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Feedback control and hybrid projective synchronization of a fractional-order Newton-Leipnik system. (English) Zbl 1248.93074
Summary: In this work, stability analysis of the fractional-order Newton-Leipnik system is studied by using the fractional Routh-Hurwitz criteria. The fractional Routh-Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria. Based on the fractional Routh-Hurwitz conditions and using specific choice of linear feedback controllers, it is shown that the Newton-Leipnik system is controlled to its equilibrium points. Moreover, the theoretical basis of hybrid projective synchronization of commensurate and incommensurate fractional-order Newton-Leipnik systems is investigated. Based on the stability theorems of fractional-order systems, the controllers for hybrid projective synchronization are derived. Numerical results show the effectiveness of the theoretical analysis.
93B52Feedback control
34H10Chaos control (ODE)
[1]Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K.: Chaos in a fractional order Chua’s system, IEEE trans circ syst I 42, 485-490 (1995)
[2]Grigorenko, I.; Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system, Phys rev lett 91 (2003)
[3]Li, C. P.; Peng, G. J.: Chaos in Chen’s system with a fractional order, Chaos solitons fract 22, 443-450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013
[4]Li, C. G.; Chen, G. R.: Chaos and hyperchaos in the fractional-order Rössler equations, Physica A 341, 55-61 (2004)
[5]Wang, X. Y.; Wang, M. J.: Dynamic analysis of the fractional-order Liu system and its synchroniztion, Chaos 17 (2007) · Zbl 1163.37382 · doi:10.1063/1.2755420
[6]Pecora, L. M.; Carrol, T. L.: Synchronization in chaotic systems, Phys rev lett 64, 821-824 (1990)
[7]Matouk, A. E.: Dynamic analysis feedback control and synchronization of Liu dynamical system, Nonlinear anal theor meth appl 69, 3213-3224 (2008) · Zbl 1176.34060 · doi:10.1016/j.na.2007.09.029
[8]Arman, Kiani-B.; Kia, Fallahi; Naser, Pariz: 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 · doi:10.1016/j.cnsns.2007.11.011
[9]Matignon D. Stability results for fractional differential equations with applications to control processing Computational Engineering in Systems and Application. In: Multiconference, IMACS, IEEE-SMC, Lille, France 1996; 2:963 – 968.
[10]Deng, W. H.; Li, C. P.: Chaos synchronization of the fractional Lü system, Physica A 353, 61-72 (2005)
[11]Li, C. P.; Deng, W. H.: Chaos synchronization of the Chua system with a fractional order, Physica A 360, 171-185 (2006)
[12]Li, C. P.; Yan, J. P.: The synchronization of three fractional differential systems, Chaos solitons fract 32, 751-757 (2007)
[13]Zhu, H.; Zhou, S. B.; He, Z.: Chaos synchronization of the fractional-order Chen’s system, Chaos, solitons fract 41, 2733-2740 (2009) · Zbl 1198.93206 · doi:10.1016/j.chaos.2008.10.005
[14]Wang, X. Y.; Song, J. M.: Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control, Commun nonlinear sci numer simulat 14, 3351-3357 (2009) · Zbl 1221.93091 · doi:10.1016/j.cnsns.2009.01.010
[15]Wang, X. Y.; He, Y. J.; Wang, M. J.: Chaos control of a fractional order modified coupled dynamos system, Nonlinear anal 71, 6126-6134 (2009) · Zbl 1187.34080 · doi:10.1016/j.na.2009.06.065
[16]Sara, Dadras; Reza, Moneni Hamid: Control of a fractional-order economical system via sliding mode, Physica A 389, 2434-2442 (2010)
[17]Matouk, A. E.: Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol-Duffing circuit, Commun nonlinear sci numer simulat 16, 975-986 (2011) · Zbl 1221.93227 · doi:10.1016/j.cnsns.2010.04.027
[18]Leipnik, R. B.; Newton, T. A.: Double strange attractors in rigid body motion with linear feedback control, Phys lett A 86, 63-67 (1981)
[19]Chen, S.; Zhang, Q.; Xie, J.: A stable-manifold-based method for chaos control and synchronization, Chaos solitons fract 20, 947-954 (2004) · Zbl 1050.93032 · doi:10.1016/j.chaos.2003.09.021
[20]Richter, H.: Controlling chaotic system with multiple strange attractors, Phys lett A 300, 182-188 (2002) · Zbl 0997.37012 · doi:10.1016/S0375-9601(02)00183-4
[21]Wang, X.; Tian, L.: Bifurcation analysis and linear control of the Newton – leipnik system, Chaos solitons fract 27, 31-38 (2006) · Zbl 1091.93031 · doi:10.1016/j.chaos.2005.04.009
[22]Sheu, L. J.; Chen, H. K.; Chen, J. H.: Chaos in a new system with a fractional order, Chaos solitons fract 31, 1203-1212 (2007)
[23]Sheu, L. J.; Chen, H. K.; Chen, J. H.: Chaos in the Newton – leipnik system with fractional order, Chaos solitons fract 36, 98-103 (2008)
[24]Jia, Q.: Chaos control and synchronization of the Newton – leipnik chaotic system, Chaos solitons fract 35, 814-824 (2008)
[25]Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, Geophys J roy astron soc 13, 529-539 (1967)
[26]Ahmed, E.; El-Sayed, A. M. A.; El-Saka, H. A. A.: On some Routh – Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems, Phys let A 358, 1-4 (2006) · Zbl 1142.30303 · doi:10.1016/j.physleta.2006.04.087
[27]Ahmed, E.; El-Sayed, A. M. A.; El-Saka, H. A. A.: Equilibrium points, stability and numerical solutions of fractional-order predator – prey and rabies models, J math anal appl 325, 542-553 (2007) · Zbl 1105.65122 · doi:10.1016/j.jmaa.2006.01.087
[28]Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order, Electron trans numer anal 5, 1-6 (1997) · Zbl 0890.65071 · doi:emis:journals/ETNA/vol.5.1997/pp1-6.dir/pp1-6.html
[29]Diethelm, K.; Ford, N. J.: Analysis of fraction differential equations, J math anal appl 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[30]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
[31]Diethelm, K.; Freed, A. D.: The fracpece subroutine for the numerical solution of differential equations of fractional order, Forschung und wissenschaftliches rechnen, 57-71 (1999)
[32]Chang, C. M.; Chen, H. K.: Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen – Lee systems, Nonlinear dyn (2010)
[33]Tavazoei, M. S.; Haeri, M.: A necessary condition for double scroll attractor existence in fractional-order systems, Phys. lett. A 367, 102-113 (2007)
[34]Tavazoei, M. S.; Haeri, M.: Chaotic attractors in incommensurate fractional order systems, Physica D 237, 2628-2637 (2008)
[35]Li, Y.; Chen, Y. Q.; Podlubny, I.: Mittag – Leffler stability of fractional order nonlinear dynamic systems, Automatica 45, 1965-1969 (2009) · Zbl 1185.93062 · doi:10.1016/j.automatica.2009.04.003
[36]Baleanu, D.; Sadati, S. J.; Ranjbar, A.; Ghaderi, R.; Abdeljawad, T.: Mittag – Leffler stability theorem for fractional nonlinear systems with delay, Abstr appl anal (2010)
[37]Baleanu, D.; Sadati, S. J.; Ghaderi, R.; Ranjbar, A.; Abdeljawad, T.; Jarad, F.: Razumikhin stability theorem for fractional systems with delay, Abstr appl anal (2010) · Zbl 1197.34157 · doi:10.1155/2010/124812