Stability and Bifurcation Analysis in a Class of Two‐Neuron Networks with Resonant Bilinear Terms

Abstract

A class of two-neuron networks with resonant bilinear terms is considered. The stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability of the zero equilibrium will cause a bifurcating periodic solution as the time delay passes through a sequence of critical values. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.

Comparative analysis on Hopf bifurcation of integer‐order and fractional‐order two‐neuron neural networks with delay

“…In details, one can see Yang et al 46 and Xu and Liao. 47 In order to derive the main results of this article, the following necessary hypothesis is prepared:…”

Dynamical Analysis of a Computer Virus Propagation Model with Delay and Infectivity in Latent Period

“…In the remainder of this section, we use the same notations as those used by Xu and He [15] and we first compute the coordinates to describe the center manifold 0 at = 0. Let be the solution of (21) when = 0.…”

“…As is known, time delay can make a dynamical system lose its stability and then lead to the occurrence of a Hopf bifurcation. Hopf bifurcation of dynamical systems with delay has been investigated by many authors [10][11][12][13][14][15][16]. In [10], Zhuang and Zhu analyzed the existence of Hopf bifurcation for an improved HIV model with time delay and cure rate by regarding the time lag from infection of cells to the cells becoming actively infected as a bifurcation parameter.…”

Dynamical Analysis of a Viral Infection Model with Delays in Computer Networks

“…Next, we can obtain the coefficients determining the properties of the Hopf bifurcation by the algorithms introduced in [17] and using a computation process similar to that in [19,20]:…”