Approximate synchronization of coupled multi-valued logical networks

This article deals with the approximate synchronization of two coupled multi-valued logical networks. According to the initial state set from which both systems start, two kinds of approximate synchronization problem, local approximate synchronization and global approximate synchronization, are proposed for the first time. Three new notions: approximate synchronization state set (ASSS), the maximum approximate synchronization basin (MASB) and the shortest approximate synchronization time (SAST) are introduced and analyzed. Based on ASSS, several necessary and sufficient conditions are obtained for approximate synchronization. MASB, the set of all possible initial states, from which the systems are approximately synchronous, is investigated combining with the maximum invariant subset. And the calculation method of the SAST, associated with transient period, is presented. By virtue of MASB, pinning control scheme is investigated to make two coupled systems achieve global approximate synchronization. Furthermore, the related theories are also applied to the complete synchronization problem of $k$-valued ($k\geq2$) logical networks. Finally, four examples are given to illustrate the obtained results.