Stability and Bifurcation Analysis of a Three-Dimensional Recurrent Neural Network with Time Delay

Abstract

We consider the nonlinear dynamical behavior of a three-dimensional recurrent neural network with time delay. By choosing the time delay as a bifurcation parameter, we prove that Hopf bifurcation occurs when the delay passes through a sequence of critical values. Applying the nor- mal form method and center manifold theory, we obtain some local bifurcation results and derive formulas for determining the bifurcation direction and the stability of the bifurcated periodic solution. Some numerical examples are also presented to verify the theoretical analysis.

1. Introduction

Starting with the work of Hopfield [1] on neural networks, recurrent neural networks including Hopfield neural networks, Cohen-Grossberg neural networks, and cellular neural networks have been used extensively in different areas such as signal processing, pattern recognition, optimization, and associative memories. Many researchers studied the dynamical behavior of Recurrent neural network systems, and most of papers are devoted to the stability of equilibrium, existence and stability of periodic solutions, bifurcation, and chaos [2–5]. In [5], Ruiz et al. considered a particular configuration of a recurrent neural network, illustrated in Figure 1. In Figure 1, 𝑢(𝑡) is the input and 𝑦(𝑡) is the output of the network. This recurrent neural network can be described by the following system:𝑥̇₁(𝑡)=−𝑥₁(𝑡)+𝑓𝑥₂(𝑡),⋮𝑥̇_𝑛−1(𝑡)=−𝑥_𝑛−1(𝑡)+𝑢(𝑡),𝑥̇_𝑛(𝑡)=−𝑥_𝑛(𝑡)+𝑤₁𝑓𝑥₁(𝑡)+⋯+𝑤_𝑛−1𝑓𝑥_𝑛−1(𝑡),𝑦(𝑡)=𝑓𝑥_𝑛(𝑡). (1.1) Here, 𝑥(𝑡)∈𝑅^𝑛 is the state, 𝑤_𝑖∈𝑅, 𝑖=1,…,𝑛−1 are the network parameters or weights, 𝑢(𝑡) is a smooth input, and 𝑦(𝑡) is the output. The transfer function of the neurons is taken as 𝑓(⋅)=tanh(⋅). A three-node network of the form (1.1) in feedback configuration, with 𝑢(𝑡)=𝑦(𝑡), has been studied in [5]; that is,𝑥̇₁(𝑡)=−𝑥₁(𝑡)+tanh𝑥₂(𝑡),𝑥̇₂(𝑡)=−𝑥₂(𝑡)+tanh𝑥₃(𝑡),𝑥̇₃(𝑡)=−𝑥₃(𝑡)+𝑤₁tanh𝑥₁(𝑡)+𝑤₂tanh𝑥₂(𝑡). (1.2) Ruiz et al. found and analyzed the Hopf bifurcation behavior in system (1.2). In [4], Maleki et al. considered system (1.1) with a transfer function 𝑓(𝑥)=∑^∞_𝑖=1𝛼_2𝑖−1𝑥^2𝑖−1, where 𝛼_2𝑖−1>0 for 𝑖 odd and 𝛼_2𝑖−1<0 for 𝑖 even. The authors analyzed the Bogdanov-Takens bifurcation in the system.

Figure 1 

Class of recurrent neural networks.

It is well known that there exist time delays in the information processing of neurons. The delayed axonal signal transmissions in the neural network models make the dynamical behaviors become more complicated and may destabilize the stable equilibria and admit periodic oscillation, bifurcation, and chaos. Therefore, the delay is an important control parameter in living nervous system: different ranges of delays correspond to different patterns of neural activities (see, e.g., [6–11]).

In the present paper, we consider the following three-dimensional recurrent neural network model with time delay 𝑥̇₁(𝑡)=−𝑥₁(𝑡)+𝑓𝑥₂(𝑡−𝜏),𝑥̇₂(𝑡)=−𝑥₂(𝑡)+𝑓𝑥₃(𝑡−𝜏),𝑥̇₃(𝑡)=−𝑥₃(𝑡)+𝑤₁𝑓𝑥₁(𝑡−𝜏)+𝑤₂𝑓𝑥₂(𝑡−𝜏). (1.3) By choosing the time delay as a bifurcation parameter, we prove that Hopf bifurcation occurs in the neuron and study the properties of periodic solutions of this model.

The organization of this paper is as follows. In Section 2, by analyzing the characteristic equation of the linearized system of system (1.3) at the equilibrium, we discuss the stability of the equilibrium and the existence of the Hopf bifurcation occurring at the equilibrium. In Section 3, the formulae determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold are obtained by using the normal form theory and the center manifold theorem due to Hassard et al. [12]. We do some computer observations to validate our theoretical results in Section 4.

2. Stability and Existence of Hopf Bifurcation

For most of the models in the literature, including the ones in [5, 7, 8], the activation function 𝑓 is 𝑓(𝑢)=tanh(𝑐𝑢). However, we only make the following assumption on function 𝑓:(H)𝑓∈𝐶³(𝑅),𝑓(0)=0,and𝑓′(0)≠0.

Clearly, (𝑥₁,𝑥₂,𝑥₃)^𝑇=(0,0,0)^𝑇 is equilibrium of system (1.3). Linearization of (1.3) at the zero equilibrium yields 𝑥̇₁(𝑡)=−𝑥₁(𝑡)+𝑓^′(0)𝑥₂(𝑡−𝜏),𝑥̇₂(𝑡)=−𝑥₂(𝑡)+𝑓^′(0)𝑥₃(𝑡−𝜏),𝑥₃̇(𝑡)=−𝑥₃(𝑡)+𝑤₁𝑓^′(0)𝑥₁(𝑡−𝜏)+𝑤₂𝑓^′(0)𝑥₂(𝑡−𝜏), (2.1) whose characteristic equation is det𝜆+1−𝑓^′(0)𝑒^−𝜆𝜏00𝜆+1−𝑓^′(0)𝑒^−𝜆𝜏−𝑤₁𝑓^′(0)𝑒^−𝜆𝜏−𝑤₂𝑓^′(0)𝑒^−𝜆𝜏𝜆+1=0, (2.2) that is, (𝜆+1)³−𝑤₂𝑓^′2(0)(𝜆+1)𝑒^−2𝜆𝜏−𝑤₁𝑓^′3(0)𝑒^−3𝜆𝜏=0. (2.3) The zero equilibrium is stable if all roots of (2.3) have negative real parts and unstable if at least one root has positive real part. Therefore, in order to study the local stability of the zero equilibrium of system (2.3), we need to investigate the distribution of the roots of (2.3).

When 𝜏=0, characteristic equation (2.3) yields (𝜆+1)³−𝑤₂𝑓^′2(0)(𝜆+1)−𝑤₁𝑓^′3(0)=0. (2.4) Let 𝑦=𝜆+1, (2.4) reduces to 𝑦³−𝑤₂𝑓^′2(0)𝑦−𝑤₁𝑓^′3(0)=0. (2.5)

Denote 𝑤Δ=²¹𝑓^′6(0)4−𝑤³₂𝑓^′6(0)127,𝜀=−2+√32𝑖,𝛼=³𝑤₁𝑓^′3(0)2+√Δ,𝛽=³𝑤₁𝑓^′3(0)2−√Δ. (2.6) From Cardano formula for the third-degree algebra equation, we have the following lemma.

Lemma 2.1. (1) If Δ>0, then (2.5) has a real root 𝛼+𝛽 and a pair of conjugate complex roots −(𝛼+𝛽)/2±𝑖(√3/2)(𝛼−𝛽). Furthermore, the roots of (2.4) are given by 𝜆₁=−1+𝛼+𝛽 and 𝜆_2,3=−1−(𝛼+𝛽)/2±𝑖(√3/2)(𝛼−𝛽).
(2) If Δ=0, then (2.5) has a simple root 2𝛼 and a multiple root −𝛼 with the multiplicity of 2. Furthermore, the roots of (2.4) are given by 𝜆₁=−1+2𝛼 and 𝜆_2,3=−1−𝛼. Meanwhile, if 𝑤₁=𝑤₂=0, that is, 𝛼=0, then (2.4) has a multiple root −1 with the multiplicity of 3.
(3) If Δ<0, then (2.5) has three real roots 2Re{𝛼}, 2Re{𝛼𝜀}, and 2Re{𝛼𝜀}. Furthermore, the roots of (2.4) are given by 𝜆₁=−1+2Re{𝛼}, 𝜆₂=−1+2Re{𝛼𝜀}, and 𝜆₃=−1+2Re{𝛼𝜀}.

Applying Lemma 2.1, we have the following lemma.

Lemma 2.2. All roots of (2.4) have negative real parts if one of the following holds.
(1)   Δ>0 and −2<𝛼+𝛽<1.
(2)   Δ=0 and −1<𝛼<1/2.
(3)   Δ<0 and max{Re{𝛼},Re{𝛼𝜀},Re{𝛼𝜀}}<1/2.

Let 𝑧=(𝜆+1)𝑒^𝜆𝜏, then (2.3) becomes 𝑧³−𝑤₂𝑓^′2(0)𝑧−𝑤₁𝑓^′3(0)=0. (2.7) We notice that (2.7) and (2.5) have the same coefficients.

Denote the three roots of (2.7) by 𝑧_𝑛=𝑅_𝑛+𝑖𝐼_𝑛(𝑛=1,2,3). Hence, (2.3) is equivalent to (𝜆+1)𝑒^𝜆𝜏=𝑧_𝑛(𝑛=1,2,3). (2.8)

Clearly, 𝑖𝜔(𝜔>0) is a root of (2.3) if and only if 𝜔 satisfies cos𝜔𝜏−𝜔sin𝜔𝜏=𝑅_𝑛,sin𝜔𝜏+𝜔cos𝜔𝜏=𝐼_𝑛, (2.9) which implies that 𝑅cos𝜔𝜏=^𝑛+𝜔𝐼_𝑛1+𝜔²𝐼,sin𝜔𝜏=^𝑛−𝜔𝑅_𝑛1+𝜔²(𝑛=1,2,3). (2.10) This yields 𝜔²=𝑅²_𝑛+𝐼²_𝑛−1(𝑛=1,2,3). (2.11) Obviously, we have the following lemma.

Lemma 2.3. Equation (2.11) is meaningless when |𝑧_𝑛|=𝑅²_𝑛+𝐼²_𝑛≤1(𝑛=1,2,3). If (2.7) has a root, denoted by 𝑧_𝑗, satisfying |𝑧_𝑛|>1, then (2.11) has a positive root given by 𝜔_𝑗=𝑅²_𝑛+𝐼²_𝑛−1. (2.12)

Summarizing the discussion above, we have the following.

Lemma 2.4. If Δ≥0, then (2.11) has at most two positive roots. If Δ<0, then (2.11) has at most three positive roots.

Without loss of generality, one assumes that (2.11) has three positive roots 𝜔_𝑛(𝑛=1,2,3).

Define 𝜏_𝑗^(𝑛)=1𝜔_𝑛𝑅arccos^𝑛+𝜔_𝑛𝐼_𝑛1+𝜔²_𝑛+2𝑗𝜋,𝑛=1,2,3,𝑗=0,1,2,…. (2.13) Then, ±𝑖𝜔_𝑛 is a pair of purely imaginary roots of (2.3) with 𝜏=𝜏_𝑗^(𝑛).

Let 𝜆(𝜏)=𝜎(𝜏)+𝑖𝜔(𝜏) be the root of (2.3) near 𝜏=𝜏^𝑛_𝑗 satisfying 𝜎𝜏^𝑛_𝑗=0,𝜔𝜏^𝑛_𝑗=𝜔_𝑛𝑛=1,2,3,𝑗=0,1,2,…. (2.14) We have the following.

Lemma 2.5. One has𝑑𝜎(𝜏)𝑑𝜏_{𝜏=𝜏𝑗}^(𝑛)>0,𝑛=1,2,3,𝑗=0,1,2,3,…. (2.15)Proof. Differentiating both sides of (2.8) with respect to 𝜏, we have 𝑑𝜆𝑑𝜏⁻¹1=−𝜆()−𝜏𝜆+1𝜆. (2.16) Note that 𝜆(𝜏_𝑗^(𝑛))=𝑖𝜔_𝑛, therefore 𝑑()signRe𝜆𝑑𝜏_{𝜏=𝜏𝑗}^(𝑛)=signRe𝑑𝜆𝑑𝜏⁻¹_{𝜆=𝑖𝜔0}=signRe−1𝑖𝜔_𝑛𝑖𝜔_𝑛1+1=sign𝜔².+1 (2.17) We have 𝑑𝜎(𝜏)𝑑𝜏_{𝜏=𝜏𝑗}^(𝑛)=𝑑(Re𝜆)𝑑𝜏_{𝜏=𝜏𝑗}^(𝑛)>0. (2.18) This completes the proof.

For convenience, we let ∪³_𝑛=1{𝜏_𝑗^(𝑛)}^+∞_𝑗=0={𝜏_𝑖}^+∞_𝑖=0, such that 𝜏₀<𝜏₁<𝜏₂<⋯<𝜏_𝑖<⋯, (2.19) where 𝜏₀=min𝜏₁⁽⁰⁾,𝜏₂⁽⁰⁾,𝜏₃⁽⁰⁾ (2.20) and 𝜏_𝑗^(𝑛) is defined in (2.13).

Applying Lemmas 2.1–2.5 and Corollary 2.4 of Ruan and Wei [13], we have the following results.

Lemma 2.6. All roots of (2.3) have negative real parts if one of the following holds:(H₁)Δ>0, −1<𝛼+𝛽<1 and (1/4)(𝛼+𝛽)²+(3/4)(𝛼−𝛽)²<1,(H₂)Δ=0 and −1<𝛼<1/2,(H₃)Δ<0 and max{Re{𝛼},Re{𝛼𝜀},Re{𝛼𝜀}}<1/2.

Lemma 2.7. Suppose that one of the following hypothesis is satisfied:(H₄)Δ>0,−2<𝛼+𝛽<−1,(H₅)Δ>0,−1<𝛼+𝛽<1 and (1/4)(𝛼+𝛽)²+(3/4)(𝛼−𝛽)²>1.Then, there exists a sequence values of 𝜏 defined by (2.19) such that all roots of (2.3) have negative real parts for all 𝜏∈[0,𝜏₀), and (2.3) has at least one root with positive real part when 𝜏>𝜏₀, and (2.3) exactly has a pair of purely imaginary roots ±𝑖𝜔_𝑛(𝑛=1,2,3) when 𝜏=𝜏_𝑗^(𝑛)(𝑛=1,2,3;𝑗=0,1,2,3,…), where 𝜔_𝑛 and 𝜏=𝜏_𝑗^(𝑛) are defined by (2.12) and (2.13), respectively.

From Lemmas 2.5–2.7 and the Hopf bifurcation theorem for functional differential equations in [14], we have the theorem.

Theorem 2.8. (1) If one of the hypothesis (𝐻₁), (𝐻₂), (𝐻₃) is satisfied, then the zero solution of system (1.3) is asymptotically stable for all 𝜏≥0.
(2) If (𝐻₄) or (𝐻₅) is satisfied, then the zero solution of system (1.3) is asymptotically stable for 𝜏∈[0,𝜏₀) and unstable for 𝜏>𝜏₀, and system (1.3) undergoes a Hopf bifurcation at the origin when 𝜏=𝜏_𝑗(𝑗=0,1,2,3,…).

3. Direction of Hopf Bifurcations and Stability of the Bifurcating Periodic Orbits

In this section, we will study the direction of the Hopf bifurcation and stability of bifurcating periodic solutions by using the normal theory and the center manifold theorem due to Hassard et al. [12].

Let 𝑢₁(𝑡)=𝑥₁(𝜏𝑡), 𝑢₂(𝑡)=𝑥₂(𝜏𝑡), 𝑢₃(𝑡)=𝑥₃(𝜏𝑡), then system (1.3) becomes functional differential equation in 𝒞=𝒞([−1,0],ℝ³) as𝑢̇₁(𝑡)𝑢̇₂(𝑡)𝑢̇₃(𝑡)=𝜏𝐵₁𝑢₁(𝑡)𝑢₂(𝑡)𝑢₃(𝑡)+𝜏𝐵₂𝑢₁()𝑢𝑡−1₂()𝑢𝑡−1₃()𝑓𝑡−1+𝜏^′′(0)𝑢2!²²()+𝑓𝑡−1^′′′(0)𝑢3!³²()𝑓𝑡−1+⋯_′′(0)𝑢2!²³()+𝑓𝑡−1^′′′(0)𝑢3!³³()𝑤𝑡−1+⋯₁𝑓^′′(0)𝑢2!²¹()+𝑤𝑡−1¹𝑓^′′′(0)𝑢3!³¹()+𝑤𝑡−1²𝑓^′′(0)𝑢2!²²()+𝑤𝑡−1²𝑓^′′′(0)𝑢3!³²(),𝑡−1+⋯ (3.1) where𝐵₁=−1000−1000−1,𝐵₂=0𝑓^′(0)000𝑓^′(0)𝑤₁𝑓^′(0)𝑤₂𝑓^′(0)0. (3.2)Setting 𝜏=𝜈+𝜏_𝑗, we know that 𝜈=0 is Hopf bifurcation value of system (3.1).

For 𝜙=(𝜙₁,𝜙₂,𝜙₃)^𝑇∈𝒞([−1,0],ℝ³), let 𝐿_𝜈𝜙=𝜏_𝑗+𝜈𝐵₁𝜙(0)+𝐵₂𝜙(−1). (3.3) By the Riesz representation theorem, there exists a function 𝜂(𝜃,𝜈) of bounded variation for 𝜃∈[−1,0], such that 𝐿_𝜈𝜙=⁰₋₁𝑑𝜂(𝜃,𝜈)𝜙(𝜃)for𝜙∈𝒞[−1,0],ℝ³. (3.4) In fact, we can choose 𝜂(𝜃,𝜈)=𝜏_𝑗+𝜈𝐵₁𝛿(𝜃)−𝜏_𝑗+𝜈𝐵₂𝛿(𝜃+1), (3.5) where 𝛿 denotes the Dirac delta function. For 𝜙∈𝒞([−1,0],ℝ³), define 𝐴(𝜈)(𝜃)𝜙=𝑑𝜙[),𝑑𝜃,𝜃∈−1,0₀₋₁()𝜙(𝑠)𝑅(𝜈)[),𝑓𝑑𝜂𝑠,𝜈,𝜃=0,𝜙=0,𝜃∈−1,0𝜈,𝜙,𝜃=0, (3.6) where 𝑓=𝜏𝜈,𝜙_𝑗×𝑓+𝜈_′′(0)𝑢2!²²()+𝑓𝑡−1^′′′(0)𝑢3!³²()𝑓𝑡−1+⋯_′′(0)𝑢2!²³()+𝑓𝑡−1^′′′(0)𝑢3!³³()𝑤𝑡−1+⋯₁𝑓^′′(0)𝑢2!²¹()+𝑤𝑡−1¹𝑓^′′′(0)𝑢3!³¹()+𝑤𝑡−1²𝑓^′′(0)𝑢2!²²()+𝑤𝑡−1²𝑓^′′′(0)𝑢3!³²().𝑡−1+⋯ (3.7) Then system (3.1) is equivalent to 𝑢̇_𝑡=𝐴(𝜈)𝑢_𝑡+𝑅(𝜈)𝑢_𝑡, (3.8) where 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡))^𝑇, 𝑢_𝑡=𝑢(𝑡+𝜃) for 𝜃∈[−1,0].

For 𝜓∈𝒞¹([0,1],(ℝ³)^∗), define 𝐴^∗𝜓(𝑠)=−𝑑𝜓(𝑠)[),𝑑𝑠,𝑠∈−1,0₀₋₁𝑑𝜂^𝑇(𝑡,0)𝜓(−𝑡),𝑠=0, (3.9) and a bilinear inner product 𝜓(𝑠),𝜙(𝜃)=𝜓(0)𝜙(0)−⁰₋₁^𝜃_𝜉=0𝜓(𝜉−𝜃)𝑑𝜂(𝜃)𝜓(𝜉)𝑑𝜉, (3.10) where 𝜂(𝜃)=𝜂(𝜃,0). Then, 𝐴(0) and 𝐴^∗ are adjoint operators. By the discussion in Section 2, we know that ±𝑖𝜔₀𝜏_𝑗 are eigenvalues of 𝐴(0) and 𝐴^∗ corresponding to 𝑖𝜔₀𝜏_𝑗 and −𝑖𝜔₀𝜏_𝑗, respectively.

Suppose 𝑞(𝜃)=(1,𝑞₁,𝑞₂)^𝑇𝑒^{𝑖𝜔0𝜏𝑗𝜃} is the eigenvectors of 𝐴(0) corresponding to 𝑖𝜔₀𝜏_𝑗, then 𝐴(0)𝑞(𝜃)=𝑖𝜔₀𝜏_𝑗𝑞(𝜃). Then from the definition of 𝐴(0) and (3.3)–(3.5), we have 𝜏_𝑗𝐵₁𝑞(0)+𝜏_𝑗𝐵₂𝑞(−1)=𝑖𝜔₀𝜏_𝑗𝑞(0). (3.11) For 𝑞(−1)=𝑞(0)𝑒^{−𝑖𝜔0𝜏𝑗}, then we obtain 𝑞₁=𝑖𝜔₀+1𝑒^{𝑖𝜔0𝜏𝑗}𝑓^′(0),𝑞₂=𝑖𝜔₀+1²𝑒^{2𝑖𝜔0𝜏𝑗}𝑓^′2(0). (3.12) Similarly, we can obtain the eigenvector 𝑞^∗(𝑠)=𝐷(1,𝑞^∗₁,𝑞^∗₂)𝑒^{𝜔0𝜏𝑗𝑠} of 𝐴^∗ corresponding to −𝑖𝜔₀𝜏_𝑗, where 𝑞^∗₁=𝑖𝜔₀−1²𝑒^{−2𝑖𝜔0𝜏𝑗}𝑓^′2(0)𝑤₁,𝑞^∗₂=1−𝑖𝜔₀𝑒^{−𝑖𝜔0𝜏𝑗}𝑓^′(0)𝑤₁. (3.13)

In order to assure ⟨𝑞^∗(𝑠),𝑞(𝜃)⟩=1, we need to determine the value of 𝐷. By (3.10), we have𝑞^∗(𝑠)(𝜃=,𝑞)𝐷1,𝑞^∗₁,𝑞^∗₂1,𝑞₁,𝑞₂^𝑇−⁰₋₁^𝜃_𝜉=0𝐷1,𝑞^∗₁,𝑞^∗₂𝑒^{−𝑖𝜔0𝜏𝑗(𝜉−𝜃)}𝑑𝜂(𝜃)1,𝑞₁,𝑞₂^𝑇𝑒^{𝑖𝜔0𝜏𝑗𝜉}=𝑑𝜉𝐷1+𝑞₁𝑞^∗₁+𝑞₂𝑞^∗₂−⁰₋₁1,𝑞^∗₁,𝑞^∗₂𝜃𝑒^{𝑖𝜔0𝜏𝑗𝜉}𝑑𝜂(𝜃)1,𝑞₁,𝑞₂^𝑇=𝐷1+𝑞₁𝑞^∗₁+𝑞₂𝑞^∗₂+𝜏_𝑗𝑓^′(0)𝑒^{−𝑖𝜔0𝜏𝑗}𝑤₁𝑞^∗₂+𝑞₁+𝑤₂𝑞₁𝑞^∗₂+𝑞₂𝑞^∗₁=1. (3.14)Therefore, we can choose 𝐷 as 1𝐷=1+𝑞₁𝑞^∗₁+𝑞₂𝑞^∗₂+𝜏_𝑗𝑓^′(0)𝑒^{𝑖𝜔0𝜏𝑗}𝑤₁𝑞^∗₂+𝑞₁+𝑤₂𝑞₁𝑞^∗₂+𝑞₂𝑞^∗₁. (3.15)

Following the algorithms given in [12] and using similar computation process in [7], we can get that the coefficients which will be used to determine the important quantities: 𝑔₂₀=𝜏_𝑗𝑓^′′(0)𝐷𝑒^{−2𝑖𝜔0𝜏𝑗}𝑤₁𝑞^∗₂+𝑞²₁+𝑤₂𝑞^∗₂𝑞²₁+𝑞^∗₁𝑞²₂,𝑔₁₁=𝜏_𝑗𝑓^′′(0)𝐷𝑤₁𝑞^∗₂+1+𝑤₂𝑞^∗₂𝑞₁𝑞₁+𝑞^∗₁𝑞₂𝑞₂,𝑔₀₂=𝜏_𝑗𝑓^′′(0)𝐷𝑒^{2𝑖𝜔0𝜏𝑗}𝑤₁𝑞^∗₂+1+𝑤₂𝑞^∗₂𝑞²₁+𝑞^∗₁𝑞²₂,𝑔₂₁=𝜏_𝑗𝑓^′′(0)𝐷𝑤₁𝑞^∗₂𝑊⁽¹⁾₂₀(−1)𝑒^{𝑖𝜔0𝜏𝑗}+2𝑊⁽¹⁾₁₁(−1)𝑒^{−𝑖𝜔0𝜏𝑗}+1+𝑤₂𝑞^∗₂𝑊⁽²⁾₂₀(−1)𝑞₁𝑒^{𝑖𝜔0𝜏𝑗}+2𝑊⁽²⁾₁₁(−1)𝑞₁𝑒^{−𝑖𝜔0𝜏𝑗}+𝑞^∗₁𝑊⁽³⁾₂₀(−1)𝑞₂𝑒^{𝑖𝜔0𝜏𝑗}+2𝑊⁽³⁾₁₁(−1)𝑞₂𝑒^{−𝑖𝜔0𝜏𝑗}, (3.16) where 𝑊₂₀(𝜃)=𝑖𝑔₂₀𝜔₀𝜏_𝑗𝑞(0)𝑒^{𝑖𝜔0𝜏𝑗𝜃}+𝑖𝑔₀₂3𝜔₀𝜏_𝑗𝑞(0)𝑒^{−𝑖𝜔0𝜏𝑗𝜃}+𝐸₁𝑒^{2𝑖𝜔0𝜏𝑗𝜃},𝑊₁₁(𝜃)=−𝑖𝑔₁₁𝜔₀𝜏_𝑗𝑞(0)𝑒^{𝑖𝜔0𝜏𝑗𝜃}+𝑖𝑔₁₁𝜔₀𝜏_𝑗𝑞(0)𝑒^{−𝑖𝜔0𝜏𝑗𝜃}+𝐸₂, (3.17) moreover 𝐸₁, 𝐸₂ satisfy the following equations, respectively, 2𝑖𝜔₀+1−𝑓^′(0)𝑒^{−2𝑖𝜔0𝜏𝑗}002𝑖𝜔₀+1−𝑓^′(0)𝑒^{−2𝑖𝜔0𝜏𝑗}−𝑤₁𝑓^′(0)𝑒^{−2𝑖𝜔0𝜏𝑗}−𝑤₂𝑓^′(0)𝑒^{−2𝑖𝜔0𝜏𝑗}2𝑖𝜔₀𝐸+1¹=𝑓^′′(0)𝑒^{−2𝑖𝜔0𝜏𝑗}𝑞²₁𝑞²₂𝑤₁+𝑤₂𝑞²₁,1−𝑓^′(0)001−𝑓′(0)−𝑤₁𝑓^′(0)−𝑤₂𝑓^′(0)1𝐸₂=𝑓^′′(0)𝑞₁𝑞₁𝑞₂𝑞₂𝑤₁+𝑤₂𝑞₁𝑞₁. (3.18) Therefore, all 𝑔_𝑖𝑗 in (3.16) can be expressed in terms of parameters. And we can compute the following values: 𝑐₁(0)=𝑖2𝜔₀𝜏_𝑗𝑔₂₀𝑔₁₁−2𝑔₁₁²−𝑔₂₁2,𝜇₂𝑐=−Re₁(0)Re𝜆^′𝜏_𝑗,𝛽₂=2Re𝑐₁(0,𝑇₂)𝑐=−Im₁(0)+𝜇₂Im𝜆^′𝜏_𝑗𝜔₀𝜏_𝑗,𝑗=0,1,2,…, (3.19) which determine the qualities of bifurcating periodic solution in the center manifold at the critical values 𝜏_𝑗, that is, 𝜇₂ determines the directions of the Hopf bifurcation: if 𝜇₂>0(𝜇₂<0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for 𝜏>𝜏_𝑗(𝜏<𝜏_𝑗); 𝛽₂ determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if 𝛽₂<0(𝛽₂>0); 𝑇₂ determines the period of the bifurcating periodic solutions: the period increases (decreases) if 𝑇₂>0(𝑇₂<0).

4. Computer Simulation

In this section, we will confirm our theoretical analysis by numerical simulation. We give an example of system (3.1) with 𝑤₁=1,𝑤₂=−1, and 𝑓(⋅)=tanh(⋅). Then, 𝑓(0)=0 and 𝑓′(0)=1.

Equation (1.3) becomes 𝑥̇₁(𝑡)=−𝑥₁(𝑡)+tanh𝑥₂(𝑡−𝜏),𝑥̇₂(𝑡)=−𝑥₂(𝑡)+tanh𝑥₃(𝑡−𝜏),𝑥̇₃(𝑡)=−𝑥₃(𝑡)+tanh𝑥₁(𝑡−𝜏)−tanh𝑥₂(𝑡−𝜏). (4.1) From (2.6), we have Δ=0.2870,𝛼=1.0118,𝛽=−0.3295. By Lemma 2.1, we know (2.7) has roots 𝑧₁=𝛼+𝛽=0.6823 and 𝑧_2,3=−(𝛼+𝛽)/2±𝑖(√3/2)(𝛼−𝛽)=−0.3412±1.1615𝑖. Clearly, |𝑧₁|=0.6823<1, |𝑧_2,3|=1.4655>1. From (2.12), it follows that 𝜔=|𝑧_2,3|²−1=0.6823, and, from (2.13), we get 𝜏₀=1.8434. Thus, the zero equilibrium is asymptotically stable when 𝜏∈[0,𝜏₀) as is illustrated by the computer simulations (see Figures 2(a)–2(d)). When 𝜏 passes through the critical value 𝜏₀, zero equilibrium loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcates from the origins (0,0,0), which are depicted in Figures 3(a)–3(d).

(a)

(b)

(c)

(d)

Figure 2 

When 𝜏 = 1.7 < 𝜏0 = 1.8434, the zero equilibrium is asymptotically stable. Here initial value is (1, 1, 1).

(a)

(b)

(c)

(d)

Figure 3 

When 𝜏 = 1.9 > 𝜏0 = 1.8434, a periodic orbit bifurcates from the zero equilibrium. Here, initial value is (1, 1, 1).

Acknowledgment

The author is very grateful to the referees for their valuable suggestions, which help to improve the paper significantly.