On the distribution of the roots of a fifth degree exponential polynomial with application to a delayed neural network model

In this paper, we mainly study the distribution of the roots of a fifth degree exponential polynomial. We obtain the sufficient and necessary conditions for the existence of purely imaginary roots of the exponential polynomial. Applying the obtained results, we consider a neural network model consisting of five neurons with delays. The sum of delays ττ is regarded as the bifurcation parameter. Under some conditions, we show that the zero solution is locally asymptotically stable when the time delay is suitably small, while change of stability of zero solution will cause a bifurcating periodic solution as the time delay ττ passes through a certain critical value. In order to illustrate our theoretical analysis, some numerical simulations are presented.