Symplectic synchronization of Lorenz-Stenflo system with uncertain chaotic parameters via adaptive control. (English) Zbl 1261.93066

Yang, Cheng-Hsiung

Abstr. Appl. Anal. 2013, Article ID 528325, 14 p. (2013).

Summary: A new symplectic chaos synchronization of chaotic systems with uncertain chaotic parameters is studied. The traditional chaos synchronizations are special cases of the symplectic chaos synchronization. A sufficient condition is given for the asymptotical stability of the null solution of error dynamics and a parameter difference. The symplectic chaos synchronization with uncertain chaotic parameters may be applied to the design of secure communication systems. Finally, numerical results are studied for symplectic chaos synchronized from two identical Lorenz-Stenflo systems in three different cases.

MSC:

93D20	Asymptotic stability in control theory
34H10	Chaos control for problems involving ordinary differential equations
90B18	Communication networks in operations research
93C40	Adaptive control/observation systems

Keywords:

synchronization of Lorenz-Stenflo system; new symplectic chaos synchronization; uncertain chaotic parameters; asymptotic stability; chaotic systems

References:

[1]	Femat, R.; Solís-Perales, G., On the chaos synchronization phenomena, Physics Letters A, 262, 1, 50-60 (1999) · Zbl 0936.37010 · doi:10.1016/S0375-9601(99)00667-2
[2]	Ge, Z.-M.; Yang, C.-H., Pragmatical generalized synchronization of chaotic systems with uncertain parameters by adaptive control, Physica D, 231, 2, 87-94 (2007) · Zbl 1167.34357 · doi:10.1016/j.physd.2007.03.019
[3]	Yang, S. S.; Duan, C. K., Generalized synchronization in chaotic systems, Chaos, Solitons & Fractals, 9, 10, 1703-1707 (1998) · Zbl 0946.34040 · doi:10.1016/S0960-0779(97)00149-5
[4]	Krawiecki, A.; Sukiennicki, A., Generalizations of the concept of marginal synchronization of chaos, Chaos, Solitons & Fractals, 11, 9, 1445-1458 (2000) · Zbl 0982.37022 · doi:10.1016/S0960-0779(99)00062-4
[5]	Ge, Z. M.; Yang, C. H.; Chen, H. H.; Lee, S. C., Non-linear dynamics and chaos control of a physical pendulum with vibrating and rotating support, Journal of Sound and Vibration, 242, 2, 247-264 (2001) · doi:10.1006/jsvi.2000.3353
[6]	Chen, M. Y.; Han, Z. Z.; Shang, Y., General synchronization of Genesio-Tesi systems, International Journal of Bifurcation and Chaos, 14, 1, 347-354 (2004) · Zbl 1085.93023 · doi:10.1142/S0218127404009077
[7]	Ge, Z. M.; Yang, C. H., Symplectic synchronization of different chaotic systems, Chaos, Solitons & Fractals, 40, 5, 2532-2543 (2009) · Zbl 1198.93200 · doi:10.1016/j.chaos.2007.10.055
[8]	Ge, Z. M.; Yang, C. H., The generalized synchronization of a Quantum-CNN chaotic oscillator with different order systems, Chaos, Solitons & Fractals, 35, 5, 980-990 (2008) · Zbl 1141.37017 · doi:10.1016/j.chaos.2006.05.090
[9]	Ge, Z.-M.; Yang, C.-H., Synchronization of complex chaotic systems in series expansion form, Chaos, Solitons & Fractals, 34, 5, 1649-1658 (2007) · Zbl 1152.37314 · doi:10.1016/j.chaos.2006.04.072
[10]	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
[11]	Stenflo, L., Generalized Lorenz equations for acoustic-gravity waves in the atmosphere, Physica Scripta, 53, 1, 83-84 (1996) · doi:10.1088/0031-8949/53/1/015
[12]	Vincent, U. E., Synchronization of identical and non-identical 4-D chaotic systems using active control, Chaos, Solitons & Fractals, 37, 4, 1065-1075 (2008) · Zbl 1153.37359 · doi:10.1016/j.chaos.2006.10.005
[13]	Liu, Z., The first integrals of nonlinear acoustic gravity wave equations, Physica Scripta, 61, 5, 526 (2000) · doi:10.1238/Physica.Regular.061a00526
[14]	Zhou, C.; Lai, C. H.; Yu, M. Y., Bifurcation behavior of the generalized Lorenz equations at large rotation numbers, Journal of Mathematical Physics, 38, 10, 5225-5239 (1997) · Zbl 0897.76038 · doi:10.1063/1.531938
[15]	Banerjee, S.; Saha, P.; Chowdhury, A. R., Chaotic scenario in the Stenflo equations, Physica Scripta, 63, 3, 177-180 (2001) · Zbl 1115.37366 · doi:10.1238/Physica.Regular.063a00177
[16]	Chen, Y.; Wu, X.; Gui, Z., Global synchronization criteria for two Lorenz-Stenflo systems via single-variable substitution control, Nonlinear Dynamics, 62, 1-2, 361-369 (2010) · Zbl 1217.34086 · doi:10.1007/s11071-010-9723-5
[17]	Ge, Z.-M.; Chen, Y.-S., Synchronization of unidirectional coupled chaotic systems via partial stability, Chaos, Solitons & Fractals, 21, 1, 101-111 (2004) · Zbl 1048.37027 · doi:10.1016/j.chaos.2003.10.004
[18]	Yang, C. H.; Li, S. Y.; Tsen, P. C., Synchronization of chaotic system with uncertain variable parameters by linear coupling and pragmatical adaptive tracking, Nonlinear Dynamics, 70, 3, 2187-2202 (2012) · Zbl 1268.34100 · doi:10.1007/s11071-012-0609-6
[19]	Ge, Z. M.; Chen, C. C., Phase synchronization of coupled chaotic multiple time scales systems, Chaos, Solitons & Fractals, 20, 3, 639-647 (2004) · Zbl 1069.34056 · doi:10.1016/j.chaos.2003.08.001
[20]	Tam, L. M.; Chen, J. H.; Chen, H. K.; Si Tou, W. M., Generation of hyperchaos from the Chen-Lee system via sinusoidal perturbation, Chaos, Solitons & Fractals, 38, 3, 826-839 (2008) · doi:10.1016/j.chaos.2007.01.039
[21]	Sebastian Sudheer, K.; Sabir, M., Modified function projective synchronization of hyperchaotic systems through Open-Plus-Closed-Loop coupling, Physics Letters A, 374, 19-20, 2017-2023 (2010) · Zbl 1236.34072 · doi:10.1016/j.physleta.2010.02.068
[22]	Sun, M.; Tian, L.; Jia, Q., Adaptive control and synchronization of a four-dimensional energy resources system with unknown parameters, Chaos, Solitons & Fractals, 39, 4, 1943-1949 (2009) · Zbl 1197.93100 · doi:10.1016/j.chaos.2007.06.117
[23]	Li, Z.; Zhao, X., Generalized function projective synchronization of two different hyperchaotic systems with unknown parameters, Nonlinear Analysis: Real World Applications, 12, 5, 2607-2615 (2011) · Zbl 1225.34061 · doi:10.1016/j.nonrwa.2011.03.009