Spatial-Temporal Synchronization Dynamics Analysis Of A Class Of Fractional Memristive Neural Networks

Fractional Order Memristive Neural Networks(FMNNs)as a combination of Fractional Order Neural Networks(FNNs)and memristor,which has shown great application value in engineering,control,automation and other fields.And the dynamic analysis of FMNNs has also become one of the most popular research topics.However,most results only focus in the meaning of time and ignore the diffusion phenomenon in space during the actual circuit simulation.Therefore,we will study the Reaction Diffusion Fractional Order Memristive Neural Networks(RD-FMNNs)model from the time-space view in this paper.The sufficient conditions to ensure the synchronization of the model are given by designing of adaptive controllers.The effectiveness of the controllers verified by MATLAB numerical simulation.In this paper,the Hopfield neural network is taken as an example to discuss the synchronization problem of RD-FMNNs model with the discontinuous activation functions.The specific work is as follows:1.The synchronization problem of RD-FMNNs with time delay term is studied.Firstly,to overcome the problem that the solution of discontinuous differential equation does not exist,we introduce the solution in the sense of Filippov according to the principle of extreme value mapping and differential inclusion.Then,for the reaction diffusion term in space and the time delay phenomenon,we use the Green's first formula and the fractional comparison theorem to transform them into a appropriate form.Based on the fractional partial differential equations and stability theory,a reasonable controller is designed,and sufficient conditions for global synchronization are given.Finally,taking the two-dimensional RD-FMNNS model as an example,the validity of the controller and the correctness of the synchronization criterion are verified by MATLAB numerical simulation.2.The synchronization problem of RD-FMNNs with unknown parameters is discussed.Firstly,for the unknown parameters in the model,we designed an adaptive controller with compensation based on the theory of dynamic compensator in control theory.By constructing appropriate auxiliary functions,we derive sufficient conditions for the synchronization of the model according to the relative techniques such as GronwallBellman inequality and the stability theory of fractional equation.Finally,taking the twodimensional RD-FMNNS model as an example,the validity of the designed controller and the correctness of the theorem are verified through MATLAB numerical simulation.3.The synchronization problem of RD-FMNNs with the actuator fails is investigated.According to the idea of robust control,the faults of actuator strength and deviation are regarded as unknown parameter disturbances in the feedback process.Based on the principle of dynamic compensation,the pinning adaptive controller is designed to realize the synchronization of coupled RD-FMNNs with actuator fault tolerance.