# Synchronization, Control and Data Assimilation of the Lorenz System
###### Abstract  
We explore several aspects of replica synchronization with the goal of retrieving the values of parameters applied to the Lorenz system. The idea is to establish a computer replica (slave) of a natural system (master, simulated in this paper), and exploit the fact that the slave synchronizes with the master only if they evolve with the same parameters. As a byproduct, in the synchronized phase, the state variables of the slave and those of the master are the same, thus allowing us to perform measurements that would be impossible in the real system. We review some aspects of master-slave synchronization using a subset of variables with intermittent coupling. We show how synchronization can be achieved when some of the state variables are available for direct measurement using a simulated annealing approach, and also when they are accessible only through a scalar function, using a pruned-enriching ensemble approach, similar to genetic algorithms without cross-over. We also explore the case of exploiting the \"gene exchange\" option among members of the ensemble. master-slave synchronization; data assimilation; parameter estimation; Lorenz system +
Footnote †: journal: _arxiv.org_  
The goal of this paper is to approach this problem from the point of view of dynamical systems [5]. In this context, the theme of synchronization has been explored [6], but it needs to be extended in order to cover the application field of data assimilation. The synchronization technique is particularly recommendable when the amount of noise (in the measure and intrinsic to the system) is small, i.e., when the target system can be assumed to be well represented by deterministic dynamics, even if these dynamics are not continuous (as in Cellular Automata [7]). We shall investigate here the application of synchronization methods to the classic Lorenz system [8]. However, the method can be applied to other chaotic systems and possibly to their implementation in electronic hardware. The Lorenz system is chosen because, for classical values of parameter, there is only one chaotic attractor. Other systems, such as Chua's [9; 10], may present the coexistence of attractors (hidden attractors) [11; 12], which may affect synchronization [13]. Our investigation is carried out considering some synchronization schemes; see Section 2. We start from the classic Pecora-Carrol master-slave synchronization scheme, recalled in Section 2.1, in which the values of some of the state variables of the master are imposed to the corresponding variables of the slave system. This choice of coupling variables is denoted \"coupling direction\" in the tangent space of the system. The synchronization threshold is connected to the value of the conditional Lyapunov exponent [14; 15], i.e., the exponential growing rate along the difference space between master and slave, as reported in Section 2.2. This scheme is then extended to partial synchronization, i.e., to the case in which only a portion of the values of the state variables of the master system signal is fed to the slave, as shown in Section 2.3. We can thus establish the minimum fraction of signal (coupling strength) able to synchronize the two replicas, which depends on the coupling direction. However, one cannot pretend to be able to perform measurements in continuous time, as achieved in the original synchronization scheme, in which the experimental reference system was a chaotic electronic circuit. Therefore, we deal with the problem of intermittency, i.e., performing measurements, and consequently applying the synchronization scheme, only at certain time intervals. We show that the synchronization scheme also works for intermittent measurements, provided that the coupling strength is increased, as shown in Section 2.4. In the case of systems with different parameters, synchronization cannot be complete, and we discuss generalized synchronization; see Section 2.5.