In the study, the finite-time synchronization (FTSN) for a kind of master-slave quaternion-valued fractional-order neural networks (MSQVFONNS) is discussed. By using the negative definition of matrix and the properties of the determinant, two novel criteria on the FTSN for the considered MSFOQVNNS are established. The negative definition of the matrix and the properties of the determinant are introduced to study the FTSN for neural networks (NNs) in our article. Since until, studies about the FTSN for the NNs are rare and researchers have only used the linear matrix inequality (LMI), finite time stability theorems (FTSTs) of fractional order and Lyapunov direct method to study the FTSN for the MSFOQVNNS, so far, our method to study the FTSN for the MSFOQVNNS is of definite significance.

In this discussion, the finite time synchronization (FTSN) of master-slave fractional-order chaotic systems (MSFOCSS) is explored. By adopting the maximum-valued method (MVM) of functions of five variables, two novel criteria on the FTSN are obtained for the MSFOCSS. So far, the studies of the FTSN of the MSFOCSS have been rare. Furthermore, the existing results on the FTSN of MSFOCSS have been achieved often by adopting the LMI method and finite time stability theorems (FTST). Thus, instead of utilizing the past research methods, adopting the MVM to study the FTSN of the MSFOCSS is an innovative study work.

In this article, the finite time anti-synchronization (FTAS) of master-slave 6D Lorenz systems (MS6DLSS) is discussed. Without using previous study methods, by introducing new study methods, namely by adopting the properties of quadratic inequalities of one variable and utilizing the negative definiteness of the quadratic form of the matrix, two criteria on the FTAS are achieved for the discussed MS6DLSS. Up to now, the existing results on FTAS of chaotic systems have been achieved often by adopting the linear matrix inequality (LMI) method and finite time stability theorems (FTST). Adopting the new study methods studies the FTAS of the MS6DLSS, and the novel results on the FTAS are gotten for the MS6DLSS, which is innovative study work.