Bifurcation and Hybrid Control for A Simple Hopfield Neural Networks with Delays

Abstract

A detailed analysis on the Hopf bifurcation of a delayed Hopfield neural network is given. Moreover, a new hybrid control strategy is proposed, in which time-delayed state feedback and parameter perturbation are used to control the Hopf bifurcation of the model. Numerical simulation results confirm that the new hybrid controller using time delay is efficient in controlling Hopf bifurcation.

1. Introduction

It is well known that neural networks are complex and large-scale nonlinear dynamical system. In the last decade, the dynamical characteristics (including stable, unstable, oscillatory, and chaotic behavior) of Hopfield neural networks (HNNs) with time delays have become a subject of intense research activities. Many stability criteria are obtained. We refer the reader to [1–8] and the references cited therein. However, the periodic nature of neural impulses is of fundamental significance in the control of regular dynamical functions such as breathing and heart beating. Neural networks involving persistent oscillations such as limit cycle may be applied to pattern recognition and associative memory. In differential equations with delays, periodic oscillatory behavior can arise through the Hopf bifurcation. Therefore, it is also very significant to study the class of problem. Olien and Bélair [9] investigated the bifurcation of the following HNNs system: 𝑥̇₁=−𝑎₁𝑥₁(𝑡)+𝑏₁₁𝑓₁𝑥₁𝑡−𝜏₁+𝑏₁₂𝑓₂𝑥₂𝑡−𝜏₂,𝑥̇₂=−𝑎₂𝑥₂(𝑡)+𝑏₂₁𝑓₁𝑥₁𝑡−𝜏₃+𝑏₂₂𝑓₂𝑥₂𝑡−𝜏₁, (1) in which 𝜏₁=𝜏₂=𝜏₃=𝜏, 𝑎_𝑖=1 and 𝑓_𝑖(0)=0, 𝑖=1,2, and Huang et al. [10] study further the bifurcation and periodic nature of system (1) with 2𝜏₁=𝜏₂+𝜏₃, 𝑎_𝑖=1 and 𝑓_𝑖(0)=0, 𝑖=1,2. Moreover, many authors also consider discrete form of system (1); we can see [11–13].

In recent years, bifurcation control has attracted many researchers from various disciplines. The aim of bifurcation control is to design a controller to modify the bifurcation properties of a given nonlinear system, thereby to achieve some desirable dynamical behaviors. After the pioneering work initiated by Ott et al. [14], there have been many ideas and methods of bifurcation control [15–20]. However, from the control theory point of view, we may classify the current methods into two main categories: the first one is feedback control where state feedback is applied to control bifurcation or chaos, and the other is nonfeedback methods. Recently, Luo et al. [21] proposed a new control strategy for period-doubling bifurcations in a discrete nonlinear dynamical system. Moreover, Liu and Chung [22] investigated the same control strategy in a continuous dynamical system without time delays. Now, we extend this strategy to deal with bifurcation control in HNNs system (1).

In the paper, we will propose a new hybrid control strategy in which the parameter perturbation and time-delayed state feedback are combined and used to control Hopf bifurcation in system (1). Simulation results demonstrate the correctness of our theoretical analysis. The comparison shows that the control strategy is effective as it meets the purpose of retarding the occurrence of bifurcation.

2. Stability and Hopf Bifurcation of System (1) without Control

In this section, we will consider system (1) with 2𝜏₁=𝜏₂+𝜏₃≜2𝜏 and 𝑓_𝑖(0)=0, 𝑖=1,2. It is obviously that (0,0) is an equilibrium point of system (1).

To simplify, here we denote 𝑐₁₁=𝑏₁₁𝑓^′₁(0), 𝑐₁₂=𝑏₁₂𝑓^′₂(0), 𝑐₂₁=𝑏₂₁𝑓^′₁(0), 𝑐₂₂=𝑏₂₂𝑓^′₂(0), 𝑎₁=𝑎₂=𝑏. Consider the linearized system of system (1) at (0,0)𝑥̇₁=−𝑏𝑥₁(𝑡)+𝑐₁₁𝑥₁(𝑡−𝜏)+𝑐₁₂𝑥₂𝑡−𝜏₂,𝑥̇₂=−𝑏𝑥₂(𝑡)+𝑐₂₁𝑥₁𝑡−𝜏₃+𝑐₂₂𝑥₂(𝑡−𝜏). (2) The characteristic equation of the linearized system (2) is (𝜆+𝑏)²−𝑐₁₁+𝑐₂₂(𝜆+𝑏)𝑒^−𝜆𝜏+𝑐₁₁𝑐₂₂−𝑐₁₂𝑐₂₁𝑒^−2𝜆𝜏=0, (3) which determines the local stability of the equilibrium solution. Thus, we will find some conditions which ensure that all roots of (3) have negative real parts. To facilitate the calculation in this paper, we rewrite the characteristic equation (3) as follows: (𝜆+𝑏)²𝑒^2𝜆𝜏−2𝑇(𝜆+𝑏)𝑒^𝜆𝜏+𝐷=0, (4) where 𝐷=𝑐₁₁𝑐₂₂−𝑐₁₂𝑐₂₁, 𝑇=(1/2)(𝑐₁₁+𝑐₂₂). Obviously, (4) is a quadratic polynomial in the variable (𝜆+𝑏)𝑒^𝜆𝜏 and has roots given by (𝜆+𝑏)𝑒^𝜆𝜏=𝑇±𝑇²−𝐷. (5)

In the following, we distinguish two cases to discuss (5).

2.1. As 𝑇²≥𝐷

In this part, we state a result due to [23] as a lemma to analyze (5), which is, for the convenience of the reader, stated as follows.

Lemma 1. For the transcendental equation 𝜆^𝑛+𝑝₁⁽⁰⁾𝜆^𝑛−1+⋯+𝑝⁽⁰⁾_𝑛−1𝜆+𝑝_𝑛⁽⁰⁾+𝑝₁⁽¹⁾𝜆^𝑛−1+⋯+𝑝⁽¹⁾_𝑛−1𝜆+𝑝_𝑛⁽¹⁾𝑒^−𝜆𝜎1+⋯+𝑝₁^(𝑚)𝜆^𝑛−1+⋯+𝑝^(𝑚)_𝑛−1𝜆+𝑝_𝑛^(𝑚)𝑒^{−𝜆𝜎𝑚}=0, (6) where 𝜎_𝑖≥0(𝑖=1,…,𝑚) and 𝑝_𝑗^(𝑖)(𝑖=1,…,𝑚;𝑗=1,…,𝑛) are constants. As (𝜎₁,…,𝜎_𝑚) varies, the sum of the orders of the zeros of (6) in the open right half-plane can change only if a zero appears on or crosses the imaginary axis.

For convenience, we make the following assumptions: (H1)𝑏>𝑇±𝑇²−𝐷.(H2)𝑏²>(𝑇±𝑇²−𝐷)².(H3)(𝑇+𝑇²−𝐷)²<𝑏²<(𝑇−𝑇²−𝐷)².(H4)𝑏²<(𝑇±𝑇²−𝐷)².

Lemma 2. If (H1) and (H2) hold, then all roots of (3) have negative real parts for every 𝜏∈[0,+∞).

Proof. For (3), when 𝜏=0, its roots can be expressed as 𝜆_1,2=−𝑏+𝑇±𝑇²−𝐷. Clearly, all roots of (3) are negative if (H1) holds. We want to determine if the real part of some root increases to reach zero and eventually becomes positive as 𝜏≠0. We can see that 𝜆 is a root of (3) if and only if 𝜆 is a root of (5).
We write 𝜆=𝜌+𝑖𝜔 for a root of the characteristic equations (5), separate the real and imaginary parts of the ensuing equations (5), and obtain 𝑒^𝜌𝜏𝜌+𝑏cos(𝜔𝜏)−𝜔sin(𝜔𝜏)=𝑇±𝑇²𝑒−𝐷,_𝜌𝜏𝜌+𝑏sin(𝜔𝜏)+𝜔cos(𝜔𝜏)=0. (7) A change in the stability of the stationary solution can only occur when 𝜌=0, that is, 𝑏cos(𝜔𝜏)−𝜔sin(𝜔𝜏)=𝑇±𝑇²()()−𝐷,𝑏sin𝜔𝜏+𝜔cos𝜔𝜏=0. (8) By (8), we have 𝜔²=𝑇±𝑇²−𝐷²−𝑏². (9) By (9), if (H2) holds, we know that (5) has no purely imaginary roots, and then applying Lemma 1 one obtains that all roots of (3) have negative real parts. This completes the proof of lemma.

Lemma 3. For (5), one obtains the following results.(1)If (H1) and (H3) hold, then (5) have a pair of purely imaginary roots ±𝑖𝜔₋ at 𝜏=𝜏_−,𝑗.(2)If (H1) and (H4) hold, then (5) have a pair of purely imaginary roots ±𝑖𝜔₊ at 𝜏=𝜏_+,𝑗 and have another pair of purely imaginary roots ±𝑖𝜔₋ at 𝜏=𝜏_−,𝑗,where 𝜔²_±=𝑇±𝑇²−𝐷²−𝑏²,𝜏_±,𝑗=1𝜔_±−𝜔arctan^±𝑏+𝑗𝜋𝜔_±𝜏,𝑗=0,1,…;₀=min𝜏_+,0,𝜏_−,0. (10)(11)

Here one denotes ±𝜔₀ especially as a pair of purely imaginary roots of (5) at 𝜏=𝜏₀. To see if 𝜏_−,0 and 𝜏₀ are bifurcation values, one needs to verify if the transversality conditions hold. In fact, one has the following.

Lemma 4. The following transversality conditions: Re𝑑𝜆𝑑𝜏_𝜏±,𝑗>0 (12) are satisfied.

Proof. By (5), we have 𝑑𝜆𝑒𝑑𝜏^𝜆𝜏+(𝜆+𝑏)𝜏𝑒^𝜆𝜏𝑑𝜆𝑑𝜏+𝜆𝑒^𝜆𝜏=0. (13) Hence, 𝑑𝜆(()𝜏)().𝑑𝜏1+𝜆+𝑏=−𝜆𝜆+𝑏 (14) Obviously, we have 𝑑𝜆𝑑𝜏⁻¹=(1+(𝜆+𝑏)𝜏)()1−𝜆𝜆+𝑏=−𝜆()−𝜏𝜆+𝑏𝜆, (15) then signRe𝑑𝜆𝑑𝜏_𝜏±,𝑗=signRe𝑑𝜆𝑑𝜏⁻¹_𝜏±,𝑗=signRe−1𝑖𝜔(𝑖𝜔+𝑏)=signRe−1−𝜔²+𝑖𝑏𝜔=sign−1⋅−𝜔²𝜔⁴+𝑏²𝜔²1=sign𝜔²+𝑏²>0. (16) We complete the proof of Lemma 4.

From Lemmas 2–4, we can obtain the following theorem about the distribution of the characteristic roots of (3).

Theorem 5. Let 𝜏_−,0, 𝜏₀ be defined by (11).(i)If (H1) and (H2) hold, then all roots of (3) have negative real parts for all 𝜏≥0.(ii)If (H1) and (H3)((H4)) hold, then when 𝜏<𝜏_−,0(𝜏<𝜏₀), all roots of (3) have negative real parts, when 𝜏=𝜏_−,0(𝜏=𝜏₀), (3) has a pair of purely imaginary roots ±𝑖𝜔₋(±𝑖𝜔₀), and when 𝜏>𝜏_−,0(𝜏>𝜏₀), (3) has at least one root with positive real part.

By using Theorem 5, the stability and bifurcation of system (1) can be summarized as the following theorem.

Theorem 6. For system (1), let (H1) hold and let 𝜏_−,0,𝜏₀ be defined by (11).(i)If (H2) holds, then the equilibrium point (0,0) is asymptotically stable for discrete delays 𝜏≥0.(ii)If (H3)((H4)) holds, there is a critical value 𝜏=𝜏_−,0(𝜏=𝜏₀) of the discrete delay so that if 𝜏<𝜏_−,0(𝜏<𝜏₀), then the equilibrium point (0,0) is asymptotically stable; if 𝜏>𝜏_−,0(𝜏>𝜏₀), then (0,0) is unstable; Hopf bifurcation occurs when 𝜏=𝜏_−,0(𝜏=𝜏₀).

2.2. As 𝑇²<𝐷

For convenience, we have the following assumptions: (H5)𝑏>𝑇.(H6)𝑏²>𝐷.(H7)𝑏²<𝐷.

Similar to the deduction of Lemma 2, we have the following result.

Lemma 7. If (H5) and (H6) hold, then all roots of (3) have negative real parts for every 𝜏∈[0,+∞).

Proof. For (3), when 𝜏=0, its roots can be expressed as 𝜆_1,2=−𝑏+𝑇±𝑇²−𝐷. Clearly, all roots of (3) have negative real parts if (H5) holds. We want to determine if the real part of some root increases to reach zero and eventually becomes positive as 𝜏≠0. We can see that 𝜆 is a root of (3) if and only if 𝜆 is a root of (5).
We write 𝜆=𝜌+𝑖𝜔 for a root of the characteristic equation (3), separate the real and imaginary parts of the ensuing equations (5), and obtain 𝑒^𝜌𝜏()(𝑒𝜌+𝑏cos𝜔𝜏−𝜔sin𝜔𝜏)=𝑇,_𝜌𝜏𝜌+𝑏sin(𝜔𝜏)+𝜔cos(𝜔𝜏)=±𝐷−𝑇². (17) A change in the stability of the equilibrium point can only occur when 𝜌=0, that is, ()()()()𝑏cos𝜔𝜏−𝜔sin𝜔𝜏=𝑇,𝑏sin𝜔𝜏+𝜔cos𝜔𝜏=±𝐷−𝑇². (18) Hence, we have 𝜔²=𝐷−𝑏². (19) By (19), if (H6) holds, we know that (5) has no purely imaginary roots, and then applying Lemma 1 one obtains that all roots of (3) have negative real parts. This completes the proof of lemma.

Lemma 8. For (3), one obtains the following results.
If (H5) and (H7) hold, then (3) have a pair of purely imaginary roots ±𝑖𝜔 at 𝜏=𝜏_±,𝑗, where 𝜔²=𝐷−𝑏²,𝜏_±,𝑗=1𝜔arccos𝑏𝑇±𝜔𝐷−𝑇²𝑏²+𝜔²+2𝑗𝜋𝜔𝜏,𝑗=0,1,…;₀=min𝜏_+,0,𝜏_−,0. (20)(21)

According to Lemma 4, one knows that 𝜏₀ is bifurcation values.

From Lemmas 1 and 2, one can obtain the following theorem about the distribution of the characteristic roots of (3).

Theorem 9. Let 𝜏₀ be defined by (21).(i)If (H5) and (H6) hold, then all roots of (3) have negative real parts for all 𝜏≥0.(ii)If (H5) and (H7) hold, then when 𝜏<𝜏₀, all roots of (3) have negative real parts, when 𝜏=𝜏₀, (3) has a pair of purely imaginary roots ±𝑖𝜔, and when 𝜏>𝜏₀, (3) has at least one root with positive real part.

By using Theorem 9, the stability and bifurcation of system (1) can be summarized as the following theorem.

Theorem 10. For system (1), let (H5) hold and let 𝜏₀ be defined by the following: (21).(i)If (H6) holds, then the equilibrium point (0,0) is asymptotically stable for discrete delays 𝜏≥0.(ii)If (H7) holds, there is a critical value 𝜏=𝜏₀ of the discrete delay so that if 𝜏<𝜏₀, then the equilibrium point (0,0) is asymptotically stable; if 𝜏>𝜏₀, then (0,0) is unstable; Hopf bifurcation occurs when 𝜏=𝜏₀.

3. Stability Analysis and Bifurcation with Hybrid Control

In this section, we will consider system (1) with hybrid control described by the following differential equation: 𝑥̇₁=𝛼−𝑎₁𝑥₁(𝑡)+𝑏₁₁𝑓₁𝑥₁𝑡−𝜏₁+𝑏₁₂𝑓₂𝑥₂𝑡−𝜏₂+𝛽𝑥₁𝑡−𝜏₁,𝑥̇₂=𝛼−𝑎₂𝑥₂(𝑡)+𝑏₂₁𝑓₁𝑥₁𝑡−𝜏₃+𝑏₂₂𝑓₂𝑥₂𝑡−𝜏₁+𝛽𝑥₂𝑡−𝜏₁, (22) where 𝛼>0 and 𝛽∈𝑅. Obviously, (0,0) is also an equilibrium point of system (22).

Linearizing the system (22) at the equilibrium point (0,0), we obtain 𝑥̇₁=−𝛼𝑏𝑥₁(𝑡)+𝑐₁₁𝛼+𝛽𝑥₁𝑡−𝜏₁+𝑐₁₂𝛼𝑥₂𝑡−𝜏₂,𝑥̇₂=−𝛼𝑏𝑥₂(𝑡)+𝑐₂₁𝛼𝑥₁𝑡−𝜏₃+𝑐₂₂𝛼+𝛽𝑥₂𝑡−𝜏₁. (23) Then the characteristic equation for the linearized system around (0,0)^𝑇 is given by (𝜆+𝛼𝑏)²𝑒^2𝜆𝜏−𝛼𝑐₁₁+𝑐₂₂+2𝛽(𝜆+𝛼𝑏)𝑒^𝜆𝜏+𝛼𝑐₁₁+𝛽𝛼𝑐₂₂+𝛽−𝛼²𝑐₁₂𝑐₂₁=0, (24) which is a quadratic polynomial in the variable (𝜆+𝛼𝑏)𝑒^𝜆𝜏 and has roots given by (𝜆+𝛼𝑏)𝑒^𝜆𝜏=𝑇^′±𝑇^′2−𝐷^′, (25) where 𝐷^′=𝛼²𝐷+2𝛼𝛽𝑇+𝛽²,𝑇^′=𝛼𝑇+𝛽. (26) By (26), we know that (𝑇^′)²−𝐷^′=𝛼²(𝑇²−𝐷), and thus, (𝑇^′)²≥𝐷^′ ((𝑇^′)²<𝐷^′) holds if and only if 𝑇²≥𝐷 (𝑇²<𝐷) holds.

In the following, we also distinguish two cases to discuss (25).

3.1. As 𝑇²≥𝐷

Corresponding to Part I of Section 2, we make the following assumptions for convenience: (H1)^′𝑏>𝑇±𝑇²−𝐷+(𝛽/𝛼).(H2)^′𝑏²>(𝑇±𝑇²−𝐷+(𝛽/𝛼))².(H3)^′(𝑇+𝑇²−𝐷+(𝛽/𝛼))²<𝑏²<(𝑇−𝑇²−𝐷+(𝛽/𝛼))².(H4)^′𝑏²<(𝑇±𝑇²−𝐷+(𝛽/𝛼))².Denote 𝜔^′_±²=𝑇^′±𝑇^′2−𝐷^′2−𝛼²𝑏²,𝜏^′_±,𝑗=1𝜔^′_±arctan−𝛼𝑏𝜔_±+𝑗𝜋𝜔^′_±𝜏,𝑗=0,1,…,_′0=min𝜏^′_+,0,𝜏^′_−,0. (27)(28) Similarly, we can obtain the following theorem.

Theorem 11. For system (22), let (𝐻1)^′ hold and let 𝜏^′_±,0, 𝜏^′₀ be defined by (28).(i)If (𝐻2)^′ holds, then the equilibrium point (0,0) is asymptotically stable for discrete delays 𝜏≥0.(ii)If (𝐻3)^′((𝐻4)^′) holds, there is a critical value 𝜏=𝜏^′_−,0(𝜏=𝜏^′₀) of the discrete delay so that if 𝜏<𝜏^′_−,0(𝜏<𝜏^′₀), then the equilibrium point (0,0) is asymptotically stable; if 𝜏>𝜏^′_−,0(𝜏>𝜏^′₀), then (0,0) is unstable; Hopf bifurcation occurs when 𝜏=𝜏^′_−,0(𝜏=𝜏^′₀).

3.2. As 𝑇²<𝐷

Similar to deduction of Section 2.2, we have the following assumptions: (H5)^′𝑏>𝑇+(𝛽/𝛼),(H6)^′𝑏²>𝐷+(2𝛽/𝛼)𝑇+(𝛽/𝛼)²,(H7)^′𝑏²<𝐷+(2𝛽/𝛼)𝑇+(𝛽/𝛼)².

In this part, we denote 𝜔^′2=𝐷^′−𝛼²𝑏²,𝜏^′_±,𝑗=1𝜔^′arccos𝛼𝑏𝑇^′±𝜔^′𝐷^′−𝑇^′2𝛼²𝑏²+𝜔^′2+2𝑗𝜋𝜔^′,𝑗=0,1,⋯;𝜏^′₀=min𝜏^′_+,𝑗,𝜏^′_−,𝑗. (29)(30) Hence, we can obtain the following theorem.

Theorem 12. For system (22), let (𝐻5)^′ hold and let 𝜏^′₀ be defined by (30).(i)If (𝐻6)^′ holds, then the equilibrium point (0,0) is asymptotically stable for discrete delays 𝜏≥0.(ii)If (𝐻7)^′ holds, there is a critical value 𝜏=𝜏^′₀ of the discrete delay so that if 𝜏<𝜏^′₀ then the equilibrium point (0,0) is asymptotically stable; If 𝜏>𝜏^′₀, then (0,0) is unstable; Hopf bifurcation occurs when 𝜏=𝜏^′₀.

Remark 13. When 𝛼=1−𝛾 and 𝛽=𝛾 in the system (22), then we obtain the same hybrid control with [22]; however, a control model based on delayed feedback is proposed in this paper; it is well know that control theory should contain delay since any control action takes effect only after a certain delay. Hence, our hybrid control is more helpful than [22].

Remark 14. When 𝛼=1 in the system (23), then we obtain a control model only based on delayed feedback in this paper, it is clear that our hybrid control is more general than control strategy proposed by [17].

Remark 15. In [10], authors investigated the Hopf bifurcation of following HNNs with 𝛼=1 and 𝛽=0: 𝑥̇₁=𝛼−𝑥₁(𝑡)−√2−1𝑓₁𝑥₁𝑡−𝜏₁+𝑏₁₂𝑓₂𝑥₂𝑡−𝜏₂+𝛽𝑥₁𝑡−𝜏₁,𝑥̇₂=𝛼−𝑥₂(𝑡)+𝑏₂₁𝑓₁𝑥₁𝑡−𝜏₃−√2−1𝑓₂𝑥₂𝑡−𝜏₁+𝛽𝑥₂𝑡−𝜏₁, (31) By choosing 𝜏₁=3𝜋/4, 𝜏₂=𝜋/8, 𝜏₃=11𝜋/8, the authors obtained that Hopf bifurcation occurs when 𝑏₁₂𝑏₂₁=1. However, if choosing the parameters 𝛼=0.8 and 𝛽=0, by the hybrid control strategy of this paper, the Hopf bifurcation in [10] will be eliminated. We can see Figures 1 and 2.

Figure 1 

The trajectory of 𝑥1(𝑡) versus time in the system (31) with 𝜏=3𝜋/4.

Figure 2 

The trajectory of 𝑥2(𝑡) versus time in the system (31) with 𝜏=3𝜋/4.

Remark 16. It is known to all that neural networks are a special case of complex networks. Thus, it is interesting and important to further study how to expand the application of theoretical results in [24–27] and any other complex networks.

4. Examples

In this section, we give two examples to illustrate our results.

Example 1. Consider the following HNNs system with hybrid control: 𝑥̇₁=𝛼−𝑎₁𝑥₁(𝑡)+𝑏₁₁𝑓₁𝑥₁𝑡−𝜏₁+𝑏₁₂𝑓₂𝑥₂𝑡−𝜏₂+𝛽𝑥₁𝑡−𝜏₁,𝑥̇₂=𝛼−𝑎₂𝑥₂(𝑡)+𝑏₂₁𝑓₁𝑥₁𝑡−𝜏₃+𝑏₂₂𝑓₂𝑥₂𝑡−𝜏₁+𝛽𝑥₂𝑡−𝜏₁, (32) where 𝜏₁=𝜏₂=𝜏₃=𝜏, 𝑎_𝑖=0.5, 𝑏_𝑖𝑗=−0.3, and 𝑓_𝑖(𝑥)=tanh(𝑥), 𝑖,𝑗=1,2. It is obvious that (0,0) is an equilibrium point of system (32). Choosing 𝛼=1,𝛽=0, by calculation, the periodic oscillatory behavior can arise through the Hopf bifurcation as 𝜏₀=7.7063; we can see Figures 3 and 5 (𝜏=7.7063, 𝜏=17.7063). However, when 𝛼/5<2𝛽<𝛼, with complicated calculation, (H2)^′ holds; by Theorem 11(i), the equilibrium point (0,0) is asymptotically stable for any discrete delays 𝜏≥𝜏₀. For the convenience of numerical simulation, here we choose 𝛼=0.75, 𝛽=0.25, and 𝜏=17.7063 as an example. it can be seen in Figure 4 (𝜏=7.7063). Fix all coefficients of system (32) and let 𝜏 vary, and the waveforms 𝑥₁(𝑡), 𝑥₂(𝑡) without and with control are shown, respectively. Obviously, we obtain that the Hopf bifurcation in (32) without hybrid control could be eliminated by hybrid control; we can see Figure 6 (𝜏=17.7063>𝜏₀).

Figure 3 

The trajectory of 𝑥1(𝑡) and 𝑥2(𝑡) versus time in the system (32) without control (𝜏=7.7063).

Figure 4 

The trajectory of 𝑥1(𝑡) and 𝑥2(𝑡) versus time in the system (32) with control (𝜏=7.7063).

Figure 5 

The trajectory of 𝑥1(𝑡) and 𝑥2(𝑡) versus time in the system (32) without control (𝜏>7.7063).

Figure 6 

The trajectory of 𝑥1(𝑡) and 𝑥2(𝑡) versus time in the system (32) with control (𝜏>7.7063).

Example 2. Consider the following HNNs system with hybrid control: 𝑥̇₁=𝛼−𝑎₁𝑥₁(𝑡)+𝑏₁₁𝑓₁𝑥₁𝑡−𝜏₁+𝑏₁₂𝑓₂𝑥₂𝑡−𝜏₂+𝛽𝑥₁𝑡−𝜏₁,𝑥̇₂=𝛼−𝑎₂𝑥₂(𝑡)+𝑏₂₁𝑓₁𝑥₁𝑡−𝜏₃+𝑏₂₂𝑓₂𝑥₂𝑡−𝜏₁+𝛽𝑥₂𝑡−𝜏₁, (33) where 𝜏₁=𝜏₂=𝜏₃=𝜏, 𝑎_𝑖=1, 𝑏_𝑖𝑗=−1, and 𝑓_𝑖(𝑥)=tanh(𝑥), 𝑖,𝑗=1,2. It is obvious that (0,0) is an equilibrium point of system (33). Choosing 𝛼=1,𝛽=0, by calculation, we know 𝜏₀=1.2092. However, when −𝛼<𝛽<𝛼, a family of periodic solutions bifurcates from (0,0) at 𝜏^′₀ (see Figure 7). Choosing 𝛼=0.75 and 𝛽=0.25, with complicated calculation, we know 𝜏^′₀=2.2143  (see Figure 8). Fix all coefficients of system (33) and let 𝜏=𝜏₀; the waveforms 𝑥₁(𝑡),𝑥₂(𝑡) without and with control are shown, respectively. However, the Hopf bifurcation in (33) could be delayed by hybrid control which could be seen by Figure 9.

Figure 7 

The trajectory of 𝑥1(𝑡) and 𝑥2(𝑡) versus time in the system (33) without control (𝜏=1.2092).

Figure 8 

The trajectory of 𝑥1(𝑡) and 𝑥2(𝑡) versus time in the system (33) with control (𝜏=1.2092).

Figure 9 

The trajectory of 𝑥1(𝑡) and 𝑥2(𝑡) versus time in the system (33) with control (𝜏=2.2143).

5. Conclusions

In this paper, the bifurcation and the bifurcation control problems have further been investigated for a HNNs model with delays. For the model, hybrid control strategy in which the parameter perturbation and time-delayed state feedback are combined and used to control various bifurcations in a continuous nonlinear dynamical system. It should be pointed out that, although Liu also have dealt with hybrid control, the time delayed feedback control used in our paper is more helpful than the controller in [22]. On the other hand, using parameter perturbation in this paper, our control strategy is more general than the other feedback control. Numerical simulations are given to justify the validity of hybrid controller in bifurcation control.

Acknowledgments

This research was supported by the Pre-research Foundation of PLA University of Science and Technology. Youth Research Foundation of College of Science of PLA University of Science and Technology.