Identification of Nonlinear Systems by Synchronization in Chaotic Systems : Index of Synchronization in Chaotic Systems

Abstract

Since it is not easy to determine precise equations of physical nonlinear systems, some studies analyze chaos only by means of its time series, ignoring the use of system equations. From a similar point of view, we have already developed a method of statistical mechanics without system equations. In contrast to the above approaches, we discuss an identification technique which can determine the precise equations of chaotic systems. If we can determine the precise equations, it becomes possible to calculate valid quantities, such as Mel'nikov's function and bifurcation sets, for estimations of chaotic parameters. For this purpose, we use synchronization in chaotic systems. We define an error in the synchronization and its index, a statistical quantity called ISCS (index of synchronization in chaotic systems). We then consider a chaotic system and its approximation which is slightly different from the chaotic system, and show that ISCS tends to zero as the difference approaches zero. Numerical results show that we can obtain the best approximation if we choose the approximation whose ISCS tends to zero.