Controlling Neimark-Sacker Bifurcation in Delayed Species Model Using Feedback Controller

Based on the stability and orthogonal polynomial approximation theory, the ordinary, dislocated, enhancing, and random feedback control methods are used to suppress the Neimark-Sacker bifurcation to fixed point in this paper. It is shown that the convergence rate of enhancing feedback control and random feedback control can be faster than those of dislocated and ordinary feedback control.The random feedback controlmethod, which does not require any adjustable control parameters of themodel, just only slightly changes the random intensity. Finally, numerical simulations are presented to verify the effectiveness of the proposed controllers.


Introduction
The studies of biological models gradually become one of hot spots in nonlinear dynamics.The biological models have great research background and actual significance; therefore, a growing number of researchers have shown great interests in the research of biological models.In many biological models and practical problems, bifurcation and chaos are undesirable behaviors.Thus, we need to control them.In 1976, a population model, is given by Ecologist May for the first time.A onedimensional deterministic delayed population model,  +1 =   (1 −  −1 ) ,  ∈ [0, 2.28] ,  ∈ (0, 1) , is investigated by Sun et al. [1].
Recently, the Hopf bifurcation has been given much attention, and those works about bifurcation mainly include the validated existence of bifurcation and its control [2][3][4][5].The aim of bifurcation control is to design a controller to modify the bifurcation properties of a given nonlinear system and then achieve the other desirable dynamical behaviors.OGY feedback control method is studied by Ott et al. [6].Chen et al. have investigated the feedback control in continuous-time systems [7,8].The control of Hopf bifurcation in time-delayed neural network system is investigated by Zhou et al. [9].Bifurcation analysis and tracking control of an epidemic model with nonlinear incidence rate are investigated by Yi et al. [10].Wen and Xu studied feedback control of Hopf-Hopf interaction bifurcation with development of torus solutions in highdimensional maps [11].Feedback control of bifurcation and chaos in dynamical systems is investigated by Abed and Wang [12].The Hopf bifurcation control via dynamic state-feedback control is studied by Nguyen and Hong in [13].Amplitude control of limit cycle from Hopf bifurcation is studied in [14,15].Hopf bifurcation control of the system based on washout filter controller is investigated by Wu and Sun [16].Liu and Xiao have studied complex dynamic behaviors of a discrete-time predator-prey system [17].
However, owing to the uncertain factors of external environment, manufacture, material, and installation, some parameters in practical model are not constant and will be characterized as bound random parameters [18].The stochastic system can accurately represent the original system better.Therefore the study of stochastic system is more meaningful than deterministic systems.The Hopf bifurcation control is investigated in stochastic system with random parameter [18][19][20].It is of interest to examine the stochastic method in biological system and explore its implications.
The rest of this letter is organized as follows.In Section 2, the conditions for the emergence of Neimark-Sacker bifurcation are reviewed.In Section 3, the ordinary, dislocated, 2 Advances in Mathematical Physics enhancing, and random feedback controls for controlling Neimark-Sacker bifurcation are proposed.And numerical simulations are presented to verify the effectiveness of the proposed bifurcation control methods.Finally, conclusions are given in Section 4.

Neimark-Sacker Bifurcation
Let us consider the logistic population model [1] for a single species: where   stands for the population size at time  and  is the growth rate.In the real environment, the population size is determined not only by the current population size but also by its size in the past.So, we consider where  −1 stands for the population size at time  − 1 and  is the growth rate.If we introduce   =  −1 in model ( 4), a two-dimensional discrete-time dynamical model [2] can be rewritten as By a simple computation with mathematical software, it is straightforward to obtain the following proposition.The Jacobian matrix of model ( 5) evaluated at the fixed point (, ) is given by and the characteristic equation of Jacobian matrix of model ( 5) can be written as where (, ) =  − , (, ) = −.
Next, according to the point of view of biology, we study the stability of the nonzero fixed points.Note that the local stability of a fixed point is determined by the modules of eigenvalues of the characteristic equation at the fixed point.From the mathematical software and Lemma 2.2 [17], the following proposition shows the local stability of the fixed point  * ( * ,  * ).

Proposition 2. (a) 𝑂
When the term (c) of Proposition 2 holds, we can obtain that the eigenvalues of the matrix  at the fixed point  * ( * ,  * ) are a pair of conjugate complex numbers, the modules of which are one.The condition in term (c) of Proposition 2 can be written as the set the fixed point  * ( * ,  * ) can undergo Neimark-Sacker bifurcation when parameters vary in the small neighborhood of   .By a simple computation, all eigenvalues of ( 7) are when  =   = 2, we can obtain eigenvalues Obviously, the transversality condition, the nondegeneracy condition, and the additional nondegeneracy condition of Neimark-Sacker bifurcation hold (see [3]).Thus, the nontrivial fixed point loses stability in the small neighborhood of   .The bifurcation diagram and phase portrait for model (5) are depicted in Figure 1.

Neimark-Sacker Bifurcation Control
Without control, model (5) undergoes Neimark-Sacker bifurcations at the point (0.5025, 0.5025) corresponding to the value of bifurcation parameter as  = 2.01, as shown in Figure 1.To control the Neimark-Sacker bifurcation to the fixed point, the ordinary, dislocated, enhancing, and random feedback control methods are introduced as shown in the following.

Ordinary Feedback Control.
For the ordinary feedback control, the system variable is often multiplied by a coefficient as the feedback gain, and the feedback gain is added to the right-hand side of the corresponding equation.Let the feedback control input  =  1 (  −  * ), and the controlled model is given by where  1 is the feedback coefficient.

Dislocated Feedback Control.
For the dislocated feedback control, a system variable multiplied by a coefficient is added to the right-hand side of another equation.Then, this method is called dislocated feedback control.Let feedback control input  =  2 (  −  * ), and the controlled model is given by where  2 is the feedback coefficient.
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Theorem 4. The necessary and sufficient condition for the controlled delayed species model (14) to be asymptotically stable at fixed point is
Proof .The Jacobi matrix of model ( 14) is and the characteristic equation of Jacobi matrix  is according to Lemma 2.2 [17], the eigenvalues lie inside unit circle if and only if (1) > 0, (−1) > 0, and ( * ,  * ) < 1.By a simple computation, we can obtain the following conditions: (a) Thus, when (a), (b), and (c) hold, the controlled delayed species model (14) will gradually converge to the fixed point.Numerical simulations are used to investigate the controlled delayed species model (14).From Theorem 3, we conclude that our model (14) will gradually converge to the point (0.5025, 0.5025) for  2 ∈ (−1, −0.00990099), when  = 2.01.The feedback coefficient is given by  2 = −0.03.The initial values in model ( 14) are taken as [(0) = 0.2, (0) = 0.2].The behaviors of the states ((), ()) of the controlled delayed species model (14) with time  are displayed in Figure 3, respectively.

Enhancing Feedback Control.
For the enhancing feedback control, it is difficult for a complex system to be controlled by only one feedback variable, and in such cases the feedback gain is always very large.So we consider using multiple variables multiplied by a coefficient as the feedback gain.This method is called enhancing feedback control.Let feedback control inputs  1 =  3 (  −  * ),  2 =  3 (  −  * ), and the controlled model is given by where  3 is the feedback coefficient.
Proof.The Jacobi matrix of system ( 17) is and the characteristic equation of Jacobi matrix  is according to Lemma 2.2 [17], the eigenvalues lie inside unit circle if and only if (1) > 0, (−1) > 0, and ( * ,  * ) < 1.By a simple computation, we can obtain the following conditions: (a) Thus, when (a), (b), and (c) hold, the controlled delayed species model (17) will gradually converge to the fixed point.

Random Feedback Control.
To achieve the control objectives, we need to adjust the control gains for the feedback control methods above.Thus, we use the random feedback control method to control them.If a system variable multiplied by a random coefficient is added to the right-hand side of equation, then this method is called random feedback control.Let   =   −  * , V  =   −  * ; after applying the coordinate transformation, the fixed point is converted to the origin (0, 0); then we have the following system: Let feedback control input  = (  + V  );  is a random parameter;  =  4 +.Taking this controller into the right side of the second equation in (20), the controlled delayed species model can be written as where  4 is the statistic parameter of ,  and  are the input direction of state variable and random variable in the controller,  is random intensity, and  is the random variable defined on nonnegative set integer with the probability density function   .According to the orthogonal polynomial approximation [18][19][20] of discrete random function in the Hilbert space, the response of ( 21) can be expressed by the following series: where   () is the th Charlier orthogonal polynomial and  is the largest order of the polynomials we have taken.Substituting ( 22) into (21), we have Using the orthogonal polynomial approximation theory, the nonlinear term in the right side of (23) can be further reduced into a linear combination of related single polynomials.It is written as We obtained The recurrence formula for Charlier polynomial is In order to facilitate the numerical analysis of this paper, we take  = 1,  = 1.Based on the orthogonality of orthogonal polynomials, we can finally obtain the equivalent deterministic equation: And the ensemble mean response of model ( 21  For the random feedback control, which does not require any adjustable feedback control parameters, we need to slightly change the random intensity of random feedback controller as the random intensity is very small.By means of numerical simulations, we find that the random feedback method to control the Neimark-Sacker bifurcation is available.Next we discuss the influence of the initial values for random feedback control.Taking  0 , V 0 as random initial values covering [0, 0.5], according to numerical simulations, we can find that random feedback control has robustness for