Multiple Periodic Solutions In Delay-Coupled System Of Neural Oscillators

In this paper, our major work is to discuss stability and bifurcation of delay-coupled system of neural oscillators. Because the system has the Dn (?) Z2 symme-try, we could use Lie group to represent its symmetry and Lie group representation theory to explore the spatio-temporal pattern of the bifurcated periodic solutions from the trivial equilibrium. For this paper, We are interested in studying how the coupling strength and the time delay can affect the stability of the zero solution of system (1.2) and the bifurcations of periodic solutions when stability is lost, considering effect of the coupling strength and the time delay on the existence, spatio-temporal patterns, and stability of Hopf bifurcating periodic solutions. We expand our work through following stages:Study the linear stability of equilibrium. Firstly, through decomposing n-dimension complex space, we obtain the characteristic equation of linearized sys-tem. Then, by analyzing the distribution of zero solutions of transcendental equa-tion, we discuss the distribution of zero solutions of characteristic equation and obtain some results on the sign properties of zero solutions. By means of linear stability theory, we have the conclusions about asymptotical stability independent of delay, asymptotical stability as well as instability of trivial solution.Secondly, whenπcrosses the critical values, namely, the stability of the trivial solution changes, we examine the conditions of Hopf bifurcation at critical values and then explore details of the bifurcated periodic solutions. We need to point out that in the presence of Dn (?) Z2 symmetry, there exists a two-dimensional or four-dimensional generalized eigenspace associated with eigenvalues of the in-finitesimal generator of the C0-semigroup generated by the linearized system. So, we distinguish two cases to discuss bifurcated periodic solutions. Noticing that the spatio-pattern of the periodic solution is determined by its isotropy subgroup, which is smaller than Dn(?)Z2×S1, we describe the spatio-patterns of two kinds of periodic solutions. By using the center-manifold theory and normal form approach combined with the representation of Lie groups, according to the standard Hopf theorem and equivariant Hopf theorem, we derive the formulas about stability and bifurcation direction of these periodic solutions.Finally, we use numerical simulations to illustrate results of the paper.