Bifurcation Analysis in a Two-Dimensional Neutral Differential Equation

Abstract

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 high-dimension 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: 𝑥̇₁(𝑡)=−𝜇𝑥₁(𝑡)+𝑎𝑓𝑥₁(𝑡−𝜏)+𝑏𝑥̇₂(),𝑥̇𝑡−𝜏₂(𝑡)=−𝜇𝑥₂(𝑡)+𝑎𝑓𝑥₂(𝑡−𝜏)+𝑏𝑥̇₁(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, 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 𝐴=−𝜇00−𝜇,𝐵=𝑎00𝑎,𝐶=0𝑏𝑏0. (3) And the corresponding characteristic equation is Δ(𝜆)=𝜆+𝜇−𝑎𝑒^−𝜆𝜏2−𝑏²𝜆²𝑒^−2𝜆𝜏=0, (4) that is, 𝜆+𝜇−(𝑎−𝑏𝜆)𝑒^−𝜆𝜏𝜆+𝜇−(𝑎+𝑏𝜆^−𝜆𝜏_def)𝑒=Δ₁(𝜆)⋅Δ₂(𝜆)=0. (5)

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−𝑏²𝜆²=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 𝑖𝑣₀+𝜇²+𝑏²𝑣²₀cos2𝑣₀𝜏−𝑖sin2𝑣₀𝜏=0. (6) Separating the real and imaginary parts gives that (𝑏²−1)𝑣²₀=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 lim_𝜆→+∞Δ_1,2(𝜆)=+∞.
When 𝑏≠0,𝑖𝑣(𝑣>0) is a root of (4) if and only if 𝑣 satisfies 𝑎sin𝑣𝜏=−𝑣(1+𝑏cos𝑣𝜏),𝑎cos𝑣𝜏=𝑢+𝑏𝑣sin𝑣𝜏 (7) or 𝑎sin𝑣𝜏=−𝑣(1−𝑏cos𝑣𝜏),𝑎cos𝑣𝜏=𝑢−𝑏𝑣sin𝑣𝜏. (8)
Applying the method in [3], we have that there exists a unique value of 𝑣, denoted by 𝑣⁺_1𝑚 (𝑣⁻_1𝑚), satisfying (7) and a unique value of 𝑣, denoted by 𝑣⁺_2𝑚 (𝑣⁻_2𝑚), satisfying (8) on the interval  (2(𝑚−1)𝜋/𝜏,2𝑚𝜋/𝜏),𝑚∈ℤ⁺ when 𝑎>0(<0). We also can show that 𝑣⁺_𝑘𝑚∈((2𝑚−1)𝜋/𝜏,2𝑚𝜋/𝜏),𝑘=1,2,𝑚∈ℤ⁺ with 𝑎>0 and 𝑣⁻_𝑘𝑚∈(2(𝑚−1)𝜋/𝜏,(2𝑚−1)𝜋/𝜏),𝑘=1,2,𝑚∈ℤ⁺ with 𝑎<0.
Define 𝑎^±_𝑘𝑚=±𝑣^±2_𝑘𝑚1−𝑏²+𝜇²,𝑘=1,2,𝑚∈ℤ⁺. (9)𝑎^±_𝑘𝑚 make sense since |𝑏|<1, and 𝑎⁺_𝑘𝑚 and 𝑎⁻_𝑘𝑚 are all dependent on 𝜏. Hence, ±𝑣⁺_𝑘𝑚 (±𝑣⁻_𝑘𝑚) are a pair of purely imaginary roots of (4) with 𝑎=𝑎⁺_𝑘𝑚(𝑎⁻_𝑘𝑚). And we have the transversal condition SignRe𝑑𝜆𝑑𝑎_𝑎=𝑎^±_𝑘𝑚=1,𝑎=𝑎⁺_𝑘𝑚,−1,𝑎=𝑎⁻_𝑘𝑚,𝑘=1,2,𝑚∈ℤ⁺. (10)
Moreover, it is fulfilled that 𝑎⁻_{𝑘×(𝑚+1)}<𝑎⁻_𝑘𝑚<0<𝑎⁺_𝑘𝑚<𝑎⁺_{𝑘×(𝑚+1)}, following the fact that 0<𝑣⁻_𝑘𝑚<𝑣⁺_𝑘𝑚<𝑣⁻_{𝑘×(𝑚+1)}<𝑣⁺_{𝑘×(𝑚+1)}. Now we can distinguish two cases. First, 𝑏>0. From (7) and (8), we can obtain that 𝑎⁻_1𝑚cos𝑣⁻_1𝑚𝜏>𝑎⁻_2𝑚cos𝑣⁻_2𝑚𝜏, which implies 𝑣⁻_1𝑚>𝑣⁻_2𝑚 and 𝑎⁻_1𝑚<𝑎⁻_2𝑚. Second, 𝑏<0. With the similar process as above, it is obtained that 𝑣⁻_1𝑚<𝑣⁻_2𝑚 and 𝑎⁻_1𝑚>𝑎⁻_2𝑚.
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∈(𝑎⁻₁(𝜏),𝜇), and (4) has at least two roots with positive real part when 𝑎∈(−∞,𝑎⁻₁(𝜏))∪(𝜇,+∞), a pair of purely imaginary roots when 𝑎=𝑎^±_𝑘𝑚(𝜏)(𝑘=1,2,𝑚∈ℤ⁺), where 𝑎⁻₁(𝜏)=𝑎⁻₁₁(𝑎⁻₂₁) 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.

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^−𝜆𝜏𝜆+𝜇−𝑎𝑓^′𝑐⁺𝑒^−𝜆𝜏−𝑏²𝜆²𝑒^−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 lim_𝜆→+∞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 Δ_{(𝑐+,𝑐+)}(𝜆,𝜏)=𝜆+𝜇−𝑎𝑓^′𝑐⁺𝑒^−𝜆𝜏2−𝑏²𝜆²𝑒^−2𝜆𝜏=0. (12) In particular, Δ_{(𝑐+,𝑐+)}(𝜆,0)=𝜆+𝜇−𝑎𝑓^′𝑐⁺²−𝑏²𝜆²=0, (13) which implies 𝜆_1,2=−𝜇+𝑎𝑓^′𝑐⁺1±𝑏<0. (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⁺sin𝜂𝜏+𝑏𝜂cos𝜂𝜏=−𝜂,𝑎𝑓^′𝑐⁺cos𝜂𝜏−𝑏𝜂sin𝜂𝜏=𝜇 (15) or 𝑎𝑓^′𝑐⁺sin𝜂𝜏−𝑏𝜂cos𝜂𝜏=−𝜂,𝑎𝑓^′𝑐⁺cos𝜂𝜏+𝑏𝜂sin𝜂𝜏=𝜇. (16) This leads to 𝜂²=𝑎𝑓^′𝑐⁺²−𝜇²1−𝑏²<0. (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∈(𝑎⁻₁(𝜏),𝜇) and unstable when 𝑎∈(−∞,𝑎⁻₁(𝜏))∪(𝜇,+∞).
(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) when 𝑎=𝑎^±_𝑘𝑚(𝜏)(𝑘=1,2,𝑚∈ℤ⁺).

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 𝜔sin𝜔𝜏=−𝑎±𝑏𝜇𝜇²+𝜔²,cos𝜔𝜏=𝑎𝜇∓𝑏𝜔²𝜇²+𝜔². (18) This leads to 𝜔²=(𝑎²−𝜇²)/(1−𝑏²), 𝜔=₀^def(𝑎²−𝜇²)/(1−𝑏²) makes sense when |𝑎|>𝜇. Define𝜏_1𝑗=1𝜔₀arccos𝑎𝜇−𝑏𝜔²₀𝜇²+𝜔²₀1+2𝑗𝜋,𝑎<−𝜇,𝜔₀−arccos𝑎𝜇−𝑏𝜔²₀𝜇²+𝜔²₀𝜋𝜏+2𝑗+1,𝑎>𝜇,𝑗=0,1,2,…,_2𝑗=1𝜔₀arccos𝑎𝜇+𝑏𝜔²₀𝜇²+𝜔²₀1+2𝑗𝜋,𝑎<−𝜇,𝜔₀−arccos𝑎𝜇+𝑏𝜔²₀𝜇²+𝜔²₀+2𝑗+1𝜋,𝑎>𝜇,𝑗=0,1,2,…. (19)Then 𝑖𝜔₀ is a purely imaginary root of (4) with 𝜏=𝜏_𝑘𝑗,𝑘=1,2 defined by (19).
Let 𝜆(𝜏)=𝛼(𝜏)+𝑖𝜚(𝜏) be the root of (4), satisfying 𝛼𝜏_𝑘𝑗=0,𝜚𝜏_𝑘𝑗=𝜔₀. (20) Differentiating both side of (4) gives that Re𝑑𝜆𝑑𝜏⁻¹_{𝜏=𝜏𝑘𝑗}=1−𝑏²𝑎²+𝑏²𝜔²₀>0. (21) This implies that 𝛼^′(𝜏_𝑘𝑗)>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=𝜏_𝑘𝑗,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.

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=𝜏_𝑘𝑗,𝑘=1,2,𝑗=0,1,2,….

3. Properties of Hopf BifurCation

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 𝑦_𝑖(𝑡)=𝑥_𝑖(𝜏𝑡),𝑖=1,2, and 𝑌(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡))^𝑇, then (1) becomes 𝑑[𝑌(𝑡)((𝑡)(𝑌(,𝑑𝑡−𝐶𝑌𝑡−1)]=𝜏𝐴𝑌+𝜏𝐵𝐹𝑡−1)) (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 𝑧+𝜇𝜏−𝑎𝜏𝑒^−𝑧2−𝑏²𝑧²𝑒^−2𝑧=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.

Lemma 6. Assume |𝑎|>𝜇.(i) If 𝜏=𝜏_𝑗,𝑗=0,1,2,…, then (23) has a pair of purely imaginary roots ±𝑖𝜔₀𝜏_𝑗, where 𝜏_𝑗=𝜏_𝑘𝑗 and 𝜔₀ are defined by (19).(ii) Let 𝑧(𝜏)=𝛾(𝜏)+𝑖𝜍(𝜏) be the root of (23), satisfying 𝛾𝜏_𝑗=0,𝜍𝜏_𝑗=𝜔₀𝜏_𝑗, (24)then 𝛾^′𝜏_𝑗=𝜏_𝑗𝛼^′𝜏_𝑗>0,𝑗=0,1,2,…. (25)(iii) Equation (23) has at least a pair of roots with positive real parts when 𝜏=𝜏_𝑗 for 𝑗≥1, and it has two positive roots when 𝑎>𝜇 and 𝜏=𝜏₀. All roots of (23) with 𝜏=𝜏₀, except ±𝑖𝜔₀𝜏₀, have negative real parts when 𝑎<−𝜇.

For convenience, denote 𝜏=𝜏_𝑗+𝜈. Then we know that (22) undergoes a Hopf bifurcation at origin when 𝜈=0. For 𝜙∈𝐶([−1,0],ℝ²), let 𝐷𝜙(0)(),𝐿𝜏=𝜙−𝐶𝜙−1𝜈,𝜙=𝐴_𝑗+𝜈𝜙(0)+𝐵𝜏_𝑗𝜙(),𝐹=𝑓+𝜈−1𝜈,𝜙^′′′(0)6𝜏_𝑗+𝜈𝐵𝜙³(−1)+⋯. (26) By the Riesz representation theorem, there exist functions 𝜂(𝜃) and 𝜇(𝜃) such that 𝐷𝜙=𝜙(0)−⁰₋₁(𝜃)𝜙(𝜃),𝐿=𝑑𝜇𝜈,𝜙⁰₋₁𝑑𝜂(𝜃)𝜙(𝜃). (27) In fact, we can choose 𝜇(𝜃)=(],𝜂(𝜃)=𝜏−𝐶,𝜃=−1,0,𝜃∈−1,0−𝐵_𝑗(),𝐴𝜏+𝜈,𝜃=−1,0,𝜃∈−1,0_𝑗+𝜈,𝜃=0. (28)

Define 𝐴(𝜈)(𝜃)𝜙=𝑑𝜙[),𝜙𝑑𝜃,𝜃∈−1,0_′(0)−𝐷𝜙^′(0)𝑅(𝜈)[),𝐹+𝐿𝜙,𝜃=0,𝜙=0,𝜃∈−1,0𝜈,𝜙,𝜃=0. (29) Then (22) can be written as 𝑌̇_𝑡=𝐴(𝜈)𝑌_𝑡+𝑅(𝜈)𝑌_𝑡. (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 lim_𝜃→0𝜙(𝜃) exists}.

The adjoint operator 𝐴^∗𝜓=−𝑑𝜓/𝑑𝑠 with domain 𝐷𝐴^∗=𝜓∈𝐶=^′def𝐶[0,1],ℝ²∶𝑑𝜓𝑑𝑠∈𝐶^′;𝐷𝑑𝜓.𝑑𝑠=−𝐿𝜓 (31) For (𝜓,𝜙)∈𝐶^′×𝐶, define a bilinear form:𝜓,𝜙=𝜓(0)𝜙(0)−⁰₋₁𝑑^𝜃_𝛼=0𝜓()(𝛼𝜙(𝜃)−𝜃−𝛼𝑑𝜇)_{0−1𝑠𝜃=0}𝜓(𝜃−𝑠)𝑑𝜂(𝑠)𝜙(𝜃)𝑑𝜃. (32)

It is not difficult to verify that 𝑞(𝜃)=𝑞₀𝑒^{𝑖𝜔0𝜏𝑗𝜃}(𝜃∈[−1,0]) and 𝑞^∗(𝑠)=𝑙𝑞^∗₀𝑒^{𝑖𝜔0𝜏𝑗𝑠}(𝑠∈[0,1]) are the eigenvectors of 𝐴(0) and 𝐴^∗ corresponding to the eigenvalues 𝑖𝜔₀𝜏_𝑗 and −𝑖𝜔₀𝜏_𝑗, respectively, where 𝑞₀=(1,−1)^𝑇𝜏_𝑗=𝜏_1𝑗,(1,1)^𝑇𝜏_𝑗=𝜏_2𝑗,𝑞^∗₀=(1,−1)𝜏_𝑗=𝜏_1𝑗,(1,1)𝜏_𝑗=𝜏_2𝑗,1𝑙=𝑞^∗₀𝐼−𝐶𝑒^{−𝑖𝜔0𝜏𝑗}+𝐵𝜏_𝑗𝑒^{−𝑖𝜔0𝜏𝑗}𝑞₀, (33) and (𝑞^∗,𝑞)=1.

Now we compute the center manifold 𝐶₀ at 𝜈=0. Define 𝑧(𝑡)=𝑞^∗,𝑌_𝑡,𝑊(𝑡,𝜃)=𝑌_𝑡(𝜃)−2Re𝑧(𝑡)𝑞(𝜃), (34) then we have 𝑧(̇𝑡)=𝑖𝜔₀𝜏_𝑗𝑧+𝑞^∗(0)𝐹0,𝑌_𝑡. (35) Equation (35) can be written in the abbreviated form as 𝑧(̇𝑡)=𝑖𝜔₀𝜏_𝑗𝑧+𝑔(𝑧,𝑧), (36) with 𝑔(𝑧,𝑧)=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2+⋯. (37) Noting that 𝑌_𝑡(𝜃)=𝑊(𝑡,𝜃)+𝑧(𝑡)𝑞(𝜃)+𝑧(𝑡)𝑞(𝜃), we have𝑌_𝑡()=𝑞−1⁰¹𝑒^{−𝑖𝜔0𝜏𝑗}𝑧+𝑞₀₁𝑒^{𝑖𝜔0𝜏𝑗}𝑧+𝑊⁽¹⁾₂₀()𝑧−1²2+𝑊⁽¹⁾₁₁(−1)𝑧𝑧+𝑊⁽¹⁾₀₂(−1)𝑧²2𝑞+⋯₀₂𝑒^{−𝑖𝜔0𝜏𝑗}𝑧+𝑞₀₂𝑒^{𝑖𝜔0𝜏𝑗}𝑧+𝑊⁽²⁾₂₀()𝑧−1²2+𝑊⁽²⁾₁₁(−1)𝑧𝑧+𝑊⁽²⁾₀₂(−1)𝑧²2+⋯, (38)where 𝑞₀=(𝑞₀₁,𝑞₀₂)^𝑇. Therefore, from (26), we have 𝐹0,𝑌_𝑡=𝑓^′′′(0)6𝜏_𝑗𝑌³_𝑡(−1)+⋯_def=𝐹_𝑧2𝑧²2+𝐹_𝑧𝑧𝑧𝑧+𝐹_𝑧²𝑧²2+𝐹_𝑧2𝑧𝑧²𝑧2+⋯. (39) Substituting the expression of 𝐹(0,𝑌_𝑡) into (35) and comparing its coefficients with that of (36) give that 𝑔₂₀=𝑔₀₂=𝑔₁₁𝑔=0,₂₁=𝑙𝑞^∗₀𝑓^′′′(0)6𝜏_𝑗𝐵𝑞²₀₁𝑞₀₁𝑞²₀₂𝑞₀₂𝑒^{−𝑖𝜔0𝜏𝑗}. (40) It is well known that the coefficient 𝑐₁(0) of third degree term of Poincaré normal form of (35) is given by (see [12]) 𝑐₁(0)=𝑖2𝜔₀𝜏_𝑗𝑔₂₀𝑔₁₁−2𝑔₁₁²−13𝑔₀₂²+𝑔₂₁2. (41) Consequently, when 𝜏=𝜏_1𝑗 or 𝜏=𝜏_2𝑗, 𝑐₁(0)=𝜏_𝑘𝑗121+∓𝑏+𝑎𝜏_𝑘𝑗𝑒^{−𝑖𝜔0𝜏𝑘𝑗2}×𝑎𝑓^′′′(0)1+∓𝑏+𝑎𝜏_𝑘𝑗𝑒^{−𝑖𝜔0𝜏𝑘𝑗}𝑒^{−𝑖𝜔0𝜏𝑘𝑗}. (42) Hence we obtain SignRe𝑐₁(0)=Sign𝑎𝑓^′′′(0)×𝑎𝜇∓𝑏𝜔²₀𝜇²+𝜔²₀+∓𝑏+𝑎𝜏_𝑘𝑗2𝑎𝜇∓𝑏𝜔²₀𝜇²+𝜔²₀²−1. (43) Summarizing the above analysis, we have the following theorem.

Theorem 7. For (22), the direction of Hopf bifurcation at 𝜏=𝜏_𝑘𝑗,𝑘=1,2,𝑗=0,1,2,…, is supercritical (subcritical), and the bifurcating periodic solutions are asymptotically stable (unstable) when  Re𝑐₁(0)<0 (Re𝑐₁(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,𝜏_𝑘𝑗),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 𝑦̇₁(𝑡)−𝑏𝑦̇₂(𝑡−1)=−𝜇𝜏𝑦₁(𝑡)+𝑎𝜏𝑓𝑦₁(),𝑦̇𝑡−1₂(𝑡)−𝑏𝑦̇₁(𝑡−1)=−𝜇𝜏𝑦₂(𝑡)+𝑎𝜏𝑓𝑦₂(𝑡−1). (44) We have known that (44) undergoes a local Hopf bifurcation at the origin when 𝜏=𝜏_𝑘𝑗,𝑘=1,2,𝑗=0,1,2,…. Now we 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: 𝑋=𝐶([−1,0],ℝ²), Σ=𝐶𝑙{(𝑌,𝜏,𝑇)∶𝑌 is a 𝑇-periodic solution of (44)}⊂𝑋×ℝ₊×ℝ₊, 𝑁={(𝑌,𝜏,𝑇)∶𝜇𝑌=𝑎𝐹(𝑌)}.

Denote 𝐶(𝟎,𝜏_𝑗,2𝜋/𝜔₀𝜏_𝑗) the connected component of (𝟎,𝜏_𝑗,2𝜋/𝜔₀𝜏_𝑗) in Σ, where 𝜔₀=(𝑎²−𝜇²)/(1−𝑏²) and 𝜏_𝑗(𝑗=0,1,2,…) are defined by Lemma 6.

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)(𝑑𝑡−𝑏𝑢𝑡−1)]=0. (45) Therefore, there exists 𝑐₁∈ℝ such that 𝑢(𝑡)=𝑐₁+𝑏𝑢(𝑡−1). (46) Which implies that when 𝑡∈[𝑛,𝑛+1], we have 𝑢(𝑡)=𝑐₁^𝑛−1_𝑖=0𝑏^𝑖+𝑏^𝑛𝑢(𝑡−𝑛). (47) Hence we obtain lim_𝑡→+∞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 lim_𝑡→+∞𝑣(𝑡)=𝑐₂/(1+𝑏).
Then the conclusion follows from lim_𝑡→+∞𝑦₁(𝑡)=(1/2)[𝑐₁/(1−𝑏)+𝑐₂/(1+𝑏)], and the proof is complete.

Lemma 9. If |𝑏|<1/2, then all periodic solution of (44) are uniformly bounded.

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 𝑢𝑡₁−𝑏𝑢𝑡₁−1=max_𝑡∈ℝ[𝑢(𝑡)(,𝑢𝑡−𝑏𝑢𝑡−1)]₂−𝑏𝑢𝑡₂−1=min_𝑡∈ℝ[𝑢(𝑡)−𝑏𝑢(𝑡−1)], (48) together with 𝑇⁰ and 𝑇₀ such that 𝑢𝑇⁰=max_𝑡∈ℝ+𝑢(𝑡),𝑢𝑇₀=min_𝑡∈ℝ+𝑢(𝑡), (49)
Now we take the case −1/2<𝑏<0 as an example, and it has the same result when 0⩽𝑏<1/2.
By (44), (48), and (H1), we have that 𝑢𝑡_𝑖=𝑎𝑢²_𝑗=1𝑓𝑦_𝑗𝑡_𝑖∈−2|𝑎|𝐿−1𝜇,2|𝑎|𝐿𝜇,𝑢𝑡𝑖=1,2,₂−𝑏𝑢𝑡₂(𝑡)()𝑡−1⩽𝑢−𝑏𝑢𝑡−1⩽𝑢₁−𝑏𝑢𝑡₁−1. (50) On one hand, we have 𝑢(𝑡)⩽𝑏𝑢(𝑡−1)+𝑢𝑡₁−𝑏𝑢𝑡₁−1⩽𝑏²𝑢(𝑡−2)+𝑏𝑢𝑡₂−𝑏𝑢𝑡₂+𝑢𝑡−1₁−𝑏𝑢𝑡₁−1⩽𝑏^2𝑚𝑢(𝑡−2𝑚)+^𝑚_𝑖=1𝑏^2𝑖−1𝑢𝑡₂−𝑏𝑢𝑡₂+−1_𝑚𝑖=1𝑏^2(𝑖−1)𝑢𝑡₁−𝑏𝑢𝑡₁⩽𝑏−11−𝑏²𝑢𝑡₂−𝑏𝑢𝑡₂+1−11−𝑏²𝑢𝑡₁−𝑏𝑢𝑡₁⩽2|𝑎|𝐿−1()𝜇−𝑏1+𝑏1−²𝑏²𝑢𝑇₀−𝑏1−𝑏²𝑢𝑇⁰. (51) And on the other hand, we can obtain 𝑢(𝑡)2|𝑎|𝐿⩾−()𝜇−𝑏1+𝑏1−𝑏²𝑢𝑇₀−𝑏²1−𝑏²𝑢𝑇⁰. (52) It follows that 𝑢𝑇₀⩾1−𝑏²1+𝑏−𝑏²−2|𝑎|𝐿()𝜇−𝑏1+𝑏1−²𝑏²𝑢𝑇⁰,𝑢𝑇⁰⩽2|𝑎|𝐿()𝜇.1+2𝑏 (53) Similarly, one can prove that 𝑢𝑇₀2|𝑎|𝐿⩾−()𝜇.1+2𝑏 (54) Which implies that −2|𝑎|𝐿()𝜇(𝑡)⩽2|𝑎|𝐿1+2𝑏⩽𝑢()𝜇.1+2𝑏 (55)
Let 𝑣(𝑡)=𝑦₁(𝑡)−𝑦₂(𝑡). In the same way, it can be verified that −2|𝑎|𝐿()𝜇(𝑡)⩽2|𝑎|𝐿1+2𝑏⩽𝑣()𝜇.1+2𝑏 (56)
Then 𝑦₁(𝑡) and 𝑦₂(𝑡) are uniformly bounded, and the proof is complete.

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)+𝑎𝜏𝑓𝑢₁(𝑡),𝑢̇₂(𝑡)−𝑏𝑢̇₁(𝑡)=−𝜇𝜏𝑢₂(𝑡)+𝑎𝜏𝑓𝑢₂(𝑡), (57) where ⋅ denotes 𝑑/𝑑𝑡. Denote 𝐺=𝑢∈ℝ²∶𝑢_𝑗∈−2|𝑎|𝐿()𝜇,2|𝑎|𝐿1+2𝑏()𝜇.1+2𝑏,𝑗=1,2 (58) Lemma 9 shows that the periodic solution of (44) belong to the region 𝐺. Clearly, (57) is equivalent to 𝑢̇₁=𝜏1−𝑏²−𝜇𝑢₁+𝑏𝑢₂+𝑎𝑓𝑢₁+𝑏𝑓_2def𝑢=𝑃𝑢₁,𝑢₂,𝑢̇₂=𝜏1−𝑏²−𝜇𝑢₂+𝑏𝑢₁+𝑎𝑓𝑢₂+𝑏𝑓_1def𝑢=𝑄𝑢₁,𝑢₂, (59) Then 𝜕𝑃𝜕𝑢₁+𝜕𝑄𝜕𝑢₂=𝜏1−𝑏²−2𝜇+𝑎𝑓^′𝑢₁+𝑓^′𝑢₂. (60) 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>𝜏_𝑗(𝑗⩾1).

Proof. First note that the stationary points (𝟎,𝜏_𝑗,2𝜋/𝜔₀𝜏_𝑗) of (44) are nonsingular and isolate centers (see [14]) under the assumption |𝑎|>𝜇 for each 𝜏>𝜏_𝑗(𝑗⩾1). By (25), there exist 𝜖>0, 𝛿>0, and a smooth curve 𝜆∶(𝜏_𝑗−𝛿,𝜏_𝑗+𝛿)→ℂ, such that Δ(𝜆(𝜏))=Δ_{(𝟎,𝜏,𝑇)}(𝜆(𝜏𝜆(𝜏)))=0,−𝑖𝜔₀𝜏_𝑗<𝜖, (61) for all 𝜏∈[𝜏_𝑗−𝛿,𝜏_𝑗+𝛿], where Δ is defined as (23), and 𝜆𝜏_𝑗=𝑖𝜔₀𝜏_𝑗,𝑑Re(𝜆(𝜏))𝑑𝜏_{𝜏=𝜏𝑗}>0. (62) Let Ω_𝜖=𝑢,𝑝∶0<𝑢<𝜖,𝑝−2𝜋𝜔₀𝜏_𝑗<𝜖. (63) Clearly, if 𝜏−𝜏_𝑗⩽𝛿 and (𝑢,𝑝)∈Ω_𝜖 such that Δ_{(0,𝜏,𝑇)}(𝑢+𝑖2𝜋/𝑝)=0, then 𝜏=𝜏_𝑗,  𝑢=0, and 𝑝=2𝜋/𝜔₀𝜏_𝑗. Moreover, if we put 𝐻^±0,𝜏_𝑗,2𝜋𝜔₀𝜏_𝑗𝑢,𝑝=Δ_{(0,𝜏±𝛿,𝑇)}𝑢+𝑖2𝜋𝑝, (64) at 𝑚=1, we have, from Re𝜆^′(𝜏_𝑗)>0, that the crossing number is 𝛾₁0,𝜏_𝑗,2𝜋𝜔₀𝜏_𝑗=deg_𝐵𝐻⁻0,𝜏_𝑗,2𝜋𝜔₀𝜏_𝑗,Ω_𝜖−deg_𝐵𝐻⁺0,𝜏_𝑗,2𝜋𝜔₀𝜏_𝑗,Ω_𝜖=−1. (65) Using the local Hopf bifurcation theorem for NFDE [14, Theorem 5.6], we conclude that the connected component 𝐶(𝟎,𝜏_𝑗,2𝜋/𝜔₀𝜏_𝑗) through (𝟎,𝜏_𝑗,2𝜋/𝜔₀𝜏_𝑗) in Σ is nonempty, and thus 𝐶(𝟎,𝜏_𝑗,2𝜋/𝜔₀𝜏_𝑗) 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,𝜏_𝑗,2𝜋/𝜔₀𝜏_𝑗) onto the 𝑌-space is bounded. Meanwhile, the projection of 𝐶(𝟎,𝜏_𝑗,2𝜋/𝜔₀𝜏_𝑗) onto 𝜏-space is bounded below from Lemma 8.
By the definition of 𝜏_𝑗, we know that 2𝜋<𝜔₀𝜏_𝑗<2𝑗+1𝜋,𝑗⩾1, (66) under the assumptions that |𝑎|>𝜇 and |𝑏|<1, which implies 1<𝑗+12𝜋𝜔₀𝜏_𝑗<1. (67)
Applying Lemma 10, one has 1/(𝑗+1)<𝑇<1 if (𝑌,𝜏,𝑇)∈𝐶(𝟎,𝜏_𝑗,2𝜋/𝜔₀𝜏_𝑗) for 𝑗⩾1, when |𝑏|<1/2 and either 𝑎<−𝜇 or 𝑎>𝜇/𝑀 holds. Thus in order for 𝐶(𝟎,𝜏_𝑗,2𝜋/𝜔₀𝜏_𝑗) to be unbounded, its projection onto the 𝜏-space must be unbounded. Consequently, the projection of 𝐶(𝟎,𝜏_𝑗,2𝜋/𝜔₀𝜏_𝑗) onto the 𝜏-space include [𝜏_𝑗,∞) 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>𝜏_𝑗(𝑗⩾1). The corresponding numerical simulation results are shown in Figures 1, 2, and 3.

(a)

(b)

(c)

Figure 1 

For system (1), the equilibrium (0,0) is asymptotically stable when 𝜏=0.55<𝜏0≈0.6378.

(a)

(b)

(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).

(a)

(b)

(c)

Figure 3 

The system (1) undergoes a Hopf bifurcation at (0,0) with sufficiently large 𝜏=50.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities, (no. DL11AB02).