Share This Article:

On Slide Mode Control of Chaotic Rikitake Two-Disk Dynamo—Chaotic Simulations of the Reversals of the Earth’s Magnetic Field

Abstract Full-Text HTML Download Download as PDF (Size:2533KB) PP. 136-143
DOI: 10.4236/ijmnta.2014.33015    3,299 Downloads   3,848 Views  


The modern nonlinear theory, bifurcation and chaos theory are used in this paper to analyze the dynamics of the Rikitake two-disk dynamo system. The mathematical model of the Rikitake system consists of three nonlinear differential equations, which found to be the same as the mathematical model of the well-known Lorenz system. The study showed that under certain value of control parameter, the system experiences a chaotic behaviour. The experienced chaotic oscillation may simulate the reversal of the Earth’s magnetic field. The main objective of this paper is to control the chaotic behaviour in Rikitake system. So, a nonlinear controller based on the slide mode control theory is designed. The study showed that the designed controller was so effective in controlling the unstable chaotic oscillations.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Harb, A. and Ayoub, N. (2014) On Slide Mode Control of Chaotic Rikitake Two-Disk Dynamo—Chaotic Simulations of the Reversals of the Earth’s Magnetic Field. International Journal of Modern Nonlinear Theory and Application, 3, 136-143. doi: 10.4236/ijmnta.2014.33015.


[1] Rikitake, T. (1958) Oscillations of a System of Disk Dynamos. Proceedings of the Cambridge Philosophical Society, 54, 89-105.
[3] Danca, M.-F. and Codreanu, S. (2011) Finding the Rikitake’s Attractors by Parameter Switching. arXiv:1102.2164v1
[4] Danca, M.-F. and Codreanu, S. (2012) Modeling Numerically the Rikitake’s Attractors. Journal of the Franklin Institute, 349, 861-878.
[5] Bullard, E.C. (1949) The Magnetic Field within the Earth. Proceedings of the Royal Society, 197, 433-453.
[6] Bullard, E.C. and Gellman, H. (1954) Homogeneous Dynamos and Terrestrial Magnetism. Philosophical Transactions of the Royal Society of London A, 247, 213-278.
[7] Bullard, E.C. (1955) The Stability of a Homopolar Dynamo. Proceedings of the Cambridge Philosophical Society, 51, 744-760.
[8] Liu, X.-J., Li, X.-F., Chang, Y.-X. and Zhang, J.-G. (2008) Chaos and Chaos Synchronism of the Rikitake Two-Disk Dynamo. 4th International Conference on Natural Computation, IEEE Computer Society, Jinan, 18-20 October 2008, 613-617.
[9] McMillen, T. (1999) The Shape and Dynamics of the Rikitake Attractor. The Nonlinear Journal, 1, 1-10.
[10] Llibre, J. and Messias, M. (2009) Global Dynamics of the Rikitake System. Physica D, 238, 241-252.
[11] Chen, C.-C. and Tseng, C.-Y. (2007) A Study of Stochastic Resonance in the Periodically Forced Rikitake Dynamo. Terrestrial, Atmospheric and Oceanic Sciences, 18, 671-680.
[12] Harb, A. and Ayoub, N. (2010) Nonlinear Control of Chaotic Rikitake Two-Disk Dynamo. International Journal of Nonlinear Science, 9, 1-8.
[13] Harb, A. and Harb, B. (2004) Chaos Control of Third-Order Phase-Locked Loops Using Backstepping Nonlinear Controller. Chaos, Solitons & Fractals, 20, 719-723.
[14] Harb, A. and Abedl-Jabbar, N. (2003) Controlling Hopf Bifurcation and Chaos in a Small Power System. Chaos, Solitons & Fractals, 18, 1055-1063.
[15] Nayfeh, A.H., Harb, A.M. and Chin, C. (1996) Bifurcation in a Power System Model. International Journal of Bifurcation and Chaos, 6.
[16] Abed, E.H., Alexander, J.C., Wang, H., Hamdan, A.H. and Lee, H.C. (1992) Dynamic Bifurcation in a Power System Model Exhibiting Voltage Collapse. Proceedings of the 1992 IEEE International Symposium on Circuits and Systems, San Diego, 10-13 May 1992, 2509-2512.
[17] Chiang, H.-D., Dobson, I., Thomas, R.J., Thorp, J.S. and Fekih-Ahmed, L. (1990) On Voltage Collapse in Electric Power Systems. IEEE Transactions on Power Systems, 5, 601-611.
[18] Chang, H.-C. and Chen, L.-H. (1984) Bifurcation Characteristics of Nonlinear Systems under Conventional PID Control. Chemical Engineering Science, 39, 1127-1142.
[19] Abed, E.H. and Fu, J.-H. (1986) Local Feedback Stabilization and Bifurcation Control, I. Hopf Bifurcation. Systems Control Letters, 7, 11-17.
[20] Jian, J., Shen, Y. and Yu, H. (2008) Synchronization of Rikitake Chaotic Attractor via Partial System States. Proceedings of the 9th International Conference for Young Computer Scientists, Hunan, 18-21 November 2008, 2919-2924.
[21] Korondi, P., Hashimoto, H. and Utkin, V. (1994) Proceedings of the Asian Control Conference, 381 Lee, S. 1995.
[22] Hung, J.Y., Gao, W. and Hung, J.C. (1993) Variable Structure Control: A Survey. IEEE Transactions on Industrial Electronics, 40, 2-22.
[23] Slotine, J.-J.E. and Sastry, S.S. (1983) Tracking Control of Non-Linear Systems Using Sliding Surfaces, with Application to Robot Manipulators. International Journal of Control, 38, 465-492.
[24] Slotine, J.-J.E. and Li, W. (1991) Applied Nonlinear Control. Prentice Hall, New Jersey.

comments powered by Disqus

Copyright © 2019 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.