Synchronization, State Bounds, And Stability Of Multi-delay Quaternion Neural Networks

The function of neural network is closely related to its dynamic behavior.Some dynamic phenomena can be detected in the simplified model,and it has the characteristics of high fault tolerance,strong self-learning ability and parallel computing.In recent years,it has developed into one of the hot fields of scientific research.With the development of neural networks,people gradually find that quaternion neural networks have better application prospects than ordinary real-valued or complexvalued neural networks.On the other hand,high-order neural networks show very good applicability in parallel computing,adaptive pattern recognition,optimization problems and other fields.In addition,neural network inevitably has time delays and is disturbed by the external environment,which makes neural network have the characteristics of complexity,multiplicity and superposition,so neural networks with time delays have also received extensive attention.Based on the multi delays high-order quaternion Hopfield neural network,this paper mainly studies global exponential synchronization,state bounding and reachable set estimation,and Lagrangian global exponential stability.The main research contents of this paper are as follows:Firstly,the global exponential synchronization of high-order quaternion Hopfield neural networks with time-varying discrete delays and unbounded distributed delays is studied.On the basis of no construction of any Lyapunov–Krasovskii(L–K)functional,the delay-dependent global exponential synchronization criterion of high-order quaternion Hopfield neural networks with multi delays is directly derived according to the definition.Two simulation examples show the effectiveness of the obtained results,and it is applied to image security transmission.Secondly,state bounding of high-order quaternion Hopfield neural networks with bounded external disturbance,time-varying discrete delays and unbounded distributed delays is studied.Similarly,without constructing any L–K functional,delay-dependent sufficient conditions for the state trajectory to be contained in a cuboid or globally exponentially converge into a cuboid are given.And reachable set estimation of higher-order quaternion Hopfield neural networks with multi delays is obtained.A simulation example shows the effectiveness of the obtained results.Finally,Lagrangian global exponential stability of quaternion memristive neural networks with leakage delays,time-varying discrete delays and unbounded distributed delays is studied.Based on the definition of Lagrangian global exponential stability,the delay-dependent Lagrangian global exponential stability criterion of quaternion memristive neural network with multiple delays is derived.A simulation example shows the effectiveness of the obtained results.The innovations of this paper are as follows:(1)The method proposed in this paper does not require to construct any L–K functional;(2)The established criterion and the state bounding sufficient conditions only need to solve simple inequalities,which is suitable for real-valued and complex-valued neural networks;(3)The proposed method can be extended to many neural network models with time delay.