Detecting Generalized Synchronization of Chaotic Dynamical Systems A Kernel-based Method and Choice of Its Parameter

Synchronization of chaotic systems has been explored extensively in recent years.1) In addition to complete synchronization between two identical chaotic systems,2) various notions of chaotic synchronization have evolved.1) Among them, the concept of generalized synchronization (GS), which refers to a situation in which the states of two systems connected each other via a continuous mapping, has been introduced in order to study coherent behavior between two systems with different dynamics.3) Experimental detection of GS from data is a challenging problem. Because the synchronization manifold of GS has a highly nonlinear structure, conventional statistical tools such as the correlation coefficient does not work. Recently, the interest in the kernel methods has been stimulated in the machine learning community for analyzing data with nonlinearity in a unified manner.4) Since the great success of Support Vector Machine, a considerable effort has been devoted to derive kernelization of various multivariate analysis methods. Therefore, it is meaningful to explore applicability of the kernel-based methods for analyzing nonlinear dynamics. In this paper, we particularly employ Kernel Canonical Correlation Analysis (Kernel CCA)5) for characterizing GS. We present an example for which Kernel CCA works successfully and also discuss how an optimal value of parameter of Kernel CCA can be chosen.