Function matrix projection synchronization for the multi-time delayed fractional order memristor-based neural networks with parameter uncertainty

Abstract

Fractional Order Memristor-Based Neural Networks (FOMBNNs) has the strong sensitivity to initial values and shows more complex paths, so its Projective Synchronization (PS) and applications have been widely used in the fields of security communication. Our main work intends to extend the scaling factor of PS to a function matrix depending on time t and proposes a new synchronization type for the first time, i.e., Function Matrix Projective Synchronization (FMPS) for FOMBNNs, whose scaling factor is highly variable over time and difficult to predict. However, the FOMBNNs is a state dependent discontinuous system and it is easy to produce complex nonlinearity, which makes the study of the FMPS becomes a challenge. Therefore, our work will commit to solving this problem and realizing the FMPS for multi-time delayed FOMBNNs with parameter uncertainty. Firstly, the error functions and FMPS are defined, which can be degenerated to matrix PS, modified PS, PS, anti-synchronization and complete synchronization. Then, for the multi-time delayed FOMBNNs with parameter uncertainty, the active controller is designed and the sufficient condition for realizing the FMPS is proved by using a Lyapunov functional and some Lemmas of fractional calculus. Finally, the FMPS of four numerical examples are given and trajectories of their synchronization errors approach to 0, which illustrate the efficiency of the proposed synchronization analysis. This research will provide a general method for studying the FMPS of other dynamical systems.