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 |
Keywords:
fractional order; hyperchaotic Chen system; Routh-Hurwitz conditions; hyperchaos; synchronizationReferences:
| [1] | Butzer, P. L.; Westphal, U., An Introduction to Fractional Calculus (2000), World Scientific: World Scientific Singapore · Zbl 0987.26005 |
| [2] | Sun, H. H.; Abdelwahab, A. A.; Onaral, B., IEEE Trans. Automat. Control, 29, 441 (1984) · Zbl 0532.93025 |
| [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) · Zbl 1140.34302 |
| [5] | Caputo, M., Geophys. J. R. Astron. Soc., 13, 529 (1967) |
| [6] | Adda, F. Ben, J. Fract. Calc., 11, 21 (1997) · Zbl 0907.26005 |
| [7] | Podlubny, I., Fract. Calc. Appl. Anal., 5, 367 (2002) · Zbl 1042.26003 |
| [8] | Matouk, A. E., Math. Probl. Eng., 2009 (2009), Article ID 572724, 11 pages · Zbl 1181.37047 |
| [9] | Li, C.; Liao, X.; Wong, K. W., Chaos Solitons Fractals, 23, 183 (2005) · Zbl 1068.94004 |
| [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.; 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) · Zbl 1221.93091 |
| [13] | Deng, H.; Li, T.; Wang, Q.; Li, H., Chaos Solitons Fractals, 41, 962 (2009) · Zbl 1198.34115 |
| [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.; 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) · Zbl 1014.34003 |
| [17] | Diethelm, K.; Ford, N. J.; Freed, A. D., Nonlinear Dynam., 29, 3 (2002) · Zbl 1009.65049 |
| [18] | Diethelm, K., Electron. Trans. Numer. Anal., 5, 1 (1997) · Zbl 0890.65071 |
| [19] | Yan, Z., Appl. Math. Comput., 168, 1239 (2005) · Zbl 1160.93384 |
| [20] | Abarbanel, H. D.I., Analysis of Observed Chaotic Data (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0875.70114 |
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