## 7 Networks
### Synchronization
#### 7.1.5 Chaotic Synchronization  
Dmitriev et al. [138] considered a synchronization scheme for two-dimensional maps that is based on transmission of information about the state of drive system by unidirectional coupling. In the proposed scheme (see Figure 3 of [138]), the signal \(x_{n}\) produced by iteration of the Lozi map is transformed by a two-level quantizer (threshold device) into a symbolic sequence \(X_{n}=sign(x_{n})\), i.e., into \(-1\) if \(x_{n}<0\) and into \(+1\) if \(x_{n}\geq 0\). Then, this binary signal is sent through the channel, where it is corrupted by noise (\(z_{n}=X_{n}+w_{n}\)). The quantizer transforms it back to the binary signal \(\hat{X}_{n}=sign(z_{n})\). The \(\pm 1\) values of the binary sequence correspond to the symbols \(L\) and \(R\) of the symbolic sequence. This symbolic sequence is used in the reconstruction block to restore the chaotic sequence. Thus, in the absence of noise, the chaotic sequence reconstructed with a high precision is fed to the input of the response system (or an estimate of this chaotic sequence if there is noise). The reconstruction algorithm provided by Dmitriev et al. is defined as follows. For a given symbolic sequence \(X_{1},\ldots,X_{n},\ldots,X_{n+k},\ldots,X_{N}\), where \(X_{k}\in\{-1,1\}\): \(N\) is the number of elements in the sequence, it is possible to evaluate \(|x_{n}|\) at the first step for all elements other than the first and the last (i.e., for \(n=2,\ldots,N-1\)) in the following way:  
\[x_{n}^{1}=X_{n}\Big{|}x_{n}^{1}\Big{|}=X_{n}\Big{|}\frac{\alpha-1}{\alpha}- \frac{1}{\alpha}X_{n+1}+\frac{\beta}{\alpha}X_{n-1}\Big{|}. \tag{109}\]  
The elements of the symbolic sequence are used at this stage as an initial approximation for the estimate. At the \(i\)th step of the procedure, the estimates obtained at the previous \((i-1)\)th step are used:  
\[x_{n}^{i}=X_{n}\Big{|}x_{n}^{i}\Big{|}=X_{n}\Big{|}\frac{\alpha-1}{\alpha}- \frac{1}{\alpha}x_{n+1}^{i-1}+\frac{\beta}{\alpha}x_{n-1}^{i-1}\Big{|}, \tag{110}\]  
where \(n=2,\ldots,N-1\). The process is complete when the difference between the estimates of \(x_{i}\) in the \(i\)th and \(i+1\)th iterations is less then some small \(\epsilon\). It has been shown that in order to achieve synchronization of both hyperbolic and non-hyperbolic maps it is sufficient to transmit information contained in the chaotic sequence and then utilize this information to reconstruct the chaotic sequence. The precision of reconstruction depends on the length of the sequence and the number of elementsof the symbolic sequence that are directly used to reconstruct the chaotic sample. The reconstructed sequence of samples ensures high-quality synchronization of drive and response systems. It has been shown that the necessary amount of information that must be delivered to the response system is determined by the rate of information production by the dynamical system. When this threshold is increased, however, arbitrarily precise synchronization can be achieved. The only price to pay for this is increased complexity of the reconstruction algorithm and delay of the reconstructed signal. This is contrary to the common belief that the synchronization error necessarily monotonically increases with the noise. Another method based on evolutionary optimization, called the Tribes optimization approach, was used by Coelho and Bernert [139] for synchronization of two identical discrete chaotic systems subject to different initial conditions. The Tribes algorithm, first introduced by Clerc [140], is an adaptive and robust parameter searching technique. Moreover, Tribes is a parameter-free particle swarm system paradigm in which the population is divided in subpopulations that each maintaining their own order and structure. The Tribes algorithm is inspired by the social behavior of bird flocking, and is an adaptive optimization procedure that does not require sociometric or swarm size parameter tuning.