Document Type : Research Article
Authors
1 Department of Mathematics, Payame Noor University (PNU), Tehran, Iran.
2 Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.
10.30473/coam.2023.65079.1211
Abstract
In this paper, we use a graphical algorithm to control and synchronization of a chaotic system. Most of the controllers designed for synchronizing chaotic systems are complex, but the controllers designed using contraction and graphical methods are often simple and linear. Therefore, we explain the relationship between contraction analysis and the graphical method for controlling and synchronizing chaotic systems. We apply this approach to control and synchronize the chaotic Genesio-Tesi system. The stability of the error system in synchronization is investigated using the contraction method. Finally, we provide numerical simulations to demonstrate the effectiveness of the proposed method.
Keywords
[1] Boutayeb, M., Darouach, M., Rafaralahy, H. (2002). “Generalized state-space observers for chaotic synchronization with applications to secure communication”, IEEE Transactions on Circuits and Systems I, 49(3), 345-349.
[2] Femat, R., Jauregui-Ortiz, R., Solys-Perales, G. (2001). “A chaos-based communication scheme via robust asymptotic feedback”, IEEE Transactions on Circuits and Systems, 48(10), 1161-1169.
[3] Femat, R., Alvarez, J., Castillo-Toledo, B., Gonzalez, J. (1999). “On robust chaos suppression in a class of no driven oscillators: Application to the Chua’s circuit”, IEEE Transactions on Circuits and Systems, 46, 1150-1152.
[4] Genesio, R., Tesi, A.A. (1992). “Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems”, Automatica, 28, 531-548.
[5] Godsil, C., Royle, G. (2001). “Algebraic graph theory”, Springer Verlag, New York.
[6] Granas A., Dugundji J. (2003). “Fixed point theory”, Springer-Verlag, New York.
[7] Grassi, G., Mascolo, S. (1997). “Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal”, IEEE Transactions on Circuits and Systems I, 44(10), 11-14.
[8] Hartman, P. (1961). “On stability in the large for systems of ordinary differential equations”, Canadian Journal of Mathematics, 13, 480-492.
[9] Hu, J., Chen, S., Chen, L. (2005). “Adaptive control for anti-synchronization of Chua’s chaotic system”, Physics Letters A, 339, 455-460.
[10] Jackson, E.A., Hubler, A. (1990). “Periodic entrainment of chaotic logistic map dynamics”, Physica D, 44, 404-409.
[11] Joshi, S.K. (2021). “Synchronization of chaotic dynamical systems”, International Journal of Dynamics and Control, 9, 1285-1302.
[12] Jouffroy, J., Slotine, J.J.E. (2004). “Methodological remarks on contraction theory”, In: Proceedings of the 43rd IEEE conference on decision and control at Atlantis, Bahamas, 2537-2543.
[13] Kapitaniak, T. (1996). “Controlling chaos: Theoretical and practical methods in non-linear dynamics”, Academic, New York.
[14] Kapitaniak, T. (2000). “Chaos for engineers: Theory, applications and control”, 2nd Editions, Springer, New York.
[15] Lewis, D.C. (1949). “Metric properties of differential equations”, American Journal of Mathematics, 71, 294-312.
[16] Li, D., Lu, JA., Wu, X. (2005). “Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems”, Chaos, Solitons & Fractals, 23, 79-85.
[17] Liao, T.L., Huang, NS. (1999). “An observer based approach for chaotic synchronization and secure communication”, IEEE Transactions on Circuits and Systems I, 46(9), 1144-1149.
[18] Lohmiller, W. (1999). “Contraction analysis of nonlinear systems”, Ph.D. Thesis, Department of Mechanical Engineering, MIT.
[19] Lohmiller W., Slotine J. J. E. (1998). “On contraction analysis for nonlinear systems”, Automatica, 34, 683-696.
[20] Lohmiller, W., Slotine, J. J. E. (2000). “Control system design for mechanical systems using contraction theory”, IEEE Transaction on Automatic Control, 45(5), 884-889.
[21] Lorenz, E. (1963). “Deterministic non periodic flow”, Atmospheric Sciences, 20, 130-141.
[22] Lu, L., Zhang, F., Zou, C. (2021). “Finite-time synchronization in the laser network based on sliding mode control technology”, Optik, 225, 165605.
[23] Njougouo, T., Camargo, V., Louodop, P., Ferreira, F.F., Talla, P. K., Cerdeira H.A. (2022).
“Synchronization in a multilevel network using the Hamilton–Jacobi–Bellman (HJB) technique”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 32, 093133, 1-14.
[24] Ogorzalek, M.J. (1993). “Taming chaos: Part II – Control scheme via robust asymptotic feedback”, IEEE Transactions on Circuits and Systems I, 40(10), 700-706.
[25] Ott, E., Grebogi, C., York, J.A. (1990). “Controlling chaos”, Physical Review Letters, 64(11), 1196-1199.
[26] Pecoram L.M., Carroll, TL. (1990). “Synchronization in chaotic systems”, Physical Review Letters, 64, 821-824.
[27] Pham, Q.C., Slotine, J.J.E. (2007). “Stable concurrent synchronization in dynamic system networks”, Neural Networks, 20, 62-77.
[28] Russo, G. (2010). “Analysis, control and synchronization of nonlinear systems and networks via contraction theory: theory and applications”, Ph.D. Thesis, Department of Systems and Computer Engineering University of Naples Federico II, Napoli, Italy.
[29] Russo, G., di Bernardo, M. (2009). “An algorithm for the construction of synthetic self-synchronizing biological circuits”, In International Symposium on Circuits and Systems, 305-308.
[30] Russo, G., di Bernardo, M. (2009). “How to synchronize biological clocks”, Journal of Computational Biology, 16, 379-393.
[31] Russo, G., di Bernardo, M. (2009). “Solving the rendezvous problem for multi-agent systems using contraction theory”, in Proceeding International Conference on Decision and Control, 5821-5826.
[32] Russo, G., di Bernardo, M., Slotine, J.J.E. (2009). “An algorithm to prove contraction, consensus, and network synchronization”, In Proceedings of the International Workshop NecSys.
[33] Russo, G., di Bernardo, M., Slotine, J.J.E. (2011). “A graphical algorithm to prove contraction of nonlinear circuits and systems”, IEEE Transactions on Circuits and Systems I, 58(2), 336-348.
[34] Sharma, B.B., Kar, I.N. (2009). “Contraction theory based adaptive synchronization of chaotic systems”, Chaos, Solitons & Fractals, 41, 2437-2447.
[35] Slotine, J.J.E., Li, W. (1990). “Applied Nonlinear Control”, Englewood Cliffs, NJ: Prentice-Hall.
[36] Tarammim, A., Akter, M.T. (2022). “A comparative study of synchronization methods of rucklidge Chaotic Systems with Design of Active Control and backstepping methods”, International Journal of Modern Nonlinear Theory and Application, 11(2), 31-51.
[37] Tsukamoto, H., Chung, S.J., Slotine, J.J.E. (2021). “Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview”, Nonlinear Analysis, Annual Reviews in Control, 52, 135-169.
[38] Wang, W., Slotine, J.J.E. (2005). “On partial contraction analysis for coupled nonlinear oscillators”, Biological Cybernetics, 92, 38-53.
[39] Wang, Y., Guan, Z.H., Wang, H.O. (2003). “Feedback and adaptive control for the synchronization of Chen system via a single variable”, Phys. Lett. A., 312, 34-40.
[40] Yassen, M.T. (2005). “Controlling chaos and synchronization for new chaotic system using linear feedback control”, Chaos, Solitons & Fractals, 26, 913-920.
[41] Zhang, M., Zang, H., Bai, L. (2022). “A new predefined-time sliding mode control scheme for synchronizing chaotic systems”, Chaos, Solitons & Fractals, 164, 112745.
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