Dynamical Analysis and Finite-Time Fast Synchronization of a Novel Autonomous Hyper-Chaotic System

Subject Areas : Renewable energy

Javad Mostafaee ¹ , Saleh Mobayen ² , Behrouz Vaseghi ³ , Mohammad Vahedi ⁴

1 - Department of Electrical Engineering- Saveh Branch, Islamic Azad University, Saveh, Iran.
2 - Department of Electrical Engineering- University of Zanjan, Zanjan, Iran.
3 - Department of Electrical Engineering- Abhar Branch, Islamic Azad University, Abhar, Iran.
4 - Department of Electrical Engineering- Saveh Branch, Islamic Azad University, Saveh, Iran.

Received: 2021-02-02 Accepted : 2021-03-25 Published : 2021-11-22

Keywords: Chaotic analysis, finite-time synchronization, New hyper-chaotic system, fast terminal sliding mode control,

Abstract :

This paper constructs a new complex hyper-chaotic system with attractive coexisting dynamic behaviors. We analyze the hyper-chaotic attractors, equilibrium points, Poincaré maps, Kaplan-York dimension, and Lyapunov exponent behaviors. The characteristics of hyper-chaotic systems include higher complexity, higher parametric resistance and sensitivity to very small changes in initial conditions. We prove that the introduced hyper–chaotic system is much more complex than the similar hyper-chaotic systems, that can suitable for use in encryption and secure communication. Next, the work describes a fast terminal sliding mode controller scheme for the fast synchronization and stability of the new complex hyper–chaotic system. It is shown that by applying uncertainty to the system, both steps of the sliding mode control have finite-time convergence properties. Next, a comparison will be made between a newly designed controller and a similar. Finally, using the MATLAB simulation, the results are confirmed for the new system. The results shown that the new hyper-chaotic system with many adsorbents is much more complex than similar systems, and the proposed controller has a faster convergence response than the similar controller.

References:

[1] X. Liu, S. Qi, R. Malekain, Z. Li , "Observer-based composite adaptive dynamic terminal sliding-mode controller for nonlinear uncertain SISO systems", International Journal of Control, Automation and Systems, vol. 17, no. 1, pp. 94-106, Jan. 2019 (doi: 10.1007/s12555-018-0117-7).

[2] S.J. Sheela, K.V. Suresh, D. Tandur, "Security of industrial wireless sensor networks: A review", Proceeding of the IEEE/ITACT, pp. 1-6, Bangalore, India, Dec. 2015 (doi: 10.1109/ITACT.2015.7492658).

[3] M. Wollschlaeger, T. Sauter, J. Jasperneite, "The future of industrial communication: automation networks in the era of the internet of things and industry 4.0", IEEE Industrial Electronics Magazine, vol. 11, no. 1, pp. 17-27, March 2017 (doi: 10.1109/MIE.2017.2649104).

[4] M. Wollschlaeger, T. Sauter, J. Jasperneite, "The future of industrial communication: Automation networks in the era of the internet of things and industry", IEEE Industrial Electronics Magazine, vol. 11, no. 1, pp. 17-27, March 2017 (doi: 10.1109/MIE.2017.2649104).

[5] M.C. Pai, "Chaos control of uncertain time‐delay chaotic systems with input dead‐zone nonlinearity", Complexity, vol. 21, no. 3, pp. 13-20, Oct. 2016 (doi: org/10.1002/cplx.21611).

[6] T.M. Hoang, "A Chaos-based image cryptosystem using nonstationary dynamics of logistic map", Proceeding of the IEEE/ICTC, pp. 591-596, Jeju Island, Korea (South), Oct. 2019 (doi: 10.1109/ICTC46­69­1.2­0­19.8­939826).

[7] E. Tlelo-Cuautle, C. Ramos-López, M. Sánchez-Sánchez, D. Pano-Azucena, A. Sánchez-Gaspariano, C. Núñez-Pérez, L. Camas-Anzueto, "Application of a chaotic oscillator in an autonomous mobile robot", Journal of Electrical Engineering, vol. 65, no. 3, pp. 157-162, June 2014 (doi: 10.2478/jee-2014-0024).

[8] L. Minati, H. Ito, A.Perinelli, L. Riccy, L. Faec, N. Yushimura, Y. Koike, L. Minati, "Connectivity influences on nonlinear dynamics in weakly-synchronized networks: Insights from rössler systems, electronic chaotic oscillators, model and biological neurons", IEEE Access, vol. 7, pp. 174793-174821, Dec. 2019 (doi: 10.1109/ACCESS.2019.2957014).

[9] A. Zhou, S. Wang, F. Wang, "Low-complexity and robust detection for hybrid chaos communication", Proceeding of the IEEE/WCSP, pp. 1-5, Xi'an, China, Dec. 2019 (doi: 10.1109/WCSP.2019.8927976).

[10] S. Zhang, T. Gao, "A coding and substitution frame based on hyper-chaotic systems for secure communica­tion", Nonlinear Dynamics, vol. 84, no. 2, pp. 833-849, Dec. 2016 (doi: 10.1007/s11071-015-2530-2).

 [11] R. Sedivy, R. M. Mader, "Fractals, chaos, and cancer: do they coincide?", Cancer Investigation, vol. 15, no. 6, pp. 601-607, June 1997 (doi: 10.3109/07357909709047603).

 [12] B. Xu, Y. Wang, L. Liu, "Twice pulse ignition boost strategy for missile guidance Based on improved particle swarm optimization algorithm", Proceeding of the IEEE/CCC, pp. 9907-9912, Wuhan, July 2018 (doi: 10.23919/ChiCC.2018.8484243).

[13] M. Sciamanna, K. A. Shore, "Physics and applications of laser diode chaos", Nature Photonics, vol. 9, no. 3, pp. 151-162, Feb. 2015 (doi: 10.1038/nphoton.2014.326).

 [14] H. Dimassi, A. Loría, "Adaptive unknown-input observers-based synchronization of chaotic systems for telecommunication", IEEE Trans. on Circuits and Systems I: Regular Papers, vol. 58, no. 4, pp. 800-812, Nov. 2010 (doi: 10.1109/TCSI.2010.2089547).

[15] O. Rossler, "An equation for hyperchaos", Physics Letters A, vol. 71, no. 2-3, pp. 155-157, April 1979 (doi: 10.1016/0375-9601(79)90150-6).

[16] E. Dong, Z. Zhang, M. Yuan, Y. Ji, X. Zhou, Z. Wang, "Ultimate boundary estimation and topological horseshoe analysis on a parallel 4D hyperchaotic system with any number of attractors and its multi-scroll", Nonlinear Dynamics, vol. 95, no. 4, pp. 3219-3236, March 2019 (doi: 10.1007/s11071-018-04751-3).

[17] A. Hajipour, M. Hajipour, D. Baleanu, "On the adaptive sliding mode controller for a hyperchaotic fractional-order financial system", Physica A: Statistical Mechanics and its Applications, vol. 497, pp. 139-153, May 2018 (doi: 10.1016/j.physa.2018.01.019).

[18] C. Zhou, C. Yang, D. Xu, C.-Y. Chen, "Dynamic analysis and finite-time synchronization of a new hyperchaotic system with coexisting attractors", IEEE Access, vol. 7, pp. 52896-52902, April 2019 (doi: 10.1­109/ACCES­S.2019.2911486).

[19] F. F. Franco, E. L. Rempel, P. R. Muñoz, "Crisis and hyperchaos in a simplified model of magnetoconvection", Physica D: Nonlinear Phenomena, Article Number: 132417, May 2020 (doi: 10.101­6/j.p­hysd.2020.132417).

[20] G. Leutcho, J. Kengne, L. K. Kengne, "Dynamical analysis of a novel autonomous 4-D hyperjerk circuit with hyperbolic sine nonlinearity: Chaos, antimonotonicity and a plethora of coexisting attractors", Chaos, Solitons & Fractals, vol. 107, pp. 67-87, Feb. 2018 (doi: 10.1016/j.chaos.2017.12.008).

[21] Z. T. Njitacke, J. Kengne, H. Fotsin, "Coexistence of multiple stable states and bursting oscillations in a 4D Hopfield neural network", Circuits, Systems, and Signal Processing, vol. 39, pp. 3424-3444, Jan. 2020 (doi: org/10.1002/cplx.21611).

[22] W. Tai, Q. Teng, Y. Zhou, J. Zhou, Z. Wang, "Chaos synchronization of stochastic reaction-diffusion time-delay neural networks via non-fragile output-feedback control", Applied Mathematics and Computation, vol. 354, pp. 115-127, Aug. 2019 (doi: 10.1016/j.amc.2019.02.028).

[23] T. Wang, D. Wang, K. Wu, "Chaotic adaptive synchronization control and application in chaotic secure communication for industrial Internet of Things", IEEE Access, vol. 6, pp. 8584-8590, Jan. 2018 (doi: 10.11­09/AC­CE­SS.2018.2797979).

[24] L. Wang, M. Ding, "Dynamical analysis and passive control of a new 4D chaotic system with multiple attractors", Modern Physics Letters B, vol. 32, no. 22, Article Number: 1850260, Dec. 2018 (doi: 10.1142­/S0­217984918502603).

[25] K. Rajagopal, H. Jahanbakhshi, M. Varan, I. Bayir, V. Pham, S. Jafari, A. Kartikian, "A hyperchaotic memristor oscillator with fuzzy based chaos control and LQR based chaos synchronization", AEU-International Journal of Electronics and Communications, vol. 94, pp. 55-68, Jan. 2018 (doi: 10.101­6/j.ae­ue.2­018.06.043).

[26] M. M. Zirkohi, "Chaos synchronization using higher-order adaptive PID controller", AEU-International Journal of Electronics and Communications, vol. 94, pp. 157-167, Sep. 2018 (doi: 10.1016/j.ae­ue.20­18.0­7.0­0­5).

[27] S. Vaidyanathan, S. T. Kingni, A. Sambas, M. A. Mohamed, M. Mamat, "A new chaotic jerk system with three nonlinearities and synchronization via adaptive backstepping control", International Journal of Engine­ering and Technology, vol. 7, no. 3, pp. 1936-1943, Dec. 2018 (doi: 10.14419/ijet.v7i3.15378).

[28] C. Huang, L. Cai, J. Cao, "Linear control for synchronization of a fractional-order time-delayed chaotic financial system", Chaos, Solitons & Fractals, vol. 113, pp. 326-332, Aug. 2018 (doi: 10.1016/j.c­hao­s.20­18.0­5.022).

[29] M.-H. Wang, S.-D. Lu, M.-J. Hsieh, "Application of extension neural network algorithm and chaos synchronization detection method to partial discharge diagnosis of power capacitor", Measurement, vol. 129, pp. 227-235, Dec. 2018 (doi: 10.1016/j.measurement.2018.07.022).

[30] A. Mohammadzadeh, S. Ghaemi, O. Kaynak, "Robust predictive synchronization of uncertain fractional-order time-delayed chaotic systems", Soft Computing, vol. 23, no. 16, pp. 6883-6898, June 2019 (doi: 10.1002/cplx.21611).

[31] Y. Yin, F. Liu, P. Shi, "Finite-time continuous gain-scheduled control on stochastic hyperchaotic systems", Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 224, no. 6, pp. 679-688, Sept. 2010 (doi: 10.1243/09596518JSCE971).

[32] E. D. Dongmo, K. S. Ojo, P. Woafo, A. N. Njah, "Difference synchronization of identical and nonidentical chaotic and hyperchaotic systems of different orders using active backstepping design", Journal of Computational and Nonlinear Dynamics, vol. 13, no. 5, April 2018 (doi: 10.1115/1.4039626).

[33] S. Vaidyanathan, L. G. Dolvis, K. Jacques, C.-H. Lien, A. Sambas, "A new five-dimensional four-wing hyperchaotic system with hidden attractor, its electronic circuit realisation and synchronisation via integral sliding mode control", International Journal of Modelling, Identification and Control, vol. 32, no. 1, pp. 30-45, June 2019 (doi: 10.1504/IJMIC.2019.101959).

[34] A. Modiri, S. Mobayen, "Adaptive terminal sliding mode control scheme for synchronization of fractional-order uncertain chaotic systems", ISA Transactions, vol. 105, pp. 33-50, Oct. 2020 (doi: 10.1016/j.isa­tra.2­020.0­5.039).

[35] S. Mobayen, S. Javadi, "Disturbance observer and finite-time tracker design of disturbed third-order nonholonomic systems using terminal sliding mode", Journal of Vibration and Control, vol. 23, no. 2, pp. 181-189, April 2017 (doi: 10.1177/1077546315576611).

[36] S. Mobayen, "An adaptive chattering-free PID sliding mode control based on dynamic sliding manifolds for a class of uncertain nonlinear systems", Nonlinear Dynamics, vol. 82, no. 1-2, pp. 53-60, May 2015 (doi: 10.1007/s11071-015-2137-7).

[37] P.A. Hosseinabadi, A.S.S. Abadi, S. Mekhilef, H.R. Pota, "Chattering-free trajectory tracking robust predefined-time sliding mode control for a remotely operated vehicle", Journal of Control, Automation and Electrical Systems, pp. 1-19, May 2020 (doi: 10.1007/s40313-020-00599-4).

[38] M. Zak, "Terminal attractors for addressable memory in neural networks", Physics Letters A, vol. 133, no. 1, pp. 18-22, Oct. 1988 (doi: 10.1016/0375-9601(88)90728-1).

[39] X. Tong, Y. Liu, M. Zhang, H. Xu, Z. Wang, "An image encryption scheme based on hyperchaotic Rabinovich and exponential chaos maps", Entropy, vol. 17, no. 1, pp. 181-196, Dec. 2015 (doi: 10­.339­0/e17­010181).

[40] P.D.K. Kuate, Q. Lai, H. Fotsin, "Complex behaviors in a new 4D memristive hyperchaotic system without equilibrium and its microcontroller-based implementation", The European Physical Journal Special Topics, vol. 228, no. 10, pp. 2171-2184, Oct. 2019 (doi: 10.1140/epjst/e2019-900032-5).

 [41] F. Nazarimehr, K. Rajagopal, J. Kengne, S. Jafari, V.T. Pham, "A new four-dimensional system containing chaotic or hyper-chaotic attractors with no equilibrium, a line of equilibria and unstable equilibria", Chaos, Solitons & Fractals, vol. 111, pp. 108-118, May 2018 (doi: 10.1016/j.chaos.2018.04.009).

[42] V. Van Huynh, A.J. M. Khalaf, A. Alsaedi, T. Hayat, H.R. Abdolmohammadi, "A new memristive chaotic flow with a line of equilibria", The European Physical Journal Special Topics, vol. 228, no. 10, pp. 2339-2349, Oct. 2019 (doi: 10.11140/cplx.900055-9)

[43] K. Sun, X. Liu, C. Zhu, J. Sprott, "Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system", Nonlinear Dynamics, vol. 69, no. 3, pp. 1383-1391, Feb. 2012 (doi: 10.1002/cplx.012-03540)

[44] S. T. Tchinda, G. Mpame, A.N. Takougang, V.K. Tamba, "Dynamic analysis of a snap oscillator based on a unique diode nonlinearity effect, offset boosting control and sliding mode control design for global chaos synchronization", Journal of Control, Automation and Electrical Systems, vol. 30, no. 6, pp. 970-984, Sept. 2019 (doi: 10.1007/s40313-019-00518-2).

[45] J. Kengne, G. D. Leutcho, A. N. K. Telem, "Reversals of period doubling, coexisting multiple attractors, and offset boosting in a novel memristive diode bridge-based hyperjerk circuit", Analog Integrated Circuits and Signal Processing, vol. 101, no. 3, pp. 379-399, Dec. 2019 (doi: 10.1007/s10470-018-1372-5).

[46] A. Jeevarekha, S. Sabarathinam, K. Thamilmaran, P. Philominathan, "Analysis of 4D autonomous system with volume-expanding phase space", Nonlinear Dynamics, vol. 84, no. 4, pp. 2273-2284, Feb. 2016 (doi: 10.1007/s11071-016-2644-1).

[47] Z. Njitacke, J. Kengne, T.F. Fozin, B. Leutcha, H. Fotsin, "Dynamical analysis of a novel 4-neurons based Hopfield neural network: Emergences of antimonotonicity and coexistence of multiple stable states", International Journal of Dynamics and Control, vol. 7, no. 3, pp. 823-841, Dec. 2019 (doi: 10.1007/s40435-019-00509-w).

[48] V.F. Signing, J. Kengne, "Reversal of period-doubling and extreme multistability in a novel 4D chaotic system with hyperbolic cosine nonlinearity", International Journal of Dynamics and Control, vol. 7, no. 2, pp. 439-451, June 2019 (doi: 10.1007/s40435-018-0452-9).

[49] M. Chen, M. Sun, B. Bao, H. Wu, Q. Xu, J. Wang, "Controlling extreme multistability of memristor emulator-based dynamical circuit in flux–charge domain", Nonlinear Dynamics, vol. 91, no. 2, pp. 1395-1412, Nov. 2018 (doi: 10.1007/s11071-017-3952-9).

[50] X. Zhang, H. Zhu, H. Yao, "Analysis of a new three-dimensional chaotic system", Nonlinear Dynamics, vol. 67, no. 1, pp. 335-343, March 2012 (doi: 10.1007/s11071-011-9981-x).

[51] P. Frederickson, J.L. Kaplan, E.D. Yorke, J.A. Yorke, "The Liapunov dimension of strange attractors", Journal of Differential Equations, vol. 49, no. 2, pp. 185-207, Aug. 1983 (doi: 10.1016/0022-0396(83­)90­01­1-­6).

[52] G. Qi, M.A. Wyk, B.J. Wyk, G. Chen, "On a new hyperchaotic system", Physics Letters A, vol. 372, no. 2, pp. 124-136, Jan. 2008 (doi: 10.1016/j.physleta.2007.10.082).

[53] F.Y. Dalkiran, J.C. Sprott, "Simple chaotic hyperjerk system", International Journal of Bifurcation and Chaos, vol. 26, no. 11, p. 1650189, Dec. 2016 (doi: 10.1142/S0218127416501893).

[54] C. Li, J. C. Sprott, W. Thio, H. Zhu, "A new piecewise linear hyperchaotic circuit", IEEE Trans. on Circuits and Systems II: Express Briefs, vol. 61, no. 12, pp. 977-981, Sept. 2014 (doi: 10.1109/TCSII.2014.235­691­2).

[55] S. Vaidyanathan, "Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system via backstepping control method", Archives of Control Sciences, vol. 26, no. 3, pp. 311-338, Jan. 2016 (doi: 10.1515/acsc-2016-0018).

[56] S. Zhang, Y. C. Zeng, Z. Jun Li, "A novel four-dimensional no-equilibrium hyper-chaotic system with grid multiwing hyper-chaotic hidden attractors", Journal of Computational and Nonlinear Dynamics, vol. 13, no. 9, Sept. 2018 (foi: 10.1115/1.4039980).

[57] S. Vaidyanathan, "Analysis, control and synchronization of a novel 4-D highly hyperchaotic system with hidden attractors", Advances in Chaos Theory and Intelligent Control: Springer, pp. 529-552, April 2016 (doi: 10.1007/978-3-319-30340-6_22).

[58] H. Lin, C. Wang, Y. Tan, "Hidden extreme multistability with hyperchaos and transient chaos in a Hopfield neural network affected by electromagnetic radiation", Nonlinear Dynamics, vol. 99, no. 3, pp. 2369-2386, Dec. 2020 (doi: 10.1007/s11071-019-05408-5).

[59] E.E. Mahmoud, "Dynamics and synchronization of new hyperchaotic complex Lorenz system", Mathematical and Computer Modelling, vol. 55, no. 7-8, pp. 1951-1962, April 2012 (doi: 10.10­16/j.m­cm.2­01­1.11.053).

[60] M. Steinberger, M. Horn, L. Fridman, "Variable-Structure Systems and Sliding-Mode Control", ed: Springer, 2020 (ISBN: 978-3-030-36621-6).

[61] A. Abdurahman, H. Jiang, Z. Teng, "Finite-time synchronization for memristor-based neural networks with time-varying delays", Neural Networks, vol. 69, pp. 20-28, Sept. 2015 (doi: 10.1016/j.neunet.2015.04.015).

[62] C. Li, F. Zhang, "A survey on the stability of fractional differential equations", The European Physical Journal Special Topics, vol. 193, no. 1, pp. 27-47, April 2011 (doi: 10.1140/epjst/e2011-01379-1).

[63] X. Yu, M. Zhihong, "Fast terminal sliding-mode control design for nonlinear dynamical systems", IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, no. 2, pp. 261-264, Aug. 2002 (doi: 10.1109/81.983876).

[64] X. Liu, S. Qi, R. Malekain, Z. Li, "Observer-based composite adaptive dynamic terminal sliding-mode controller for nonlinear uncertain SISO systems", International Journal of Control, Automation and Systems, vol. 17, no. 1, pp. 94-106, January 2019 (doi: 10.1007/s12555-018-0117-7).

[65] H. Bouslehi, H. Seddik, "A new rapid hyperchaotic system for more efficient 2D data encryption", Multimedia Tools and Applications, vol. 77, no. 6, pp. 7741-7762, May 2018 (doi: 10.1007/s11042-017-4675-0).