Synchronization chaotic

Cuomo, K. M., and Oppenheim, A. V. (1992) Synchronized chaotic circuits and systems for communications. MIT Research Laboratory of Electronics Technical ReportNo. 575. 

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... 

In this section we consider a model of interactions between the Kerr oscillators applied by J. Fiurasek et al. [139] and Perinova and Karska [140]. Each Kerr oscillator is externally pumped and damped. If the Kerr nonlinearity is turned off, the system is linear. This enables us to perform a simple comparison of the linear and nonlinear dynamics of the system, and we have found a specific nonlinear version of linear filtering. We study numerically the possibility of synchronization of chaotic signals generated by the Kerr oscillators by employing different feedback methods. 

E. Mosekilde, Yu. Maistrenko, and D. Post-nov Chaotic synchronization - applications to living systems. World Scientific, Singapore, 2002. 

B. Lading, E. Mosekilde, S. Yanchuk, and Yu. Maistrenko Chaotic synchronization between coupled pancreatic b-cells. Prog. Theor. Phys. 2000,139[Suppl.] 164-177. 

S. Yanchuk, Yu. Maistrenko, B. Lading, and E. Mosekilde Chaotic synchronization in time-continuous systems. 

Network Synchronization in Tonic, Chaotic and Bursting Regimes... 

Fig. 12.11 Two-mode oscillatory behavior in the single nephron model. Black colored regions correspond to a chaotic solution. The figure shows different regions in which 1 4, 1 5 and 1 6 synchronization occurs in the interaction between the fast myogenic oscillations...

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. 

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],... 

Let us consider the case of a = 30 corresponding to a weakly developed chaotic attractor in the individual nephron. The coupling strength y = 0.06 and the delay time T2 in the second nephron is considered as a parameter. Three different chaotic states can be identified in Fig. 12.16. For the asynchronous behavior both of the rotation numbers ns and n f differ from 1 and change continuously with T2. In the synchronization region, the rotation numbers are precisely equal to 1. Here, two cases can be distinguished. To the left, the rotation numbers ns and n/ are both equal to unity and both the slow and the fast oscillations are synchronized. To the right (T2 > 14.2 s), while the slow mode of the chaotic oscillations remain locked, the fast mode drifts randomly. In this case the synchronization condition is fulfilled only for one of oscillatory modes, and we speak of partial synchronization. A detailed analysis of the experimental data series reveals precisely the same phenomena [31]. 

Figures 12.18a and b show examples of the tubular pressure variations in pairs of neighboring nephrons for hypertensive rats. These oscillations are significantly more irregular than the oscillations displayed in Fig. 12.17 and, as previously discussed, it is likely that they can be ascribed to a chaotic dynamics. In spite of this irregularity, however, one can visually observe a certain degree of synchronization between the interacting nephrons.

The experiments were carried out on arrays consisting of from 1 to 64 nickel electrodes. The anodic dissolution of Ni in sulfuric acid is a HN-NDR oscillator that exhibits also chaotic oscillations. Its dynamics has been thoroughly studied by Sheintuch and coworkers [201-203] (see also the discussion in Ref. [9]). The synchronization experiments were carried out with periodically and chaotically oscillating individual electrodes. 

At the root of these behaviors is the phenomenon of synchronization of a macroscopic oscillating system. This topic has been discussed occasionally in the literature but has rarely been explicitly treated. In principle there are several stages or hierarchical levels on which oscillations can occur (1) the single-crystal plane, (2) the catalytically active metal crystallite, (3) the catalyst pellet, (4) arrangements of several pellets in one layer of a flow reactor or a CSTR, and finally (5) the catalytic packed-bed reactor (327). On each of these levels, different types of oscillations may exist, but to become observable on the next level oscillations on the respective sublevels must be synchronized. For example, if oscillations of the CO/O2 reaction on a Pt(lOO) face of a Pt crystallite supported on a pelletized support material in a packed-bed reactor occur, the reaction on the (100) facet as a whole must oscillate in synchrony, other (100) facets of the crystallite have to synchronize, other crystallites in the pellet must couple to the first crystallite, and, finally, all pellets in one layer of the bed must display oscillations in synchrony. If the synchronization on one of these levels fails, different oscillators will superimpose and their effects will cancel. One would then only observe a possible increased level of noise in the measured conversion. On the other hand, if synchronization occurs independently over several regions of a system, then it might exhibit apparently chaotic behavior caused by incomplete coupling. 

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. 

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