Finite-Time Synchronization of the Rabinovich and Rabinovich-Fabrikant Chaotic Systems for Different Evolvable Parameters
(1) * Edwin A. Umoh Mail (Department of Electrical and Electronic Engineering Technology, Federal Polytechnic, Kaura Namoda, Nigeria)
(2) Alfian Ma'arif Mail (Department of Electrical Engineering, Universitas Ahmad Dahlan, Yogyakarta, Indonesia)
(3) Omokhafe J. Tola Mail (Department of Electrical and Electronic Engineering, Federal University of Technology, Minna, Nigeria)
(4) Iswanto Suwarno Mail (Department of Electrical Engineering, Universitas Muhammadiyah Yogyakarta, Yogyakarta, Indonesia)
(5) Muhammed N. Umar Mail (Department of Electrical and Electronic Engineering Technology, Federal Polytechnic, Kaura Namoda, Nigeria)
*corresponding author

Abstract


This paper addresses the challenge of synchronizing the dynamics of two distinct 3D chaotic systems, specifically the Rabinovich and Rabinovich-Fabrikant systems, employing a finite-time synchronization approach. These chaotic systems exhibit diverse characteristics and evolving chaotic attractors, influenced by specific parameters and initial conditions. Our proposed low-cost finite-time synchronization method leverages the signum function's tracking properties to facilitate controlled coupling within a finite time frame. The design of finite-time control laws is rooted in Lyapunov stability criteria and lemmas. Numerical experiments conducted within the MATLAB simulation environment demonstrate the successful asymptotic synchronization of the master and slave systems within finite time. To assess the global robustness of our control scheme, we applied it across various system parameters and initial conditions. Remarkably, our results reveal consistent synchronization times and dynamics across these different scenarios. In summary, this study presents a finite-time synchronization solution for non-identical 3D chaotic systems, showcasing the potential for robust and reliable synchronization under varying conditions.


Keywords


Chaos; Finite-Time Controller; Lyapunov Stability Criteria; Synchronization

   

DOI

https://doi.org/10.31763/ijrcs.v3i4.1149 		      

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References


[1] C. Nwachioma, J. H. Perez-Cruz, A. Jimenez, M. Ezuma, and R. Rivera-Blas, “A New Chaotic Oscillator—Properties, Analog Implementation, and Secure Communication Application,” IEEE Access, vol. 7, pp. 7510–7521, 2019, https://doi.org/10.1109/ACCESS.2018.2889964.

[2] C. Nwachioma and J. H. Perez-Cruz, “Analysis of a new chaotic system, electronic realization and use in navigation of differential drive mobile robot,” Chaos, Solitons and Fractals, vol. 144, p. 110684, 2021, https://doi.org/10.1016/j.chaos.2021.110684.

[3] S. Nasr, A. Abadi, K. Bouallegue, and H. Mekki, “Chaos engineering and control in mobile robotics applications,” 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018), vol. 2, pp. 364–371, 2018, https://doi.org/10.5220/0006867103640371.

[4] Q. Hu, “Chaotic Oscillation Control Model of Power System Under Electromechanical Power Disturbance,” Front. Energy Res., vol. 10, p. 887561, 2022, https://doi.org/10.3389/fenrg.2022.887561.

[5] A. J. M. Khalaf, T. Kapitaniak, K. Rajagopal, A. Alsaedi, T. Hayat, and V.-T. Pham, “A new three-dimensional chaotic flow with one stable equilibrium: dynamic properties and complexity analysis,” Open Phys., vol. 16, no. 1, pp. 260–265, 2018, https://doi.org/10.1515/phys-2018-0037.

[6] E. A. Umoh and O. N. Iloanusi, “Algebraic structure, dynamics and electronic circuit realization of a novel reducible hyperchaotic system,” 2017 IEEE 3rd International Conference on Electro-Technology for National Development (NIGERCON 2017), pp. 483–490, 2017, https://doi.org/10.1109/NIGERCON.2017.8281917.

[7] Y. Wang, X. Li, X. Li, Y. Guang, Y. Wu, and Q. Ding, “FPGA-Based Implementation and Synchronization Design of a New Five-Dimensional Hyperchaotic System,” Entropy, vol. 24, no. 9, p. 1179, 2022, https://doi.org/10.3390/e24091179.

[8] L. Yi, W. Xiao, W. Yu, and B. Wang, “Dynamical analysis, circuit implementation and deep belief network control of new six-dimensional hyperchaotic system,” Journal of Algorithms & Computational Technology, vol. 12, no. 4, pp. 361-375, 2018, https://doi.org/10.1177/1748301818788649.

[9] W. Yu et al., “Design of a new seven-dimensional hyperchaotic circuit and its application in secure communication,” IEEE Access, vol. 7, pp. 125586–125608, 2019, https://doi.org/10.1109/ACCESS.2019.2935751.

[10] S. N. Lagmiri, M. Amghar, and N. Sbiti, “Hyperchaos based cryptography: new seven dimensional system to secure communication,” Circ. Comput. Sci., vol. 2, no. 2, pp. 20–30, 2017, https://doi.org/10.22632/ccs-2017-251-59.

[11] Q. Yang, D. Zhu, and L. Yang, “A New 7D Hyperchaotic System with Five Positive Lyapunov Exponents Coined,” Int. J. Bifurc. Chaos, vol. 28, no. 5, p. 1850057, 2018, https://doi.org/10.1142/S0218127418500578.

[12] K. Benkouider et al., “A new 10-D hyperchaotic system with coexisting attractors and high fractal dimension: Its dynamical analysis, synchronization and circuit design,” Plus One, vol. 17, no. 4, p. e0266053, 2022, https://doi.org/10.1371/journal.pone.0266053.

[13] R. T. Fotsa, A. R. Tchamda, A. S. K. Tsafack, and S. T. Kingni, “Microcontroller Implementation, Chaos Control, Synchronization and Antisynchronization of Josephson Junction Model,” Int. J. Robot. and Control System, vol. 1, no. 2, pp. 198–208, 2021, https://doi.org/10.31763/ijrcs.v1i2.354.

[14] L. Makouo, A. S. K. Tsafack, M. M. Tingue, A. R. Tchamda, and S. F. Kingni, “Synchronization and chaos control using a single controller of five-dimensional autonomous homopolar disc dynamo,” Int. J. Robot. Control Syst., vol. 1, no. 3, pp. 244–255, 2021, https://doi.org/10.31763/ijrcs.v1i3.380.

[15] P. P. Singh and B. K. Roy, “Comparative performances of synchronisation between different classes of chaotic systems using three control techniques,” Annu. Rev. Control, vol. 45, pp. 152–165, 2018, https://doi.org/10.1016/j.arcontrol.2018.03.003.

[16] U. E. Kocamaz, Y. Uyaroglu, and H. Kizmaz, “Controlling Hyperchaotic Rabinovich System with Single State Controllers : Comparison of Linear Feedback , Sliding Mode , and Passive Control Methods,” Optik, vol. 130, pp. 914–921, 2017, https://doi.org/10.1016/j.ijleo.2016.11.006.

[17] P. P. Singh, J. P. Singh, and B. K. Roy, “Nonlinear Active Control Based Hybrid Synchronization between Hyperchaotic and Chaotic Systems,” IFAC Proceedings Volumes, vol. 47, no. 1, pp. 287-291, 2014, https://doi.org/10.3182/20140313-3-IN-3024.00068.

[18] S. Li, Y. Wu, and G. Zheng, “Adaptive synchronization of the hyperchaotic Liu system,” Front. Phys., vol. 9, p. 812048, 2021, https://doi.org/10.3389/fphy.2021.812048.

[19] E. A. Umoh, “Adaptive Hybrid Synchronization of Lorenz-84 System with Uncertain Parameters,” TELKOMNIKA Indones. J. Electr. Eng., vol. 12, no. 7, pp. 5251-5260, 2014, https://doi.org/10.11591/telkomnika.v12i7.5853.

[20] Z. Xianren, Y. Shihui, Z. Wentao, L. Zhenbo, and L. Linmei, “Function Projective Synchronization of Chaotic Systems with a New Kind of Scaling Function,” Chem. Eng. Trans., vol. 75, pp. 583–588, 2019, https://doi.org/10.3303/CET1975098.

[21] S. Çiçek, U. E. Kocamaz, and Y. Uyaroğlu, “Secure chaotic communication with jerk chaotic system using sliding mode control method and its real circuit implementation,” Iranian journal of science and technology, transactions of electrical engineering, vol. 43, pp. 687-698, 2019, https://doi.org/10.1007/s40998-019-00184-9.

[22] E. A. Umoh and O. J. Tola, “Robust global synchronization of a hyperchaotic system with wide parameter space via integral sliding mode control technique,” Int. J. Robot. Control Syst., vol. 1, no. 4, pp. 453–465, 2021, https://doi.org/10.31763/ijrcs.v1i4.485.

[23] J. Zaqueros-Martinez, G. Rodriguez-Gomez, E. Tlelo-Cuautle, and F. Orihuela-Espina, “Fuzzy Synchronization of Chaotic Systems with Hidden Attractors,” Entropy, vol. 25, no. 3, p. 495, 2023, https://doi.org/10.3390/e25030495.

[24] G. Fedele, “Invariant Ellipsoids Method for Chaos Synchronization in a Class of Chaotic Systems,” Int. J. Robot. Control Syst., vol. 2, no. 1, pp. 57–66, 2022, https://doi.org/10.31763/ijrcs.v2i1.533.

[25] A. S. Pikovsky, M. I. Rabinovich, and V. Y. Traktengerts, “Onset of stochasticity in decay confinement of parametric instability,” Sov. Physics. JETP, vol. 47, no. 4, pp. 715–719, 1978, https://www.osti.gov/biblio/6426388.

[26] O. Chis and M. Puta, “The dynamics of Rabinovich system,” Differ. Geom. - Dyn. Syst., vol. 10, pp. 91–98, 2007, https://doi.org/10.48550/arXiv.0710.4583.

[27] E. A. Umoh, "Chaos control of the complex Rabinovich system via Takagi-Sugeno fuzzy controller," 2013 IEEE International Conference on Emerging & Sustainable Technologies for Power & ICT in a Developing Society (NIGERCON), pp. 217-222, 2013, https://doi.org/10.1109/NIGERCON.2013.6715659.

[28] Y. F. Alharbi, M. A. Sohaly, M. A. Abdelrahman, “Fundamental solutions to the stochastic perturbed nonlinear Schrödinger’s equation via gamma distribution,” Results in Physics, vol. 25, p. 104249, 2021, https://doi.org/10.1016/j.rinp.2021.104249.

[29] M. F. Danca and G. Chen, “Bifurcation and chaos in a complex model of dissipative medium,” Int. J. Bifurc. Chaos, vol. 14, no. 10, pp. 3409–3447, 2004, https://doi.org/10.1142/S0218127404011430.

[30] M.-F. Danca, M. Fec̆kan, N. Kuznetsov, and G. Chen, “Looking More Closely at the Rabinovich–Fabrikant System,” Int. J. Bifurc. Chaos, vol. 26, no. 2, p. 1650038, 2016, https://doi.org/10.1142/S0218127416500383.

[31] M. F. Danca, N. Kuznetsov, and G. Chen, “Unusual dynamics and hidden attractors of the Rabinovich–Fabrikant system,” Nonlinear Dyn., vol. 88, pp. 791–805, 2017, https://doi.org/10.1007/s11071-016-3276-1.

[32] M. F. Danca, “Lyapunov exponents of a class of piecewise continuous systems of fractional order,” Nonlinear dynamics, vol. 81, pp. 227-237, 2015, https://doi.org/10.1007/s11071-015-1984-6.

[33] E. A. Umoh and O. N. Iloanusi, “Synchronized Dynamics of Hyperchaotic Flows via an Improved Finite-Time Adaptive Controller Design,” Int. J. Eng. Res. Africa, vol. 48, pp. 49–62, 2020, https://doi.org/10.4028/www.scientific.net/JERA.48.49.

[34] L. Dekui and W. Xingmin, “Finite-Time Generalized Synchronization and Parameter Identification of Chaotic Systems with Different Dimensions,” Wuhan Univ. J. Nat. Sci., vol. 27, no. 2, pp. 135–141, 2022, https://doi.org/10.1051/wujns/2022272135.

[35] S. Pang, Y. Feng, and Y. Liu, “Finite-Time Synchronization of Chaotic Systems with Different Dimension and Secure Communication,” Mathematical Problems in Engineering, vol. 2016, 2016, https://doi.org/10.1155/2016/7693547.

[36] J. Li and J. Zheng, “Finite‑time synchronization of different dimensional chaotic systems with uncertain parameters and external disturbances,” Sci. Rep., vol. 12, p. 15407, 2022, https://doi.org/10.1038/s41598-022-19659-7.

[37] C. Chantawat and T. Botmart, “Finite-time H-inf synchronization control for coronary artery chaos system with input and state time-varying delays,” PLoS One, vol. 17, no. 4, p. e0266706, 2022, https://doi.org/10.1371/journal.pone.0266706.

[38] L. Wang, T. Dong, and M.-F. Ge, “Finite-time synchronization of memristor chaotic systems and its application in image encryption,” Appl. Math. Comput., vol. 347, pp. 293–305, 2019, https://doi.org/10.1016/j.amc.2018.11.017.

[39] S. P. Bhat and D. S. Bernstein, “Finite-time stability of continuous autonomous systems,” SIAM J. Control Optim., vol. 38, no. 3, pp. 751–766, 2000, https://doi.org/10.1137/S0363012997321358.

[40] M. Defoort, K. Veluvolu, M. Djemai, A. Polyakov, and G. Demesure, “Leader-follower fixed-time consensus for multi-agent systems with unknown non-linear inherent dynamics,” IET Control Theory Appl., vol. 9, no. 14, pp. 2165–2170, 2015, https://doi.org/10.1049/iet-cta.2014.1301.

[41] Y. Feng, L. Sun and X. Yu, "Finite time synchronization of chaotic systems with unmatched uncertainties," 30th Annual Conference of IEEE Industrial Electronics Society, 2004. IECON 2004, vol. 3, pp. 2911-2916, 2004, https://doi.org/10.1109/IECON.2004.1432272.

[42] H. Wang, Z. Han, and Q. Xie, “Finite-time chaos control of unified chaotic systems with uncertain parameters,” Nonlinear Dyn., vol. 55, pp. 323–328, 2009, https://doi.org/10.1007/s11071-008-9364-0.

[43] M. P. Aghababa, S. Khanmohammadi, and G. Alizadeh, “Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique,” Appl. Math. Model., vol. 35, no. 6, pp. 3080–3091, 2011, https://doi.org/10.1016/j.apm.2010.12.020.

[44] G. McCartney, F. Popham, R. McMaster, and A. Cumbers, “Defining health and health inequalities,” Public health, vol. 172, pp. 22-30, 2019, https://doi.org/10.1016/j.puhe.2019.03.023.


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International Journal of Robotics and Control Systems
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