Synchronization of Non-Identical Chaotic Systems with Different Orders Based On Vector Norms Approach

Authors: Rihab Gam, Anis Sakly, Faouzi M'sahli

Abstract:

A new strategy of control is formulated for chaos synchronization of non-identical chaotic systems with different orders using the Borne and Gentina practical criterion associated with the Benrejeb canonical arrow form matrix, to drift the stability property of dynamic complex systems. The designed controller ensures that the state variables of controlled chaotic slave systems globally synchronize with the state variables of the master systems, respectively. Numerical simulations are performed to illustrate the efficiency of the proposed method.

Keywords: Synchronization, Non-identical chaotic systems, Different orders, Arrow form matrix.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1075098

References:

[1] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990:821-4.
[2] Bai EW, Lonngren KE. Sequential synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 2000:1041-4.
[3] S. Hammami, K. Ben Saad, M. Benrejeb. On the synchronization of identical and non-identical 4-D chaotic systems using arrow form matrix. Chaos, Solitons and Fractals 42 (2009) 101-112.
[4] Shih-Yu Li, Zheng-Ming Ge. Generalized synchronization of chaotic systems with different orders by fuzzy logic constant controller. Expert systems with application 38 (2011) 2302-2310.
[5] Gentina JC, Borne P. Sur une condition d-application du critère de stabilité linéaires ├á certaines classes de systèmes continus non linéaires. CRAS, Paris, T. 275; 1972, p. 401-404.
[6] Benrejeb M, Gasmi M. On the use of an arrow form matrix for modelling and stability analysis of singularly perturbed non linear systems. SAMS 2001 40:509-25.
[7] S. Hammami, M. Benrejeb, M. Feki, P. Borne. Feedback control design for Rössler and Chen chaotic systems anti-synchronization. Physics Letters A 374 (2010) 2835-2840.
[8] Chen HK. Synchronization of two different chaotic systems: a new system and each of the dynamical systems Lorenz, Chen and Liu. Chaos, Solitons & Fractals 2005:1049-56.
[9] U.E. Vincent. Synchronization of identical and non-identical 4-D chaotic systems using active control. Chaos, Solitons and Fractals 37 (2008) 1065-1075.
[10] Juhn-Horng Chen a, Hsien-Keng Chen b,*, Yu-Kai Lin a. Synchronization and anti-synchronization coexist in Chen-Lee chaotic systems. Chaos, Solitons and Fractals 39 (2009) 707-716.
[11] Y.J. Sun, Solution bounds of generalized Lorenz chaotic systems, Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.08.015.
[12] Nijmeijer H, Mareels IMY. An observer looks at synchronization. IEEE Trans Circ Syst I 1997:882-90.
[13] Mbouna Ngueuteu GS, Yamapi R, Woafo P. Effects of higher nonlinearity on the dynamics and synchronization of two coupled electromechanical devices. Commun Nonlinear Sci Numer Simul 2006:1213-40.
[14] Osipov G et al. Phase synchronization effects in a lattice of non-identical Rössler attractors. Phys Rev E 1997:2353-61.
[15] Ho MC, Hung YC. Synchronization of two different systems by using generalized active control. Phys Lett A 2002:424-8.
[16] Benrejeb M, Borne P, Laurent F. Sur une application de la représentation en flèche ├á l-analyse des processus. RAIRO Automatique/Systems Analysis and Control 1982;16(2):133-46.
[17] Borne P, Benrejeb M. On the representation and the stability study of large scale systems. In: Proceedings of ICCCC conference, Bãile Felix Spa-Oradea, Romania; May 2008.
[18] M. Chen, D. Zhou, Y. Shang, A new observer-based synchronization scheme for private communication, Chaos, Solitons & Fractals 24 (2005) 1025-1030.
[19] S.H. Chen, Q. Yang, C.P. Wang, Impulsive control and synchronization of unified chaotic system, Chaos, Solitons & Fractals 20 (2004) 751- 758.
[20] D.V. Efimov, Dynamical adaptive synchronization, International Journal of Adaptive Control and Signal Processing 20 (2006) 491-507.
[21] H.T. Yau, C.S. Shieh, Chaos synchronization using fuzzy logic controller, Nonlinear Analysis: Real World Applications (2007), doi:10.1016/j.nonrwa.2007.05.009.
[22] Y.P. Tian, X. Yu, Stabilizing unstable periodic orbits of chaotic systems via an optimal principle, Journal of the Franklin Institute 337 (2000) 771–779.
[23] S.M. Guo, L.S. Shieh, G. Chen, C.F. Lin, Effective chaotic orbit tracker: a prediction-based digital redesign approach, IEEE Transactions on Circuits and Systems I 47 (2000) 1557–1560.