Dynamic Analysis And Bifurcation Control Of Double-ring Systems With Multiple Time Delays

Ring networks exist in many neural structures,and the geometry of the rings has been widely used in physiological and biochemical modeling research.Many researchers have analyzed the ring model in the context of neural network theory.Ring neural network belongs to a class of feedback system,which has important biological significance.Ring networks are regarded as the cornerstones of connected topological networks,and the experimentation on rings supports people in their effort to further comprehend the dynamic behavior of cyclic networks.Therefore,it is very eventful to study the stability of the ring neural network.The interaction between two coupled neurons is non-transient,and the time delay explains this non-transient characteristic.Therefore,the time delay must be premeditated when detecting the stability of the neural network.Hopf bifurcation is a limit cycle generated when ordinary differential equations change their stability through a pair of pure imaginary eigenvalues in the equilibrium state,which is closely related to the oscillation phenomenon in engineering.Therefore,the dynamic analysis of the double-ring neural network with time delays has great potential in both theoretical and practical research fields.Build on precursory research results,this paper further gives the bifurcation theory and control strategy of the double-ring neural network model with multiple time delays.The specific work is as follows:1.A hybrid control strategy is proposed for the double-ring neural network model with multiple time delays.By utilizing the sum of time delays as the bifurcation parameter and extrapolating the stability of the controlled system,the stability of the system and sufficient conditions to generate Hopf bifurcation are derived.By regulating the controller parameters,the dynamic behavior of the controlled system can be fruitfully optimized.2.Under the ring topology,a double-ring high-dimensional neural network model is established.The dynamic characteristics of a high-dimensional neural network model in which two rings share a node are studied.The existence and stability of the equilibrium point of the network model are probed.The effect of the sum of time delays and the number of neurons on the stability of the periodic solution is also deliberated.The results demonstrate that the dynamics of neural networks are intently related to the sum of delays and the number of neurons.3.Introducing the fractional order into the double-ring controlled neural network.Based on the theoretical analysis of the complex dynamics of the neuron model built,the local stability and Hopf bifurcation conditions of the controlled system are proved.The influence of system parameters and order on the critical value of bifurcation is explored.The study found that the fractional order has a huge impact on the stability of the system.