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.
Highlights
Based on the assumption that the elements in the network can respond to and communicate with each other instantaneously without time delays, Hopfield proposed Hopfield neural networks HNNs model in 1980s 1, 2
When τ passes through the critical value τ0, the equilibrium E∗ 0, 0 loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcations from the equilibrium E∗ 0, 0
We note that, if the two-neuron networks with resonant bilinear terms begin with a stable equilibrium, but become x2 (t)
Summary
Based on the assumption that the elements in the network can respond to and communicate with each other instantaneously without time delays, Hopfield proposed Hopfield neural networks HNNs model in 1980s 1, 2. The appearance of a cycle bifurcating from an equilibrium of an ordinary or a delayed neural network with a single parameter, which is known as a Hopf bifurcation, has attracted much attention see 3–13. X 1 t α1 a f x1 α2 b f x2 cx1x2, 1.1 x 2 t α2 − b f x1 α1 − a f x2 dx1x2, Abstract and Applied Analysis where xi t i 1, 2 represents the state of the ith neuron at time t, f xi i 1, 2 is the connection function between two neurons, and α1, α2, a, b, c, d are real parameters, and obtained a sufficient condition for a Bautin bifurcation to occur for system 1.1 by using the standard normal form theory and with Maple software.
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