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Dynamical behaviors and synchronization in the fractional order hyperchaotic Chen system. (English) Zbl 1234.34036
Summary: Some dynamical behaviors are studied in the fractional order hyperchaotic Chen system which shows hyperchaos with order less than 4. The analytical conditions for achieving synchronization in this system via linear control are investigated theoretically by using the Laplace transform theory. Routh-Hurwitz conditions and numerical simulations are used to show the agreement between the theoretical and numerical results. To the best of our knowledge this is the first example of a hyperchaotic system synchronizable just in the fractional order case, using a specific choice of controllers.

MSC:
34D06 Synchronization of solutions to ordinary differential equations
26A33 Fractional derivatives and integrals
34C28 Complex behavior and chaotic systems of ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N35 Dynamical systems in control
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[1] Butzer, P.L.; Westphal, U., An introduction to fractional calculus, (2000), World Scientific Singapore · Zbl 0987.26005
[2] Sun, H.H.; Abdelwahab, A.A.; Onaral, B., IEEE trans. automat. control, 29, 441, (1984)
[3] Ahmed, E.; Elgazzar, A.S., Physica A, 379, 607, (2007)
[4] El-Sayed, A.M.A.; El-Mesiry, A.E.M.; El-Saka, H.A.A., Appl. math. lett., 20, 817, (2007)
[5] Caputo, M., Geophys. J. R. astron. soc., 13, 529, (1967)
[6] Adda, F. Ben, J. fract. calc., 11, 21, (1997)
[7] Podlubny, I., Fract. calc. appl. anal., 5, 367, (2002)
[8] Matouk, A.E., Math. probl. eng., 2009, (2009), Article ID 572724, 11 pages
[9] Li, C.; Liao, X.; Wong, K.W., Chaos solitons fractals, 23, 183, (2005)
[10] Li, C.; Chen, G., Physica A, 341, 55, (2004)
[11] J. Liu, X. Li, Synchronization of fractional hyperchaotic Lü system via unidirectional coupling method, in: Proceedings of the 7th World Congress on Intelligent Control and Automation, 2008, p. 4653.
[12] Wang, X.Y.; Song, J.M., Commun. nonlinear sci. numer. simul., 14, 3351, (2009)
[13] Deng, H.; Li, T.; Wang, Q.; Li, H., Chaos solitons fractals, 41, 962, (2009)
[14] Matouk, A.E., Phys. lett. A, 373, 2166, (2009) · Zbl 1229.34099
[15] D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in System Application, Lille, France, vol. 2, 1996, p. 963.
[16] Diethelm, K.; Ford, N.J., J. math. anal. appl., 265, 229, (2002)
[17] Diethelm, K.; Ford, N.J.; Freed, A.D., Nonlinear dynam., 29, 3, (2002)
[18] Diethelm, K., Electron. trans. numer. anal., 5, 1, (1997)
[19] Yan, Z., Appl. math. comput., 168, 1239, (2005)
[20] Abarbanel, H.D.I., Analysis of observed chaotic data, (1996), Springer-Verlag New York · Zbl 0875.70114
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