The Influence Of Initial Value Function On The Domain Of Attraction Of Several Kinds Of Multi-stable Systems With Time-delay

Multi-steady state is a typical nonlinear phenomenon.In a multi-stable system,each stable equilibrium point has its corresponding attraction domain.Studying the shape and size of the attracting domain of each stable equilibrium point can reveal the relevant dynamic characteristics,which has important theoretical and practical significance.Since finite information transmission and processing speed,time delay is inevitable in active controlled system.The initial value space of the time-delay system is in the infinite dimension space.Researching the shape and size of the attraction of each equilibrium point of the time-delay system is still a challenging research topic.In this paper,a proper definition of domain of attraction in its physical state space is introduced.Then,study the shape and size of the attraction domain of multi-steady state Duffing system and Hopfield neural network system with time-delay.Finally,research the influence of different initial value functions on domain of attraction for each stable equilibrium point.Through the stability switching analysis,the stability and stability domain of each equilibrium point of the time-delay Duffing system is obtained,and the area where the multi-steady state phenomenon exists is given.In the case of multi-steady state,the numerical integration method is used for different initial value functions,and the attraction domains of various stable equilibrium points in the physical space are obtained.The results show that the initial value function has a certain influence on the structure of the attraction domain.The width and shape of the attraction domain corresponding to different initial value functions are not the same,and it can also lead to irregular structures.First,for a multi-stability Hopfield neural network system,the number of stable equilibrium points and the structure of the attraction domain with parameters are analyzed.In the case of multi-steady state,the stability switching analysis shows that the stability of the equilibrium point of the system with time delay does not change.For a given time delay,the numerical integration method is used to obtain the attraction domain corresponding to different initial value functions.The results show that the boundaries of the attraction domain corresponding to different initial value functions are not the same;in the physical space,there is also a phenomenon that a stable equilibrium point is in the attraction domain of another stable equilibrium point.