Stabilization of a 5-D hyperchaotic Rikitake system with unknown parameters

Document Type : Full Length Article


School of Mechanical Engineering, Shiraz University, Shiraz, Islamic Republic of Iran



In this paper, a nonlinear 5-D hyperchaotic Rikitake dynamic system has been taken into consideration. The hyperchaotic behavior of the model was proved, and the response of the system has been shown. Besides, in the case of existing parametric uncertainties in the system, it shows even more complex behavior. An adaptive control strategy to have stable behavior is synchronized for an uncertain hyperchaotic system with an identical 5-D system. The stability of the control law has been identified by using the Lyapunov stability theory. The numerical simulations are presented for the hyperchaotic Rikitake system with unknown parameters and a system with time-varying parameters to indicate the effectiveness of the proposed algorithm for a class of complex systems. Moreover, since there are often lags between the signals gained by the system and the signals that the controller receives, the control input with the time delay parameter is implemented in the model. Also, the results show the gradual transformation from an unstable system into a stable one.


  • Phase portrait and time evolutions of the 5-D hyperchaotic Rikitake system is depicted.
  • Adaptive synchronization for autonomous and non-autonomous Rikitake model is proposed.
  • Time- delay existence  in control input is investigated.
  • Stability of the systems is guaranteed by the proposed strategy.


Main Subjects

[1] L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Physical Review Letters, 64 (1990) 821-824.
[2] B. Vaseghi, M.A. Pourmina, S. Mobayen, Finite-time chaos synchronization and its application in wireless sensor networks, Transactions of the Institute of Measurement and Control, 40 (2018) 3788-3799.
[3] A. Khan, S. Kumar, Measuring chaos and synchronization of chaotic satellite systems using sliding mode control, Optimal Control Applications and Methods, 39 (2018) 1597-1609.
[4] M. Bettayeb, U.M. Al–Saggaf, S. 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 (2018) 3651-3664.
[5] L. Liu, G. Xie, R. Li, Synchronization stability analysis of medical cyber-physical cloud system considering multi-closed-loops, Journal of Circuits, Systems and Computers, 28 (2019) 1950198.
[6] A.T. Azar, S. Vaidyanathan, Computational intelligence applications in modeling and control, Springer, 2014.
[7] A.T. Azar, S. Vaidyanathan, Chaos modeling and control systems design, Springer, 2015.
[8] E.N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963) 130-141.
[9] O.E. Rössler, An equation for continuous chaos, Physics Letters A, 57 (1976) 397-398.
[10] G. Chen, T. Ueta, Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 9 (1999) 1465-1466.
[11] H.-K. Chen, C.-I. Lee, Anti-control of chaos in rigid body motion, Chaos, Solitons & Fractals, 21 (2004) 957-965.
[12] G. Cai, Z. Tan, Chaos synchronization of a new chaotic system via nonlinear control, Journal of Uncertain systems, 1 (2007) 235-240.
[13] V. Sundarapandian, Analysis and anti-synchronization of a novel chaotic system via active and adaptive controllers, Journal of Engineering Science and Technology Review, 6 (2013) 45-52.
[14] A. Arneodo, P. Coullet, C. Tresser, Possible new strange attractors with spiral structure, Communications in Mathematical Physics, 79 (1981) 573-579.
[15] J.C. Sprott, Some simple chaotic flows, Physical Review E, 50 (1994) R647.
[16] V.-T. Pham, S. Vaidyanathan, C. Volos, S. Jafari, S.T. Kingni, A no-equilibrium hyperchaotic system with a cubic nonlinear term, Optik-International Journal for Light and Electron Optics, 127 (2016) 3259-3265.
[17] S. Vaidyanathan, C.K. Volos, V.-T. Pham, Analysis, Adaptive Control and Adaptive Synchronization of a Nine-Term Novel 3-D Chaotic System with Four Quadratic Nonlinearities and its Circuit Simulation, Journal of Engineering Science & Technology Review, 8 (2015).
[18] S. Vaidyanathan, C. Volos, Advances and Applications in Chaotic systems, Springer, 2016.
[19] S. Vaidyanathan, A.T. Azar, Adaptive backstepping control and synchronization of a novel 3-D jerk system with an exponential nonlinearity, in:  Advances in chaos theory and intelligent control, Springer, 2016, pp. 249-274.
[20] S. Vaidyanathan, Hyperchaos, qualitative analysis, control and synchronisation of a ten-term 4-D hyperchaotic system with an exponential nonlinearity and three quadratic nonlinearities, International Journal of Modelling, Identification and Control, 23 (2015) 380-392.
[21] V.-T. Pham, S. Vaidyanathan, C. Volos, S. Jafari, F.E. Alsaadi, F.E. Alsaadi, Chaos in a simple snap system with only one nonlinearity, its adaptive control and real circuit design, Archives of Control Sciences, 29 (2019).
[22] R. Behinfaraz, S. Ghaemi, S. Khanmohammadi, Adaptive synchronization of new fractional‐order chaotic systems with fractional adaption laws based on risk analysis, Mathematical Methods in the Applied Sciences, 42 (2019) 1772-1785.
[23] S. Vaidyanathan, S. Sampath, Anti-synchronisation of identical chaotic systems via novel sliding control and its application to a novel chaotic system, International Journal of Modelling, Identification and Control, 27 (2017) 3-13.
[24] X. Chen, T. Huang, J. Cao, J.H. Park, J. Qiu, Finite-time multi-switching sliding mode synchronisation for multiple uncertain complex chaotic systems with network transmission mode, IET Control Theory & Applications, 13 (2019) 1246-1257.
[25] X. Yang, Q. Song, J. Cao, J. Lu, Synchronization of coupled Markovian reaction–diffusion neural networks with proportional delays via quantized control, IEEE transactions on neural networks and learning systems, 30 (2018) 951-958.
[26] I. Ahmad, A.B. Saaban, A.B. Ibrahim, M. Shahzad, Global chaos synchronization of new chaotic system using linear active control, Complexity, 21 (2015) 379-386.
[27] A. Kazemy, M. Farrokhi, Synchronization of chaotic Lur’e systems with state and transmission line time delay: a linear matrix inequality approach, Transactions of the Institute of Measurement and Control, 39 (2017) 1703-1709.
[28] Y.-J. Liu, S. Tong, Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems, Automatica, 76 (2017) 143-152.
[29] S.-Y. Li, M.A.B. Hernández, Robust synchronization of chaotic systems with novel fuzzy rule-based controllers, Information Sciences, 481 (2019) 604-615.
[30] W. Ao, T. Ma, R.-V. Sanchez, H. Gan, Finite-time and fixed-time impulsive synchronization of chaotic systems, Journal of the Franklin Institute, (2019).
[31] Y. Wang, D. Tong, Q. Chen, W. Zhou, Exponential synchronization of chaotic systems with stochastic perturbations via quantized feedback control, Circuits, Systems, and Signal Processing, 39 (2020) 474-491.
[32] M. Asadollahi, A.R. Ghiasi, M.A. Badamchizadeh, Adaptive synchronization of chaotic systems with hysteresis quantizer input, ISA transactions, 98 (2020) 137-148.
[33] I. Ahmad, M. Shafiq, Oscillation free robust adaptive synchronization of chaotic systems with parametric uncertainties, Transactions of the Institute of Measurement and Control, (2020) 0142331220903668.
[34] T. Rikitake, Oscillations of a system of disk dynamos, in:  Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 1958, pp. 89-105.
[35] X. Yu, Y. Song, Chaos synchronization via controlling partial state of chaotic systems, International Journal of Bifurcation and Chaos, 11 (2001) 1737-1741.
[36] S. Vaidyanathan, V.-T. Pham, C. Volos, A 5-D hyperchaotic Rikitake dynamo system with hidden attractors, The European Physical Journal Special Topics, 224 (2015) 1575-1592.