Projective synchronization of different fractional-order chaotic systems with non-identical orders. (English) Zbl 1257.34040

Summary: This paper investigates the projective synchronization (PS) of different fractional order chaotic systems while the derivative orders of the states in drive and response systems are unequal. Based on some essential properties of fractional calculus and the stability theorems of fractional-order systems, we propose a general method to achieve the PS in such cases. The fractional operators are introduced into the controller to transform the problem into synchronization problem between chaotic systems with identical orders, and the nonlinear feedback controller is proposed based on the concept of active control technique. The method is both theoretically rigorous and practically feasible. We present two examples that illustrate the effectiveness and applications of the method, which include the PS between two 3-D commensurate fractional-order chaotic systems and the PS between two 4-D fractional-order hyperchaotic systems with incommensurate and commensurate orders, respectively. Abundant numerical simulations are given which agree well with the analytical results. Our investigations show that PS can also be achieved between different chaotic systems with non-identical orders. We have further reviewed and compared some relevant methods on this topic reported in several recent papers. A discussion on the physical implementation of the proposed method is also presented in this paper.


34D06 Synchronization of solutions to ordinary differential equations
34A08 Fractional ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
34H10 Chaos control for problems involving ordinary differential equations
Full Text: DOI


[1] Pecora, L.M.; Carroll, T.L., Synchronization in chaotic systems, Phys. rev. lett., 64, 821-824, (1990) · Zbl 0938.37019
[2] Lakshmanan, M.; Murali, K., Chaos in nonlinear oscillators: controlling and synchronization, (1996), World Scientific Pub Co Inc · Zbl 0868.58058
[3] Blasius, B.; Huppert, A.; Stone, L., Complex dynamics and phase synchronization in spatially extended ecological systems, Nature, 399, 354-359, (1999)
[4] Wu, C.W.; Chua, L.O., A simple way to synchronize chaotic systems with applications to secure communication systems, Internat. J. bifur. chaos, 3, 1619-1627, (1993) · Zbl 0884.94004
[5] Chee, C.Y.; Xu, D., Secure digital communication using controlled projective synchronisation of chaos, Chaos solitons fractals, 23, 1063-1070, (2005) · Zbl 1068.94010
[6] Rosenblum, M.; Pikovsky, A.; Kurths, J., Phase synchronization of chaotic oscillators, Phys. rev. lett., 76, 1804-1807, (1996)
[7] Shahverdiev, E.M.; Sivaprakasam, S.; Shore, K.A., Lag synchronization in time-delayed systems, Phys. lett. A, 292, 320-324, (2002) · Zbl 0979.37022
[8] Li, C.; Liao, X.; Wong, K., Lag synchronization of hyperchaos with application to secure communications, Chaos solitons fractals, 23, 183-193, (2005) · Zbl 1068.94004
[9] Rulkov, N.; Sushchik, M.; Tsimring, L.; Abarbanel, H., Generalized synchronization of chaos in directionally coupled chaotic systems, Phys. rev. E, 51, 980-994, (1995)
[10] Yang, S.; Duan, C., Generalized synchronization in chaotic systems, Chaos solitons fractals, 9, 1703-1707, (1998) · Zbl 0946.34040
[11] Mainieri, R.; Rehacek, J., Projective synchronization in three-dimensional chaotic systems, Phys. rev. lett., 82, 3042-3045, (1999)
[12] Hu, M.; Xu, Z., Adaptive feedback controller for projective synchronization, Nonlinear anal. real world appl., 9, 1253-1260, (2008) · Zbl 1144.93364
[13] Chen, J.; Jiao, L.; Wu, J.; Wang, X., Projective synchronization with different scale factors in a driven-response complex network and its application in image encryption, Nonlinear anal. real world appl., 11, 3045-3058, (2010) · Zbl 1214.93014
[14] Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D.; Zhou, C., The synchronization of chaotic systems, Phys. rep., 366, 1-101, (2002) · Zbl 0995.37022
[15] Li, C.; Liao, X.; Yu, J., Synchronization of fractional order chaotic systems, Phys. rev. E, 68, 67203, (2003)
[16] Erjaee, G.; Momani, S., Phase synchronization in fractional differential chaotic systems, Phys. lett. A, 372, 2350-2354, (2008) · Zbl 1220.34004
[17] Ping, Z.; Xue-Feng, C.; Nian-Ying, Z., Generalized synchronization between different fractional-order chaotic systems, Commun. theor. phys., 50, 931, (2008) · Zbl 1392.34065
[18] Peng, G.; Jiang, Y.; Chen, F., Generalized projective synchronization of fractional order chaotic systems, Phys. A, 387, 3738-3746, (2008)
[19] Wang, X.; He, Y., Projective synchronization of fractional order chaotic system based on linear separation, Phys. lett. A, 372, 435-441, (2008) · Zbl 1217.37035
[20] Wang, J.; Zhang, Y., Designing synchronization schemes for chaotic fractional-order unified systems, Chaos solitons fractals, 30, 1265-1272, (2006) · Zbl 1142.37332
[21] Odibat, Z.; Corson, N.; Aziz-Alaoui, M.; Bertelle, C., Synchronization of chaotic fractional-order systems via linear control, Internat. J. bifur. chaos, 20, 81-97, (2010) · Zbl 1183.34095
[22] Xin, B.; Chen, T.; Liu, Y., Synchronization of chaotic fractional-order WINDMI systems via linear state error feedback control, Math. probl. eng., 2010, (2010) · Zbl 1205.93054
[23] Tavazoei, M.; Haeri, M., Synchronization of chaotic fractional-order systems via active sliding mode controller, Phys. A, 387, 57-70, (2008)
[24] Wang, J.W.; Chen, A.M., A new scheme to projective synchronization of fractional-order chaotic systems, Chin. phys. lett., 27, 110501, (2010)
[25] Bhalekar, S.; Daftardar-Gejji, V., Synchronization of different fractional order chaotic systems using active control, Commun. nonlinear sci. numer. simul., 15, 3536-3546, (2010) · Zbl 1222.94031
[26] Wang, M.J.; Wang, X.Y.; Niu, Y.J., Projective synchronization of a complex network with different fractional order chaos nodes, Chin. phys. B, 20, 010508, (2011)
[27] Zhou, P.; Zhu, W., Function projective synchronization for fractional-order chaotic systems, Nonlinear anal. real world appl., 12, 811-816, (2011) · Zbl 1209.34065
[28] Zhou, S.; Lin, X.; Li, H., Chaotic synchronization of a fractional order system based on washout filter control, Commun. nonlinear sci. numer. simul., 16, 1533-1540, (2011)
[29] Zhou, P.; Kuang, F., Synchronization between fractional-order chaotic system and chaotic system of integer orders, Acta phys. sin., 59, 6851-6858, (2010)
[30] Zhou, P.; Cheng, Y.M.; Kuang, F., Synchronization between fractional-order chaotic systems and integer orders chaotic systems (fractional-order chaotic systems), Chin. phys. B, 19, 090503, (2010)
[31] Zhou, P.; Ding, R., Chaotic synchronization between different fractional-order chaotic systems, J. franklin inst., (2011) · Zbl 1254.93078
[32] Zhou, P.; Cao, Y.X., Function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems, Chin. phys. B, 19, 100507, (2010)
[33] Odibat, Z.M., Adaptive feedback control and synchronization of non-identical chaotic fractional order systems, Nonlinear dyn., 60, 479-487, (2010) · Zbl 1194.93105
[34] Jia, L.X.; Dai, H.; Hui, M., Nonlinear feedback synchronisation control between fractional-order and integer-order chaotic systems, Chin. phys. B, 19, 110509, (2010)
[35] Si, G.Q.; Sun, Z.Y.; Zhang, Y.B., A general method for synchronizing an integer-order chaotic system and a fractional-order chaotic system, Chin. phys. B, 20, 080505, (2011) · Zbl 1274.37030
[36] Femat, R.; Solis-Perales, G., Synchronization of chaotic systems with different order, Phys. rev. E, 65, 36226, (2002)
[37] Shi, X.; Wang, Z., Adaptive added-order anti-synchronization of chaotic systems with fully unknown parameters, Appl. math. comput., 215, 1711-1717, (2009) · Zbl 1179.65158
[38] Al-Sawalha, M.; Noorani, M., Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters, Commun. nonlinear sci. numer. simul., 15, 3022-3034, (2010) · Zbl 1222.34059
[39] Bowong, S.; McClintock, P., Adaptive synchronization between chaotic dynamical systems of different order, Phys. lett. A, 358, 134-141, (2006) · Zbl 1142.93405
[40] Li, S.; Ge, Z., Generalized synchronization of chaotic systems with different orders by fuzzy logic constant controller, Expert syst. appl., 38, 2302-2310, (2011)
[41] Zhu, F., Full-order and reduced-order observer-based synchronization for chaotic systems with unknown disturbances and parameters, Phys. lett. A, 372, 223-232, (2008) · Zbl 1217.37038
[42] Ling, L.; Zhi-An, G., Synchronization of chaotic systems with different orders, Chin. phys., 16, 346, (2007)
[43] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[44] Diethelm, K., The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type, (2010), Springer Berlin · Zbl 1215.34001
[45] D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Proceeeding of IMACS, IEEE-SMC, Lille, France, 1996, pp. 963-968.
[46] Deng, W.; Li, C.; Lu, J., Stability analysis of linear fractional differential system with multiple time delays, Nonlinear dyn., 48, 409-416, (2007) · Zbl 1185.34115
[47] Agiza, H.; Yassen, M., Synchronization of Rössler and Chen chaotic dynamical systems using active control, Phys. lett. A, 278, 191-197, (2001) · Zbl 0972.37019
[48] Vincent, U., Synchronization of identical and non-identical 4-D chaotic systems using active control, Chaos solitons fractals, 37, 1065-1075, (2008) · Zbl 1153.37359
[49] Oustaloup, A.; Levron, F.; Mathieu, B.; Nanot, F.M., Frequency-band complex noninteger differentiator: characterization and synthesis, IEEE trans. circuit syst. I, 47, 25-39, (2000)
[50] D. Valério, J. Sá da Costa, Ninteger: a non-integer control toolbox for MatLab, in: Fractional derivatives and applications, IFAC, Bordeaux, France, 2004, pp. 208-213.
[51] Petráš, I., Chaos in the fractional-order volta’s system: modeling and simulation, Nonlinear dyn., 57, 157-170, (2009) · Zbl 1176.34050
[52] Petráš, I., Modeling and numerical analysis of fractional-order Bloch equations, Comput. math. appl., 61, 341-356, (2011) · Zbl 1211.65096
[53] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys. rev. lett., 91, 34101, (2003)
[54] Hartley, T.T.; Lorenzo, C.F.; Qammer, K., Chaos in a fractional order chua’s system, IEEE trans. circuit syst. I, 42, 485-490, (1995)
[55] Pan, L.; Zhou, W.; Zhou, L.; Sun, K., Chaos synchronization between two different fractional-order hyperchaotic systems, Commun. nonlinear sci. numer. simul., 16, 2628-2640, (2011) · Zbl 1221.37220
[56] Lu, J.J.; Liu, C.X., Realization of fractional-order Liu chaotic system by circuit, Chin. phys., 16, 1586-1590, (2007)
[57] Chen, X.R.; Liu, C.X.; Wang, F.Q., Circuit realization of the fractional-order unified chaotic system, Chin. phys. B, 17, 1664-1669, (2008)
[58] Zhang, R.X.; Yang, S.P., Chaos in fractional-order generalized Lorenz system and its synchronization circuit simulation, Chin. phys. B, 18, 3295-3303, (2009)
[59] Lorenzo, C.F.; Hartley, T.T., Variable order and distributed order fractional operators, Nonlinear dyn., 29, 57-98, (2002) · Zbl 1018.93007
[60] Podlubny, I.; Petráš, I.; Vinagre, B.M.; O’leary, P.; Dorčák, L’, Analogue realizations of fractional-order controllers, Nonlinear dyn., 29, 281-296, (2002) · Zbl 1041.93022
[61] Charef, A., Analogue realisation of fractional-order integrator, differentiator and fractional \(\operatorname{PI} \lambda \operatorname{D} \mu\) controller, IEE proc. control theory appl., 153, 714-720, (2006)
[62] J.L. Zhou, Y.F. Pu, X. Yuan, K. Liao, Any fractional order H type analog fractance circuit, in: ASIC, 2005. ASICON 2005. 6th International Conference On IEEE, Shanghai, China, 2006, pp. 1098-1101.
[63] Jiang, C.; Carletta, J.; Hartley, T.T., Implementation of fractional-order operators on field programmable gate arrays, (), 333-346 · Zbl 1129.93014
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