Phase and antiphase synchronization of two identical hyperchaotic complex nonlinear systems.(English)Zbl 1204.93096

Summary: A scheme is designed to achieve phase synchronization (PS) and antiphase synchronization (APS) for an $$n$$-dimensional hyperchaotic complex nonlinear system. For this scheme, we have used the idea of an active control technique based on Lyapunov stability analysis to determine analytically the complex control functions which are needed to achieve PS and APS. We applied this scheme, as an example, to study PS and APS of hyperchaotic attractors of two identical hyperchaotic complex Lorenz systems. These complex systems appear in many important fields of physics and engineering. Our scheme can also be applied to two different hyperchaotic complex systems, for which PS and APS have not been investigated, as far as we know, in the literature. Numerical results are plotted to show phases and amplitudes of these hyperchaotic attractors, thus demonstrating that PS and APS are achieved. The bifurcation diagrams are computed for a wide range of parameters of the system parameters and are found to be symmetrical about the horizontal axis for APS, while they lack any symmetry for PS.

MSC:

 93D15 Stabilization of systems by feedback 34C28 Complex behavior and chaotic systems of ordinary differential equations 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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 [1] DeShazer, D.J., Breban, R., Ott, E., Roy, R.: Detecting phase synchronization in a chaotic laser array. Phys. Rev. Lett. 87, 044101 (2001) · Zbl 1060.78527 [2] Liu, W., Xiao, J., Qian, X., Yang, J.: Antiphase synchronization in coupled chaotic oscillators. Phys. Rev. E 73, 057203 (2006) [3] Rosa, E. Jr., Ticos, C.M., Pardo, W.B., Walkenstein, J.A., Monti, M., Kurths, J.: Experimental Chua-plasma phase synchronization of chaos. Phys. Rev. E 68, 025202(R) (2003) [4] Baptista, M.S., Silva, T.P., Sartorelli, J.C., Caldas, I.I., Rosa, E.: Phase synchronization in the perturbed Chua circuit. Phys. Rev. E 67, 056212 (2003) [5] Parlitz, U., Junge, L., Lauterborn, W.: Experimental observation of phase synchronization. Phys. Rev. E 54(2), 2115–2117 (1996) [6] Tass, P., Rosenblum, M.G., Weule, J., Kurths, J., Pikovsky, A., Volkmann, J., Schnitzler, A., Freund, H.-J.: Detection of n:m phase locking from noisy data: Application to magnetoencephalography. Phys. Rev. Lett. 81, 3291–3294 (1998) [7] Yalcinkaya, T., Lai, Y.C.: Phase characterization of chaos. Phys. Rev. Lett. 79, 3885–3888 (1997) [8] Koronovskii, A.A., Kurovskaya, M.K., Hramov, A.E.: Relationship between phase synchronization of chaotic oscillators and time scale synchronization. Tech. Phys. Lett. 31(10), 847–850 (2005) [9] Kim, C.M., Rim, S., Kye, W.H., Ryu, J.W., Park, Y.J.: Anti-synchronization of chaotic oscillators. Phys. Lett. A 320, 39–46 (2003) · Zbl 1098.37521 [10] Maza, D., Vallone, A., Mancini, H., Boccaletti, S.: Experimental phase synchronization of a chaotic convective flow. Phys. Rev. Lett. 85, 5567–5570 (2000) [11] Schäfer, C., Rosenblum, M.G., Abel, H.H., Kurths, J.: Synchronization in human cardiorespiratory system. Phys. Rev. E 60, 857–870 (1999) [12] Blasius, B., Huppert, A., Stone, L.: Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 354–359 (1999) [13] Tallon-Baudry, C., Bertrand, O., Delpuech, C., Pernier, J.: Stimulus specificity of phase-locked and non-phase-locked 40 Hz visual responses in human. J. Neurosci. 16(13), 4240–4249 (1996) [14] Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, Berlin (1984) · Zbl 0558.76051 [15] Feng, X.-Q., Shen, K.: Phase synchronization and anti-phase synchronization of chaos for degenerate optical parametric oscillator. Chin. Phys. 14, 1526–1532 (2005) [16] Luo, A.C.J.: A theory for synchronization of dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 14, 1901–1951 (2009) · Zbl 1221.37218 [17] Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., Zhou, C.S.: The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002) · Zbl 0995.37022 [18] Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76(11), 1804–1807 (1996) [19] Pikovsky, A.S., Rosenblum, M.G., Kurths, J.: Synchronization in a population of globally coupled chaotic oscillators. Europhys. Lett. 34(3), 165–170 (1996) [20] Pereira, T., Baptista, M.S., Kurths, J.: Phase and average period of chaotic oscillators. Phys. Lett. A 362, 159–165 (2007) · Zbl 1197.70015 [21] Juan, M., Xing-yuan, W.: Nonlinear observer based phase synchronization of chaotic systems. Phys. Lett. A 369, 294–298 (2007) · Zbl 1209.34076 [22] Ho, M.C., Hung, Y.C., Chou, C.H.: Phase and anti-phase synchronization of two chaotic systems by using active control. Phys. Lett. A 296, 43–48 (2002) · Zbl 1098.37529 [23] Mahmoud, G.M., Bountis, T., Mahmoud, E.E.: Active control and global synchronization of the complex Chen and Lü systems. Int. J. Bifurc. Chaos 17(12), 4295–4308 (2007) · Zbl 1146.93372 [24] Mahmoud, G.M., Al-Kashif, M.A., Aly, S.A.: Basic properties and chaotic synchronization of complex Lorenz system. Int. J. Mod. Phys. C 18(10), 253–265 (2007) · Zbl 1115.37035 [25] Mahmoud, G.M., Aly, S.A., Al-Kashif, M.A.: Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system. Nonlinear Dyn. 51, 171–181 (2008) · Zbl 1170.70365 [26] Mahmoud, G.M., Al-Kashif, M.A., Farghaly, A.A.: Chaotic and hyperchaotic attractors of a complex nonlinear system. J. Phys. A, Math. Theor. 41(5), 055104 (2008) · Zbl 1131.37036 [27] Mahmoud, G.M., Aly, S.A., Farghaly, A.A.: On chaos synchronization of a complex two coupled dynamos system. Chaos Solitons Fractals 33, 178–187 (2007) · Zbl 1152.37317 [28] Mahmoud, G.M., Bountis, T., AbdEl-Latif, G.M., Mahmoud, E.E.: Chaos synchronization of two different complex Chen and Lü systems. Nonlinear Dyn. 55, 43–53 (2009) · Zbl 1170.70011 [29] Mahmoud, G.M., Bountis, T.: The dynamics of systems of complex nonlinear oscillators: A Review. Int. J. Bifurc. Chaos 14(11), 3821–3846 (2004) · Zbl 1091.34524 [30] Mahmoud, G.M., Ahmed, M.E., Mahmoud, E.E.: Analysis of hyperchaotic complex Lorenz systems. Int. J. Mod. Phys. C 19(10), 1477–1494 (2008) · Zbl 1170.37311
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