Chaos synchronization

Ditto W L and Showalter K (eds) 1997 Control and synchronization of chaos focus issue Chaos 7 509-687... 

Zhou, C., and Kurths, J. Noise-induced synchronization and coherence resonance of a Hodgkin-Huxley model of thermally sensitive neurons. Chaos 2003,13 401— 409. 

Zhou, C and Kurths,). Hierarchical synchronization in complex networks with heterogeneous degrees. Chaos 2006, 16 015104. 

Fig. 12.12 Internal rotation number as a function of the parameter a calculated from the single-nephron model. Inserts present phase projections for typical regimes. Note how the intra-nephron synchronization is maintained through a complete period-doubling cascade to chaos.

In a review article on oscillatory reactions (294), Sheintuch discusses the effect of introducing a heat balance for the catalyst rather than a mass balance for the reactor into the differential equation system for a surface reaction with oxidation/reduction cycles. Although the coverage equations alone can yield oscillatory behavior, as was the case for the models discussed in the previous section, Sheintuch s model is discussed in this section because introduction of the heat balance adds qualitatively new features. In this extended system complex, multiple peak behavior and quasiperiodicity was observed as shown in Fig. 8. Sheintuch also investigated the interaction of two oscillators. This work, however, will be treated in detail in Section V, were synchronization and chaos are discussed. 

Kevin Cuomo and Alan Oppenheim (1992, 1993) have implemented a new approach to this problem, building on Pecora and Carroll s (1990) discovery of synchronized chaos. Here s the strategy When you transmit the message to your friend, you also mask it with much louder chaos. An outside listener only hears the chaos, which sounds like meaningless noise. But now suppose that your friend has a magic receiver that perfectly reproduces the chaos—then he can subtract off the chaotic mask and listen to the message ... 

The hard part is to make a receiver that can synchronize perfectly to the chaotic transmitter. In Cuomo s set-up, the receiver is an identical Lorenz circuit, driven in a certain clever way by the transmitter. We ll get into the details later, but for now let s content ourselves with the experimental fact that synchronized chaos does occur. Figure 9.6.2 plots thereceivervariables u t) and v t) against their transmitter counterparts u(t) and v(z). 

Cuomo, K. M., and Oppenheim, A. V. (1993) Circuit implementation of synchronized chaos, with applications to communications. Phys. Rev. Lett. 71, 65. 

All the complex behavior described so far in this Chapter arises from the diffusive coupling of the local dynamics which in the homogeneous case have simple fixed points as asymptotic states. If the local dynamics becomes more complex, the range of possible dynamic behavior in the presence of diffusion becomes practically unlimited. It is clear that coupling chaotic subsystems could produce an extremely rich dynamics. But even the case of periodic local dynamics does so. Diffusively coupled chemical or biological oscillators may become synchronized (Pikovsky et ah, 2003), or rather additional instabilities may arise from the spatial coupling. This may produce target waves, spiral patterns, front instabilities and several different types of spatiotemporal chaos or phase turbulence (Kuramoto, 1984). 

Y. Zhang, J. Kastrup, R. Klann, K. H. Ploog, and H. T. Grahn Synchronization and chaos induced by resonant tunneling in GaAs/AlAs superlattices, Phys. Rev. Lett. 77, 3001 (1996). 

Fischer, Y. Liu, and P. Davis. Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication. Phys. Rev. A, 62 011801, 2000. 

H. G. Winful and L. Rahman. Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers. Phys. Rev. Lett, 65 1575, 1990. 

S. Yanchuk, Yu. Maistrenko, and E. Mosekilde. Synchronization of time-continuous chaotic oscillators. Chaos, 13 388-400, 2003. 

B. Blasius L. Stone. Chaos and phase synchronization in ecological sys-... 

E. Montbrio B. Blasius. Using nonisochronicity to control synchronization in ensembles of non-identical oscillators. Chaos 13, 291-308 (2003). 

Rate oscillations, spatiotemporal patterns and chaos, e.g. dissipative structures were also observed in heterogeneous catalytic reactions. If compared with pattern formation in homogeneous systems, the surface studies introduced new aspects, like anisotropic diffusion, and the possibility of global synchronization via the gas phase. Application of field electron and field ion microscopy to the study of oscillatory surface reactions provided the capability of obtaining images with near-atomic resolution. The most extensively studied reaction is CO oxidation, which is catalyzed by group VIII noble metals. 

Rul kov, N.F., A.R. Volkovskii, A. Rodriguez-Lozano, E. Del Rio M.G. Velarde. 1992. Mutual synchronization of chaotic self-oscillators with dissipative coupling. Int. J. Bif. Chaos 2 669-76. 

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