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Lag, anticipated, and complete synchronization and cascade control in the dynamical systems. (English) Zbl 1407.34088

Summary: We obtain the lag, anticipated, and complete hybrid projective synchronization control (LACHPS) of dynamical systems to study the chaotic attractors and control problem of the chaotic systems. For illustration, we take the Colpitts oscillators as an example to achieve the analytical expressions of control functions. Numerical simulations are used to show the effectiveness of the proposed technique.

MSC:

34H10 Chaos control for problems involving ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
93C15 Control/observation systems governed by ordinary differential equations
Full Text: DOI

References:

[1] Lü, J.; Chen, G., A new chaotic attractor coined, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 12, 3, 659-661 (2002) · Zbl 1063.34510 · doi:10.1142/S0218127402004620
[2] Lü, J.; Chen, G.; Cheng, D.; Celikovsky, S., Bridge the gap between the Lorenz system and the Chen system, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 12, 12, 2917-2926 (2002) · Zbl 1043.37026 · doi:10.1142/S021812740200631X
[3] Zhou, T.; Chen, G.; Yang, Q., Constructing a new chaotic system based on the Silnikov criterion, Chaos, Solitons and Fractals, 19, 4, 985-993 (2004) · Zbl 1053.37015 · doi:10.1016/S0960-0779(03)00251-0
[4] Zhou, T.; Tang, Y.; Chen, G., Chen’s attractor exists, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 14, 9, 3167-3177 (2004) · Zbl 1129.37326 · doi:10.1142/S0218127404011296
[5] Zhou, T.; Tang, Y.; Chen, G., Complex dynamical behaviors of the chaotic Chen’s system, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 13, 9, 2561-2574 (2003) · Zbl 1046.37018 · doi:10.1142/S0218127403008089
[6] Zhou, T.; Chen, G., Classification of chaos in 3-D autonomous quadratic systems. I. Basic framework and methods, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 16, 9, 2459-2479 (2006) · Zbl 1185.37092 · doi:10.1142/S0218127406016203
[7] Qi, G.; Chen, G.; Du, S.; Chen, Z.; Yuan, Z., Analysis of a new chaotic system, Physica A, 352, 2-4, 295-308 (2005) · doi:10.1016/j.physa.2004.12.040
[8] Yang, Q.; Chen, G.; Zhou, T., A unified Lorenz-type system and its canonical form, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 16, 10, 2855-2871 (2006) · Zbl 1185.37088 · doi:10.1142/S0218127406016501
[9] Yang, Q.; Chen, G.; Huang, K., Chaotic attractors of the conjugate Lorenz-type system, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 17, 11, 3929-3949 (2007) · Zbl 1149.37308 · doi:10.1142/S0218127407019792
[10] Yang, Q.; Chen, G., A chaotic system with one saddle and two stable node-foci, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 18, 5, 1393-1414 (2008) · Zbl 1147.34306 · doi:10.1142/S0218127408021063
[11] Elabbasy, E. M.; Agiza, H. N.; El-Dessoky, M. M., Adaptive synchronization for four-scroll attractor with fully unknown parameters, Physics Letters A, 349, 1-4, 187-191 (2006) · doi:10.1016/j.physleta.2005.09.018
[12] Hoang, T. M.; Nakagawa, M., Projective-lag synchronization of coupled multidelay feedback systems, Journal of the Physical Society of Japan, 75, 9 (2006) · doi:10.1143/JPSJ.75.094801
[13] Miao, Q.; Tang, Y.; Lu, S.; Fang, J., Lag synchronization of a class of chaotic systems with unknown parameters, Nonlinear Dynamics, 57, 1-2, 107-112 (2009) · Zbl 1176.34097 · doi:10.1007/s11071-008-9424-5
[14] Li, C.; Liao, X.; Wong, K.-w., Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication, Physica D, 194, 3-4, 187-202 (2004) · Zbl 1059.93118 · doi:10.1016/j.physd.2004.02.005
[15] Wang, D.; Zhong, Y.; Chen, S., Lag synchronizing chaotic system based on a single controller, Communications in Nonlinear Science and Numerical Simulation, 13, 3, 637-644 (2008) · Zbl 1130.34322 · doi:10.1016/j.cnsns.2006.05.005
[16] Mahmoud, G. M.; Bountis, T., The dynamics of systems of complex nonlinear oscillators: a review, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 14, 11, 3821-3846 (2004) · Zbl 1091.34524 · doi:10.1142/S0218127404011624
[17] Yan, Z., Q-S synchronization in 3D Henon-like map and generalized Henon map via a scalar controller, Physics Letters A, 342, 309-317 (2005) · Zbl 1222.37093
[18] Yan, Z., Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic systems-A symbolic-numeric computation approach, Chaos, 15, 2 (2005) · Zbl 1080.93008 · doi:10.1063/1.1876612
[19] Yan, Z., Controlling hyperchaos in the new hyperchaotic Chen system, Applied Mathematics and Computation, 168, 2, 1239-1250 (2005) · Zbl 1160.93384 · doi:10.1016/j.amc.2004.10.016
[20] Yan, Z.; Yu, P., Hyperchaos synchronization and control on a new hyperchaotic attractor, Chaos, Solitons and Fractals, 35, 2, 333-345 (2008) · Zbl 1132.93042 · doi:10.1016/j.chaos.2006.05.027
[21] Hu, M.; Xu, Z.; Zhang, R.; Hu, A., Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order, Physics Letters A, 365, 4, 315-327 (2007) · doi:10.1016/j.physleta.2007.01.038
[22] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Physical Review Letters, 64, 8, 821-824 (1990) · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821
[23] Al-sawalha, M. M.; Noorani, M. S. M., Anti-synchronization of chaotic systems with uncertain parameters via adaptive control, Physics Letters A, 373, 32, 2852-2857 (2009) · Zbl 1233.93056 · doi:10.1016/j.physleta.2009.06.008
[24] Li, G.-H., Generalized projective synchronization between Lorenz system and Chen’s system, Chaos, Solitons and Fractals, 32, 4, 1454-1458 (2007) · Zbl 1129.37013 · doi:10.1016/j.chaos.2005.11.073
[25] Li, G.-H., Modified projective synchronization of chaotic system, Chaos, Solitons and Fractals, 32, 5, 1786-1790 (2007) · Zbl 1134.37331 · doi:10.1016/j.chaos.2005.12.009
[26] Chicone, C., Ordinary Differential Equations with Applications. Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34 (2006), New York, NY, USA: Springer, New York, NY, USA · Zbl 1120.34001
[27] Fečkan, M., Topological Degree Approach to Bifurcation Problems. Topological Degree Approach to Bifurcation Problems, Topological Fixed Point Theory and Its Applications, 5 (2008), New York, NY, USA: Springer, New York, NY, USA · Zbl 1153.37028 · doi:10.1007/978-1-4020-8724-0
[28] Wang, Z.-L., Projective synchronization of hyperchaotic Lü system and Liu system, Nonlinear Dynamics, 59, 3, 455-462 (2010) · Zbl 1183.70055 · doi:10.1007/s11071-009-9552-6
[29] Li, G.-H., Projective lag synchronization in chaotic systems, Chaos, Solitons and Fractals, 41, 5, 2630-2634 (2009) · Zbl 1198.93202 · doi:10.1016/j.chaos.2008.09.042
[30] Luo, A. C. J., A theory for synchronization of dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 14, 5, 1901-1951 (2009) · Zbl 1221.37218 · doi:10.1016/j.cnsns.2008.07.002
[31] Mahmoud, G. M.; Aly, S. A.; Farghaly, A. A., On chaos synchronization of a complex two coupled dynamos system, Chaos, Solitons and Fractals, 33, 1, 178-187 (2007) · Zbl 1152.37317 · doi:10.1016/j.chaos.2006.01.036
[32] Mahmoud, G. M.; Al-Kashif, M. A.; Aly, S. A., Basic properties and chaotic synchronization of complex Lorenz system, International Journal of Modern Physics C, 18, 2, 253-265 (2007) · Zbl 1115.37035 · doi:10.1142/S0129183107010425
[33] Mahmoud, G. M.; Bountis, T.; Mahmoud, E. E., Active control and global synchronization of the complex Chen and Lü systems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 17, 12, 4295-4308 (2007) · Zbl 1146.93372 · doi:10.1142/S0218127407019962
[34] Mahmoud, G. M.; Mahmoud, E. E.; Ahmed, M. E., A hyperchaotic complex Chen system and its dynamics, International Journal of Applied Mathematics & Statistics, 12, D07, 90-100 (2007) · Zbl 1136.37327
[35] Mahmoud, G. M.; Ahmed, M. E.; Mahmoud, E. E., Analysis of hyperchaotic complex Lorenz systems, International Journal of Modern Physics C, 19, 10, 1477-1494 (2008) · Zbl 1170.37311 · doi:10.1142/S0129183108013151
[36] Mahmoud, G. M.; Al-Kashif, M. A.; Farghaly, A. A., Chaotic and hyperchaotic attractors of a complex nonlinear system, Journal of Physics A, 41, 5 (2008) · Zbl 1131.37036 · doi:10.1088/1751-8113/41/5/055104
[37] Mahmoud, G. M.; Aly, S. A.; AL-Kashif, M. A., Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system, Nonlinear Dynamics, 51, 1-2, 171-181 (2008) · Zbl 1170.70365 · doi:10.1007/s11071-007-9200-y
[38] Mahmoud, G. M.; Bountis, T.; AbdEl-Latif, G. M.; Mahmoud, E. E., Chaos synchronization of two different chaotic complex Chen and Lü systems, Nonlinear Dynamics, 55, 1-2, 43-53 (2009) · Zbl 1170.70011 · doi:10.1007/s11071-008-9343-5
[39] Han, M., On Hopf cyclicity of planar systems, Journal of Mathematical Analysis and Applications, 245, 2, 404-422 (2000) · Zbl 1054.34052 · doi:10.1006/jmaa.2000.6758
[40] Han, M., The Hopf cyclicity of Lienard systems, Applied Mathematics Letters, 14, 2, 183-188 (2001) · Zbl 0997.34026 · doi:10.1016/S0893-9659(00)00133-6
[41] Han, M.; Chen, G.; Sun, C., On the number of limit cycles in near-Hamiltonian polynomial systems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 17, 6, 2033-2047 (2007) · Zbl 1157.34025 · doi:10.1142/S0218127407018208
[42] Li, C.; Yan, J., Generalized projective synchronization of chaos: the cascade synchronization approach, Chaos, Solitons & Fractals, 30, 1, 140-146 (2006) · Zbl 1144.37370 · doi:10.1016/j.chaos.2005.08.155
[43] Chen, Y.; Li, X., Function projective synchronization between two identical chaotic systems, International Journal of Modern Physics C, 18, 5, 883-888 (2007) · Zbl 1139.37301 · doi:10.1142/S0129183107010607
[44] Li, X.; Chen, Y., Function projective synchronization of two identical new hyperchaotic systems, Communications in Theoretical Physics, 48, 5, 864-873 (2007) · doi:10.1088/0253-6102/48/5/022
[45] Mahmoud, G. M.; Mahmoud, E. E., Modified projective lag synchronization of two nonidentical hyperchaotic complex nonlinear systems, International Journal of Bifurcation and Chaos, 21, 8, 2369-2379 (2011) · Zbl 1248.34080 · doi:10.1142/S0218127411029859
[46] Li, Y.; Chen, Y.; Li, B., Adaptive control and function projective synchronization in 2d discrete-time chaotic systems, Communications in Theoretical Physics, 51, 2, 270-278 (2009) · Zbl 1171.93340 · doi:10.1088/0253-6102/51/2/17
[47] Li, Y.; Li, B.; Chen, Y., Adaptive function projective synchronization of discrete-time chaotic systems, Chinese Physics Letters, 26, 4 (2009) · doi:10.1088/0256-307X/26/4/040504
[48] Li, Y.; Li, B.; Chen, Y., Anticipated function synchronization with unknown parameters of discrete-time chaotic systems, International Journal of Modern Physics C, 20, 4, 597-608 (2009) · Zbl 1179.37051 · doi:10.1142/S0129183109013820
[49] Li, Y.; Li, B., Chaos control and projective synchronization of a chaotic Chen-Lee system, Chinese Journal of Physics, 47, 3, 261-270 (2009)
[50] Fotsin, H. B.; Daafouz, J., Adaptive synchronization of uncertain chaotic colpitts oscillators based on parameter identification, Physics Letters A, 339, 3-5, 304-315 (2005) · Zbl 1145.93313 · doi:10.1016/j.physleta.2005.03.049
[51] Li, Y.; Zheng, C. L., The complex network synchronization via chaos control nodes, Journal of Applied Mathematics, 2013 (2013) · Zbl 1266.93121 · doi:10.1155/2013/823863
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