Finite-time synchronization of a new five-dimensional hyper-chaotic system via terminal sliding mode control

Document Type : Article


1 Department of Electrical Engineering, Saveh Branch, Islamic Azad University, Saveh, Iran

2 - Department of Electrical Engineering, University of Zanjan, Zanjan, Iran - Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliu, Yunlin 64002, Taiwan, R.O.C.

3 Department of Electrical Engineering, Abhar Branch, Islamic Azad University, Abhar, Iran


This paper constructs a new five–dimensional hyper–chaotic system with complex dynamic behaviors. It also analyzes the chaotic attractor, bifurcation diagram, equilibrium points, Poincare map, Kaplan–Yorke dimension and Lyapunov exponent behaviors. We prove that the introduced new hyper-chaotic system has complex and nonlinear behaviors. Next, the work describes fast terminal sliding mode control scheme for the control and synchronization of the new hyper–chaotic system. Stability analysis is performed using the Lyapunov stability theory. For the synchronization, both master and slave systems are perturbed by different parameter and model uncertainties. Both steps of the sliding mode controller have finite–time convergence properties. Subsequently, it has been shown that the state paths of both master–slave systems can reach each other in a finite time. One of the main features of the proposed controller is the finite time stability of the terminal sliding surface designed with high–order power function of error and derivative of error. Finally, using the MATLAB simulation, the results are confirmed for the new hyper–chaotic system.


1. Vaseghi, B., Pourmina, M.A., and Mobayen, S. "Secure communication in wireless sensor networks based on chaos synchronization using adaptive sliding mode control", Nonlinear Dynamics, 89(3), pp. 1689-1704 (2017).
2. Coatrieux, G., Matre, H., Sankur, B., et al. "Relevance of watermarking in medical imaging", International Conference on Information Technology Applications in Biomedicine, IEEE, pp. 250-255 (2000).
3. Bouslimi, D., Coatrieux, G., and Roux, C. "A joint encryption/watermarking algorithm for verifying the reliability of medical images: Application to echographic images", Computer Methods and Programs in Biomedicine., 106(1), pp. 47-54 (2012).
4. Fridrich, J. "Symmetric ciphers based on twodimensional chaotic maps", International Journal of Bifurcation and Chaos., 8(06), pp. 1259-1284 (1998).
5. Pai, M.C. "Chaos control of uncertain time-delay chaotic systems with input dead-zone nonlinearity", Complexity, 21(3), pp. 13-20 (2012).
6. Al-sawalha, M.M. and Al-Dababseh, A.F. "Nonlinear anti-synchronization of two Hyper chaotic systems", Applied Mathematical Sciences, 5(38), pp. 1849-1856 (2011).
7. Zhou, X., Wang, W., Liu, Z., et al. "Impact angle constrained three-dimensional integrated guidance and control based on fractional integral terminal sliding mode control", IEEE Access, 7, pp. 126857-126870 (2019).
8. Hoang, T.M. "A chaos-based Image cryptosystem using nonstationary dynamics of logistic map", International Conference on Information and Communication Technology Convergence (ICTC), IEEE, pp. 591-596 (2019).
9. Tlelo-Cuautle, E., Ramos-Lopez, H.C., Sanchez- Sanchez, M., et al. "Application of a chaotic oscillator in an autonomous mobile robot", Journal of Electrical Engineering, 65(3), pp. 157-162 (2014).
10. Minati, L., Ito, H., Perinelli, A., et al. "Connectivity influences on nonlinear dynamics in weaklysynchronized networks: Insights from Rossler Systems, Electronic Chaotic Oscillators, Model and Biological Neurons", IEEE Access, 7, pp. 174793-174821 (2019).
11. Bao, H. and Cao, J. "Finite-time generalized synchronization of nonidentical delayed chaotic systems", Nonlinear Analysis: Modelling and Control, 21(3), pp.306-324 (2016). 
12. Huang, Y. and Yang, X.-S. "Hyperchaos and bifurcation in a new class of four-dimensional Hopfield neural networks", Neurocomputing, 69(13-15), pp. 1787-1795 (2006).
13. Chen, G. and Ueta, T. "Yet another chaotic attractor", International Journal of Bifurcation and Chaos, 9(07), pp. 1465-1466 (1999).
14. Lu, J. and Chen, G. "A new chaotic attractor coined", International Journal of Bifurcation and Chaos, 12(03), pp. 659-661 (2002).
15. Qi, G., Chen, G., Du, S., et al. "Analysis of a new chaotic system", Physica A: Statistical Mechanics and its Applications, 352(2-4), pp. 295-308 (2005).
16. Rossler, O. "An equation for hyperchaos", Physics Letters A., 71(2-3), pp. 155-157 (1979).
17. Dimassi, H. and Loria, A. "Adaptive unknown-input observers-based synchronization of chaotic systems for telecommunication", IEEE Transactions on Circuits and Systems I: Regular Papers, 58(4), pp. 800-812 (2010).
18. Gao, T., Chen, Z., Yuan, Z., et al. "A hyperchaos generated from Chen's system", International Journal of Modern Physics C., 17(04), pp. 471-478 (2006).
19. Yanchuk, S. and Kapitaniak, T. "Chaos-hyperchaos transition in coupled Rossler systems", Physics Letters A, 290(3-4), pp. 139-144 (2001).
20. Barboza, R. "Dynamics of a hyperchaotic Lorenz system", International Journal of Bifurcation and Chaos, 17(12), pp. 4285-4294 (2007).
21. Lorenz, E.N. "Deterministic nonperiodic  flow", Journal of the Atmospheric Sciences., 20(2), pp. 130-141 (1963).
22. Qi, G. and Liang, X. "Force analysis of Qi chaotic system", International Journal of Bifurcation and Chaos, 26(14), p. 1650237 (2016).
23. Jalnine, A.Y. "Generalized synchronization of identical chaotic systems on the route from an independent dynamics to the complete synchrony", Regular and Chaotic Dynamics, 18(3), pp. 214-225 (2013).
24. Gonchenko, A.S., Gonchenko, S.V., and Kazakov, A.O. "Richness of chaotic dynamics in nonholonomic models of a Celtic stone", Regular and Chaotic Dynamics, 18(5), pp. 521-538 (2013).
25. Tian, X., Yang, Z., and Fei, S. "Adaptive synchronization of fractional order chaotic systems based on modified feedback approach", 37th Chinese Control Conference (CCC), IEEE, pp. 10121-10126 (2018).
26. Canyelles-Pericas, P., Dai, X., Binns, R., et al. "Decomposing chaos into a harmonic oscillator with nonlinear feedback using pole placement methods", 56th Annual Conference on Decision and Control (CDC), IEEE, pp. 2078-2082 (2017).
27. Yang, C., Xiong, Z., and Yang, T. "Finite-time synchronization of coupled inertial memristive neural networks with mixed delays via nonlinear feedback control", Neural Processing Letters, 12, pp. 1-18 (2020).
28. Meng, Z., Xia, Z., Yu, H., et al. "Neural adaptive synchronization control of chaotic FitzHugh-Nagumo neurons in the external electrical stimulation", Chinese Control Conference (CCC), IEEE, pp. 2731-2736 (2019).
29. Tan, F., Zhou, L., Chu, Y., et al. "Fixed-time stochastic outer synchronization in double-layered multiweighted coupling networks with adaptive chatteringfree control", Neurocomputing, 29, pp. 18-35 (2020).
30. Singh, P.P., Singh, J.P., and Roy, B. "Tracking control and synchronization of Bhalekar-Gejji chaotic systems using active backstepping control", International Conference on Industrial Technology (ICIT), IEEE, pp. 322-326 (2018).
31. Liu, Z. "Design of nonlinear optimal control for chaotic synchronization of coupled stochastic neural networks via Hamilton-Jacobi-Bellman equation", Neural Networks, 99, pp. 166-177 (2018).
32. Nguyen, V., Johnson, J., and Melkote, S. "Active vibration suppression in robotic milling using optimal control", International Journal of Machine Tools and Manufacture, 152, p. 103541 (2020).
33. Wang, Y. and Yu, H. "Fuzzy synchronization of chaotic systems via intermittent control", Chaos, Solitons & Fractals, 106, pp. 154-160 (2018).
34. Yau, H.-T., Wu, C.-H., Liang, Q.-C., et al. "Implementation of optimal PID control for chaos synchronization by FPGA chip", International Conference on Fluid Power and Mechatronics IEEE, pp. 56-61 (2011).
35. Chen, Y., Zhang, X., Shi, G., et al. "Synchronization of chaotic delayed neural networks via sampled-data feedback control with stochastic sampling", Recent Advances in Computer Science and Information Engineering, 28, pp. 217-222 (2012).
36. Wu, J. and Li, X. "Global stochastic synchronization of kuramoto-oscillator networks with distributed control", IEEE Transactions on Cybernetics, 135, pp. 1- 11 (2020).
37. Henein, M.M., Sayed, W.S., Radwan, A.G., et al. "Switched active control synchronization of three fractional order chaotic systems", International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON), IEEE, pp. 1-6 (2016).
38. Li, X.F. "Synchronization of chaotic permanent magnet synchronous motor system via sliding mode control", International Conference on Systems Engineering (ICSEng), IEEE, pp. 1-4 (2018).
39. Tirandaz, H., Aminabadi, S.S., and Tavakoli, H. "Chaos synchronization and parameter identification of a finance chaotic system with unknown parameters, a linear feedback controller", Alexandria Engineering Journal, 57(3), pp. 1519-1524 (2018).
40. Wu, X., Zhu, C., and Kan, H. "An improved secure communication scheme based passive synchronization of hyperchaotic complex nonlinear system", Applied Mathematics and Computation, 252, pp. 201-214 (2015).
41. Sun, K., Zhu, S., Wei, Y., et al. "Finite-time synchronization of memristor-based complex-valued neural networks with time delays", Physics Letters A., 383(19), pp. 2255-2263 (2019).
42. Liu, X. "Adaptive finite time stability of delayed systems with applications to network synchronization", Available: arXiv:2002.00145 (2020).
43. Yan, J.-J., Yang, Y.-S., Chiang, T.-Y., et al. "Robust synchronization of unified chaotic systems via sliding mode control", Chaos, Solitons & Fractals, 34(3), pp. 947-954 (2007).
44. Yin, L., Deng, Z., Huo, B., et al. "Finite-time synchronization for chaotic gyros systems with terminal sliding mode control", IEEE Transactions on Systems, 49(6), pp. 1131-1140 (2017).
45. Mobayen, S. and Javadi, S. "Disturbance observer and finite-time tracker design of disturbed third-order nonholonomic systems using terminal sliding mode", Journal of Vibration and Control, 23(2), pp. 181-189 (2017).
46. Slotine, J.-J.E. and Li, W. "Applied nonlinear control", Prentice Hall Englewood Cliffs., pp. 1-461, New Jersey, US (1991).
47. Feng, Y., Zhou, M., Han, F., et al. "Speed control of induction motor servo drives using terminal sliding-mode controller", Advances in Variable Structure Systems, 36, pp. 341-356 (2018).
48. Pisano, A. and Usai, E. "Output-feedback control of an underwater vehicle prototype by higher-order sliding modes", Automatica, 40(9), pp. 1525-1531 (2004). 
49. Besnard, L., Shtessel, Y.B., and Landrum, B. "Quadrotor vehicle control via sliding mode controller driven by sliding mode disturbance observer", Journal of the Franklin Institute, 349(2), pp. 658-684 (2012).
50. Ghamati, M. and Balochian, S. "Design of adaptive sliding mode control for synchronization Genesio-Tesi chaotic system", Chaos, Solitons & Fractals, 75, pp. 111-117 (2015).
51. Lu, L., Yu, M., and Luan, L. "Synchronization between uncertain chaotic systems with a diverse structure based on a second-order sliding mode control", Nonlinear Dynamics, 70(3), pp. 1861-1865 (2012).
52. Li, H., Liao, X., Li, C., et al. "Chaos control and synchronization via a novel chatter free sliding mode control strategy", Neurocomputing, 74(17), pp. 3212- 3222 (2011).
53. Ouannas, A., Grassi, G., and Azar, A.T. "A new generalized synchronization scheme to control fractional chaotic systems with non-identical dimensions and different orders", International Conference on Advanced Machine Learning Technologies and Applications, Springer, pp. 415-424 (2019).
54. Wu, X., Wang, H., and Lu, H. "Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication", Nonlinear Analysis: Real World Applications, 13(3), pp. 1441-1450 (2012).
55. Abadi, A.S.S., Hosseinabadi, P.A., Mekhilef, S., et al. "Chattering-free fixed-time sliding mode control for bilateral teleoperation under unknown time-varying delay via disturbance and state observers", Advanced Control for Applications: Engineering and Industrial Systems, 2(4), p. e52 (2020).
56. Hosseinabadi, P.A., Abadi, A.S.S., Mekhilef, S., et al. "Chattering-free trajectory tracking robust predefinedtime sliding mode control for a remotely operated vehicle", Journal of Control, Automation and Electrical Systems, 31, pp. 1-19 (2020).
57. Benkouider, K., Bouden, T., and Halimi, M. "Dynamical analysis, synchronization and circuit implementation of a new hyperchaotic system with line equilibrium", International Conference on Control, Decision and Information Technologies (CoDIT), IEEE, pp. 1717-1722 (2019).
58. Lin, S. and Zhang, W. "Chattering reduced sliding mode control for a class of chaotic systems", Nonlinear Dynamics, 93(4), pp. 2273-2282 (2018).
59. Zirkohi, M.M. "An efficient approach for digital secure communication using adaptive backstepping fast terminal sliding mode control", Computers & Electrical Engineering, 76, pp. 311-322 (2019).
60. Abadi, A.S.S., Hosseinabadi, P.A., and Mekhilef, S. "Two novel approaches of NTSMC and ANTSMC synchronization for smart grid chaotic systems", Technology and Economics of Smart Grids and Sustainable Energy, 3(1), p. 14 (2018).
61. Khan, A., Singh, S., Azar, A.T., et al. "Synchronization between a novel integer-order hyperchaotic system and a fractional-order hyperchaotic system using tracking control", International Conference on Modelling, Identification and Control (ICMIC), IEEE, pp. 1-8 (2018).
62. Wang, L., Dong, T., and Ge, M.-F. "Finite-time synchronization of memristor chaotic systems and its application in image encryption", Applied Mathematics and Computation, 347, pp. 293-305 (2019).
63. Sabaghian, A. and Balochian, S. "Parameter estimation and synchronization of hyper chaotic Lu system with disturbance input and uncertainty using two under-actuated control signals", Transactions of the Institute of Measurement and Control, 41(6), pp. 1729-1739 (2019).
64. Sabaghian, A., Balochian, S., and Yaghoobi, M. "Synchronisation of 6D hyper-chaotic system with unknown parameters in the presence of disturbance and parametric uncertainty with unknown bounds", Connection Science, 32(4), pp. 362-383 (2020).
65. Wiggins, S. "Introduction to applied nonlinear dynamical systems and chaos", Springer Science & Business Media, pp. 1-842, College Park, MD (2003).
66. Wolf, A., Swift, J.B., Swinney, H.L., et al. "Determining Lyapunov exponents from a time series", Physica D: Nonlinear Phenomena., 16(3), pp. 285-317 (1985).
67. Li, C., Gong, Z., Qian, D., et al. "On the bound of the Lyapunov exponents for the fractional differential systems", Chaos: An Interdisciplinary Journal of Nonlinear Science, 20(1), p. 013127 (2010).
68. Grassberger, P. and Procaccia, I. "Characterization of strange attractors", Physical Review Letters, 50(5), p. 346 (1983).
69. Grassberger, P. and Procaccia, I. "Measuring the strangeness of strange attractors", Physica D: Nonlinear Phenomena, 9(1-2), pp. 189-208 (1983).
70. Mahmoud, E.E. "Generation and suppression of a new hyperchaotic nonlinear model with complex variables", Applied Mathematical Modelling, 38(17-18), pp. 4445- 4459 (2014).
71. Mahmoud, E.E. "Dynamics and synchronization of new hyperchaotic complex Lorenz system", Mathematical and Computer Modelling, 55(7-8), pp. 1951-1962 (2012).
72. Yang, X. and Cao, J. "Finite-time stochastic synchronization of complex networks", Applied Mathematical Modelling, 34(11), pp. 3631-3641 (2010).
73. Abdurahman, A., Jiang, H., and Teng, Z. "Finite-time synchronization for memristor-based neural networks with time-varying delays", Neural Networks, 69, pp. 20-28 (2015).
74. Moulay, E. and Perruquetti, W. "Finite time stability and stabilization of a class of continuous systems", Journal of Mathematical Analysis and Applications, 323(2), pp. 1430-1443 (2006).
75. Efe, M. O. "Fractional fuzzy adaptive sliding-mode control of a 2-DOF direct-drive robot arm", IEEE Transactions on Systems, 38(6), pp. 1561-1570 (2008).