Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose

Mathematical Citation Quotient (MCQ) 2018: 0.62

Open Access
See all formats and pricing
More options …

Chaos synchronization of a fractional nonautonomous system

Zakia Hammouch
  • Corresponding author
  • E3MI Group Department of Mathematics„ FSTE Moulay Ismail University„ BP 509 Boutalamine Errachidia 52000, Morocco
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Toufik Mekkaoui
  • E3MI Group Department of Mathematics„ FSTE Moulay Ismail University„ BP 509 Boutalamine Errachidia 52000, Morocco
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2014-03-20 | DOI: https://doi.org/10.2478/msds-2014-0001


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 efiectiveness of the proposed methods

Keywords: Chaos; Fractional-order system; Active control; PLL; Synchronization


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

  • [2] A. Blokh, C. Cleveland, M. Misiurewicz, Expanding polymodials. Modern dynamical systems and applications, 253-270, Cambridge Univ. Press, Cambridge, 2004.Google Scholar

  • [3] R.Caponetto, G.Dongola, and L.Fortuna, Fractional order systems: Modeling and control application, World Scientifc, Singapore, 2010.Google Scholar

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

  • [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.Web of ScienceGoogle Scholar

  • [6] K.Diethelm and N.Ford, Analysis of fractional diferential equations. Journal of Mathematical Analysis and Applications, 265 (2002) pp.229-248.Google Scholar

  • [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.CrossrefGoogle Scholar

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

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

  • [10] M. Haeri and A. Emadzadeh, Synchronizing diferent chaotic systems using active sliding mode control, Chaos, Solitons and Fractals. 31 (2007) pp.119-129.CrossrefGoogle Scholar

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

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

  • [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.Web of ScienceCrossrefGoogle Scholar

  • [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.Google Scholar

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

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

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

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

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

  • [20] K. B. Oldham and J. Spanier, The Fractional Calculus. Academic Press, New York, NY, USA, 1974.Google Scholar

  • [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.121-128Google Scholar

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

  • [23] L.M. Pecora and T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64, (1990) pp.821-824.CrossrefGoogle Scholar

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

  • [25] A.Pikovsky, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press 2011.Google Scholar

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

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

  • [28] A. Ucar, K.E. Lonngren and E.W. Bai, Synchronization of the unifed chaotic systems via active control, Chaos, Solitons and Fractals. 27 (2006) pp.1292-97.CrossrefGoogle Scholar

  • [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.Google Scholar

  • [30] G. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2008. Google Scholar

About the article

Received: 2013-12-30

Accepted: 2014-02-14

Published Online: 2014-03-20

Published in Print: 2014-01-01

Citation Information: Nonautonomous Dynamical Systems, Volume 1, Issue 1, ISSN (Online) 2353-0626, DOI: https://doi.org/10.2478/msds-2014-0001.

Export Citation

© 2014 Zakia Hammouch, Toufik Mekkaoui. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Kashif Ali Abro, Ali Asghar Memon, and Anwar Ahmed Memon
Analog Integrated Circuits and Signal Processing, 2018
C. J. Zuñiga-Aguilar, J. F. Gómez-Aguilar, R. F. Escobar-Jiménez, and H. M. Romero-Ugalde
The European Physical Journal Plus, 2018, Volume 133, Number 1

Comments (0)

Please log in or register to comment.
Log in