The Hopf Bifurcation Of A N Dimensional Neural Network System

This paper studies the stability and Hopf bifurcation of the equilibria in the n-dimensional neural network modeled by Lotka-Volterra system. Lotka-Volterra system is one of the most important nonlinear dynamic systems. In recent years, many results have been made on Lotka-Volterra system, which have been significantly applied in many areas such as the neural network, biology, economy and physics. For convenient analysis, the paper mainly concerns the reduced four-dimensional systemThis paper first studies the stability of critical equilibria of the system. Through theoretical proof, we know they are unstable. For the positive equilibria, they are di-vided into two categories, respectively, the form (r, r, r2, r2) and (bi,si,bisi,si2)(i= 1,2). For the equilibrium (r,r,r2,r2), Through theoretical analysis and proof,we know it is asymptotically stable when and only when K>0, T>0,n≥3 and k< nr or k>nr and T<k(n-1)/k-nr.But when k> nr,T>k(n-1)/k-nr, (r, r, r2,r2) is unstable.Then based on the Hopf bifurcation theory, and through theoretical analysis, we prove that when the parameter changes, the Hopf bifurcation occurs at the first positive equilib-rium, and get its Hopf bifurcation condition, that is, for any T= T*=k(n-1)/(k-nr), the Hopf bifurcation will occur at (r, r, r2, r2) in the reduced system. In this paper, we also apply the Normal form theory to prove the direction and sta-bility of the Hopf bifurcation, setting n=3, k 2.222222222 and T*=36.10214169, we find the super Hopf bifurcation occurs at the equilibrium (r, r,r2,r2), and we give the corresponding bifurcation diagram by using Maple software. For the other positive equilibria (bi,si,bisi,si2), the equilibrium (b2,S2,b2S2,s22) is always unstable/Thus the nearby Hopf bifurcation period orbits do not exist. however for the equilibrium (b1,s1,b1s1,s12),we study it mainly by numerical experiments, and find that in certain parameters, the Hopf bifurcation also occurs at the equilibrium point. What's more, though numerical example analysis, we find that when n=4, k=3.141621149, T*= 16.4823, the Hopf bifurcation can occur simultaneously in the equilibria (r, r, r2, r2) and (b1,s1, b1S1,S12). This result is of significance for the study of the whole model.