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

It is well known that neural networks are complex and largescale 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:


Introduction
It is well known that neural networks are complex and largescale 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][2][3][4][5][6][7][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: +  22  2 ( 2 ( −  1 )) , (1) in which  1 =  2 =  3 = ,   = 1 and   (0) = 0,  = 1, 2, and Huang et al. [10] study further the bifurcation and periodic nature of system (1) with 2 1 =  2 +  3 ,   = 1 and   (0) = 0,  = 1, 2.Moreover, many authors also consider discrete form of system (1); we can see [11][12][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][16][17][18][19][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 perioddoubling 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.

2.1.
As  2 ≥ .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.
Proof.For (3), when  = 0, its roots can be expressed as  1,2 = − +  ± √  2 − .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  ̸ = 0. We can see that  is a root of (3) if and only if  is a root of (5).
We write  =  +  for a root of the characteristic equations (5), separate the real and imaginary parts of the ensuing equations ( 5), and obtain A change in the stability of the stationary solution can only occur when  = 0, that is, By (8), we have 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.where Here one denotes ± 0 especially as a pair of purely imaginary roots of ( 5) at  =  0 .To see if  −,0 and  0 are bifurcation values, one needs to verify if the transversality conditions hold.In fact, one has the following.

Lemma 4. The following transversality conditions:
are satisfied.
Proof.By (5), we have Hence, Obviously, we have then 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  −,0 ,  0 be defined by (11).
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  −,0 ,  0 be defined by (11).
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  ∈ [0, +∞).
Proof.For (3), when  = 0, its roots can be expressed as  1,2 = −+± √  2 − .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  ̸ = 0. We can see that  is a root of (3) if and only if  is a root of (5).
We write  =  +  for a root of the characteristic equation ( 3), separate the real and imaginary parts of the ensuing equations ( 5 Hence, we have 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.

Mathematical Problems in Engineering
From Lemmas 1 and 2, one can obtain the following theorem about the distribution of the characteristic roots of (3).Theorem 9. Let  0 be defined by (21).
By using Theorem 9, the stability and bifurcation of system (1) can be summarized as the following theorem.

Stability Analysis and Bifurcation with Hybrid Control
In this section, we will consider system (1) with control described by the following differential equation: where  > 0 and  ∈ .Obviously, (0, 0) is also an equilibrium point of system (22).Linearizing the system (22) at the equilibrium point (0, 0), we obtain Then the characteristic equation for the linearized system around (0, 0)  is given by which is a quadratic polynomial in the variable ( + )  and has roots given by where By (26), we know that (  ) 2 −   =  2 ( 2 − ), and thus, (  ) 2 ≥   ((  ) 2 <   ) holds if and only if  2 ≥  ( 2 < ) holds.
In the following, we also distinguish two cases to discuss (25).

3.1.
As  2 ≥ .Corresponding to Part I of Section 2, we make the following assumptions for convenience: Similarly, we can obtain the following theorem.

3.2.
As  2 < .Similar to deduction of Section 2.2, we have the following assumptions: In this part, we denote Hence, we can obtain the following theorem.
Remark 13.When  = 1 −  and  =  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  = 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 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][25][26][27] and any other complex networks.

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

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.