Dynamical Analysis Of Several Delayed Fractional Order Reaction-diffusion Neural Networks

Diffusion is widespread phenomenon in nature.As one of the powerful tools to characterize abnormal diffusion,reaction-diffusion equation is widely used in physics,biology and economics.In addition,fractional-order neural networks can better reflect the process of information dependence,so this paper conducts in-depth exploration on the dynamical behavior of several types of fractional-order reaction-diffusion neural networks.With the help of the theory of partial functional differential equations,Green’s theorem,Jensen’s integral inequality,Filippov differential inclusion theory,fractional-order Razumikhin theory,Lyapunov functional method,the influences of order,diffusion coefficient and space position on the dynamical behavior of network system are discussed,and some meaningful results are obtained.This dissertation mainly includes the following three parts:1、A class of fractional-order nonidentical delayed neural network model with Caputo-type fractional partial differential operator and reaction-diffusion term is established.By constructing Caputo differential inequality,a new finite-time convergence lemma is proposed.With the help of the Filippov differential inclusion theory,Green’s theorem,fractional Razumikhin theory and novel finite-time convergence lemma,some synchronization criteria for Caputo-type fractional nonidentical reaction-diffusion neural networks with time delay are established under the Neumann boundary conditions and two hybrid controllers with time delay and sign function.The derived criteria are given in the form of inequalities,which reduces the computational complexity of testing synchronization performance.2、A class of fractional-order reaction-diffusion neural networks with leakage and discrete delays in the double-layer network structure is explored.By designing a hybrid controller and applying the Green’s theorem,Jensen’s integral inequality and Lyapunov functional method,several synchronization criteria for fractional bidirectional associative memory reaction-diffusion neural networks are derived under the Dirichlet boundary conditions.The presented criteria are given in the form of eigenvalues of the system parameter matrix,and the influence of the parameter matrix on the synchronization performance is revealed.3、A class of fractional-order bidirectional associative memory reaction-diffusion neural networks with the control input and output is discussed.By means of the Lyapunov function and the existing passivity definition(integral form or fractional integral form),four finite time passivity definition of fractional derivative form under the double-layer network structure are given.By selecting a hybrid controller with sign function and combining the Green’s theorem,Jensen’s integral inequality,Lyapunov functional method and four new definitions of passivity,several passivity criteria for fractional bidirectional associative memory reaction-diffusion neural networks are obtained.The conditions satisfied are given as matrix inequality form,which can be obtained a group of feasible solutions by Yalmip toolbox,and the conservatism of the results is lower than that of the algebraic form.