Estimation of optimal coupling functions for synchronization of coupled chaotic oscillators

Synchronization phenomena are very important not only because they are observed in many different systems but also have rich potential for application, e.g., to data assimilation. For the application, searching for the best coupling function between systems is necessary. For example, in the Lorenz system, it is well known that coupling y component can give better synchronization than coupling x component. It is also very important for the purpose of the supermodeling, which combines different imperfect models and get better prediction by synchronizing these imperfect models. Therefore, we need to find out the coupling function which achieves synchronization with minimum coupling coefficient. We consider unidirectionally coupled identical chaotic systems. The purpose of this study is to find the optimal coupling function which enables to synchronize two systems with the smallest coupling constant. The best way to synchronize the system would be to introduce the coupling function corresponding to the Lyapunov vector. However, it is in general very difficult to obtain such vectors, especially from the experimental data. Therefore a data-driven method to give a good coupling function is required. A good candidate for such a method is to use the superposition of principal component analysis (PCA) vectors. The feasibility of such methods will be studied. In the present contribution, we compare different choice of coupling schemes mainly in low dimensional dynamical systems such as the Lorenz system. In such small systems we can obtain synchronization error for all possible coupling functions, and we can estimate their performance. We measure the synchronization error, and introduce a quantity which quantifies how small signal can efficiently synchronize the system. This comparison will be made in other hyperchaotic systems.