# Synchronization, Control and Data Assimilation of the Lorenz System
## 2 Master-Slave Synchronization  
### Pecora-Carrol Synchronization  
In 1990, [PERSON] and [PERSON] introduced the idea of master-slave synchronization [14]. They studied simulated systems, such as the Lorenz [8] and Rossler [16] ones, and also experimental electronic circuits [17]. They considered two replicas of the same system with the same parameters. One of the replicas (the master) was left to evolve unperturbed. Some of the state variables of the master replaced the corresponding variables of the slave system, which was therefore no longer autonomous. This topic was further explored in a 2015 paper [15]. In order to explain the main idea of this master-slave (or replica) synchronization, let us consider first a one-dimensional, time-discrete dynamical system, such as, for instance, the logistic map [18]. It is defined by an equation of the kind  
\[x_{n+1}=f(x_{n}), \tag{1}\]  
where \(n\) labels the time interval. The usual Lyapunov exponent \(\lambda\) is defined as  
\[\lambda=\lim_{N\to\infty}\sum_{n=1}^{N}\log(f^{\prime}(x_{n})), \tag{2}\]  
where \(f^{\prime}(x)=\mathrm{d}f/\mathrm{d}x\). Let us now introduce a replica \(X\) such that  
\[ \begin{split} x_{n+1}&=f(x_{n});\\ X_{n+1}&=(1-p)f(X_{n})+pf(x_{n}),\end{split}  \tag{3}\]  
where \(p\) is the coupling parameter (coupling strength). For \(p=0\), the two maps are uncoupled, and because they are assumed to be chaotic, they generally take different values. For \(p=1\), map \(X\) immediately synchronizes with map \(x\). There is a critical value \(p_{c}\) of the coupling parameter for which the synchronized state is stable, and it is related to the Lyapunov exponent \(\lambda\),  
\[p_{c}=1-\exp(-\lambda), \tag{4}\]  
as can be seen by taking the linearized difference among the maps. This scenario holds also for higher-dimensional (\(K\)) maps, which can be written as  
\[\mathbf{r}_{n+1}=\mathbf{F}(\mathbf{r}_{n}), \tag{5}\]  
where \(\mathbf{r}_{n}\) denotes the state of the system at iteration \(n\) and has \(K\) components. The system now has \(K\) Lyapunov exponents, of which at least one, \(\lambda_{\text{MAX}}\), is assumed to be positive. The synchronization procedure in this case is  
\[ \begin{split}\mathbf{r}_{n+1}&=\mathbf{F}(\mathbf{r}_{n});\\ \mathbf{R}_{n+1}&=(\mathbb{I}-p\mathcal{C})\mathbf{F}(\mathbf{R }_{n})+p\mathcal{C}\mathbf{F}(\mathbf{r}_{n}),\end{split}  \tag{6}\]  
where the coupling is now implemented by means of a diagonal matrix \(C\), with diagonal entries equal to zero or one, defining the coupling directions. In the following, we shall indicate the diagonal of \(C\) as \(\mathbf{C}=[c_{x},c_{y},c_{z}]\). In the case in which \(C=\mathbb{I}\), i.e., all entries are mixed with the same proportion, the stability of the synchronized phase is again related to the maximum Lyapunov exponent \(\lambda_{\text{MAX}}\),  
\[p_{c}=1-\exp(-\lambda_{\text{MAX}}), \tag{7}\]  
because the evolution of an infinitesimal difference among the replicas, \(\delta\), is given by  
\[\delta_{n+1}=(1-p)J(x)\delta_{n}, \tag{8}\]  
where \(J\) is the Jacobian of \(F\)  
\[J_{ij}=\left(\frac{\partial F_{i}}{\partial x_{j}}\right)_{x=x_{n}}. \tag{9}\]This scheme can be extended to continuous-time systems by replacing the entries from one system in the coupled differential equation, so that  
\[\dot{\mathbf{r}} =\mathbf{F}(\mathbf{r}); \tag{10}\] \[\dot{\mathbf{R}} =(\mathbb{I}-p\mathbf{C})\mathbf{F}(\mathbf{R})+p\mathbf{C}\mathbf{F}(\mathbf{r}).\]  
where the Lyapunov exponents of the original system are now given by the eigenvalues of the (symmetrized) time-average of Jacobian \(\Lambda\)  
\[\Lambda=\lim_{t\to\infty}\frac{1}{2t}\Bigg{(}\int_{0}^{t}J(\mathbf{r}(t))+\left( \int_{0}^{t}J(\mathbf{r}(t))\right)^{T}\Bigg{)}. \tag{11}\]  
In practice, however, the differential Equation (10) is implemented as a map by discretizing the time, and therefore, Lyapunov exponents are computed as in the previous case.