| Citation: |
M. M. El-Dessoky, E. O. Alzahrani, N. A. Almohammadi. CONTROL AND ADAPTIVE MODIFIED FUNCTION PROJECTIVE SYNCHRONIZATION OF LIU CHAOTIC DYNAMICAL SYSTEM[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 601-614. doi: 10.11948/2156-907X.20180119
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CONTROL AND ADAPTIVE MODIFIED FUNCTION PROJECTIVE SYNCHRONIZATION OF LIU CHAOTIC DYNAMICAL SYSTEM
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Abstract
In this work, the feedback control method is proposed to control the behaviour of Liu chaotic dynamical system. The controlled system is stable under some conditions on the parameters of the system determined by Routh-Hurwitz criterion. This paper also presents the adaptive modified function projective synchronization (AMFPS) between two identical Liu chaotic dynamical systems. Based on the Lyapunov stability theorem, adaptive control laws are designed to achieving the AMFPS. Finally, some numerical simulations are obtained to validate the proposed methods. -
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References
[1] H. N. Agiza, On the analysis of stability, bifurcation, chaos and chaos control of kopel map, Chaos, Solitons & Fractals, 1999, 10(11), 1909-1916. [2] S. K. Agrawal and S. Das, Function projective synchronization between four dimensional chaotic systems with uncertain parameters using modified adaptive control method, J. Process Control, 2014, 24(5), 517-530. doi: 10.1016/j.jprocont.2014.02.013 [3] E. Bai and K. Lonngren, Sequential synchronization of two Lorenz system using active control, Chaos, Solitons & Fractals, 2000, 11(7), 1041-1044. [4] N. Cai, Y. Jing and S. Zhang, Modified projective synchronization of chaotic systems with disturbances via active sliding mode control, Commun. Nonlinear Sci. Numer. Simul., 2010, 15(6), 1613-1620. doi: 10.1016/j.cnsns.2009.06.012 [5] T. L. Carroll and L. M. Perora, Synchronizing chaotic circuits, IEEE Transactions on Circuits and Systems, 1991, 38(4), 453-456. doi: 10.1109/31.75404 [6] G. Chen, Chaos on some controllability conditions for chaotic dynamics control, Chaos, Solitons & Fractals, 1997, 8(9), 1461-1470. [7] Y. Chen and X. Li, Function projective synchronization between two identical chaotic systems, Int. J. Mod. Phys. C, 2007, 18(5), 883-888. doi: 10.1142/S0129183107010607 [8] S. Dadras and H. Momeni, Control of a fractional-order economical system via sliding mode, Physica A, 2010, 389(12), 2434-2442. doi: 10.1016/j.physa.2010.02.025 [9] H. Du, Q. Zeng and C. Wang, Function projective synchronization of different chaotic systems with uncertain parameters, Phys. Lett. A, 2008, 372(33), 5402-5410. doi: 10.1016/j.physleta.2008.06.036 [10] E. M. Elabbasy, H. N. Agiza and M. M. El-Dessoky, Global chaos synchronization for four-scroll attractor by nonlinear control, Sci. Res. Essays, 2006, 1(3), 65-71. [11] E. M. Elabbasy and M. M. El-Dessoky, Adaptive coupled synchronization of coupled chaotic dynamical systems, Trends Applied Sci. Res., 2007, 2(2), 88-102. doi: 10.3923/tasr.2007.88.102 [12] E. M. Elabbasy and M. M. El-Dessoky, Synchronization of Van Der Pol oscillator and chen chaotic dynamical system, Chaos, Solitons & Fractals, 2008, 36(5), 1425-1435. [13] M. M. El-Dessoky, Synchronization and anti-synchronization of a hyperchaotic Chen system, Chaos, Solitons & Fractals, 2009, 39(4), 1790-1797. [14] M. M. El-Dessoky, Anti-synchronization of four scroll attractor with fully unknown parameters, Nonlinear Anal.-Real World Appl., 2010, 11(2), 778-783. doi: 10.1016/j.nonrwa.2009.01.048 [15] M. M. El-Dessoky, E. O. Alzahrany, and N. A. Almohammadi. Function Projective Synchronization for Four Scroll Attractor by Nonlinear Control, Appl. Math. Sci., 2017, 11(26), 1247-1259. [16] M. M. El-Dessoky and M. T. Yassen, Adaptive feedback control for chaos control and synchronization for new chaotic dynamical system, Math. Probl. Eng., 2012, Vol. 2012, Article ID 347210, 12 pages. [17] A. Hegazi, H. N. Agiza and M. M. El-Dessoky, Controlling chaotic behaviour for spin generator and Rossler dynamical systems with feedback control, Chaos, Solitons & Fractals, 2001, 12(4), 631-658. [18] J. Huang, Adaptive synchronization between different hyperchaotic systems with fully uncertain parameters, Phys. Lett. A, 2008, 372(27-28), 4799-4804. doi: 10.1016/j.physleta.2008.05.025 [19] C. Hwang, J. Yuan, J. Hsieh and R. Lin, A linear continuous feedback control of chua's circuit, Chaos, Solitons & Fractals, 1997, 8(9), 1507-1515. [20] G. Li, Modified projective synchronization of chaotic system, Chaos, Solitons & Fractals, 2007, 32(5), 1786-1790. [21] G. Li, Generalized synchronization of chaos based on suitable separation, Chaos, Solitons & Fractals, 2009, 39(5), 2056-2062. [22] C. Liu, T. Liu, L. Liu, and K. Liu, A new chaotic attractor, Chaos, Solitons & Fractals, 2004, 22(5), 1031-1038. [23] A. Loria, Master-slave synchronization of fourth order Lu chaotic oscillators via linear output feadback, IEEE Trans. Circuits Syst. Ⅱ-Express Briefs, 2010, 57(3), 213-217. doi: 10.1109/TCSII.2010.2040303 [24] K. Ojo, S. Ogunjo and A. Olagundoye, Projective synchronization via active control of identical chaotic oscillators with parametric and external excitation, International Journal of Nonlinear Science, 2017, 24(2), 76-83. [25] E. Ott, C. Grebogi and J. Yorke, Controlling chaos, Phys. Rev. Lett., 1999, 64(11), 1179-1184. [26] J. Park, Adaptive modified projective synchronization of a unified chaotic system with an uncertain parameter, Chaos, Solitons & Fractals, 2007, 34(5), 1552-1559. [27] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 1990, 64(8), 821-824. doi: 10.1103/PhysRevLett.64.821 [28] J. Petereit and A. Pikovsky, Chaos synchronization by nonlinear coupling, Commun. Nonlinear Sci. Numer. Simul., 2017, 44(C), 344-351. [29] N. Rulkov, M. Sushchik, L. Tsimring and H. Abarbanel, Generalized synchronization of chaos in directionally coupled chaotic systems, Phys. Rev. Lett., 1995, 51(2), 980-994. [30] L. Runzi and W. Zhengmin, Adaptive function projective synchronization of unified chaotic systems with uncertain parameters, Chaos, Solitons & Fractals, 2009, 42(2), 1266-1272. [31] A. Singh and S. Gakkhar, Controlling chaos in a food chain model, Math. Comput. Simul., 2015, 115(C), 24-36. [32] Y. Tang and J. Fang, General method for modified projective synchronization of hyperchaotic systems with known or unknown parameter, Phys. Lett. A, 2008, 372(11), 1816-1826. doi: 10.1016/j.physleta.2007.10.043 [33] K. Vishal and S. Agrawal, On the dynamics, existence of chaos, control and synchronization of a novel complex chaotic system, Chin. J. Phys., 2017, 55(2), 519-532. doi: 10.1016/j.cjph.2016.11.012 [34] X. Xu, Generalized function projective synchronization of chaotic systems for secure communication, EURASIP J. Adv. Signal Process., 2011, 2011(1), 6180-6187. [35] C.-H. Yang and C.-L. Wu, Nonlinear dynamic analysis and synchronization of four-dimensional Lorenz-Stenflo system and its circuit experimental implementation, Abstract Appl. Anal., 2014, Vol. 2014, Article ID 213694, 17 pages. [36] S. Yang and C. Duan, Generalized synchronization in chaotic systems, Chaos, Solitons & Fractals, 1998, 9(10), 1703-1707. [37] X. Yang, A framework for synchronization theory Chaos, Solitons & Fractals, 2000, 11(9), 1365-1368. [38] Y. Yua and H. Li, Adaptive generalized function projective synchronization of uncertain chaotic systems, Nonlinear Anal.-Real World Appl., 2010, 11(4), 2456-2464. doi: 10.1016/j.nonrwa.2009.08.002 [39] S. Zheng, Adaptive modified function projective synchronization of unknown chaotic systems with different order, Appl. Math. Comput., 2011, 218(10), 5891-5899. [40] S. Zheng, G. Dong and Q. Bi, Adaptive modified function projective synchronization of hyperchaotic systems with unknown parameters, Commun. Nonlinear Sci. Numer. Simul., 2010, 15(11), 3547-3556. doi: 10.1016/j.cnsns.2009.12.010 -
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M. M. El-Dessoky, E. O. Alzahrani, N. A. Almohammadi. CONTROL AND ADAPTIVE MODIFIED FUNCTION PROJECTIVE SYNCHRONIZATION OF LIU CHAOTIC DYNAMICAL SYSTEM[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 601-614. doi: 10.11948/2156-907X.20180119
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Figure 1. Liu chaotic system in (a)
$3$ -dimensional (b)$x$ -$z$ plane and (c)$x$ -$y$ plane where$a=10$ ,$b=40$ ,$c=2.5$ ,$k=1$ and$h=4$ -
Figure 2. The time responses for the states of the controlled Liu system to a fixed point
$(x_{1}, y_{1}, z_{1})$ -
Figure 3. The time responses for the states of the controlled Liu system to a fixed point
$(x_{2}, y_{2}, z_{2})$ -
Figure 4. The time responses for the states of the controlled Liu system to a fixed point
$(x_{3}, y_{3}, z_{3})$ -
Figure 5. The behaviour of the trajectories
$e_{1}, \ e_{1}\ and\ e_{1}$ of the error system tends to zero for AMFPS -
Figure 6. The behaviour of the trajectories
$e_{1}, e_{1}\ \mbox{and}\ e_{1}$ of the error system tends to zero for MFPS -
Figure 7. The behaviour of the trajectories
$e_{1}, e_{1}\ \mbox{and} \ e_{1}$ of the error system tends to zero for AMPS -
Figure 8. The behaviour of the trajectories
$e_{1}, e_{1}\ \mbox{and}\ e_{1}$ of the error system tends to zero for MPS -
Figure 9. The behaviour of the trajectories
$e_{1}, \ e_{1}\ \mbox{and} \ e_{1}$ of the error system tends to zero for complete synchronization -
Figure 10. The behaviour of the trajectories
$e_{1}, \ e_{1} \ \mbox{and}\ e_{1}$ of the error system tends to zero for anti phase synchronization