), 140–157.
[6] Boiko, I., Fridman, I., Iriarte, R., Pisano, A. and Usai, E. Parameter tuning of second-order sliding mode controllers for linear plants with dynamic actuators, Automatica, 42 (2006), 833–839.
[7] Bouzeriba, A., Boulkroune, A. and Bouden, T. Projective synchroniza-tion of two different fractional-order chaotic systems via adaptive fuzzy control, Neural. Comput. Appl. 27 (2016), 1349–1360.

[8] Bowonga S., Kakmenib, M. and Koinac, R. Chaos synchronization and duration time of a class of uncertain systems, Math. Comput. Simul. 71(3) (2006), 212–228.
[9] Chai, Y., Chen, L., Wu, R. and Dai, J. Q − S synchronization of the fractional-order unified system, Pramana, 80 (2013), 449–461.
[10] Cook, A.E. and Roberts, P.H. The Rikitake two disk dynamo system, Math. Proc. Cambridge, 68(2), (1970), 547–569.
[11] Danca, M.F. and Kuznetsov, N. MATLAB code for Lyapunov expo-nents of fractional-order systems, Int. J. Bifurcat. Chaos, 28 (5) (2018) 1850067.
[12] Diethelm K. and Ford, N.J. Analysis of fractional differential equations, J. Math Anal. Appl. 265(2) (2002), 229–248.
[13] Dong, C. and Wang, J. Hidden and coexisting attractors in a novel 4 − D hyperchaotic system with no equilibrium point, Fractal Fract. 6(6) (2022), 306.
[14] Du, C., Liu, L. and Shi, S. Synchronization of fractional-order complex chaotic system using active control method, IEEE UEMCON. (2019), 0817–0823.
[15] El-Shahed, M. MHD of a fractional viscoelastic fluid in a circular tube, Mech. Res. Commun. 33(2) (2006), 261–268.
[16] Flores-Tlacuahuac, F. and Biegler, L.T. Optimization of fractional-order dynamic chemical processing systems, Ind. Eng. Chem. Res. 53(13) (2014), 5110–5127.
[17] Hartley, T.T., Lorenzo, C.F. and Qammar, H.K. Chaos in a fractional order Chua system, IEEE Trans. Circuits Syst. I, 42(8) (1996), 485–490.
[18] Jesus, I.S. and Machado, J.T. Fractional control of heat diffusion sys-tems, Nonlinear Dyn. 54(3) (2008), 263–282.
[19] Kaouache, S. Projective synchronization of the modified fractional-order hyperchaotic Rossler system and its application in secure communication, Univ. J. Math. Appl. (4) (2021), 50–58.

[20] Kaouache, S. General method for hybrid projective combination synchro-nization of a class of nonlinear fractional-order chaotic systems, Int. J. Appl. Math. 36(4) (2023), 509–520.
[21] Kaouache, S. and Abdelouahab, M.S. Modified projective synchroniza-tion between integer order and fractional order hyperchaotic systems, JARDCS 10(5) (2018), 96–104.
[22] Kaouache, S. and Abdelouahab, M.S. Generalized synchronization be-tween two chaotic fractional non-commensurate order systems with dif-ferent dimensions, Nonlinear Dyn. Syst. Theory. 18 (2018), 273–284.
[23] Kaouache, S. and Abdelouahab, M.S. Inverse matrix projective syn-chronization of novel hyperchaotic system with hyperbolic sine function non-linearity, Dyn. Contin. Discrete Impuls. Syst. Ser. B: Appl. Alg. 27 (2020), 145–154.
[24] Kaouache, S., Abdelouahab, M.S. and Bououden, R. Reduced generalized combination synchronization between two n-dimentional integer-order hyperchaotic systems and one m-dimentional fractional-order chaotic system, Aust. J. Math. Anal. Appl. 17(2) (2020), Art. 19, 8 pp.
[25] Kaouache, S. and Bouden, T. Modified hybrid synchronization of iden-tical fractional hyperchaotic systems with incommensurate order, Dyn. Contin. Discrete Impuls. Syst. Ser. A: Math. Anal. 28 (2021), 25–36.
[26] Kaouache, S., Hamri, N.D., Hacinliyan, A.S., Kandiran, E., Deruni, B. and Keles, A.C. Increaszd order generalized combination synchronization of non identical dimensional fractional-order systems by introducing dif-ferent observable variable functions, Nonlinear Dyn. Syst. Theory, 20 (2020), 307–315.
[27] Kiani-B, A., Fallahi, L., Pariz, K. and Leung, H. A chaotic secure com-munication scheme using fractional chaotic systems based on an ex-tended fractional Kalman filter, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 863–879.
[28] Kilbas, A., Srivastava H. and Trujillo J. Theory and applications of fractional differential equations, Elsevier, 2006.

[29] Labed, B., Kaouache, S. and Abdelouahab, M.S. Control of a novel class of uncertain fractional-order hyperchaotic systems with external disturbances via sliding mode controller, Nonlinear Dyn. Syst. Theory, 20(2) (2020), 203–213.
[30] Li, C. and Chen, G. Chaos in the fractional order Chen system and its control, Chaos Solitons Fract. 22(3) (2004), 549–554.
[31] Li, C. and Chen, G. Chaos and hyperchaos in the fractional-order Rossler equations, Phys. A, 341(1) (2004), 55–61.
[32] Li, C. and Deng, W.H. Chaos synchronization of fractional-order differ-ential systems, Int. J. Mod. Phys. B, 20(7) (2006), 791–803.
[33] Lin, J., Yan, J. and Liao, T. Chaotic synchronization via adaptive sliding mode observers subject to input nonlinearity, Chaos Solitons Fract. 24 (2005), 371–381.
[34] Lin, L., Zhuang, Y., Xu, Z., Yang, D. and Wu, D. Encryption algo-rithm based on fractional order chaotic system combined with adaptive predefined time synchronization, Front. Phys. 11 (2023), 1202871.
[35] Lu, J. Chaotic dynamics of the fractional-order Lu system and its syn-chronization, Phys. Lett. A, 354(4) (2006), 305–311.
[36] Lu, L. Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal, Chaos Solitons Fract. 27 (2006), 519–525.
[37] Muthukumar, P., Balasubramaniam, P. and Ratnavelu, K. Synchro-nization of a novel fractional order stretch-twist-fold (STF) flow chaotic system and its application to a new authenticated encryption scheme (AES), Nonlinear Dyn. 77(4) (2014), 1547–1559.
[38] Pecora, L.M. and Carroll, T.L. Synchronization in chaotic systems, Phys. Rev. Lett. 64(8) (1990), 821–824.
[39] Pham, V.T., Rahma, F., Frasca, M. and Fortuna, L. Dynamics and synchronization of a novel hyperchaotic system without equilibrium, Int. J. Bifurcat. Chaos, 24(6) (2014), 1450087.

[40] Pikovsky, A.S., Rosenblum, M.G., Osipov, G.V. and Kurths, J. Phase synchronization of chaotic oscillators by external driving, Phys. D, 104(3) (1997), 219–238.
[41] Rikitake, T. Oscillations of a system of disk dynamos, Math. Proc. Cam-bridge, 54(1) (1958), 89–105.
[42] Rossler, O. An equation for hyperchaos, Phys. Lett. A, 71(2-3) (1979), 155–157.
[43] Sau, N.H., Binh, T.N., Thanh, N.T. and Thuan, M.V. Event-triggered H∞ controller design for uncertain fractional-order systems with time-varying delays, J. Appl. Math. Comput. 70(3) (2024), 1813–1835.
[44] Shao, S. and Chen, M. Fractional-order control for a novel chaotic system without equilibrium, J. Autom. Sin. 6(4) (2016), 1000–1009.
[45] Sheu, L. A speech encryption using fractional chaotic systems, Nonlinear Dyn. 65 (2011), 103–108.
[46] Tabasi, M., Hosseini, S.A. and Houshmand, M. Stability analysis of two fractional-order Duffing-Holmes chaotic systems using a generalized pre-dictive controller, Int. J. Dyn. Control. (2024), 1–12.
[47] Vafaei, V., Jodayree Akbarfam, A. and Kheiri, H. A new synchronisa-tion method of fractional-order chaotic systems with distinct orders and dimensions and its application in secure communication, Int. J. Syst. Sci. 52(16) (2021), 3437–3450.
[48] Vaidyanathan, S., Dolvis, L. G., Jacques, K., Lien, C. H. and Sambas, A. A new five-dimensional four-wing hyperchaotic system with hidden attractor, its electronic circuit realisation and synchronisation via inte-gral sliding mode control, Int. J. Model. Identif. Control, 32(1) (2019), 30–45.
[49] Vaidyanathan, S., Volos, Ch.K. and Pham, V.T. Analysis, control, syn-chronization and SPICE implementation of a novel 4 − D hyperchaotic Rikitake Dynamo system without equilibrium, J. Eng. Technol. Rev. 8(2) (2015), 232–244.

[50] Volos, Ch.K., Kyprianidis, I.M. and Stouboulos, I.N. Image encryption process based on chaotic synchronization phenomena, Signal Proc. 93(5) (2013), 1328–1340.
[51] Wang, Z., Cang, S., Ochola, E.O. and Sun, Y. A hyperchaotic system without equilibrium, Nonlinear Dyn. 69 (2012), 531–537.
[52] Wei, J., Zhang, C., Guo, Y. and Wang, F. Cluster synchronization of stochastic two-layer delayed neural networks via pinning impulsive con-trol, J. Appl. Math. Comput. 70(2) (2024), 1193–1210.
[53] Wu, X. and Lu, Y. Generalized projective synchronization of the fractional-order Chen hyperchaotic system, Nonlinear Dyn. 57 (2009), 25–35.
[54] Yang, N. and Liu, C. A novel fractional-order hyperchaotic system sta-bilization via fractional sliding-mode control, Nonlinear Dyn. 74 (2013), 721–732.
[55] Zhang, C., Zhang, C., Zhang, X. and Liang, Y. Sampling-based event-triggered control for cluster synchronization in two-layer nonlinear net-works, J. Appl. Math. Comput. 69(5) (2023), 3969–3986.
[56] Zhang, R. and Yang, S. Adaptive synchronization of fractional-order chaotic systems via a single driving variable, Nonlinear Dyn. 66 (2011), 831–837.
[57] Zhang, S. and Zeng, Y. A simple Jerk-like system without equilibrium: Asymmetric coexisting hidden attractors, bursting oscillation and double full Feigenbaum remerging trees, Chaos Solitons Fract. 120 (2019), 25–40.
[58] Zhou, J. and Bao, H. Fixed-time synchronization for competitive neural networks with Gaussian-wavelet-type activation functions and discrete delays, J. Appl. Math. Comput. 64(1) (2020), 103–118.