Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 367589, 9 pages
http://dx.doi.org/10.1155/2013/367589
Research Article
Bifurcation Analysis in a Two-Dimensional Neutral
Differential Equation
Ming Liu and Xiaofeng Xu
Department of Mathematics, Northeast Forestry University, Harbin 150040, China
Correspondence should be addressed to Xiaofeng Xu; xxf ray@163.com
Received 2 February 2013; Accepted 5 April 2013
Academic Editor: Chunrui Zhang
Copyright © 2013 M. Liu and X. Xu. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The dynamics of a 2-dimensional neural network model in neutral form are investigated. We prove that a sequence of Hopf
bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating
periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions
is established using a global Hopf bifurcation result of Krawcewicz et al. Finally, some numerical simulations are carried out to
support the analytic results.
1. Introduction
The fundamental theories and applications of neutral functional differential equations have been largely developed and
there are a great deal of important conclusions on the stability,
bifurcation theory, and numerical solutions of neutral delay
differential equations during the last few decades. It is well
known that Wei and Ruan [1] have established a basic theory
to analyze the distribution of the zeros of general transcendental equations and applied it to the neutral functional
differential equations. One can use the theory to study the
existence of Hopf bifurcation to a neutral delay differential
equation. In [2], Wang and Wei extend the computation of
the properties of Hopf bifurcation, such as the direction of
bifurcation and stability of bifurcating periodic solutions, of
DDE introduced by Kazarinoff et al. Based on combining the
global Hopf bifurcation theory of neutral equations due to
Krawcewicz et al. and the higher dimensional Bendixson’s
criterion for ordinary differential equations due to Li and
Muldowney, the global existence of periodic solutions of
neutral differential equations have been studied by Qu et
al. [3]. But research findings of bifurcation about highdimension neutral differential equations can rarely be found.
In order to determine the dynamics of artificial neural
network, the two-dimensional delay neural network without
self-feedback is studied, and some significant results are
obtained (see [4–8]). In this paper, we consider the following
system:
𝑥1̇ (𝑡) = −𝜇𝑥1 (𝑡) + 𝑎𝑓 (𝑥1 (𝑡 − 𝜏)) + 𝑏𝑥2̇ (𝑡 − 𝜏) ,
𝑥2̇ (𝑡) = −𝜇𝑥2 (𝑡) + 𝑎𝑓 (𝑥2 (𝑡 − 𝜏)) + 𝑏𝑥1̇ (𝑡 − 𝜏) ,
(1)
where 𝜇, 𝑎, and 𝑏 are real numbers with 𝜇 > 0 and |𝑏| < 1.
𝑓 is assumed adequately smooth, for example, 𝑓 ∈ 𝐶3 , and
satisfies the following condition:
(H1) 𝑓(0) = 0, 𝑓󸀠 (0) = 1, and there exists 𝐿 > 0, such that
|𝑓(𝑥)| ⩽ 𝐿 for all 𝑥 ∈ R,
(H2) 𝑓󸀠 (𝑥) > 0 for all 𝑥 ∈ R, 𝑥𝑓󸀠󸀠 (𝑥) < 0 for all 𝑥 ≠ 0.
In the present paper, we provide a detailed analysis of
this equation. By applying the local Hopf bifurcation theory
(see [9] and also [1, 2, 10, 11]), we investigate the existence of
stable periodic oscillations for (1). More specially, we prove
that the equilibrium (0, 0) loses its stability as 𝜏 increases,
and a sequence of Hopf bifurcations occurs at the origin.
Whereafter, based on the normal form and center manifold
theory due to [12, 13], by using the method introduced
in Wang and Wei [2], we derive a sufficient condition
for determining the direction of Hopf bifurcation and the
stability of the bifurcating periodic solutions. Furthermore,
2
Abstract and Applied Analysis
the existence of periodic solutions for 𝜏 far away from the
Hopf bifurcation values is also established, by using a global
Hopf bifurcation result due to Krawcewicz et al. [14] (also see
[15, 16]).
The rest of this paper is organized as follows: in Section 2,
taking 𝑎 and 𝜏 as parameters, we give the analysis of
stability and bifurcations at equilibria. Section 3 is devoted to
establishing the direction and stability of Hopf bifurcation.
Finally, a global Hopf bifurcation is established, and some
numerical simulations are carried out to illustrate the analytic
results in Section 4.
2. Stability and Bifurcation Analysis
In this section, we will investigate the stability and bifurcations of the equilibria by taking 𝑎 and 𝜏 as bifurcation
parameters.
Under the given hypothesis (H1), it is easy to check that
(0, 0) is an equilibrium point of system (1). At this trivial
equilibrium, the linearization of system (1) is
𝑋̇ (𝑡) = 𝐴𝑋 (𝑡) + 𝐵𝑋 (𝑡 − 𝜏) + 𝐶𝑋̇ (𝑡 − 𝜏) ,
(2)
where
−𝜇 0
𝐴=(
),
0 −𝜇
𝑎 0
𝐵=(
),
0 𝑎
𝐶=(
0 𝑏
).
𝑏 0
(3)
And the corresponding characteristic equation is
2
Δ (𝜆) = (𝜆 + 𝜇 − 𝑎𝑒−𝜆𝜏 ) − 𝑏2 𝜆2 𝑒−2𝜆𝜏 = 0,
(4)
that is,
(𝜆 + 𝜇 − (𝑎 − 𝑏𝜆) 𝑒−𝜆𝜏 ) (𝜆 + 𝜇 − (𝑎 + 𝑏𝜆) 𝑒−𝜆𝜏 )
(5)
def
= Δ 1 (𝜆) ⋅ Δ 2 (𝜆) = 0.
Case 1. Choose 𝑎 as parameter.
For convenience, we give two claims at first.
Claim 1. All the roots of (4) with 𝑎 = 0 have negative real
parts for any 𝜏 ⩾ 0.
Proof. When 𝑎 = 0 and 𝜏 = 0, (4) becomes (𝜆+𝜇)2 − 𝑏2 𝜆2 = 0,
and the roots are 𝜆 1,2 = −𝜇/(1 ± 𝑏) < 0.
𝑖V0 (V0 > 0) is a zero of Δ(𝜆) with 𝑎 = 0 if and only if V0
solves
2
(𝑖V0 + 𝜇) + 𝑏2 V02 (cos 2V0 𝜏 − 𝑖 sin 2V0 𝜏) = 0.
(6)
Separating the real and imaginary parts gives that (𝑏2 −1)V02 =
𝜇2 , which contradicts with |𝑏| < 1. Then the conclusion
follows Wei and Ruan [1], and the proof is complete.
Claim 2. Equation (4) has at least two positive roots when
𝑎 > 𝜇.
In fact, the conclusion follows from Δ 1,2 (0) = 𝜇 − 𝑎 < 0
and lim𝜆 → +∞ Δ 1,2 (𝜆) = +∞.
When 𝑏 ≠ 0, 𝑖V (V > 0) is a root of (4) if and only if V
satisfies
𝑎 sin V𝜏 = −V (1 + 𝑏 cos V𝜏) ,
𝑎 cos V𝜏 = 𝑢 + 𝑏V sin V𝜏
(7)
or
𝑎 sin V𝜏 = −V (1 − 𝑏 cos V𝜏) ,
𝑎 cos V𝜏 = 𝑢 − 𝑏V sin V𝜏.
(8)
Applying the method in [3], we have that there exists a
+
−
(V1𝑚
), satisfying (7) and
unique value of V, denoted by V1𝑚
+
−
), satisfying (8) on
a unique value of V, denoted by V2𝑚 (V2𝑚
the interval (2(𝑚 − 1)𝜋/𝜏, 2𝑚𝜋/𝜏), 𝑚 ∈ Z+ when 𝑎 > 0(<
+
∈ ((2𝑚 − 1)𝜋/𝜏, 2𝑚𝜋/𝜏), 𝑘 =
0). We also can show that V𝑘𝑚
+
−
∈ (2(𝑚 − 1)𝜋/𝜏, (2𝑚 −
1, 2, 𝑚 ∈ Z with 𝑎 > 0 and V𝑘𝑚
+
1)𝜋/𝜏), 𝑘 = 1, 2, 𝑚 ∈ Z with 𝑎 < 0.
Define
2
±
±
(1 − 𝑏2 ) + 𝜇2 ,
= ±√V𝑘𝑚
𝑎𝑘𝑚
𝑘 = 1, 2, 𝑚 ∈ Z+ .
(9)
±
+
−
make sense since |𝑏| < 1, and 𝑎𝑘𝑚
and 𝑎𝑘𝑚
are all
𝑎𝑘𝑚
+
−
dependent on 𝜏. Hence, ±V𝑘𝑚 (±V𝑘𝑚 ) are a pair of purely
+
−
(𝑎𝑘𝑚
). And we have the
imaginary roots of (4) with 𝑎 = 𝑎𝑘𝑚
transversal condition
𝑑𝜆 󵄨󵄨󵄨
Sign Re ( )󵄨󵄨󵄨
±
𝑑𝑎 󵄨󵄨𝑎=𝑎𝑘𝑚
(10)
+
1,
𝑎 = 𝑎𝑘𝑚
,
+
={
𝑘 = 1, 2, 𝑚 ∈ Z .
−
−1, 𝑎 = 𝑎𝑘𝑚
,
−
−
+
Moreover, it is fulfilled that 𝑎𝑘×(𝑚+1)
< 𝑎𝑘𝑚
< 0 < 𝑎𝑘𝑚
<
+
−
+
−
, following the fact that 0 < V𝑘𝑚
< V𝑘𝑚
< V𝑘×(𝑚+1)
<
𝑎𝑘×(𝑚+1)
+
. Now we can distinguish two cases. First, 𝑏 > 0. From
V𝑘×(𝑚+1)
−
−
−
−
cos V1𝑚
𝜏 > 𝑎2𝑚
cos V2𝑚
𝜏,
(7) and (8), we can obtain that 𝑎1𝑚
−
−
−
−
which implies V1𝑚 > V2𝑚 and 𝑎1𝑚 < 𝑎2𝑚 . Second, 𝑏 < 0. With
−
−
< V2𝑚
and
the similar process as above, it is obtained that V1𝑚
−
−
𝑎1𝑚 > 𝑎2𝑚 .
Summarizing the discussion above and applying
Claims 1 and 2, we have the following.
Lemma 1. There exist sequence values of 𝑎 defined by (9)
such that all the roots of (4) have negative real parts when
𝑎 ∈ (𝑎1− (𝜏), 𝜇), and (4) has at least two roots with positive
real part when 𝑎 ∈ (−∞, 𝑎1− (𝜏)) ∪ (𝜇, +∞), a pair of purely
±
(𝜏) (𝑘 = 1, 2, 𝑚 ∈ Z+ ), where
imaginary roots when 𝑎 = 𝑎𝑘𝑚
−
−
−
𝑎1 (𝜏) = 𝑎11 (𝑎21 ) when 𝑏 < 0(> 0).
Lemma 2. Equation (1) undergoes a pitchfork bifurcation at
(0, 0) when 𝑎 = 𝜇.
Proof. Let ℎ(𝑥) = −𝜇𝑥 + 𝑎𝑓(𝑥). Then we have ℎ(0) = 0,
ℎ(±∞) = ∓∞, and ℎ󸀠 (𝑥) = −𝜇 + 𝑎𝑓󸀠 (𝑥). From (H1) and (H2),
there exists a unique zero 𝑥 = 0 of ℎ(𝑥), and (1) has only one
steady-state 𝑋 = (0, 0) when 𝑎 ≤ 𝜇. On the other hand, when
𝑎 > 𝜇, we have ℎ󸀠 (0) = 𝑎 − 𝜇 > 0 and 𝑥ℎ󸀠󸀠 (𝑥) < 0 when
𝑥 ≠ 0. Thus, there exists a unique 𝑐+ ∈ R+ (𝑐− ∈ R− ) satisfying
ℎ(𝑥) = 0, and (1) has exactly nonzero equilibria (𝑐± , 0), (0, 𝑐± ),
and (𝑐± , 𝑐± ). This completes the proof.
Abstract and Applied Analysis
3
In the following, we will investigate the stability of (𝑐± , 0),
(0, 𝑐± ), and (𝑐± , 𝑐± ).
Firstly, we consider the stability of (𝑐+ , 0) when 𝑎 > 𝜇. (It
has similar results with (𝑐− , 0) and (0, 𝑐± ).)
The characteristic equation of linearization of (1) at (𝑐+ , 0)
is
Δ (𝑐+ ,0) (𝜆) = (𝜆 + 𝜇 − 𝑎𝑒−𝜆𝜏 ) (𝜆 + 𝜇 − 𝑎𝑓󸀠 (𝑐+ ) 𝑒−𝜆𝜏 )
− 𝑏2 𝜆2 𝑒−2𝜆𝜏 = 0.
(11)
Equation (11) has at least one positive root for any 𝜏 ≥ 0. In
fact, we have ℎ󸀠 (𝑐± ) < 0 and 0 < 𝑎𝑓󸀠 (𝑐± ) < 𝜇 from the proof
of Lemma 2. Then the conclusion follows from Δ (𝑐+ ,0) (0) =
(𝜇−𝑎)(𝜇−𝑎𝑓󸀠 (𝑐+ )) < 0 and lim𝜆 → +∞ Δ (𝑐+ ,0) (𝜆) = +∞, which
implies that (𝑐+ , 0) is unstable.
Secondly, we consider the stability of (𝑐+ , 𝑐+ ) when 𝑎 > 𝜇.
(It has similar results with (𝑐+ , 𝑐− ) and (𝑐− , 𝑐± )).
The characteristic equation of linearization of (1) at
(𝑐+ , 𝑐+ ) is
2
Δ (𝑐+ ,𝑐+ ) (𝜆, 𝜏) = (𝜆 + 𝜇 − 𝑎𝑓󸀠 (𝑐+ ) 𝑒−𝜆𝜏 )
2 2 −2𝜆𝜏
−𝑏 𝜆 𝑒
In particular,
2
(13)
which implies
𝜆 1,2 =
−𝜇 + 𝑎𝑓󸀠 (𝑐+ )
< 0.
1±𝑏
(14)
Moreover, (12) has no zero root from 𝑎𝑓󸀠 (𝑐+ ) < 𝜇.
In order to prove that (𝑐+ , 𝑐+ ) is asymptotically stable for
all 𝜏 ∈ R+ , what we need is to verify that (12) has no purely
imaginary roots. Let 𝑖𝜂 (𝜂 > 0) be the root of (12), then 𝜂
solves
𝑎𝑓󸀠 (𝑐+ ) sin 𝜂𝜏 + 𝑏𝜂 cos 𝜂𝜏 = −𝜂,
𝑎𝑓󸀠 (𝑐+ ) cos 𝜂𝜏 − 𝑏𝜂 sin 𝜂𝜏 = 𝜇
(15)
𝑎𝑓󸀠 (𝑐+ ) cos 𝜂𝜏 + 𝑏𝜂 sin 𝜂𝜏 = 𝜇.
(16)
This leads to
2
𝜂2 =
(𝑎𝑓󸀠 (𝑐+ )) − 𝜇2
1 − 𝑏2
< 0.
sin 𝜔𝜏 = −
𝜔 (𝑎 ± 𝑏𝜇)
,
𝜇2 + 𝜔2
cos 𝜔𝜏 =
(17)
The assertion follows.
Applying Lemmas 1 and 2, we have the following results.
Theorem 3. (i) The zero solution of (1) is asymptotically stable
when 𝑎 ∈ (𝑎1− (𝜏), 𝜇) and unstable when 𝑎 ∈ (−∞, 𝑎1− (𝜏)) ∪
(𝜇, +∞).
𝑎𝜇 ∓ 𝑏𝜔2
.
𝜇2 + 𝜔2
(18)
def
This leads to 𝜔2 = (𝑎2 −𝜇2 )/(1−𝑏2 ), 𝜔0 = √(𝑎2 − 𝜇2 )/(1 − 𝑏2 )
makes sense when |𝑎| > 𝜇. Define
𝜏1𝑗
𝑎𝜇 − 𝑏𝜔0
1
{
{
{ 𝜔0 (arccos 𝜇2 + 𝜔02 + 2𝑗𝜋) ,
={
{1
𝑎𝜇 − 𝑏𝜔02
{
(−arccos 2
+ 2 (𝑗 + 1) 𝜋) ,
𝜔
𝜇 + 𝜔02
{ 0
𝜏2𝑗
2
𝑎𝜇 + 𝑏𝜔0
1
{
{
{ 𝜔0 (arccos 𝜇2 + 𝜔02 + 2𝑗𝜋) ,
={
2
{
{ 1 (−arccos 𝑎𝜇 + 𝑏𝜔0 + 2 (𝑗 + 1) 𝜋) ,
2
2
𝜇 + 𝜔0
{ 𝜔0
𝑎 < −𝜇,
𝑗 = 0, 1, 2, . . . ,
𝑎 > 𝜇,
𝑎 < −𝜇,
𝑗 = 0, 1, 2, . . . .
𝑎 > 𝜇,
(19)
Then 𝑖𝜔0 is a purely imaginary root of (4) with 𝜏 = 𝜏𝑘𝑗 , 𝑘 =
1, 2 defined by (19).
Let 𝜆(𝜏) = 𝛼(𝜏) + 𝑖󰜚(𝜏) be the root of (4), satisfying
𝛼 (𝜏𝑘𝑗 ) = 0,
󰜚 (𝜏𝑘𝑗 ) = 𝜔0 .
(20)
Differentiating both side of (4) gives that
Re (
or
𝑎𝑓󸀠 (𝑐+ ) sin 𝜂𝜏 − 𝑏𝜂 cos 𝜂𝜏 = −𝜂,
Case 2. Regard 𝜏 as parameter.
First of all, we know that the root of (4) with 𝜏 = 0 satisfies
that 𝜆 1,2 = (𝑎 − 𝜇)/(1 ± 𝑏) > 0 when 𝑎 > 𝜇, and 𝜆 1,2 =
(𝑎 − 𝜇)/(1 ± 𝑏) < 0 when 𝑎 < 𝜇.
Let 𝑖𝜔 (𝜔 > 0) be a root of (4); then it follows that
2
(12)
= 0.
Δ (𝑐+ ,𝑐+ ) (𝜆, 0) = (𝜆 + 𝜇 − 𝑎𝑓󸀠 (𝑐+ )) − 𝑏2 𝜆2 = 0,
(ii) Equation (1) undergoes a pitchfork bifurcation at (0, 0)
when 𝑎 = 𝜇. More precisely, some new equilibria bifurcate
from zero and (𝑐± , 0), (0, 𝑐± ) are unstable, and (𝑐± , 𝑐± ) are
asymptotically stable for 𝜏 ∈ R+ when 𝑎 > 𝜇.
(iii) Equation (1) undergoes a Hopf bifurcation at (0, 0)
±
(𝜏) (𝑘 = 1, 2, 𝑚 ∈ Z+ ).
when 𝑎 = 𝑎𝑘𝑚
󵄨
𝑑𝜆 −1 󵄨󵄨󵄨
1 − 𝑏2
= 2
> 0.
) 󵄨󵄨󵄨
𝑑𝜏
󵄨󵄨𝜏=𝜏𝑘𝑗 𝑎 + 𝑏2 𝜔02
(21)
This implies that 𝛼󸀠 (𝜏𝑘𝑗 ) > 0, 𝑘 = 1, 2, 𝑗 = 0, 1, 2, . . . .
Summarizing the discussions above, one can obtain the
following.
Lemma 4. (i) If |𝑎| < 𝜇, then all roots of (4) have negative
real parts.
(ii) If |𝑎| > 𝜇, then there exist a sequence values of 𝜏 defined
by (19) such that (4) has a pair of purely imaginary roots ±𝑖𝜔0
when 𝜏 = 𝜏𝑘𝑗 , 𝑘 = 1, 2. Additionally, if 𝑎 < −𝜇, then all roots
of (4) have negative real parts when 𝜏 ∈ [0, 𝜏0 ), all roots of
(4), except ±𝑖𝜔0 , have negative real parts when 𝜏 = 𝜏0 , and (4)
has at least a pair of root with positive real parts when 𝜏 > 𝜏0 ,
where 𝜏0 = 𝜏10 (𝜏20 ) when 𝑏 < 0 (> 0); if 𝑎 > 𝜇, then (4) has at
least two positive roots.
4
Abstract and Applied Analysis
Spectral properties in Lemma 4 immediately lead to the
dynamics near the origin described by the following theorem.
Theorem 5. For (1), the following holds.
(i) If |𝑎| < 𝜇, then (0, 0) is asymptotically stable for all
𝜏 > 0.
(ii) If 𝑎 < −𝜇, then (0, 0) is asymptotically stable when 𝜏 ∈
(0, 𝜏0 ), and unstable when 𝜏 > 𝜏0 .
(iii) If 𝑎 > 𝜇, then (0, 0) is always unstable for all 𝜏 > 0.
(iv) If |𝑎| > 𝜇, then (1) undergoes a Hopf bifurcation at
(0, 0) when 𝜏 = 𝜏𝑘𝑗 , 𝑘 = 1, 2, 𝑗 = 0, 1, 2, . . ..
For convenience, denote 𝜏 = 𝜏𝑗 + ]. Then we know that
(22) undergoes a Hopf bifurcation at origin when ] = 0. For
𝜙 ∈ 𝐶([−1, 0], R2 ), let
𝐷 (𝜙) = 𝜙 (0) − 𝐶𝜙 (−1) ,
𝐿 (], 𝜙) = 𝐴 (𝜏𝑗 + ]) 𝜙 (0) + 𝐵 (𝜏𝑗 + ]) 𝜙 (−1) ,
By the Riesz representation theorem, there exist functions
𝜂(𝜃) and 𝜇(𝜃) such that
0
𝐷 (𝜙) = 𝜙 (0) − ∫ 𝑑𝜇 (𝜃) 𝜙 (𝜃) ,
−1
Theorems 3(iii) and 5(iv) in the previous section give the
sufficient conditions for (1) to undergo Hopf bifurcations with
𝑎 and 𝜏 as bifurcation parameters. In this section, we will
investigate the direction of Hopf bifurcations and stability of
bifurcating periodic solutions, following the same algorithms
as Wang and Wei’s recent work and using a computation
process similar to that in [2] (see also [17]). We should
mention that we will choose 𝜏 as bifurcation parameter, and
the similar results follow when choosing other coefficients as
bifurcation parameters.
Let 𝑦𝑖 (𝑡) = 𝑥𝑖 (𝜏𝑡), 𝑖 = 1, 2, and 𝑌(𝑡) = (𝑦1 (𝑡), 𝑦2 (𝑡))𝑇 ,
then (1) becomes
(22)
where 𝐴, 𝐵, and 𝐶 are denoted by (2) and 𝐹(𝑌(𝑡 − 1)) =
(𝑓(𝑦1 (𝑡 − 1)), 𝑓(𝑦2 (𝑡 − 1)))𝑇 . And the corresponding characteristic equation around 𝑌 = (0, 0)𝑇 is
2
(𝑧 + 𝜇𝜏 − 𝑎𝜏𝑒−𝑧 ) − 𝑏2 𝑧2 𝑒−2𝑧 = 0,
Lemma 6. Assume |𝑎| > 𝜇.
(i) If 𝜏 = 𝜏𝑗 , 𝑗 = 0, 1, 2, . . ., then (23) has a pair of purely
imaginary roots ±𝑖𝜔0 𝜏𝑗 , where 𝜏𝑗 = 𝜏𝑘𝑗 and 𝜔0 are
defined by (19).
(ii) Let 𝑧(𝜏) = 𝛾(𝜏) + 𝑖𝜍(𝜏) be the root of (23), satisfying
𝜍 (𝜏𝑗 ) = 𝜔0 𝜏𝑗 ,
(24)
𝐿 (], 𝜙) = ∫ 𝑑𝜂 (𝜃) 𝜙 (𝜃) .
−1
In fact, we can choose
−𝐶,
𝜇 (𝜃) = {
0,
𝑗 = 0, 1, 2, . . . .
(25)
𝜃 = −1,
𝜃 ∈ (−1, 0] ,
−𝐵 (𝜏𝑗 + ]) , 𝜃 = −1,
{
{
𝜂 (𝜃) = {0,
𝜃 ∈ (−1, 0) ,
{
{𝐴 (𝜏𝑗 + ]) , 𝜃 = 0.
(28)
Define
{ 𝑑𝜙 (𝜃) ,
𝐴 (]) 𝜙 = { 𝑑𝜃
󸀠
󸀠
{𝜙 (0) − 𝐷𝜙 (0) + 𝐿𝜙,
𝜃 ∈ [−1, 0) ,
𝜃 = 0,
(29)
0,
𝜃 ∈ [−1, 0) ,
𝑅 (]) 𝜙 = {
𝐹 (], 𝜙) , 𝜃 = 0.
Then (22) can be written as
𝑌𝑡̇ = 𝐴 (]) 𝑌𝑡 + 𝑅 (]) 𝑌𝑡 .
(30)
Clearly, (30) is an abstract ODE on the phase space 𝐵𝐶󸀠 of
(22), where 𝐵𝐶󸀠 = {𝜙 : [−1, 0] → C2 , 𝜙 is continuous on
[−1, 0), and lim𝜃 → 0 𝜙(𝜃) exists}.
The adjoint operator 𝐴∗ 𝜓 = −𝑑𝜓/𝑑𝑠 with domain
𝐷 (𝐴∗ )
= {𝜓 ∈ 𝐶󸀠 = 𝐶 ([0, 1] , R2 ) :
def
then
𝛾󸀠 (𝜏𝑗 ) = 𝜏𝑗 𝛼󸀠 (𝜏𝑗 ) > 0,
(27)
0
(23)
Comparing (23) with (4), if is found that 𝑧 = 𝜆𝜏 for 𝜏 ≠ 0.
Therefore, combining this fact with Lemma 4, one has the
following.
𝛾 (𝜏𝑗 ) = 0,
𝑓󸀠󸀠󸀠 (0)
(𝜏𝑗 + ]) 𝐵𝜙3 (−1) + ⋅ ⋅ ⋅ .
6
𝐹 (], 𝜙) =
3. Properties of Hopf BifurCation
𝑑
[𝑌(𝑡) − 𝐶𝑌 (𝑡 − 1)] = 𝜏𝐴𝑌 (𝑡) + 𝜏𝐵𝐹 (𝑌 (𝑡 − 1)) ,
𝑑𝑡
(26)
𝑑𝜓
𝑑𝜓
∈ 𝐶󸀠 ; 𝐷
= −𝐿𝜓} .
𝑑𝑠
𝑑𝑠
(31)
For (𝜓, 𝜙) ∈ 𝐶󸀠 × 𝐶, define a bilinear form:
(𝜓, 𝜙) = 𝜓 (0) 𝜙 (0)
(iii) Equation (23) has at least a pair of roots with positive
real parts when 𝜏 = 𝜏𝑗 for 𝑗 ≥ 1, and it has two positive
roots when 𝑎 > 𝜇 and 𝜏 = 𝜏0 . All roots of (23) with
𝜏 = 𝜏0 , except ±𝑖𝜔0 𝜏0 , have negative real parts when
𝑎 < −𝜇.
0
𝜃
−1
𝛼=0
− ∫ 𝑑 [∫
0
−∫ ∫
𝑠
−1 𝜃=0
𝜓 (𝜃 − 𝛼) 𝑑𝜇 (𝛼)] 𝜙 (𝜃)
𝜓 (𝜃 − 𝑠) 𝑑𝜂 (𝑠) 𝜙 (𝜃) 𝑑𝜃.
(32)
Abstract and Applied Analysis
5
It is not difficult to verify that 𝑞(𝜃) = 𝑞0 𝑒𝑖𝜔0 𝜏𝑗 𝜃 (𝜃 ∈ [−1, 0])
and 𝑞∗ (𝑠) = 𝑙𝑞0∗ 𝑒𝑖𝜔0 𝜏𝑗 𝑠 (𝑠 ∈ [0, 1]) are the eigenvectors of 𝐴(0)
and 𝐴∗ corresponding to the eigenvalues 𝑖𝜔0 𝜏𝑗 and −𝑖𝜔0 𝜏𝑗 ,
respectively, where
𝑧 (𝑡) = (𝑞∗ , 𝑌𝑡 ) ,
𝑧̇ (𝑡) = 𝑖𝜔0 𝜏𝑗 𝑧 + 𝑞∗ (0) 𝐹 (0, 𝑌𝑡 ) .
(35)
Equation (35) can be written in the abbreviated form as
1
𝑧̇ (𝑡) = 𝑖𝜔0 𝜏𝑗 𝑧 + 𝑔 (𝑧, 𝑧) ,
,
𝑞0∗ (𝐼 − 𝐶𝑒−𝑖𝜔0 𝜏𝑗 + 𝐵𝜏𝑗 𝑒−𝑖𝜔0 𝜏𝑗 ) 𝑞0
(36)
with
(33)
and (𝑞∗ , 𝑞) = 1.
𝑔 (𝑧, 𝑧) = 𝑔20
𝑧2
𝑧2
𝑧2 𝑧
+ 𝑔11 𝑧𝑧 + 𝑔02 + 𝑔21
+ ⋅⋅⋅ .
2
2
2
(37)
Noting that 𝑌𝑡 (𝜃) = 𝑊(𝑡, 𝜃) + 𝑧(𝑡)𝑞(𝜃) + 𝑧(𝑡)𝑞(𝜃), we have
𝑌𝑡 (−1) = (
(1)
𝑞01 𝑒−𝑖𝜔0 𝜏𝑗 𝑧 + 𝑞01 𝑒𝑖𝜔0 𝜏𝑗 𝑧 + 𝑊20
(−1)
−𝑖𝜔0 𝜏𝑗
𝑞02 𝑒
𝑖𝜔0 𝜏𝑗
𝑧 + 𝑞02 𝑒
𝑧+
(2)
𝑊20
where 𝑞0 = (𝑞01 , 𝑞02 )𝑇 . Therefore, from (26), we have
𝑧2
𝑧2
𝑧2 𝑧
= 𝐹𝑧2 + 𝐹𝑧𝑧 𝑧𝑧 + 𝐹𝑧 2 + 𝐹𝑧2 𝑧
+ ⋅⋅⋅ .
2
2
2
(39)
𝑔20 = 𝑔02 = 𝑔11 = 0,
2
𝑞01
𝑞01
(40)
𝑓 (0)
) 𝑒−𝑖𝜔0 𝜏𝑗 .
𝜏𝑗 𝐵 ( 2
𝑞02 𝑞02
6
It is well known that the coefficient 𝑐1 (0) of third degree term
of Poincaré normal form of (35) is given by (see [12])
𝑖
󵄩 󵄩2
𝑐1 (0) =
(𝑔20 𝑔11 − 2󵄩󵄩󵄩𝑔11 󵄩󵄩󵄩 −
2𝜔0 𝜏𝑗
𝑔
1 󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑔02 󵄩󵄩 ) + 21 . (41)
3
2
Consequently, when 𝜏 = 𝜏1𝑗 or 𝜏 = 𝜏2𝑗 ,
(38)
2
𝑎𝜇 ∓ 𝑏𝜔02
𝑎𝜇 ∓ 𝑏𝜔02
+
(∓𝑏
+
𝑎𝜏
)
(2(
) − 1)] .
𝑘𝑗
𝜇2 + 𝜔02
𝜇2 + 𝜔02
(43)
Summarizing the above analysis, we have the following
theorem.
Theorem 7. For (22), the direction of Hopf bifurcation at 𝜏 =
𝜏𝑘𝑗 , 𝑘 = 1, 2, 𝑗 = 0, 1, 2, . . ., is supercritical (subcritical),
and the bifurcating periodic solutions are asymptotically stable
(unstable) when Re 𝑐1 (0) < 0 (Re 𝑐1 (0) > 0).
4. Global Hopf Bifurcation Analysis
Our objective in this section is to obtain the global continuation of periodic solution bifurcating from the point
(0, 𝜏𝑘𝑗 ), 𝑘 = 1, 2, 𝑗 = 0, 1, 2, . . ., for (1) by using a global Hopf
bifurcation theorem given by Krawcewicz et al. [14]. For the
reader’s convenience, we copy (22), which is equivalent to (1),
as follows
𝑦1̇ (𝑡) − 𝑏𝑦2̇ (𝑡 − 1) = −𝜇𝜏𝑦1 (𝑡) + 𝑎𝜏𝑓 (𝑦1 (𝑡 − 1)) ,
𝜏𝑘𝑗
󵄩2
󵄩
12󵄩󵄩󵄩󵄩1 + (∓𝑏 + 𝑎𝜏𝑘𝑗 ) 𝑒−𝑖𝜔0 𝜏𝑘𝑗 󵄩󵄩󵄩󵄩
󸀠󸀠󸀠
),
= Sign 𝑎𝑓󸀠󸀠󸀠 (0)
×[
Substituting the expression of 𝐹(0, 𝑌𝑡 ) into (35) and comparing its coefficients with that of (36) give that
𝑔21 = 𝑙 𝑞∗0
𝑧2
𝑧2
(2)
(2)
+ 𝑊11
+ ⋅⋅⋅
(−1)
(−1) 𝑧𝑧 + 𝑊02
(−1)
2
2
SignRe 𝑐1 (0)
def
󸀠󸀠󸀠
𝑧2
𝑧2
(1)
(1)
+ 𝑊11
+ ⋅⋅⋅
(−1) 𝑧𝑧 + 𝑊02
(−1)
2
2
Hence we obtain
𝑓󸀠󸀠󸀠 (0)
𝐹 (0, 𝑌𝑡 ) =
𝜏𝑗 𝑌𝑡3 (−1) + ⋅ ⋅ ⋅
6
𝑐1 (0) =
(34)
𝑊 (𝑡, 𝜃) = 𝑌𝑡 (𝜃) − 2 Re 𝑧 (𝑡) 𝑞 (𝜃) ,
then we have
(1, −1)𝑇 𝜏𝑗 = 𝜏1𝑗 ,
𝑞0 = {
𝜏𝑗 = 𝜏2𝑗 ,
(1, 1)𝑇
(1, −1) 𝜏𝑗 = 𝜏1𝑗 ,
∗
𝑞0 = {
𝜏𝑗 = 𝜏2𝑗 ,
(1, 1)
𝑙=
Now we compute the center manifold 𝐶0 at ] = 0. Define
(42)
−𝑖𝜔0 𝜏𝑘𝑗
× 𝑎𝑓 (0) [1 + (∓𝑏 + 𝑎𝜏𝑘𝑗 ) 𝑒
−𝑖𝜔0 𝜏𝑘𝑗
]𝑒
.
𝑦2̇ (𝑡) − 𝑏𝑦1̇ (𝑡 − 1) = −𝜇𝜏𝑦2 (𝑡) + 𝑎𝜏𝑓 (𝑦2 (𝑡 − 1)) .
(44)
We have known that (44) undergoes a local Hopf bifurcation
at the origin when 𝜏 = 𝜏𝑘𝑗 , 𝑘 = 1, 2, 𝑗 = 0, 1, 2, . . .. Now we
6
Abstract and Applied Analysis
begin to show that each bifurcation branch can be continued
with respect to the parameter 𝜏 under certain conditions. To
bring out the ideas in the results of subsequent part, it is
convenient to introduce the following notations:
By (44), (48), and (H1), we have that
𝑢 (𝑡𝑖 ) =
𝑎 2
2 |𝑎| 𝐿 2 |𝑎| 𝐿
,
],
∑ 𝑓 (𝑦𝑗 (𝑡𝑖 − 1)) ∈ [−
𝑢 𝑗=1
𝜇
𝜇
𝑋 = 𝐶([−1, 0], R2 ),
𝑖 = 1, 2,
Σ = 𝐶𝑙{(𝑌, 𝜏, 𝑇) : 𝑌 is a 𝑇-periodic solution of (44)} ⊂
𝑋 × R+ × R+ ,
𝑢 (𝑡2 ) − 𝑏𝑢 (𝑡2 − 1) ⩽ 𝑢 (𝑡) − 𝑏𝑢 (𝑡 − 1)
⩽ 𝑢 (𝑡1 ) − 𝑏𝑢 (𝑡1 − 1) .
̂ 𝜏, 𝑇) : 𝜇𝑌
̂ = 𝑎𝐹(𝑌)}.
̂
𝑁 = {(𝑌,
(50)
Denote 𝐶(0, 𝜏𝑗 , 2𝜋/𝜔0 𝜏𝑗 ) the connected component of
√(𝑎2
(0, 𝜏𝑗 , 2𝜋/𝜔0 𝜏𝑗 ) in Σ, where 𝜔0 =
−
𝜏𝑗 (𝑗 = 0, 1, 2, . . .) are defined by Lemma 6.
𝜇2 )/(1
−
𝑏2 )
and
Lemma 8. Equation (44) has no periodic nonconstant solution when 𝜏 = 0.
Proof. Let 𝑢(𝑡) = 𝑦1 (𝑡) + 𝑦2 (𝑡). When 𝜏 = 0, we from (44)
have
𝑑
[𝑢 (𝑡) − 𝑏𝑢 (𝑡 − 1)] = 0.
𝑑𝑡
On one hand, we have
𝑢 (𝑡) ⩽ 𝑏𝑢 (𝑡 − 1) + [𝑢 (𝑡1 ) − 𝑏𝑢 (𝑡1 − 1)]
⩽ 𝑏2 𝑢 (𝑡 − 2) + 𝑏 [𝑢 (𝑡2 ) − 𝑏𝑢 (𝑡2 − 1)]
+ [𝑢 (𝑡1 ) − 𝑏𝑢 (𝑡1 − 1)]
𝑚
⩽ 𝑏2𝑚 𝑢 (𝑡 − 2𝑚) + ∑𝑏2𝑖−1 [𝑢 (𝑡2 ) − 𝑏𝑢 (𝑡2 − 1)]
𝑖=1
𝑚
+ ∑𝑏2(𝑖−1) [𝑢 (𝑡1 ) − 𝑏𝑢 (𝑡1 − 1)]
(45)
Therefore, there exists 𝑐1 ∈ R such that
⩽
𝑢 (𝑡) = 𝑐1 + 𝑏𝑢 (𝑡 − 1) .
(46)
⩽
𝑛−1
(47)
𝑖=0
𝑏
[𝑢 (𝑡2 ) − 𝑏𝑢 (𝑡2 − 1)]
1 − 𝑏2
+
Which implies that when 𝑡 ∈ [𝑛, 𝑛 + 1], we have
𝑢 (𝑡) = 𝑐1 ∑ 𝑏𝑖 + 𝑏𝑛 𝑢 (𝑡 − 𝑛) .
1
[𝑢 (𝑡1 ) − 𝑏𝑢 (𝑡1 − 1)]
1 − 𝑏2
2 |𝑎| 𝐿
𝑏
𝑏2
𝑢 (𝑇0 ) −
𝑢 (𝑇0 ) .
−
1 − 𝑏2
(1 + 𝑏) 𝜇 1 − 𝑏2
And on the other hand, we can obtain
Hence we obtain lim𝑡 → +∞ 𝑢(𝑡) = 𝑐1 /(1 − 𝑏).
Let V(𝑡) = 𝑦1 (𝑡) − 𝑦2 (𝑡). With the similar process as above,
it is obtained that there exists 𝑐2 ∈ R such that lim𝑡 → +∞ V(𝑡) =
𝑐2 /(1 + 𝑏).
Then the conclusion follows from lim𝑡 → +∞ 𝑦1 (𝑡) =
(1/2)[𝑐1 /(1 − 𝑏) + 𝑐2 /(1 + 𝑏)], and the proof is complete.
𝑢 (𝑡) ⩾ −
2 |𝑎| 𝐿
𝑏2
𝑏
𝑢
(𝑇
)
−
𝑢 (𝑇0 ) .
−
0
1 − 𝑏2
(1 + 𝑏) 𝜇 1 − 𝑏2
Proof. Let 𝑌(𝑡) = (𝑦1 (𝑡), 𝑦2 (𝑡))𝑇 be a periodic solution of (44)
and 𝑢(𝑡) = 𝑦1 (𝑡) + 𝑦2 (𝑡). Then there exist 𝑡1 and 𝑡2 such that
𝑢 (𝑡1 ) − 𝑏𝑢 (𝑡1 − 1) = max [𝑢 (𝑡) − 𝑏𝑢 (𝑡 − 1)] ,
𝑢 (𝑡2 ) − 𝑏𝑢 (𝑡2 − 1) = min [𝑢 (𝑡) − 𝑏𝑢 (𝑡 − 1)] ,
(48)
𝑢 (𝑇0 ) ⩾
1 − 𝑏2
2 |𝑎| 𝐿
𝑏2
[−
𝑢 (𝑇0 )] ,
−
1 + 𝑏 − 𝑏2
(1 + 𝑏) 𝜇 1 − 𝑏2
2 |𝑎| 𝐿
𝑢 (𝑇 ) ⩽
.
(1 + 2𝑏) 𝜇
together with 𝑇0 and 𝑇0 such that
𝑢 (𝑇0 ) = max+ 𝑢 (𝑡) ,
𝑡∈R
𝑢 (𝑇0 ) = min+ 𝑢 (𝑡) ,
𝑡∈R
(49)
Now we take the case −1/2 < 𝑏 < 0 as an example, and it
has the same result when 0 ⩽ 𝑏 < 1/2.
(53)
0
Similarly, one can prove that
𝑢 (𝑇0 ) ⩾ −
2 |𝑎| 𝐿
.
(1 + 2𝑏) 𝜇
(54)
Which implies that
−
𝑡∈R
(52)
It follows that
Lemma 9. If |𝑏| < 1/2, then all periodic solution of (44) are
uniformly bounded.
𝑡∈R
(51)
𝑖=1
2 |𝑎| 𝐿
2 |𝑎| 𝐿
⩽ 𝑢 (𝑡) ⩽
.
(1 + 2𝑏) 𝜇
(1 + 2𝑏) 𝜇
(55)
Let V(𝑡) = 𝑦1 (𝑡) − 𝑦2 (𝑡). In the same way, it can be verified
that
2 |𝑎| 𝐿
2 |𝑎| 𝐿
−
⩽ V (𝑡) ⩽
.
(56)
(1 + 2𝑏) 𝜇
(1 + 2𝑏) 𝜇
Then 𝑦1 (𝑡) and 𝑦2 (𝑡) are uniformly bounded, and the
proof is complete.
Abstract and Applied Analysis
7
1.2
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
1
0.8
0.6
0.4
0.2
0
−0.2
0
10
20
−2
10
0
𝑥1
𝑥2
(a)
(b)
−2
−0.5
20
0
0.5
1
(𝑥1 , 𝑥2 )
(c)
Figure 1: For system (1), the equilibrium (0, 0) is asymptotically stable when 𝜏 = 0.55 < 𝜏0 ≈ 0.6378.
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
4
2
−0.8
0
−1
0
50
100
0
50
𝑥1
𝑥2
(a)
(b)
100
−5
−2
0
−4
5
(c)
Figure 2: The system (1) undergoes a Hopf bifurcation at (0, 0), and the bifurcating periodic solution is asymptotically stable when 𝜏 = 2.5 ∈
(𝜏0 , 𝜏1 ).
Lemma 10. If 𝑎 < −𝜇 or 𝑎 > 𝜇/𝑀, then (44) has
no periodic nonconstant solution of period 1, where 𝑀 =
min{𝑓󸀠 (−2|𝑎|𝐿/(1 + 2𝑏)𝜇), 𝑓󸀠 (2|𝑎|𝐿/(1 + 2𝑏)𝜇)}.
Proof. Let (𝑦1 (𝑡), 𝑦2 (𝑡)) be a periodic solution to (44) of
period 1. Then it is a periodic solution to the following system
of ordinary differential equations:
𝑢̇1 (𝑡) − 𝑏𝑢̇2 (𝑡) = −𝜇𝜏𝑢1 (𝑡) + 𝑎𝜏𝑓 (𝑢1 (𝑡)) ,
𝑢̇2 (𝑡) − 𝑏𝑢̇1 (𝑡) = −𝜇𝜏𝑢2 (𝑡) + 𝑎𝜏𝑓 (𝑢2 (𝑡)) ,
(57)
where ⋅ denotes 𝑑/𝑑𝑡. Denote
𝐺 = {𝑢 ∈ R2 : 𝑢𝑗 ∈ [−
2 |𝑎| 𝐿
2 |𝑎| 𝐿
,
] , 𝑗 = 1, 2} .
(1 + 2𝑏) 𝜇 (1 + 2𝑏) 𝜇
(58)
8
Abstract and Applied Analysis
4
4
3
3
2
2
1
1
0
0
0.2
−1
−1
−0.2
−2
−2
−3
−3
1
0.8
0.6
0.4
0
−4
0
1000
2000
−4
−0.4
−0.6
5
−0.8
0
−1
0
1000
𝑥1
𝑥2
(a)
(b)
−5
2000
0
5
−5
(c)
Figure 3: The system (1) undergoes a Hopf bifurcation at (0, 0) with sufficiently large 𝜏 = 50.
Lemma 9 shows that the periodic solution of (44) belong to
the region 𝐺. Clearly, (57) is equivalent to
𝑢̇1 =
Δ (𝜆 (𝜏)) = Δ (0,𝜏,𝑇) (𝜆 (𝜏)) = 0,
𝜏
[−𝜇 (𝑢1 + 𝑏𝑢2 ) + 𝑎 (𝑓 (𝑢1 ) + 𝑏𝑓 (𝑢2 ))]
1 − 𝑏2
= 𝑃 (𝑢1 , 𝑢2 ) ,
𝜏
[−𝜇 (𝑢2 + 𝑏𝑢1 ) + 𝑎 (𝑓 (𝑢2 ) + 𝑏𝑓 (𝑢1 ))]
1 − 𝑏2
(61)
󵄨
󵄨󵄨
󵄨󵄨𝜆 (𝜏) − 𝑖𝜔0 𝜏𝑗 󵄨󵄨󵄨 < 𝜖,
󵄨
󵄨
def
𝑢̇2 =
𝜖 > 0, 𝛿 > 0, and a smooth curve 𝜆 : (𝜏𝑗 − 𝛿, 𝜏𝑗 + 𝛿) → C,
such that
(59)
for all 𝜏 ∈ [𝜏𝑗 − 𝛿, 𝜏𝑗 + 𝛿], where Δ is defined as (23), and
𝜆 (𝜏𝑗 ) = 𝑖𝜔0 𝜏𝑗 ,
𝑑 Re(𝜆(𝜏)) 󵄨󵄨󵄨󵄨
> 0.
󵄨󵄨
󵄨󵄨𝜏=𝜏𝑗
𝑑𝜏
(62)
def
= 𝑄 (𝑢1 , 𝑢2 ) ,
Then
𝜕𝑃
𝜕𝑄
𝜏
+
=
[−2𝜇 + 𝑎 (𝑓󸀠 (𝑢1 ) + 𝑓󸀠 (𝑢2 ))] . (60)
𝜕𝑢1 𝜕𝑢2 1 − 𝑏2
From (H2), it is easy to see that 𝜕𝑃/𝜕𝑢1 + 𝜕𝑄/𝜕𝑢2 < 0 when
𝑎 < −𝜇 and 𝜕𝑃/𝜕𝑢1 + 𝜕𝑄/𝜕𝑢2 > 0 when 𝑎 > 𝜇/𝑀. Thus,
the classical Bendixson’s criterion implies that (57) has no
nonconstant periodic solution in the region 𝐺, and the proof
is complete.
Up to now, we have prepared sufficiently to state the
following global Hopf bifurcation results.
Theorem 11. If |𝑏| < 1/2 and either 𝑎 < −𝜇 or 𝑎 > 𝜇/𝑀 holds,
then (44) has at least one periodic solution for 𝜏 > 𝜏𝑗 (𝑗 ⩾ 1).
Proof. First note that the stationary points (0, 𝜏𝑗 , 2𝜋/𝜔0 𝜏𝑗 ) of
(44) are nonsingular and isolate centers (see [14]) under the
assumption |𝑎| > 𝜇 for each 𝜏 > 𝜏𝑗 (𝑗 ⩾ 1). By (25), there exist
Let
󵄨
󵄨󵄨
󵄨󵄨
2𝜋 󵄨󵄨󵄨󵄨
󵄨
Ω𝜖 = {(𝑢, 𝑝) : 0 < 𝑢 < 𝜖, 󵄨󵄨𝑝 −
󵄨 < 𝜖} .
󵄨󵄨
𝜔0 𝜏𝑗 󵄨󵄨󵄨󵄨
󵄨
(63)
Clearly, if 𝜏 − 𝜏𝑗 ⩽ 𝛿 and (𝑢, 𝑝) ∈ Ω𝜖 such that Δ (0,𝜏,𝑇) (𝑢 +
𝑖2𝜋/𝑝) = 0, then 𝜏 = 𝜏𝑗 , 𝑢 = 0, and 𝑝 = 2𝜋/𝜔0 𝜏𝑗 . Moreover,
if we put
𝐻± (0, 𝜏𝑗 ,
2𝜋
𝑖2𝜋
) (𝑢, 𝑝) = Δ (0,𝜏±𝛿,𝑇) (𝑢 +
),
𝜔0 𝜏𝑗
𝑝
(64)
at 𝑚 = 1, we have, from Re 𝜆󸀠 (𝜏𝑗 ) > 0, that the crossing
number is
𝛾1 (0, 𝜏𝑗 ,
2𝜋
2𝜋
) = deg𝐵 (𝐻− (0, 𝜏𝑗 ,
) , Ω𝜖 )
𝜔0 𝜏𝑗
𝜔0 𝜏𝑗
− deg𝐵 (𝐻+ (0, 𝜏𝑗 ,
2𝜋
) , Ω𝜖 ) = −1.
𝜔0 𝜏𝑗
(65)
Using the local Hopf bifurcation theorem for NFDE [14,
Theorem 5.6], we conclude that the connected component
Abstract and Applied Analysis
9
𝐶(0, 𝜏𝑗 , 2𝜋/𝜔0 𝜏𝑗 ) through (0, 𝜏𝑗 , 2𝜋/𝜔0 𝜏𝑗 ) in Σ is nonempty,
and thus 𝐶(0, 𝜏𝑗 , 2𝜋/𝜔0 𝜏𝑗 ) is unbounded by the global Hopf
bifurcation theorem given by Krawcewicz et al. [14, Theorem
5.14].
Lemma 9 implies that the projection of 𝐶(0, 𝜏𝑗 , 2𝜋/𝜔0 𝜏𝑗 )
onto the 𝑌-space is bounded. Meanwhile, the projection
of 𝐶(0, 𝜏𝑗 , 2𝜋/𝜔0 𝜏𝑗 ) onto 𝜏-space is bounded below from
Lemma 8.
By the definition of 𝜏𝑗 , we know that
2𝜋 < 𝜔0 𝜏𝑗 < 2 (𝑗 + 1) 𝜋,
𝑗 ⩾ 1,
(66)
under the assumptions that |𝑎| > 𝜇 and |𝑏| < 1, which implies
1
2𝜋
< 1.
<
𝑗 + 1 𝜔0 𝜏𝑗
(67)
Applying Lemma 10, one has 1/(𝑗 + 1) < 𝑇 < 1 if
(𝑌, 𝜏, 𝑇) ∈ 𝐶(0, 𝜏𝑗 , 2𝜋/𝜔0 𝜏𝑗 ) for 𝑗 ⩾ 1, when |𝑏| < 1/2
and either 𝑎 < −𝜇 or 𝑎 > 𝜇/𝑀 holds. Thus in order for
𝐶(0, 𝜏𝑗 , 2𝜋/𝜔0 𝜏𝑗 ) to be unbounded, its projection onto the 𝜏space must be unbounded. Consequently, the projection of
𝐶(0, 𝜏𝑗 , 2𝜋/𝜔0 𝜏𝑗 ) onto the 𝜏-space include [𝜏𝑗 , ∞) for 𝑗 ⩾ 1
when |𝑏| < 1/2 and either 𝑎 < −𝜇 or 𝑎 > 𝜇/𝑀 holds. The
proof is complete.
Next we carry out some numerical simulations for (1).
Let 𝑓(𝑥) = arctan(𝑥), and assume that 𝜇 = 1.2, 𝑎 = −3,
and 𝑏 = 0.2. From Lemma 4, it is obtained that 𝜔0 ≈ 2.8062
and 𝜏0 ≈ 0.6378. Accordingly, it is known that (0, 0) is
asymptotically stable for 𝜏 ∈ (0, 𝜏0 ) and unstable for 𝜏 >
𝜏0 , and Hopf bifurcation at the origin occurs when 𝜏 = 𝜏0
by Theorem 5. From Theorem 7, the direction of the Hopf
bifurcation at 𝜏 = 𝜏0 is supercritical, and the bifurcating
periodic solutions are asymptotically stable. Furthermore,
according to Theorem 11, (1) with this set of parameters has
at least one periodic solution when 𝜏 > 𝜏𝑗 (𝑗 ⩾ 1). The
corresponding numerical simulation results are shown in
Figures 1, 2, and 3.
Acknowledgment
This work is supported by the Fundamental Research Funds
for the Central Universities, (no. DL11AB02).
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