Adaptive modified function projective synchronization for a chaotic complex system

Subject Areas : Statistics

Moosarreza Shamsyeh Zahedi ¹

1 - Department of Mathematics, Payame Noor University (PNU), P.O. BOX 19395-4697, Tehran, Iran

Received: 2021-03-07 Accepted : 2021-07-06 Published : 2022-01-21

Keywords: پایداری لیاپانف, دستگاه آشوبناک با متغیرهای مختلط&rlm, , &lrm, &lrm, &rlm, همزمان&lrm, سازی تابع تصویری&rlm, ,

Abstract :

‎This ‎paper ‎presents ‎the‎ modified function projective synchronization (MFPS) in a system with complex variables related to the permanent magnet synchronous motor (PMSM) with unknown parameters.‎ In this method, by defining a suitable adaptive controller and other desired functions and with the aid of Lyapunov stability theory, the asymptotic stability of the error function can be proved. This can lead to a modified function projective synchronization between two complex chaotic systems with completely unknown parameters. Also, the error of estimating the parameters and the value of the coupling strength is accurately expressed using the proposed method.‎‎Because in the modified projective synchronization (MPS) and function projective synchronization (FPS) methods, fixed coupling strength is considered, this limits the range of applications of these methods. MFPS is, therefore, a more general definition of MPS and FPS when the scaling function matrix is chosen by a constant matrix and a scaling function, respectively. This method can be used on other complex dynamic systems. Numerical simulations are presented to show the effectiveness and validity of the above theoretical results.

References:

[1] A.C. Fowler, J.D. Gibbon, M.J. McGuinness,The complex Lorenz equations. Physica D. 4(2)(1982), 139-163.

 

[2] G.M. Mahmoud, T. Bountis, E.E. Mahmoud,Active control and global synchronization for complex Chen and Lu systems. Int. J. Bifurc.Chaos. 17 (2007), 4295-4308.

 

[3] G.M. Mahmoud, M.E. Ahmed, Modified projective synchronization and control of complex Chen and Lu systems. J. Vib. Control. 17 (2010), 1184-1194.

 

[4] G.M. Mahmoud, E.E. Mahmoud, M.E. Ahmed,On the hyperchaotic complex Lu system. Nonlinear Dyn. 58 (2009), 725-738.

 

[5] H.Y. Sun, N. Li, D.P. Zhao, Q.L. Zhang, Synchronization of Complex Networks with Coupling Delays via Adaptive Pinning Intermittent Control, International Journal of Automation and Computing. 10 (4) (2013), 312-318.

 

[6] Y. Dai, Y. Z. Cai, X. M. Xu. Synchronization and exponential estimates of complex networks with mixed time-varying coupling delays. International Journal of Automation and Computing, 6(3) (2009), 301-307.

 

[7] C.K. Ahn, An answer to the open problem of synchronization for time-delayed chaotic systems. Eur. Phys. J. Plus 127(2) (2012), 1-9.

 

[8] G.A. Leonov, V.B. Smirnova, Mathematical problems of synchronization Theory. St. Petersburg: Nauka, 2000.

[9] G.A. Leonov, Lyapunov functions in the attractors dimension theory. J. Appl. Math. Mech. 76 (2012), 129-141.

 

[10] R. Mainieri, J. Rehacek, Projective synchronization in three-dimensioned chaotic systems. Phys Rev Lett. 82 (1999), 3042-3045.

 

[11] W. L. Guo, M. Z. Mao. Projective lag synchronization and parameter identification of a new hyperchaotic system. International Journal of Automation and Computing, 10(3) (2013), 256-259.

 

[12] D.L. Xu, W.L. Ong, Z.G. Li, Criteria for the occurrence of projective synchronization in chaotic systems of arbitrary dimension. Phys Lett A. 305 (2002), 167-172.

 

[13] G.H. Li, Modified projective synchronization of chaotic system. Chaos Solitons Fractals. 32 (2007), 1786-1790.

 

[14] H.Y. Du, Q.S. Zeng, C.H. Wang, Function projective synchronization of different chaotic systems with uncertain parameters. Phys Lett A. 372 (2008), 5402-5410.

 

[15] H.Y. Du, Q.S. Zeng, C.H. Wang, Modified function projective synchronization of chaotic system. Chaos Solition Fractals. 42(4) (2009),

2399-2404.

 

[16] S. Zheng, G. Dong, Q. Bi, Adaptive modified function projective synchronization of hyperchaotic systems with unknown parameters. Commun Nonlinear Sci Numer Simulat. 15 (2010), 3547-3556.

 

[17] H. Saberi Nik, J. Saberi-Nadjafi, S. Effati, R.A. Van Gorder, Hybrid projective synchronization and control of the Baier-Sahle hyperchaotic flow in arbitrary dimensions with unknown parameters, Appl Math comput. 248 (2014), 55-69.

 

[18] H. Saberi Nik, R.A. Van Gorder, Competitive modes for the Baier-Sahle hyperchaotic flow in arbitrary dimensions, Nonlinear Dyn. 74 (2013)

581-590.

 

[19] M. Karabacak, H.I. Eskikurt, Speed and current regulation of a permanent magnet synchronous motor via nonlinear and adaptive backstepping control. Math. Comput. Model. 53 (2011), 2015-2030.

 

[20] S.X. Jing, Design and simulation of PMSM feedback linearization control system. TELKOMNIKA. 11(3), (2013), 1245-1250.

 

[21] Z. Jing, C. Yu, G. Chen, Complex dynamics in a permanent-magnet synchronous motor model. Chaos Solitons Fractals. 22 (2004), 831-848.

 

[22] X. Wang, H. Zhang, Backstepping-based lag synchronization of a complex permanent magnet synchronous motor system. Chin. Phys. B. 22(4), (2013), 048902.