˖ڍመڙጲ
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The NNs including both discrete and distributed delay have been investigated in [16, 17]. More
recently, approximation of the stability region of a Hopfield NNs with identical neurons and
the general distributed delays has been investigated in [18].
In this paper, we consider a three-neuron Hopfield NN ring with unidirectional coupling
and distributed delays described by the following equations
dx
1
dt
= −x
1
(t) + a tanh[
R
t
−∞
κ(t − s)x
3
(s)ds],
dx
2
dt
= −x
2
(t) + b tanh[
R
t
−∞
κ(t − s)x
1
(s)ds],
dx
3
dt
= −x
3
(t) + c tanh[
R
t
−∞
κ(t − s)x
2
(s)ds],
(1)
where a, b, c describe the coupling strength between the connected neurons and κ is a weighting
factor which indicates how much emphasis should be given to the size of the population at
earlier times to determine the present effect on resource availability. Usually the delay kernel
is normalized so that
R
∞
0
κ(s)ds = 1. The quantity τ =
R
∞
0
tκ(t)dt is usually called the average
delay for the kernel. If κ(t) is the Dirac function δ(t − σ), we have τ = σ and tanh[
R
t
−∞
δ(t −
τ − s)x
i
(s)ds] = tanh[x
i
(t − τ)], i = 1, 2, 3, and then system (1) becomes the case of discrete
delay, which has been studied by Wei et al. [11].
In the biological system, the following Gamma distribution delay kernel is often used
κ(s) = β
n+1
s
n
e
−βs
n!
, s ∈ (0, ∞), n = 0, 1, 2,
where β > 0 is a parameter denoting the rate of decay of effects of past memories and is also
known as exponentially fading memory, and n is an integer. For this Gamma distribution
delay kernel, we have the average delay τ = (n + 1)/β. Generally speaking, n = 0 and n = 1
represent weak delay kernel and strong delay kernel, respectively. In this paper we only consider
the case of the weak kernel, that is, κ is of the form: κ(s) = βe
−βs
, s ∈ (0, ∞). It is easy to see
that the weak kernel κ(s) decreases with the increasing of s, which means that the maximum
weighted response of the growth rate is due to current population density while past densities
have (exponentially) decreasing influence. In this paper we investigate the stability and Hopf
bifurcation induced by the average delay.
1 Local stability and delay-induced Hopf Bifurcations
In this section, we investigate the existence of equilibria of system (1), analyze the influ-
ence of the average delay τ on the stability and obtain conditions for the occurrence of Hopf
bifurcations. We first have the following lemma on the existence of the equilibrium of system
(1).
Theorem 1. System (1) has an unique equilibrium if abc ≤ 1; otherwise it has three equilibria.
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