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Phase and anti-phase synchronization of fractional order chaotic systems via active control. (English) Zbl 1221.65320
Summary: This paper is devoted to investigate the phase and anti-phase synchronization between two identical and non-identical fractional order chaotic systems using techniques from active control theory. The techniques are applied to fractional order chaotic Lü and Liu systems. Numerical results demonstrate the effectiveness and feasibility of the proposed control techniques.
MSC:
65P20Numerical chaos
References:
[1]Pecora, L. M.; Carroll, T. L.: Synchronization in chaotic systems, Phys rev lett 64, 821-824 (1990)
[2]Kocarev, L.; Parlitz, U.: General approach for chaotic synchronization with applications to communication, Phys rev lett 74, 5028-5031 (1995)
[3]Ma, J.; Ying, H. P.; Pu, Z. S.: An anti-control scheme for spiral under Lorenz chaotic signal, Chin phys lett 22, No. 5, 1065-1068 (2005)
[4]Corron, N. J.; Hahs, D. W.: A new approach to communications using chaotic signals, IEEE trans circ syst 44, 373-382 (1997) · Zbl 0902.94003 · doi:10.1109/81.572333
[5]Park, J. H.: Adaptive synchronization of hyperchaotic Chen system with uncertain parameters, Chaos soliton fract 26, 959-964 (2005) · Zbl 1093.93537 · doi:10.1016/j.chaos.2005.02.002
[6]Xian, W. Y.; Hongmin, Z.: Synchronization of two hyperchaotic systems via adaptive control, Chaos soliton fract 39, 2268-2273 (2009) · Zbl 1197.37046 · doi:10.1016/j.chaos.2007.06.100
[7]Shihua, C.; Lü, J.: Parameters identification and synchronization of chaotic systems based upon adaptive control, Phys lett A 299, 353-358 (2002) · Zbl 0996.93016 · doi:10.1016/S0375-9601(02)00724-7
[8]Lu, J.; Wu, X.; Han, X.; Lü, J.: Adaptive feedback synchronization of a unified chaotic system, Phy lett A 329, 327-333 (2004) · Zbl 1209.93119 · doi:10.1016/j.physleta.2004.07.024
[9]Huang, J.: Chaos synchronization between two novel differently perchaotic systems with unknown parameters, Nonlinear anal: theory methods appl. 69, 4174-4181 (2008) · Zbl 1161.34338 · doi:10.1016/j.na.2007.10.045
[10]Chen, S.; Lü, J.: Synchronization of an uncertain unified chaotic system via adaptive control, Chaos soliton fract. 14, 643-647 (2002) · Zbl 1005.93020 · doi:10.1016/S0960-0779(02)00006-1
[11]Lü, J.; Lu, J.: Controlling uncertain Lü system using linear feedback, Chaos soliton fract. 17, 127-133 (2003) · Zbl 1039.37019 · doi:10.1016/S0960-0779(02)00456-3
[12]Chen, H. K.: Global chaos synchronization of new chaotic systems via nonlinear control, Chaos soliton fract. 23, 1245-1251 (2005) · Zbl 1102.37302 · doi:10.1016/j.chaos.2004.06.040
[13]Zhang, Q.; Lu, J.: Chaos synchronization of a new chaotic system via nonlinear control, Chaos soliton fract. 37, 175-179 (2008)
[14]Park, J. H.: Chaos synchronization of a chaotic system via nonlinear control, Chaos soliton fract. 25, 579-584 (2005) · Zbl 1092.37514 · doi:10.1016/j.chaos.2004.11.038
[15]Bai, E. W.; Lonngen, K. E.: Synchronization of two Lorenz systems using active control, Chaos soliton fract. 8, 51-58 (1997) · Zbl 1079.37515 · doi:10.1016/S0960-0779(96)00060-4
[16]Li, G. H.; Zhou, S. P.: Anti-synchronization in different chaotic systems, Chaos soliton fract 32, 516-520 (2007)
[17]Yassen, M. T.: Chaos synchronization between two different chaotic systems using active control, Chaos soliton fract 23, 131-140 (2005) · Zbl 1091.93520 · doi:10.1016/j.chaos.2004.03.038
[18]Wu, X.; Lu, J.: Parameter identification and backstepping control of uncertain Lü system, Chaos soliton fract 18, 721-729 (2003) · Zbl 1068.93019 · doi:10.1016/S0960-0779(02)00659-8
[19]Yang, S. S.; Juan, C. K.: Generalized synchronization in chaotic systems, Chaos soliton fract 9, 1703-1707 (1998) · Zbl 0946.34040 · doi:10.1016/S0960-0779(97)00149-5
[20]Yu, H. J.; Liu, Y. Z.: Chaotic synchronization based on stability criterion of linear systems, Phys lett A 314, 292-298 (2003) · Zbl 1026.37024 · doi:10.1016/S0375-9601(03)00908-3
[21]Rosenblum, M. G.; Pikovsky, A. S.; Kurths, J.: From phase to lag synchronization in coupled chaotic oscillators, Phys rev lett 78, 4193-4196 (1997)
[22]Park, E. H.; Zaks, M. A.; Kurths, J.: Phase synchronization in the forced Lorenz system, Phys rev E 60, 6627-6638 (1999) · Zbl 1062.37502 · doi:10.1103/PhysRevE.60.6627
[23]Liu, W. Q.: Anti-phase synchronization in coupled chaotic oscillators, Phys rev E 73, 57203-57204 (2006)
[24]Erjaee, G. H.; Atabakzadeh, M.; Saha, L. M.: Interesting synchronization-like behavior, Int J bifur chaos 14, 1447-1453 (2004) · Zbl 1084.37506 · doi:10.1142/S0218127404009934
[25]Zhang, Y.; Sun, J.: Chaotic synchronization and anti-synchronization based on suitable separation, Phys lett A 330, 442-447 (2004) · Zbl 1209.37039 · doi:10.1016/j.physleta.2004.08.023
[26]Kim, C. M.; Rim, S.; Kye, W. H.; Ryu, J. W.; Park, Y. J.: Anti-synchronization of chaotic oscillators, Phys lett A 320, 39-49 (2003) · Zbl 1098.37521 · doi:10.1016/j.physleta.2003.10.051
[27]Li, C.; Liao, X.: Anti-synchronization of a class of coupled chaotic systems via linear feedback control, Int J bifur chaos 16, 1041-1047 (2006) · Zbl 1097.94037 · doi:10.1142/S0218127406015295
[28]Idowu, B. A.; Vincent, U. E.; Njah, A. N.: Anti-synchronization of chaos in nonlinear gyros via active control, J math control sci appl 1, 191-200 (2007)
[29]Vincent, U. E.; Laoye, J. A.: Synchronization, anti-synchronization and current transport in non-identical chaotic ratchets, Physica A 384, 230-240 (2007)
[30]Wang, Z. L.; Shi, X. R.: Anti-synchronization of Liu system and Lorenz system with known or unknown parameters, Nonlinear dyn 57, 425-430 (2009) · Zbl 1176.70037 · doi:10.1007/s11071-008-9452-1
[31]Pan L, Zhou W, Fang J, Li D. A novel active pinning control for synchronization and anti-synchronization of new uncertain unified chaotic systems. Nonlinear Dyn, doi:10.1007/s11071-010-9728-0.
[32]Li, C. P.; Peng, C. J.: Chaos in Chen’s system with a fractional order, Chaos soliton fract 22, 443-450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013
[33]Li, C. G.; Chen, G.: Chaos in the fractional order Chen system and its control, Chaos soliton fract 22, 549-554 (2004) · Zbl 1069.37025 · doi:10.1016/j.chaos.2004.02.035
[34]Yan, J. P.; Li, C. P.: On synchronization of three chaotic systems, Chaos soliton fract 23, 1683-1688 (2005) · Zbl 1068.94535 · doi:10.1016/j.chaos.2004.06.055
[35]Erjaee, G. H.: On analytical justification of phase synchronization in different chaotic systems, Chaos soliton fract 39, 1195-1202 (2009) · Zbl 1197.37027 · doi:10.1016/j.chaos.2007.06.006
[36]Erjaee, G. H.; Alnasr, M.: Phase synchronization in coupled sprott chaotic systems presented by fractional differential equations, Disc dyn nat soc (2009) · Zbl 1187.37044 · doi:10.1155/2009/753746
[37]Podlubny, I.: Fractional differential equations, (1999)
[38]Hifer, R.: Applications of fractional calculus in physics, (2001)
[39]Koeller, Rc.: Application of fractional calculus to the theory of viscoelasticity, J appl mech 51, No. 2, 299-307 (1984) · Zbl 0544.73052 · doi:10.1115/1.3167616
[40]Sun, Hh.; Abdelwahad, Aa.; Onaral, B.: Linear approximation of transfer function with a pole of fractional order, Ieeetrans automat control 29, No. 5, 441-444 (1984) · Zbl 0532.93025 · doi:10.1109/TAC.1984.1103551
[41]Ichise, M.; Nagayanagi, Y.; Kojima, T.: An analog simulation of noninteger order transfer functions for analysis of electrode process, J electroanal chem 33, 253-265 (1971)
[42]O. Heaviside, Electromagnetic theory, Chelsea, New York, 1971.
[43]Oustaloup, A.; Sabatier, J.; Lanusse, P.: From fractal robustness to CRONE control, Fract calculus appl anal 2, No. 1, 1-30 (1999) · Zbl 1111.93310
[44]Hartley, T. T.; Lorenzo, C. F.: Dynamics and control of initialized fractional-order systems, Nonlinear dyn 29, No. 1 – 4, 201-233 (2002) · Zbl 1021.93019 · doi:10.1023/A:1016534921583
[45]Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K.: Chaos in a fractional order Chua’s system, IEEE trans circuits syst – I 42, No. 8, 485-490 (1995)
[46]Arena, P.; Caponetto, R.; Fortuna, L.; Porto, D.: Bifurcation and chaos in noninteger order cellular neural networks, Int J bifur chaos 8, No. 7, 1527-1539 (1998) · Zbl 0936.92006 · doi:10.1142/S0218127498001170
[47]Grigorenko, I.; Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system, Phys rev lett 91, No. 3, 034101 (2003)
[48]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[49]Diethelm, K.; Ford, N. J.; Freed, A. D.: A predictor – corrector approach for the numerical solution of fractional differential equations, Nonlinear dyn 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[50]Lorenz, E. N.: Deterministic nonperiodic flow, J atmos sci 20, No. 2, 130-141 (1963)
[51]Chen, G.; Ueta, T.: Yet another chaotic attractor, Int J bifur chaos 9, No. 7, 1465-1476 (1999) · Zbl 0962.37013 · doi:10.1142/S0218127499001024
[52]Lü, J.; Chen, G.; Zhang, S.: A new chaotic attractor coined, Int J bifur chaos 12, 1001-1015 (2002)
[53]Lü, Jg.: Chaotic dynamics of the fractional order Lü system and its synchronization, Phys lett A 354, No. 4, 305-311 (2006)
[54]Liu, C.; Liu, L.; Liu, T.: A novel three-dimensional autonomous chaos system, Chaos solition fract 39, No. 4, 1950-1958 (2009) · Zbl 1197.37039 · doi:10.1016/j.chaos.2007.06.079
[55]Daftardar-Gejji, V.; Bhalekar, S.: Choas in fractional order Liu system, Comput math appl 593, 1117-1127 (2010) · Zbl 1189.34081 · doi:10.1016/j.camwa.2009.07.003
[56]Matignon D. Stability results of fractional differential equations with applications to control processing. In: IEEE-SMC proceedings of the Computational engineering in systems and application multiconference, IMACS, Lille, France, July, vol. 2; 1996. p. 963 – 8.
[57]Erjaee, G. H.; Momani, S.: Phase synchronization in fractional differential chaotic systems, Phys lett A 372, 2350-2354 (2008) · Zbl 1220.34004 · doi:10.1016/j.physleta.2007.11.065