Synchronization of different fractional order chaotic systems using active control. (English) Zbl 1222.94031
Summary: Synchronization of fractional order chaotic dynamical systems is receiving increasing attention owing to its interesting applications in secure communications of analog and digital signals and cryptographic systems. In this article we utilize active control technique to synchronize different fractional order chaotic dynamical systems. Further we investigate the interrelationship between the (fractional) order and synchronization in different chaotic dynamical systems. It is observed that synchronization is faster as the order tends to one.
MSC:
| 94A60 | Cryptography |
| 37N35 | Dynamical systems in control |
| 34A08 | Fractional ordinary differential equations |
| 34H10 | Chaos control for problems involving ordinary differential equations |
Keywords:
Caputo derivative; synchronization; fractional order dynamical systems; active control; predictor; corrector methodReferences:
| [1] | Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys Rev Lett, 64, 8, 821 (1990) · Zbl 0938.37019 |
| [2] | Pecora, L. M.; Carroll, T. L., Driving systems with chaotic signals, Phys Rev A, 44, 2374 (1991) |
| [3] | Kocarev, L.; Parlitz, U., General approach for chaotic synchronization with applications to communication, Phys Rev Lett, 74, 5028 (1995) |
| [4] | Carroll, T. L.; Heagy, J. F.; Pecora, L. M., Transforming signals with chaotic synchronization, Phys Rev E, 54, 5, 4676 (1996) |
| [5] | Hilfer, R., Applications of fractional calculus in physics (2001), World Scientific: World Scientific USA · Zbl 0998.26002 |
| [6] | He, R.; Vaidya, P. G., Implementation of chaotic cryptography with chaotic synchronization, Phys Rev E, 57, 2, 1532 (1998) |
| [7] | Huang, L.; Feng, R.; Wang, M., Synchronization of chaotic systems via nonlinear control, Phys Lett A, 320, 271 (2004) · Zbl 1065.93028 |
| [8] | Liao, T. L., Adaptive synchronization of two Lorenz systems, Chaos Solitons Fractals, 9, 1555 (1998) · Zbl 1047.37502 |
| [9] | Yassen, M. T., Adaptive control and synchronization of a modified Chua’s circuit system, Appl Math Comput, 135, 1, 113 (2001) · Zbl 1038.34041 |
| [10] | Wang, Y.; Guan, Z.; Wen, X., Adaptive synchronization for Chen chaotic system with fully unknown parameters, Chaos Solitons Fractals, 19, 899-903 (2004) · Zbl 1053.37528 |
| [11] | Park, J. H., Adaptive synchronization of Rossler system with uncertain parameters, Chaos Solitons Fractals, 25, 333-338 (2005) · Zbl 1125.93470 |
| [12] | Bowong, S.; Moukam Kakmeni, F., Synchronization of uncertain chaotic systems via backstepping approach, Chaos Solitons Fractals, 21, 999-1011 (2004) · Zbl 1045.37011 |
| [13] | Zhang, J.; Li, C.; Zhang, H.; Yu, J., Chaos synchronization using single variable feedback based on backstepping method, Chaos Solitons Fractals, 21, 1183-1193 (2004) · Zbl 1129.93518 |
| [14] | Bai, E. W.; Lonngren, K. E., Synchronization of two Lorenz systems using active control, Chaos Solitons Fractals, 8, 51-58 (1997) · Zbl 1079.37515 |
| [15] | Bai, E. W.; Lonngren, K. E., Sequential synchronization of two Lorenz systems using active control, Chaos Solitons Fractals, 11, 1041-1044 (2000) · Zbl 0985.37106 |
| [16] | Agiza, H. N.; Yassen, M. T., Synchronization of Rossler and chaotic dynamical systems using active control, Phys Lett A, 278, 191-197 (2001) · Zbl 0972.37019 |
| [17] | Ho, M. C.; Hung, Y. C., Synchronization of two different systems by using generalized active control, Phys Lett A, 301, 424 (2002) · Zbl 0997.93081 |
| [18] | Codreanu, S., Synchronization of spatiotemporal nonlinear dynamical systems by an active control, Chaos Solitons Fractals, 15, 507-510 (2003) · Zbl 1098.93509 |
| [19] | Zhang, H.; Ma, X. K., Synchronization of uncertain chaotic systems with parameters perturbation via active control, Chaos Solitons Fractals, 21, 39-47 (2004) · Zbl 1048.37031 |
| [20] | Ucar, A.; Lonngren, K. E.; Bai, E. W., Synchronization of the coupled FitzHugh0Nagumo systems, Chaos Solitons Fractals, 20, 1085-1090 (2004) · Zbl 1050.93064 |
| [21] | Yassen, M. T., Chaos synchronization between two different chaotic systems using active control, Chaos Solitons Fractals, 23, 131-140 (2005) · Zbl 1091.93520 |
| [22] | Lei, Y.; Xu, W.; Shen, J.; Fang, T., Global synchronization of two parametrically excited systems using active control, Chaos Solitons Fractals, 28, 428-436 (2006) · Zbl 1084.37029 |
| [23] | Wu, X.; Lu, H.; Shen, S., Synchronization of a new fractional order hyperchaotic system, Phys Lett A, 373, 2329-2337 (2009) · Zbl 1231.34091 |
| [24] | Shahiri, M.; Ghaderi, R.; Ranjbar, A.; Hosseinnia, S. H.; Momani, S., Chaotic fractional-order Coullet system: synchronization and control approach, Commun Nonlinear Sci Numer Simulat, 15, 665-674 (2010) · Zbl 1221.37222 |
| [25] | Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010 |
| [26] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives: theory and applications (1993), Gordon and Breach: Gordon and Breach Yverdon · Zbl 0818.26003 |
| [27] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003 |
| [28] | Sabatier, J.; Poullain, S.; Latteux, P.; Thomas, J.; Oustaloup, A., Robust speed control of a low damped electromechanical system based on CRONE control: application to a four mass experimental test bench, Nonlinear Dyn, 38, 383-400 (2004) · Zbl 1142.93389 |
| [29] | Caputo, M.; Mainardi, F., A new dissipation model based on memory mechanism, Pure Appl Geophys, 91, 134-147 (1971) · Zbl 1213.74005 |
| [30] | Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus Appl Anal, 4, 2, 153-192 (2001) · Zbl 1054.35156 |
| [31] | Agrawal, O. P., Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dyn, 29, 145-155 (2002) · Zbl 1009.65085 |
| [32] | Daftardar-Gejji, V.; Jafari, H., Boundary value problems for fractional diffusion-wave equations, Aust J Math Anal Appl, 3, 1-18 (2006) |
| [33] | Daftardar-Gejji, V.; Bhalekar, S., Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method, Appl Math Comput, 202, 113-120 (2008) · Zbl 1147.65106 |
| [34] | Sun, H. G.; Chen, W.; Chen, Y. Q., Variable-order fractional differential operators in anomalous diffusion modeling, Physica A, 388, 4586-4592 (2009) |
| [35] | Jesus, I. S.; Machado, J. A.T., Fractional control of heat diffusion systems, Nonlinear Dyn, 54, 3, 263-282 (2008) · Zbl 1210.80008 |
| [36] | Jesus IS, Machado JAT, Barbosa RS. Control of a heat diffusion system through a fractional order nonlinear algorithm. Comput Math Appl, in press. doi: 10.1016/j.camwa.2009.08.010; Jesus IS, Machado JAT, Barbosa RS. Control of a heat diffusion system through a fractional order nonlinear algorithm. Comput Math Appl, in press. doi: 10.1016/j.camwa.2009.08.010 · Zbl 1189.93047 |
| [37] | Anastasio, T. J., The fractional-order dynamics of brainstem vestibulo-oculomotor neurons, Biol Cybern, 72, 69-79 (1994) |
| [38] | Ortigueira, M. D.; Machado, J. A.T., Fractional calculus applications in signals and systems, Signal Process, 86, 10, 2503-2504 (2006) · Zbl 1172.94301 |
| [39] | Magin, R. L., Fractional calculus in bioengineering (2006), Begll House Publishers: Begll House Publishers USA |
| [40] | Daftardar-Gejji, V.; Babakhani, A., Analysis of a system of fractional differential equations, J Math Appl Anal, 293, 2, 511-522 (2004) · Zbl 1058.34002 |
| [41] | Daftardar-Gejji, V.; Jafari, H., Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, J Math Appl Anal, 328, 2, 1026-1033 (2007) · Zbl 1115.34006 |
| [42] | Matignon D. Stability results for fractional differential equations with applications to control processing. In: Computational engineering in systems and application multiconference, IMACS, IEEE-SMC proceedings, Lille, France, July, vol. 2; 1996. p. 963-8.; Matignon D. Stability results for fractional differential equations with applications to control processing. In: Computational engineering in systems and application multiconference, IMACS, IEEE-SMC proceedings, Lille, France, July, vol. 2; 1996. p. 963-8. |
| [43] | Kiani-B, A.; Fallahi, K.; Pariz, N.; Leung, H., A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter, Commun Nonlinear Sci Numer Simulat, 14, 863-879 (2009) · Zbl 1221.94049 |
| [44] | Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys Rev Lett, 91, 3, 034101 (2003) |
| [45] | Li, C.; Peng, G., Chaos in Chens system with a fractional order, Chaos Solitons Fractals, 22, 443-450 (2004) · Zbl 1060.37026 |
| [46] | Lü, J. G., Chaotic dynamics of the fractional order Lü system and its synchronization, Phys Lett A, 354, 4, 305-311 (2006) |
| [47] | Li, C.; Chen, G., Chaos and hyperchaos in the fractional order Rossler equations, Phys A Stat Mech Appl, 341, 55-61 (2004) |
| [48] | Daftardar-Gejji V, Bhalekar S. Chaos in fractional ordered Liu system. Comput Math Appl, in press. doi: 10.1016/j.camwa.2009.07.003; Daftardar-Gejji V, Bhalekar S. Chaos in fractional ordered Liu system. Comput Math Appl, in press. doi: 10.1016/j.camwa.2009.07.003 · Zbl 1189.34081 |
| [49] | Bhalekar, S.; Daftardar-Gejji, V., Fractional ordered Liu system with time-delay, Commun Nonlinear Sci Numer Simulat, 15, 8, 2178-2191 (2010) · Zbl 1222.34005 |
| [50] | Deng, W. H.; Li, C. P., Chaos synchronization of the fractional Lü system, Physica A, 353, 61-72 (2005) |
| [51] | Deng, W. H.; Li, C. P., Synchronization of chaotic fractional Chen system, J Phys Soc Jpn, 74, 1645-1648 (2005) · Zbl 1080.34537 |
| [52] | Zhou, T.; Li, C., Synchronization in fractional-order differential systems, Physica D, 212, 111-125 (2005) · Zbl 1094.34034 |
| [53] | Lia, C. P.; Deng, W. H.; Xu, D., Chaos synchronization of the Chua system with a fractional order, Physica A, 360, 171-185 (2006) |
| [54] | Wang, J.; Xionga, X.; Zhang, Y., Extending synchronization scheme to chaotic fractional-order Chen systems, Physica A, 370, 279-285 (2006) |
| [55] | Wang, J.; Zhang, Y., Designing synchronization schemes for chaotic fractional-order unified systems, Chaos Solitons Fractals, 30, 1265-1272 (2006) · Zbl 1142.37332 |
| [56] | Xingyuan, W.; Yijie, H., Projective synchronization of fractional order chaotic system based on linear separation, Phys Lett A, 372, 435-441 (2008) · Zbl 1217.37035 |
| [57] | Yu, Y.; Li, H., The synchronization of fractional-order Rossler hyperchaotic systems, Physica A, 387, 1393-1403 (2008) |
| [58] | Tavazoei, M. S.; Haeri, Mohammad, Synchronization of chaotic fractional-order systems via active sliding mode controller, Physica A, 387, 57-70 (2008) |
| [59] | Matouk, A. E., Chaos synchronization between two different fractional systems of Lorenz family, Math Prob Eng, 2009 (2009), [Article ID 572724] · Zbl 1181.37047 |
| [60] | Hu, J.; Han, Y.; Zhao, L., Synchronizing chaotic systems using control based on a special matrix structure and extending to fractional chaotic systems, Commun Nonlinear Sci Numer Simulat, 15, 115-123 (2010) · Zbl 1221.37212 |
| [61] | Luchko, Y.; Gorenflo, R., An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math Vietnamica, 24, 207-233 (1999) · Zbl 0931.44003 |
| [62] | 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 |
| [63] | Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Elec Trans Numer Anal, 5, 1-6 (1997) · Zbl 0890.65071 |
| [64] | Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J Math Anal Appl, 265, 229-248 (2002) · Zbl 1014.34003 |
| [65] | Wu, X.; Shen, S., Chaos in the fractional-order Lorenz system, Int J Comput Math, 86, 7, 1274-1282 (2009) · Zbl 1169.65115 |
| [66] | Liu, C.; Liu, L.; Liu, T., A novel three-dimensional autonomous chaos system, Chaos Solitons Fractals, 39, 4, 1950-1958 (2009) · Zbl 1197.37039 |
| [67] | Tavazoei, M. S.; Haeri, M., Chaotic attractors in incommensurate fractional order systems, Physica D, 237, 2628-2637 (2008) · Zbl 1157.26310 |
| [68] | Tavazoei, M. S.; Haeri, M., A necessary condition for double scroll attractor existence in fractional order systems, Phys Lett A, 367, 102-113 (2007) · Zbl 1209.37037 |
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