Chaos synchronization of a fractional nonautonomous system
Zakia Hammouch ; Toufik Mekkaoui
Nonautonomous Dynamical Systems, Tome 1 (2014), / Harvested from The Polish Digital Mathematics Library
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In this paper we investigate the dynamic behavior of a nonautonomous fractional-order biological system.With the stability criterion of active nonlinear fractional systems, the synchronization of the studied chaotic system is obtained. On the other hand, using a Phase-Locked-Loop (PLL) analogy we synchronize the same system. The numerical results demonstrate the effectiveness of the proposed methods.

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:267097 
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     author = {Zakia Hammouch and Toufik Mekkaoui},
     title = {Chaos synchronization of a fractional nonautonomous system},
     journal = {Nonautonomous Dynamical Systems},
     volume = {1},
     year = {2014},
     zbl = {1298.34093},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_msds-2014-0001}
}

Zakia Hammouch; Toufik Mekkaoui. Chaos synchronization of a fractional nonautonomous system. Nonautonomous Dynamical Systems, Tome 1 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_msds-2014-0001/

[1] E.W. Bai, K. E. Lonngren, Synchronization of two Lorenz systems using active control, Chaos, Solitons Fractals, 8 (1997) pp.51-58. [Crossref] | Zbl 1079.37515

[2] A. Blokh, C. Cleveland, M. Misiurewicz, Expanding polymodials. Modern dynamical systems and applications, 253–270, Cambridge Univ. Press, Cambridge, 2004. | Zbl 1147.37335

[3] R.Caponetto, G.Dongola, and L.Fortuna, Fractional order systems: Modeling and control application, World Scientific, Singapore, 2010. | Zbl 1207.82058

[4] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, J. Roy. Astral.Soc. 13(1967) pp.529- 539. [Crossref]

[5] A. Chamgoué, R. Yamapi and P. Woafo, Bifurcations in a birhythmic biological system with time-delayed noise, Nonlinear Dynamics. 73, (2013) pp.2157-2173. [WoS]

[6] K.Diethelm and N.Ford, Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 265 (2002) pp.229-248. | Zbl 1014.34003

[7] K.Diethelm, N.Ford, A.Freed and Y.Luchko, Algorithms for the fractional calculus: a selection of numerical method, Computer Methods in Applied Mechanics and Engineering, 94 (2005) pp.743-773. [Crossref] | Zbl 1119.65352

[8] H. Frohlich. Long Range Coherence and energy storage in a Biological systems. Int. J. Quantum Chem. 641 (1968) pp.649- 652.

[9] H. Frohlich, Quantum Mechanical Concepts in Biology. Theoretical Physics and Biology.(1969).

[10] M. Haeri and A. Emadzadeh, Synchronizing different chaotic systems using active sliding mode control, Chaos, Solitons and Fractals. 31 (2007) pp.119-129. [WoS][Crossref] | Zbl 1142.93394

[11] G.He and M.Luo, Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control, Appl. Math. Mech. Engl. Ed., 33 (2012), pp.567-582. [Crossref] | Zbl 1266.34022

[12] W. Hongwu and M. Junhai, Chaos Controland Synchronization of a Fractional-order Autonomous System, WSEAS Trans. on Mathematics. 11, , (2012) pp. 700-711.

[13] H.G.Kadji, J.B.Orou, R. Yamapi and P. Woafo, Nonlinear Dynamics and Strange Attractors in the Biological System. Chaos Solitons and Fractals. 32 (2007) pp.862–882. [Crossref][WoS] | Zbl 1138.37050

[14] F. Kaiser, Coherent Oscillations in Biological Systems I. Bifurcations Phenomena and Phase transitions in enzymesubstrate reaction with Ferroelectric behaviour. Z Naturforsch A. 294 (1978) pp.304-333.

[15] F. Kaiser, Coherent Oscillations in Biological Systems II. Lecture Notes in Mathematics, 1907. Springer, Berlin, (2007).

[16] E.N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Science. 20, (1963) pp.130-141.

[17] L. Lu, C. Zhang and Z.A. Guo, Synchronization between two different chaotic systems with nonlinear feedback control, Chinese Physics, 16(2007) pp.1603-1607.

[18] D. Matignon. Stability results for fractional differential equations with applications to control processing.Proceedings Comp. Eng. Sys. Appl., 963-968, 1996.

[19] K.S Miller and B. Rosso, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York 1993.

[20] K. B. Oldham and J. Spanier, The Fractional Calculus. Academic Press, New York, NY, USA, 1974. | Zbl 0292.26011

[21] O.Olusola, E. Vincent,N. Njah and E. Ali, Control and Synchronization of Chaos in Biological Systems Via Backsteping Design. International Journal of Nonlinear Science . 11 (2011) pp.121-128 | Zbl 1237.93061

[22] V.T.Pham, M.Frasca, R.Caponetto, T.M.Hoang and Luigi Fortuna, Control and synchronization of fractional-order differential equations of phase-locked-loop. Chaotic Modeling and Simulation (CMSIM), 4. (2012) pp.623-631.

[23] L.M. Pecora and T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64, (1990) pp.821-824. [Crossref][PubMed] | Zbl 0938.37019

[24] I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, 2011.

[25] A.Pikovsky, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press 2011. | Zbl 0993.37002

[26] I.Podlubny, Fractional Differential Equations: Mathematics in Science and Engineering. Academic Press-USA 1999.

[27] S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Books Pub., 1994.

[28] A. Ucar, K.E. Lonngren and E.W. Bai, Synchronization of the unified chaotic systems via active control, Chaos, Solitons and Fractals. 27 (2006) pp.1292-97. [Crossref][WoS] | Zbl 1091.93030

[29] Y. Wang, Z.H. Guan and H.O. Wang, Feedback an adaptive control for the synchronization of Chen system via a single variable, Phys. Lett A. 312 (2003) pp.34-40. | Zbl 1024.37053

[30] G. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2008. | Zbl 1152.37001