Synchronization chaotic phase

Fig. 12.15 (a) Phase plot for one of the nephrons and (b) temporal variation of the tubular pressures for both nephrons in a pair of coupled chaotically oscillating units, a = 32, T = 16 s, and e = y = 0.2. The figure illustrates the phenomenon of chaotic phase synchronization. By virtue of their mutual coupling the two chaotic oscillators adjust their (average) periods to be identical. The amplitudes, however, vary incoherently and in a chaotic manner [27],... 

The local dynamics of tire systems considered tluis far has been eitlier steady or oscillatory. However, we may consider reaction-diffusion media where tire local reaction rates give rise to chaotic temporal behaviour of tire sort discussed earlier. Diffusional coupling of such local chaotic elements can lead to new types of spatio-temporal periodic and chaotic states. It is possible to find phase-synchronized states in such systems where tire amplitude varies chaotically from site to site in tire medium whilst a suitably defined phase is synclironized tliroughout tire medium 51. Such phase synclironization may play a role in layered neural networks and perceptive processes in mammals. Somewhat suriDrisingly, even when tire local dynamics is chaotic, tire system may support spiral waves... 

Let us examine the situation for large values of a where the individual nephron exhibits chaotic dynamics. Figure 12.15a shows a phase plot for one of the nephrons in our two-nephron model for a = 32, T = 16 s, e = 0.0, and y = 0.2. Here we have introduced a slight mismatch AT = 0.2 s in the delay times between the two nephrons and, as illustrated by the tubular pressure variations of Fig. 12.15b, the nephrons follow different trajectories. However, the average period is precisely the same. This is a typical example of phase synchronization of two chaotic oscillators. 

Chaotic behavior and synchronization in heterogeneous catalysis are closely related. Partial synchronization can lead to a complex time series, generated by superposition of several periodic oscillators, and can in some cases result in deterministically chaotic behavior. In addition to the fact that macroscopically observable oscillations exist (which demonstrates that synchronization occurs in these systems), a number of experiments show the influence of a synchronizing force on all the hierarchical levels mentioned earlier. Sheintuch (294) analyzed on a general level the problem of communication between two cells. He concluded that if the gas-phase concentration is the autocatalytic variable, then synchronization is attained in all cases. However, if the gas-phase concentration were the nonautocatalytic variable, then this would lead to symmetry breaking and the formation of spatial structures. When surface variables are the model variables, the existence of synchrony is dependent upon the size scale. Only two-variable models were analyzed, and no such strict analysis has been provided for models with two or more surface concentrations, mass balances, or heat balances. There are, however, several studies that focused on a certain system and a certain synchronization mechanism. 

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). 

A. Pikovsky and M. Rosenblum. Phase synchronization of regular and chaotic self-sustained oscillators. In A. Pikovsky and Yu. Maistrenko, editors, Synchronization Theory and Application, pages 187-219. Kluwer, Dordrecht, 2003. 

Chaotic systems. Here the mere notion of synchrony is non-trivial, and several concepts have been developed. The effect of phase synchronization is a direct extension of the classical theory to the case of a subclass of self-sustained continuous time chaotic oscillators which admit a description in terms of phase. Synchronization of these systems can be described as a phase and frequency locking, in analogy to the theory of synchronization of noisy systems. An alternative approach considers a synchronization of arbitrary chaotic systems as a coincidence of their state variables (complete synchronization) or as an onset of a functional relationship between state variables of two unidirection-ally coupled systems (generalized synchronization). Although physical mechanisms behind the two latter phenomena essentially differ from the mechanisms of phase and frequency locking, all these effects constitute the field of application of the modern synchronization theory. 

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