# Synchronization, Control and Data Assimilation of the Lorenz System
## 4 Pruned-Enriching Approach  
The previous scheme cannot be always followed, because we assumed the ability to measure some of the variables of the master system and inject this value (at least intermittently) into the slave one. However, this might be impossible, either because the state variables \(x,y,z\) are not accessible individually, or because those that are accessible do not allow synchronization (for instance, if only the \(z\) variable is accessible, as illustrated in Section 2.3). We can benefit from the [PERSON] theorem [21], which states that the essential features (among which is the maximum Lyapunov exponent) of a strange attractor \(\{\mathbf{r}(t)\}_{t\to\infty}\) can be reconstructed by intermittent observations \(w_{n}=f(\mathbf{r}(n\Delta T))\) using the time series \(w_{n},w_{n-1},w_{n-2},\dots\) as surrogate data, provided that their number is larger than the dimensionality of the attractor [22]. Other conditions are that the observation interval \(\Delta T\) must be large enough to have observations \(w_{n}\) sufficiently spaced, but not so large that the \(w_{n}\) are scattered along the attractor, making the reconstruction of the trajectory impossible. It is therefore generally convenient to take an interval \(\Delta t\) substantially larger than the minimum \(\Delta t=\tau\), but of the same order of magnitude. An interesting point is that one can choose for \(f\) an arbitrary function of the original variables, provided that their correspondence is smooth. We therefore employed a method inspired by the pruned-enriching technique [23; 24], or genetic algorithm without cross-over [25]. We assume that we can only measure a function \(f(x,y,z)=f(\mathbf{r})\) of the master system, at time intervals \(\tau\). The master system evolves with parameters \(\mathbf{q}=(\sigma,\rho,\beta)\). We simulate an ensemble \(\{\mathbf{R}_{i}\}_{i=1,\dots,h}\) composed by \(h\) replicas, each one starting from state variables \(\mathbf{R}_{i}(0)\) (\(i=1,\dots,h\)) and evolving with the same equation as the master one, with parameters \(\{\mathbf{Q}_{i}\}_{i=1,\dots,h}\), where \(\mathbf{Q}_{i}=(\sigma_{i},\rho_{i},\beta_{i})\). At the beginning, \(\mathbf{R}_{i}(0)\) and \(\mathbf{Q}_{i}\) are random quantities. We let the ensemble evolve for a time interval \(T\), and then we measure the distance \(d_{i}=|f(\mathbf{r}(T))-f(\mathbf{R}_{i}(T))|\). We sort the replicas according to \(d_{i}\) and we replace half of them with larger distances, following an evolutionary law based on a cloning and perturbation scheme, as shown in Figure 6. For a more detailed description of the procedure, we can refer to Algorithms 1 and 2. Figure 6: The schematic of the pruned-enriching method. Lines denotes schematically the trajectories of replicas in the space of state variables and parameters. The dashed lines marks the trajectory of master system. Disks marks the elimination of replicas which are farther from the master one. Black dotted lines marks the pruning and enriching times, and the duplication of replicas are marked by the dashed colored lines with arrows. The variation of the duplication of the nearer replicas is either on one of the state variables or one of parameters. We assume that we have at our disposal a time series measured on some experimental system. For this simulation, we generate the time series by taking a set of measures \(\big{\{}f(\mathbf{r}(t_{n}))\big{\}}=\{f(\mathbf{r}_{n})\}\), for \(t_{0}=0,t_{1}=\tau,\ldots,t_{n}=n\tau,\ldots,t_{N}=T=N\tau\), where \(N=T/\tau\) is the length of the simulated time series. We then map the time series in an embedding space of size \(d_{e}\) defined by the vector \(\mathbf{w}_{n}=\big{(}f(\mathbf{r}_{n}),f(\mathbf{r}_{n-1}),\ldots,f(\mathbf{r}_{n-d_{e}+1}) \big{)}\), \(n\geq d_{e}\). We randomly initialize the state values of the \(h\) replicas, \(\mathbf{R}_{0}^{(i)}\) and the initial values of their parameters \(\mathbf{Q}^{(i)}\), \(i=1,\ldots,h\), in a range that is consistent with the physical values.