Synchronization of a new fractional-order hyperchaotic system. (English) Zbl 1231.34091

Summary: In this letter, a new fractional-order hyperchaotic system is proposed. By utilizing the fractional calculus theory and computer simulations, it is found that hyperchaos exists in the new fractional-order four-dimensional system with order less than 4. The lowest order to have hyperchaos in this system is 2.88. The results are validated by the existence of two positive Lyapunov exponents. Using the pole placement technique, a nonlinear state observer is designed to synchronize a class of nonlinear fractional-order systems. The observer method is used to synchronize two identical fractional-order hyperchaotic systems. In addition, the active control technique is applied to synchronize the new fractional-order hyperchaotic system and the fractional-order Chen hyperchaotic system. The two schemes, based on the stability theory of the fractional-order system, are rather simple, theoretically rigorous and convenient to realize synchronization. They do not require the computation of the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the proposed synchronization schemes.


34D06 Synchronization of solutions to ordinary differential equations
34K37 Functional-differential equations with fractional derivatives
34C28 Complex behavior and chaotic systems of ordinary differential equations
34H10 Chaos control for problems involving ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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[1] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[2] Hifer, R., Applications of Fractional Calculus in Physics (2001), World Scientific: World Scientific New Jersey
[3] Koeller, R. C., J. Appl. Mech., 51, 299 (1984)
[4] Heaviside, O., Electromagnetic Theory (1971), Chelsea: Chelsea New York · JFM 30.0801.03
[5] Levie, R. De., J. Electroanal. Chem., 281, 1 (1990)
[6] Westerlund, S., Phys. Scr., 43, 174 (1991)
[7] El-Sayed, A. M.A., Int. J. Theor. Phys., 35, 311 (1996)
[8] Ichise, M.; Nagayanagi, Y.; Kojima, T., J. Electroanal. Chem., 33, 253 (1971)
[9] Chen, G.; Friedman, G., IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 24, 170 (2005)
[10] Jenson, V. G.; Jeffreys, G. V., Mathematical Methods in Chemical Engineering (1977), Academic Press: Academic Press New York · Zbl 0413.00002
[12] Laskin, N., Physica A, 287, 482 (2000)
[13] Oustaloup, A., La Dérivation Non Entière: Théorie, Synthèse et Applications (1995), Editions Hermès: Editions Hermès Paris, France · Zbl 0864.93004
[14] Mandelbrot, B.; Van Ness, J. W., SIAM Rev., 10, 422 (1968)
[15] Podlubny, I., IEEE Trans. Automat. Control, 44, 208 (1999)
[16] Hartley, T. T.; Lorenzo, C. F., Nonlinear Dyn., 29, 201 (2002)
[17] Li, C. P.; Peng, G. J., Chaos Solitons Fractals, 22, 443 (2004)
[18] Wu, X.; Lu, Y., Nonlinear Dyn. (2008)
[19] Li, C. G.; Chen, G., Phys. A, 341, 55 (2004)
[20] Deng, W.; Li, C. P., Physica A, 353, 61 (2005)
[21] Ge, Z. M.; Zhang, A. R., Chaos Solitons Fractals, 32, 1791 (2007)
[22] Gao, T.; Chen, Z.; Yuan, Z.; Yu, D., Chaos Solitons Fractals, 33, 922 (2007)
[23] Li, C. G.; Liao, X. X.; Yu, J. B., Phys. Rev. E, 68, 067203 (2003)
[24] Zhou, T. S.; Li, C. P., Phys. D, 212, 111 (2005)
[25] Gao, X.; Yu, J. B., Chaos Solitons Fractals, 26, 141 (2005)
[26] Lu, J. G., Chaos Solitons Fractals, 26, 1125 (2005)
[27] Peng, G.; Jiang, Y.; Chen, F., Phys. A, 387, 3738 (2008)
[28] Peng, G.; Jiang, Y., Phys. Lett. A, 372, 3963 (2008)
[29] Yan, J. P.; Li, C. P., Chaos Solitons Fractals, 32, 725 (2007)
[30] Caputo, M., Geophys. J. R. Astron. Soc., 13, 529 (1967)
[31] Samko, S. G.; Klibas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordan and Breach: Gordan and Breach Amsterdam
[32] Keil, F.; Mackens, W.; Werther, J., Scientific Computing in Chemical Engineering II - Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties (1999), Springer-Verlag: Springer-Verlag Heidelberg
[33] Butzer, P. L.; Westphal, U., An Introduction to Fractional Calculus (2000), World Scientific: World Scientific Singapore · Zbl 0987.26005
[34] Charef, A.; Sun, H. H.; Tsao, Y. Y.; Onaral, B., IEEE Trans. Automat. Control, 37, 1465 (1992)
[35] Ahmad, W. M.; Sprott, J. C., Chaos Solitons Fractals, 16, 339 (2003)
[36] Diethelm, K.; Trans, Elec., Numer. Anal., 5, 1 (1997)
[37] Diethelm, K.; Ford, N. J.; Freed, A. D., Nonlinear Dyn., 29, 3 (2002)
[38] Diethelm, K.; Ford, N. J., J. Math. Anal. Appl., 265, 229 (2002)
[39] Diethelm, K.; Ford, N. J.; Freed, A. D., Numer. Algorithms, 36, 31 (2004)
[40] Chen, G.; Ueta, T., Int. J. Bifur. Chaos, 9, 1465 (1999)
[41] Wolf, A.; Swinney, J. B.; Swinney, H. L.; Vastano, J. A., Phys. D, 16, 285 (1985)
[43] Chen, C. T., Linear System Theory and Design (1984), Holt, Rinehart & Winston: Holt, Rinehart & Winston New York
[44] Bai, E. W.; Lonngren, K. E., Chaos Solitons Fractals, 11, 1041 (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.