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Parameter identification and synchronization of fractional-order chaotic systems. (English) Zbl 1245.93039
Summary: The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.

93B30 System identification
93C15 Control/observation systems governed by ordinary differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
Full Text: DOI
[1] Li, L.; Wang, L.; Liu, L., An effective hybrid PSOSA strategy for optimization and its application to parameter estimation, Appl math comput, 179, 135-146, (2006) · Zbl 1100.65052
[2] Tang, Y.; Guan, X., Parameter estimation for time-delay chaotic system by particle swarm optimization, Chaos solitons fract, 40, 1391-1398, (2009) · Zbl 1197.93155
[3] Modares, H.; Alfi, A.; Fateh, M., Parameter identification of chaotic dynamics systems through an improved particle swarm optimization, Expert syst appl, 37, 3714-3720, (2010)
[4] He, Q.; Wang, L.; Liu, B., Parameter estimation for chaotic systems by particle swarm optimization, Chaos solitons fract, 34, 654-661, (2007) · Zbl 1152.93504
[5] Tang, Y.; Guan, X., Parameter estimation of chaotic system with time-delay: A differential evolution approach, Chaos solitons fract, 42, 3132-3139, (2009) · Zbl 1198.93222
[6] Chang, W., Parameter identification of Chen and Lü systems: A differential evolution approach, Chaos solitons fract, 32, 1469-1476, (2007) · Zbl 1129.93022
[7] Chang, W., Parameter identification of rossler’s chaotic system by an evolutionary algorithm, Chaos solitons fract, 29, 1047-1053, (2006)
[8] Hu, M.; Xu, Z.; Zhang, R.; Hu, A., Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems, Phys lett A, 361, 231-237, (2007) · Zbl 1170.93365
[9] Huang, L.; Wang, M.; Feng, R., Parameters identification and adaptive synchronization of chaotic systems with unknown parameters, Phys lett A, 342, 299-304, (2005) · Zbl 1222.93203
[10] Zhang, R.; Tian, G.; Li, P.; Yang, S., Adaptive synchronization of a class of chaotic systems with uncertain parameters, Acta phys sin, 57, 2073-2080, (2008), (in Chinese) · Zbl 1174.93585
[11] Zhang, R.; Yang, S., Adaptive generalized projective synchronization of two different chaotic systems with unknown parameters, Chin phys B, 17, 4073-4079, (2008)
[12] Ma, J.; Zhang, A.; Xia, Y.; Zhang, L., Optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems, Appl math comput, 215, 3318-3326, (2010) · Zbl 1181.93032
[13] Ma, J.; Su, W.; Gao, J., Optimization of self-adaptive synchronization and parameters estimation in chaotic hindmarsh – rose neuron model, Acta phys sin, 59, 1554-1561, (2010), (in Chinese) · Zbl 1224.93031
[14] Pourmahmood, M.; Khanmohammadi, S.; Alizadeh, G., Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller, Commun nonlinear sci numer simul, 16, 2869-2879, (2011)
[15] Sun, K.; Sprott, J., Bifurcations of fractional-order diffusionless Lorenz system, Electronic J theor phys, 22, 123-134, (2009)
[16] Li, C.; Cheng, G., Chaos and hyperchaos in the fractional-order Rössler equations, Physica A, 341, 55-61, (2004)
[17] Li, C.; Peng, G., Chaos in chen’s system with a fractional order, Chaos solitons fract, 22, 443-450, (2004) · Zbl 1060.37026
[18] Lu, J.; Chen, G., A note on the fractional-order Chen system, Chaos solitons fract, 27, 685-688, (2006) · Zbl 1101.37307
[19] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys rev lett, 91, 034101, (2003)
[20] Yu, Y.; Li, H.; Wang, S.; Yu, J., Dynamic analysis of a fractional-Lorenz chaotic system, Chaos solitons fract, 42, 1181-1189, (2009) · Zbl 1198.37063
[21] Wu, X.; Shen, S., Chaos in the fractional-order Lorenz system, Int J comput math, 86, 1274-1282, (2009) · Zbl 1169.65115
[22] Yang, Q.; Zeng, C., Chaos in fractional conjugate Lorenz system and its scaling attractors, Commun nonlinear sci numer simul, 15, 4041-4051, (2010) · Zbl 1222.37037
[23] Bhalekar, S.; Daftardar-Gejji, V., Fractional ordered Liu system with time-delay, Commun nonlinear sci numer simul, 15, 2178-2191, (2009) · Zbl 1222.34005
[24] Varsha, D.; Sachin, B., Chaos in fractional ordered Liu system, Comput math appl, 59, 1117-1127, (2010) · Zbl 1189.34081
[25] Diethelm, K.; Ford, N.; Freed, A., A predictor – corrector approach for the numerical solution of fractional differential equations, Nonlinear dyn, 29, 3-22, (2002) · Zbl 1009.65049
[26] Matignon D. Stability results for fractional differential equations with applications to control processing. In: IEEE-SMC Proceedings on Computational Engineering in Systems and Application Multi-Conference, vol. 2, IMACS, Lille, France, 1996, p. 963-8.
[27] Tavazoei, M.; Haeri, M., Chaotic attractors in incommensurate fractional order systems, Physcia D, 237, 2628-2637, (2008) · Zbl 1157.26310
[28] Tavazoei, M.; Haeri, M., A necessary condition for double scroll attractor existence in fractional-order systems, Phys lett A, 367, 102-113, (2007) · Zbl 1209.37037
[29] Deng, W.; Lü, J., Design of multidirectional multi-scroll chaotic attractors based on fractional differential systems via switching control, Chaos, 16, 043120, (2006) · Zbl 1146.37316
[30] Deng, W.; Lü, J., Generating multidirectional multi-scroll chaotic attractors via a fractional differential hysteresis system, Phys lett A, 369, 438-443, (2007) · Zbl 1209.37032
[31] Diethelm, K.; Ford, J., Analysis of fractional differential equations, J math anal appl, 265, 229-248, (2002) · Zbl 1014.34003
[32] Deng, W.; Li, C.; Lü, J., Stability analysis of linear fractional differential system with multiple time delays, Nonlinear dyn, 48, 409-416, (2007) · Zbl 1185.34115
[33] Odibat, Z.; Corson, N.; Aziz-Alaoui, M.; Bertelle, C., Synchronization of chaotic fractional-order systems via linear control, Int J bifurcat chaos, 20, 81-97, (2010) · Zbl 1183.34095
[34] Song, L.; Yang, J.; Xu, S., Chaos synchronization for a class of nonlinear oscillators with fractional order, Nonlinear anal-theor, 72, 2326-2336, (2010) · Zbl 1187.34066
[35] 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
[36] Zhang, R.; Yang, S., Chaos in fractional-order generalized Lorenz system and its synchronization circuit simulation, Chin phys B, 18, 3295-3303, (2009)
[37] Zhang, R.; Yang, S., Adaptive synchronisation of fractional-order chaotic systems, Chin phys B, 19, 020510, (2010)
[38] Wu, X.; Li, J.; Chen, G., Chaos in the fractional order unified system and its synchronization, J Frank instit, 345, 392-401, (2008) · Zbl 1166.34030
[39] Zhang, R.; Yang, Y.; Yang, S., Adaptive synchronization of the fractional-order unified chaotic system, Acta phys sin, 58, 6039-6044, (2009), (in Chinese) · Zbl 1212.37058
[40] Deng, W.; Li, C., Chaos synchronization of the fractional Lü system, Physica A, 353, 61-72, (2005)
[41] Deng, H.; Li, T.; Wang, Q.; Li, H., A fractional-order hyperchaotic system and its synchronization, Chaos solitons fract, 41, 962-969, (2009) · Zbl 1198.34115
[42] Li, C.; Deng, W.; Xu, D., Chaos synchronization of the Chua system with a fractional order, Physica A, 360, 171-185, (2006)
[43] Sparrow, C., The Lorenz equations: bifurcations, chaos and strange attractors, (1982), Springer Verlag, New York · Zbl 0504.58001
[44] Kennedy J, Eberhart R. Particle swarm optimization. In: Proc IEEE Int Conf Neural Networks, vol. 4, 1995, p. 1942-8.
[45] Kennedy, J.; Eberhart, R.; Shi, Y., Swarm intelligence, (2001), CA: Morgan Kaufman San Francisco
[46] Jiang, Y.; Hu, T.; Huang, C.; Wu, X., An improved particle swarm optimization algorithm, Appl math comput, 193, 231-239, (2007) · Zbl 1193.90220
[47] Chang, W.; Shih, S., PID controller design of nonlinear systems using an improved particle swarm optimization approach, Commun nonlinear sci numer simul, 15, 3632-3639, (2010) · Zbl 1222.90083
[48] Daras, S.; Momeni, H., A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors, Phys lett A, 373, 3637-3642, (2009) · Zbl 1233.37022
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