Bifurcation of a Cohen-Grossberg Neural Network with Discrete Delays

Abstract

A simple Cohen-Grossberg neural network with discrete delays is investigated in this paper. The existence of local Hopf bifurcations is first considered by choosing the appropriate bifurcation parameter, and then explicit formulas are given to determine the direction of Hopf bifurcation and stability of the periodic solutions. Moreover, a set of sufficient conditions are given to guarantee the global Hopf bifurcation. Numerical simulations are given to illustrate the obtained results.

1. Introduction

In 1983, Cohen and Grossberg [1] proposed a kind of neural networks, which are now called Cohen-Grossberg neural networks. The networks have been successfully applied to signal processing, pattern recognition, optimization, and associative memories.

It is well known that the analysis of the dynamical behaviors is a necessary step for practical design of neural networks since their applications heavily depend on the dynamical behaviors. The research on dynamical behaviors of neural networks involves not only the dynamic analysis of equilibrium but also that of periodic solution, bifurcation, and chaos, especially, the periodic oscillatory behavior of the neural networks is of great interest in many applications [2, 3]. Since periodic oscillatory can arise through the Hopf bifurcation in different system with or without time delays, it is very important to discuss the Hopf bifurcation of neural networks. Up to now, to the best of the author’s knowledge, bifurcation of Hopfield neural networks has been discussed by many researchers [4–12], but only a few results on the bifurcation of Cohen-Grossberg neural networks have been obtained. Zhao discussed the bifurcation of a discrete-time Cohen-Grossberg neural network in [13] and the bifurcation of a continuous-time Cohen-Grossberg neural network with distributed delays in which kernel function is 𝛼e^−𝛼𝑠 in [14]; Liu discussed the local Hopf bifurcation of the following Cohen-Grossberg neural network with discrete delays by regarding time delay as the bifurcation parameter in [15]: 𝑥̇₁(𝑡)=−𝑎₁𝑥₁(𝑡)𝑏₁𝑥₁(𝑡)−𝑑₁𝑓₂𝑥₂𝑡−𝜏₂,𝑥̇₂(𝑡)=−𝑎₂𝑥₂(𝑡)𝑏₂𝑥₂(𝑡)−𝑑₂𝑓₁𝑥₁𝑡−𝜏₁, (1.1) where 𝑥_𝑖(𝑡) denotes the state variable associated with the 𝑖th neuron; 𝑎_𝑖(⋅) represents amplification function; 𝑓_𝑖(⋅) denotes the signal function of the 𝑖th neuron at time 𝑡; 𝑏_𝑖(⋅) is appropriately behaved function; 𝑑_𝑖(⋅) is connection weight of the neural network; discrete delay 𝜏_𝑖 corresponds to the finite speed of the axonal signal transmission at time 𝑡, 𝑖=1,2.

On the other hand, realistic modelling of neural networks inevitably depend on careful design and variation of the connection weight; in [16], Liu discussed the Neimark-Sacker bifurcation of a discrete-time version of the neural network (1.1) by regarding the connection weight as the bifurcation parameter. The objective of this paper is to discuss Hopf bifurcation of continuous-time system (1.1) by regarding the connection weight as the bifurcation parameter while time delays are fixed. The rest of this paper is organized as follows. Stability property, existence of Hopf bifurcation, and the stability of bifurcating periodic solutions for system (1.1) are obtained in Section 2. Global existence of Hopf bifurcation for system (1.1) is obtained in Section 3. An example is given in Section 4 to demonstrate the main results.

2. Stability Analysis and Existence of Local Hopf Bifurcation

Throughout this paper, we assume that (H₁)𝑏₁(0)=𝑏₂(0)=0,𝑓₁(0)=𝑓₂(0)=0;(H₂) there exist constants 𝑎_𝑖,𝑎_𝑖 such that 0<𝑎_𝑖≤𝑎_𝑖(⋅)≤𝑎_𝑖 for 𝑖=1,2;(H₃) there exist constants 𝑏_𝑖 such that 𝑏^′_𝑖(⋅)≥𝑏_𝑖>0 for 𝑖=1,2.

Let 𝑢₁(𝑡)=𝑥₁(𝑡−𝜏₁) and 𝑢₂(𝑡)=𝑥₂(𝑡); we transform system (1.1) into the following system𝑢̇₁(𝑡)=−𝑎₁𝑢₁(𝑡)𝑏₁𝑢₁(𝑡)−𝑑₁𝑓₂𝑢₂(𝑡−𝜏),𝑢̇₂(𝑡)=−𝑎₂𝑢₂(𝑡)𝑏₂𝑢₂(𝑡)−𝑑₂𝑓₁𝑢₁(𝑡), (2.1) in which 𝜏=𝜏₁+𝜏₂.

Obviously, system (2.1) has an equilibrium (0,0) under conditions (H₁) and (H₂).

The linearized system of system (2.1) at (0,0) is𝑢̇₁(𝑡)=−𝑎₁(0)𝑏^′₁(0)𝑢₁(𝑡)+𝑎₁(0)𝑑₁𝑓^′₂(0)𝑢₂(𝑡−𝜏),𝑢̇₂(𝑡)=−𝑎₂(0)𝑏^′₂(0)𝑢₂(𝑡)+𝑎₂(0)𝑑₂𝑓^′₁(0)𝑢₁(𝑡). (2.2)

The associated characteristic equation of system (2.2) is𝜆²+𝑎𝜆+𝑏𝑐−𝑑𝑒^−𝜆𝜏=0, (2.3) in which𝑎=𝑎₁(0)𝑏^′₁(0)+𝑎₂(0)𝑏^′₂(0),𝑏=𝑎₁(0)𝑎₂(0),𝑐=𝑏^′₁(0)𝑏^′₂(0),𝑑=𝑑₁𝑑₂𝑓^′₁(0)𝑓^′₂(0). (2.4)

If 𝜏=0, it is easy to know that (2.3) has two roots with negative real parts when 𝑑<𝑐. Equation (2.10) has at least one positive root when 𝑑>𝑐.

If 𝜏≠0, let us assume that 𝜆=𝑖𝜔(𝜔>0) is a root of (2.3); we have𝑏𝑐−𝜔²=𝑏𝑑cos𝜔𝜏,𝑎𝜔=−𝑏𝑑sin𝜔𝜏. (2.5) Adding up the squares of the corresponding sides of the above equation (2.5), we obtain𝜔⁴+𝑘𝜔²+𝑏²𝑐²−𝑑²=0, (2.6) where 𝑘=(𝑎₁(0)𝑏^′₁(0))²+(𝑎₂(0)𝑏^′₂(0))².

Denote 𝑧=𝜔²; then (2.6) becomes𝑧²+𝑘𝑧+𝑏²𝑐²−𝑑²=0. (2.7) Since 𝑘>0, (2.7) has no positive real roots when |𝑑|≤𝑐. Equation (2.7) has a positive real roots when |𝑑|>𝑐, which means (2.3) has a pair of purely imaginary root for every |𝑑|>𝑐.

We have from (2.5) that𝑎𝜔𝜔²−𝑏𝑐=tan𝜔𝜏. (2.8) It has roots as follows [4, 12]:𝜔⁺_𝑗∈2𝑗−1𝜋𝜏,2𝑗−1𝜋𝜏,𝜔⁻_𝑗∈2𝑗−1𝜋𝜏,4𝑗−1𝜋2𝜏,𝑗=1,2,…. (2.9) From (2.5) and the discussion of (2.7), we obtain that𝑑⁻_𝑗=−𝑎𝜔⁺_𝑗𝑏sin𝜔⁺_𝑗𝜏<−𝑏^′₁(0)𝑏^′₂(0)<0,𝑑⁺_𝑗=−𝑎𝜔⁻_𝑗𝑏sin𝜔⁻_𝑗𝜏>𝑏^′₁(0)𝑏^′₂(0)>0, (2.10) that is, (2.3) has a pair of purely imaginary roots ±𝑖𝜔₀ when 𝑑=𝑑^±_𝑗.

On the other hand, we have𝜆′(𝑑)⁻¹=2𝜆+𝑎𝑏𝑒^−𝜆𝜏+𝜏𝑑=2𝜆+𝑎𝜆²+𝑎𝜆+𝑏𝑐+𝜏𝑑, (2.11) thus we haveRe𝜆^′(𝑑)⁻¹∣_{𝑑=𝑑±𝑗}=𝑎𝑏𝑐+𝑎𝜔^±_𝑗²𝑏𝑐−𝜔^±_𝑗²+𝑎²𝜔^±_𝑗²+𝜏𝑑^±_𝑗; (2.12)which meansRe𝜆^′(𝑑)⁻¹∣_{𝑑=𝑑+𝑗}>0,Re𝜆^′(𝑑)⁻¹∣_{𝑑=𝑑−𝑗}<0, (2.13) and note that at 𝑑=𝑏₁(0)𝑏₂(0)=𝑐, (2.3) has a root 𝜆(𝑐)=0, and 𝜆^′(𝑐)⁻¹=(𝑎/𝑏+𝜏)𝑐>0 according to (2.11).

From the above discusses and Hopf bifurcation theorem in [17] for functional differential equations, we have the following results.

Theorem 2.1. Under assumptions (H₁)–(H₃), we have
(1)if 𝑑₁𝑑₂𝑓^′₁(0)𝑓^′₂(0)∈(𝑑⁻,𝑏^′₁(0)𝑏^′₂(0)), the equilibrium (0,0) of system (1.1) is asymptotically stable;(2)if 𝑑₁𝑑₂𝑓^′₁(0)𝑓^′₂(0)∉[𝑑⁻,𝑏^′₁(0)𝑏^′₂(0)], the equilibrium (0,0) of system (1.1) is unstable;(3)𝑑^±_𝑗(𝑗=0,1,2,…) are Hopf bifurcation values for system (1.1),where 𝑑^±_𝑗 is given by (2.10) and 𝑑⁻=max_𝑗≥1{𝑑⁻_𝑗}.

From discussion above, we obtain some conditions under which system (1.1) undergoes local Hopf bifurcation near 𝑑=𝑑^±_𝑗(𝑗=1,2,…). In addition, we can obtain explicit formulas for determining the properties of the Hopf bifurcation at critical 𝑑=𝑑^±_𝑗.

Based on the normal form theory and the center manifold theorem [18], similar to discussion in [15], we can compute the following quantities:𝐶₁(0)=𝑖2𝜔^∓_𝑗𝜏𝑔₂₀𝑔₁₁−2𝑔₁₁²−𝑔₂₀²3+𝑔₂₁2,𝜇₂𝐶=−Re₁(0)Re𝜆^′𝑑^±_𝑗,𝛽₂=2Re𝐶₁(0,𝑇₂)𝐶=−Im₁(0)+𝜇₂Im𝜆^′𝑑^±_𝑗𝜏𝜔^∓_𝑗, (2.14) where the explicit formulas for 𝑔_𝑖𝑗 in (2.14) can be obtained when we replace 𝜏^𝑗 and 𝜔₀ in 𝑔_𝑖𝑗 by 𝜏 and 𝜔^∓_𝑗 in [15], respectively.

It is well known that 𝜇₂ determines the direction of the Hopf bifurcation and 𝛽₂ determines the stability of the bifurcating periodic solutions [18]. Since Re{𝜆^′(𝑑⁺_𝑗)}>0, we know if 𝜇₂>0(𝜇₂<0), then the Hopf bifurcation is supercritical(subcritical), the bifurcating periodic solutions exist for 𝑑>𝑑⁺_𝑗(𝑑<𝑑⁺_𝑗), and the bifurcating periodic solutions are stable(unstable), and due to Re{𝜆^′(𝑑⁻_𝑗)}<0, we know if 𝜇₂<0(𝜇₂>0), then the Hopf bifurcation is supercritical(subcritical), the bifurcating periodic solutions exist for 𝑑<𝑑⁻_𝑗(𝑑>𝑑⁻_𝑗), and the bifurcating periodic solutions are stable(unstable). 𝑇₂ determines the period of the bifurcating periodic solutions: the period increases (decreases) if 𝑇₂>0(𝑇₂<0).

Remark 2.2. The main results above coincide with the main results in [16].

3. Global Existence of Hopf Bifurcation for the Model

In this section, we mainly prove that the local Hopf bifurcation of (1.1) can be extended for large values by applying the global Hopf bifurcation theorem in [19].(H₄) There exist constants 𝑀_𝑖>0 such that |𝑓_𝑖(⋅)|≤𝑀_𝑖 and 𝑓′(⋅)≥0, 𝑢_𝑖𝑓^′′(𝑢_𝑖)<0 for 𝑖=1,2.

Remark 3.1. Assumption (H₄) is suitable, for example, 𝑓_𝑖(𝑢_𝑖)=tanh𝑢_𝑖 and 𝑓_𝑖(𝑢_𝑖)=(1−𝑒^−𝑢𝑖)/(1+𝑒^−𝑢𝑖) satisfy (H₄).

Theorem 3.2. Under assumptions (H₁)–(H₄) and either (𝑑₁,𝑓₁(0),𝑓₂(0)) or (𝑑₂,𝑓₁(0),𝑓₂(0)) is fixed, system (1.1) has at least a periodic solution for 𝑑<𝑑⁻_𝑗.

Proof. First, we can prove that system (1.1) has a unique equilibrium (0,0) under conditions (H₁)–(H₄) when 𝑑<𝑑⁻.
The equilibria of system (1.1) satisfy the following equation according to condition (H₂): 𝑏₁𝑢₁−𝑑₁𝑓₂𝑢₂𝑏=0,₂𝑢₂−𝑑₂𝑓₁𝑢₁=0. (3.1) From the first equation of (3.1), we have 𝑢₁=𝑏⁻₁(𝑑₁𝑓₂(𝑢₂)); substitute it into the second equation of (3.1); we obtain 𝑏₂𝑢₂−𝑑₂𝑓₁𝑏⁻₁𝑑₁𝑓₂𝑢₂=0. (3.2) Denote 𝑔𝑢₂=𝑏₂𝑢₂−𝑑₂𝑓₁𝑏⁻₁𝑑₁𝑓₂𝑢₂, (3.3) and we have 𝑔^′𝑢₂=𝑏^′₂𝑢₂1−𝑑₁𝑑₂𝑓^′₁𝑏⁻₁𝑑₁𝑓₂𝑢₂𝑓^′₂𝑢₂𝑏^′₂𝑢₂𝑏^′₁𝑑₁𝑓₂𝑢₂. (3.4) When 𝑑₁𝑑₂<0, obviously, 𝑔^′(𝑢₂)>0 due to 𝑓^′_𝑖(⋅)>0 according to (H₄). When 𝑑₁𝑑₂>0, we obtain that 𝑓^′_𝑖(0)=max_{𝑢𝑖∈𝑅}𝑓^′_𝑖(𝑢_𝑖),𝑖=1,2, due to 𝑢_𝑖𝑓_𝑖^′′(𝑢_𝑖)<0 according to (H₄), combined with 𝑑=𝑑₁𝑑₂𝑓^′₁(0)𝑓^′₂(0)<𝑑⁻<0 and condition (H₃); we obtain 𝑑₁𝑑₂𝑓^′₁(𝑏⁻₁(𝑑₁𝑓₂(𝑢₂)))𝑓^′₂(𝑢₂)<𝑑⁻₁<0; consequently, 𝑔^′(𝑢₂)>0. Hence 𝑔(𝑢₂) is an increasing bijective function, since 𝑔₂(0)=0, 𝑢₂=0 is a unique solution of (3.3), and (3.1) has a unique solution (0,0), that is, (0,0) is a unique equilibrium of system (1.1) when 𝑑<𝑑⁻.
Second, let us prove the global existence of Hopf bifurcation of system (1.1). Following the work of Wu [19], we make the following definitions.
Denote the equivalent system (2.1) of system (1.1) as follows: 𝑢̇=𝐹𝑢_𝑡(𝜃),𝜏 (3.5) in which 𝑢=(𝑢₁,𝑢₂), 𝑢_𝑡(𝜃)=𝑢(𝑡+𝜃)∈𝐶([−𝜏,0],𝑅²).
And denote 𝑋=C[−𝜏,0],𝑅²,Σ=cl𝑢(𝑡),𝑑,𝑝∈𝑋×𝑅×𝑅⁺, (3.6) where 𝑢(𝑡) is a 𝑝-periodic solution of (2.1). Let ℓ(0,𝑑⁻_𝑗,2𝜋/𝜔⁺_𝑗) denote the connected component through (0,𝑑⁻_𝑗,2𝜋/𝜔⁺_𝑗) in Σ, where 𝑑⁻_𝑗 and 𝜔⁺_𝑗 are given by (2.9) and (2.10), respectively. From Theorem 2.1, we know that (0,𝑑⁻_𝑗,2𝜋/𝜔⁺_𝑗) is nonempty, and all centers are isolated center. At fact, sets of centers are 0,𝑑⁻_𝑗,2𝜋𝜔⁺_𝑗,𝑗=1,2,…. (3.7) The characteristic function Δ0,𝑑,𝑝(𝜆)=𝜆²+𝑎𝜆+𝑏𝑐−𝑑𝑒^−𝜆𝜏 (3.8) is continuous in (𝑑,𝑝,𝜆)∈𝑅×𝑅×𝑅×𝐶. Theorem 2.1 above ensures that there exist 𝜀>0, 𝜖>0 and a smooth curve 𝜆∶(𝑑⁻_𝑗−𝛿,𝑑⁻_𝑗+𝛿)→𝐶 such that Δ(𝜆(𝑑))=0,|𝜆(𝑑)−𝑖𝜔⁺_𝑗|<𝜖 for all 𝑑∈[𝑑⁻_𝑗−𝛿,𝑑⁻_𝑗+𝛿], and 𝜆(𝑑⁻_𝑗)=𝑖𝜔⁺_𝑗,Re(𝜆^′(𝑑)⁻¹)|_{𝑑=𝑑−j}<0. Define Ω_𝜀={(𝑢,𝑝)∶0<𝑢<𝜖,|𝑝−2𝜋/𝜔⁺_𝑗|<𝜀. It is not difficult to show that Δ(0,𝑑,𝑝)(𝑢+𝑖2𝜋/𝑝)=0 on [𝑑⁻_𝑗−𝛿,𝑑⁻_𝑗+𝛿]×𝜕Ω_𝜖 if and only if 𝑑=𝑑⁻_𝑗,𝑢=0,𝑝=2𝜋/𝜔⁺_𝑗.
Moreover, if we define 𝐻^±0,𝑑⁻_𝑗,2𝜋𝜔⁺_𝑗𝑢,𝑝=Δ0,𝑑⁻_𝑗±𝛿,𝑝𝑢+𝑖2𝜋𝑝, (3.9) and we can compute the crossing number of every isolated center (0,𝑑⁻_𝑗,2𝜋/(𝜔⁺_𝑗)) as follows: 𝛾0,𝑑⁻_𝑗,2𝜋𝜔⁺_𝑗=deg_𝐵𝐻⁻0,𝑑⁻_𝑗,2𝜋𝜔⁺_𝑗,Ω_𝜀−deg_𝐵𝐻⁺0,𝑑⁻_𝑗,2𝜋𝜔⁺_𝑗,Ω_𝜀=−1, (3.10) hence we can obtain from Theorem 3.3 in [19] that the connected component ℓ(0,𝑑⁻_𝑗,2𝜋/(𝜔⁺_𝑗)) in Σ is unbounded.
On the other hand, we can prove that all nontrivial periodic solutions of system (2.1) are uniformly bounded.
Let 𝑉(𝑡)=𝑢²₁+𝑢²₂, we have 𝑉̇(𝑡)=1𝑉(𝑡)−𝑎₁𝑢₁(𝑡)𝑏₁𝑢₁(𝑡)𝑢₁(𝑡)+𝑑₁𝑓₂𝑢₂(𝑡−𝜏)𝑢₁(𝑡)−𝑎₂𝑢₂(𝑡)𝑏₂𝑢₂(𝑡)𝑢₂(𝑡)+𝑑₂𝑓₁𝑢₁(𝑡)𝑢₂(𝑡)≤1𝑉(𝑡)−𝑎₁𝑏₁𝑢₁(𝑡)+𝑑₁𝑀₂𝑢₁(𝑡)−𝑎₂𝑏₂𝑢₂(𝑡)+𝑑₂𝑀₁𝑢₂(𝑡≤1)𝑉(𝑡)−𝑘₁𝑉²(𝑡)+√2𝑘₂𝑉(𝑡)≤−𝑘₁𝑉(𝑡)+√2𝑘₂, (3.11) where 𝑘₁=max{𝑎₁𝑏₁,𝑎₂𝑏₂} and 𝑘₂=max{|𝑑₁|𝑀₁,|𝑑₂|𝑀₂}. It follows that 𝑉(𝑡)≤𝑘+𝑉(0)𝑒^{−𝑘𝑡𝑡→+∞}√−−−−→𝑘,𝑘=2𝑘₂𝑘₁. (3.12) Hence, all nontrivial periodic solutions of system (2.1) are uniformly bounded.
At the same time, we can prove that the period 𝑝 of a periodic solution of system (2.1) with 𝑑<𝑑⁻ on ℓ(0,𝑑⁻_𝑗,2𝜋/𝜔⁺_𝑗) is also uniformly bounded. In fact, system has no nontrivial 𝜏-periodic solution. For a contradiction, suppose that system (2.1) has a nontrivial 𝜏-periodic solution, then the following differential equation also has a nontrivial 𝜏-periodic solution 𝑢̇₁(𝑡)=−𝑎₁𝑢₁(𝑡)𝑏₁𝑢₁(𝑡)−𝑑₁𝑓₂𝑢₂(𝑡)=𝑃𝑢₁,𝑢₂,𝑢̇₂(𝑡)=−𝑎₂𝑢₂(𝑡)𝑏₂𝑢₂(𝑡)−𝑑₂𝑓₁𝑢₁(𝑡)=𝑄𝑢₁,𝑢₂. (3.13)
On the other hand, let 𝐵𝑢₁,𝑢₂=1𝑎₁𝑢₁𝑎₂𝑢₂,𝜕(𝐵𝑃)𝜕𝑢₁+𝜕(𝐵𝑄)𝜕𝑢₂𝑏=−^′1𝑢₁𝑎₂𝑢₂+𝑏^′₂𝑢₂𝑎₁𝑢₁𝑏≤−¹𝑎₂+𝑏₂𝑎₂<0. (3.14) Thus, we conclude that system (2.6) has no 𝜏-periodic solution according to Bendixson-Dulac criterion. Consequently, system (2.1) has no 𝜏/𝑛-periodic solution for positive integer 𝑛≥1. From definition of 𝜔⁺_𝑗, we have 𝜏𝑚<𝑝<2𝜋𝜔⁺₁, (3.15) where 𝑚 is a positive integer. Therefore, the projection of ℓ(0,𝑑⁻_𝑗,2𝜋/𝜔⁺_𝑗) on 𝑝-space is bounded.
Hence, we obtain that the projection ℓ(0,𝑑⁻_𝑗,2𝜋/𝜔⁺_𝑗) on 𝑑-space must be unbounded. As a result, for any 𝑑<𝑑⁻_𝑗, system (2.1) has at least a periodic solution with a period in (𝜏/𝑚,2𝜋/𝜔⁺₁) and the proof of Theorem 3.2 is complete.

Remark 3.3. The discussions of bifurcations of system (1.1) at 𝑑=𝑏^′₁(0)𝑏^′₂(0) are a very cockamamie case. It is difficult to obtain good conditions to guarantee the global existence of periodic solutions of system (1.1), but if 𝑎₁(𝑥₁(𝑡))=𝑎₂(𝑥₂(𝑡))=1, 𝑏₁(𝑡,𝑥₁(𝑡))=𝜇₁𝑥₁(𝑡), and 𝑏₂(𝑡,𝑥₂(𝑡))=𝜇₂(𝑡)𝑥₂(𝑡), system (1.1) reduces to the following Hopfield neural network: 𝑥̇₁(𝑡)=−𝜇₁𝑥₁(𝑡)+𝑑₁𝑓₂𝑥₂𝑡−𝜏₂,𝑥̇₂(𝑡)=−𝜇₁𝑥₁(𝑡)+𝑑₂𝑓₁𝑥₁𝑡−𝜏₁. (3.16) A supercritical pitchfork of system (3.16) can occur at 𝑑=𝜇₁𝜇₂ under assumptions (H₁)–(H₄) [11], and the periodic solutions of system (2.2) are also globally existent [12]. Of course, Theorems 2.2 and Theorem 3.1 for (3.16) in [12] are special cases of Theorems 2.1 and 3.2 in this paper.

4. A Numerical Example

Consider the following Cohen-Grossberg neural network with discrete delays:𝑥̇₁(𝑡)=−6+cos𝑥₁(𝑡)𝑥₁(𝑡)−𝑑₁tanh𝑥₂(𝑡−2),𝑥̇₂(𝑡)=−42𝑥₂(𝑡)+tanh𝑥₁(𝑡−1). (4.1) Since 𝑓′(0)=1, 𝑏^′₁(0)=1, 𝑏^′₂(0)=2, 𝑎₁(0)=7, and 𝑎₂(0)=4, 𝑑₂=1, using MATLAB 7.0, we can obtain that𝑏^′₁(0)𝑏^′₂(0)=2,𝑑=𝑑₁𝑑₂𝑓^′₁(0)𝑓^′₂(0)=𝑑₁,𝜔⁺₁≈0.9618,𝑑⁻₁𝜔≈−2.0334,₊₂≈2.8952,𝑑⁻₂𝜔≈−2.3023,₊₃≈4.8522,𝑑⁻₃𝑑≈−2.8461,…,₋=𝑑⁻₁≈2.0334. (4.2)

Moreover, due to 𝑓^′′(0)=0, 𝑓^′′′(0)=2, 𝑎^′₁(0)=𝑎^′₂(0)=𝑎₂^′′(0)=0, 𝑎₁^′′(0)=1, and 𝑏^′₁(0)=𝑏^′₂(0)=0, it is easy to know that 𝑔₂₀=𝑔₁₁=𝑔₀₂=0 and𝑔₂₁=3𝐷14𝑑₁𝜌𝜌²𝑒^{−𝑖3𝜔+𝑗}+8. (4.3)

Case 1. Let 𝑑₁=−1.8. Note that 𝑑=−1.8∈(−2.0334,2), system (4.1) satisfies Theorem 2.1, and the equilibrium (0,0) of system (4.1) is asymptotically stable. Figure 1 shows the dynamic behaviors of system (4.1) with initial condition (0.1,0.2).

(a)

(b)

Figure 1 

(a) Phase plot in space (𝑥1,𝑥2) for system (4.1) with 𝑑1=−1.8. (b) Phase plot in space (𝑥1,𝑥2) for system (4.1) with 𝑑1=−1.8.

Case 2. Let 𝑑₁=−2.1 and −3.0, respectively. Note that 𝑑=−2.1,−3.0<𝑑⁻=−2.0334, the equilibrium (0,0) loses its stability, and Hopf bifurcations occurs, that is, periodic solutions bifurcate from (0,0) according to Theorem 3.2. Furthermore, we can from (4.3) obtain that 𝑔₂₁(𝑑⁻₁)=−70.7796−1.8445𝑖, 𝑔₂₁(𝑑⁻₂)=−9.7399−1.1614𝑖, 𝑔₂₁(𝑑⁻₃)=−5.6912−0.9305𝑖,…; so, 𝜇₂<0 for 𝑑⁻_𝑗,𝑗=1,2,3. Hence, the periodic solutions are stable for 𝑑₁=−2.1,−3.0. Figures 2 and 3 show the dynamic behaviors of system (4.1) with initial condition (0.1,0.2).

(a)

(b)

Figure 2 

(a) Phase plot in space (𝑥1,𝑥2) for system (4.1) with 𝑑1=−2.1. (b) Phase plot in space (𝑥1,𝑥2) for system (4.1) with 𝑑1=−2.1.

(a)

(b)

Figure 3 

(a) Phase plot in space (𝑥1,𝑥2) for system (4.1) with 𝑑1=−3.0. (b) Phase plot in space (𝑥1,𝑥2) for system (4.1) with 𝑑1=−3.0.