HOPF BIFURCATION ANALYSIS AND DESIGN OF HYBRID CONTROL FOR GROWTH MODEL WITH DELAY

In this paper, we investigate the problem of bifurcation control for a delayed logistic growth model. By choosing the timedelay as the bifurcation parameter, we present a Hybrid controler to control Hopf bifurcation. We show that the onset of Hopf bifurcation can be delayed or advanced via a hybeid controller by setting proper controling parameter. Under consideration model as operator Equation, apply orthogonal decomposition, compute the center manifold and normal form we determined the direction and stability of bifurcating periodic solutions. Therefore the Hopf bifurcation of the model became controllable to achieve desirable behaviors which are applicable in certain circumstances. AMS Subject Classification: 34H20, 34D23


Introduction
The single-species logistic growth model governed by delay differential (and integro-differential) equations plays an important role in population dynamics and ecology that has been investigated in-depth involving the stability, persistent, oscillations and chaotic behavior of solutions [3]- [8].Gopalsamy and Weng [4] considered the following control system: where a, b, c, d, e, k, r ∈ (0, +∞) and τ ∈ [0, +∞).The authors presented some sufficient conditions for the global asymptotic stability of the positive equilibrium of the system.On one hand, in Song et al. [6], the authors considered the Hopf bifurcation for a regulated logistic growth model which is a special case of (1.1) as follows: The authors gave the explicit algorithm determining the direction of Hopf bifurcations and the stability of the periodic solutions, while they didnt discuss the existence of stability switches of this system.On the other hands, Gopalsamy and Weng [4] investigate the following control system: the initial conditions for the system (1.3) take the form of n(s The solutions of (1.3) are defined for all t > 0 and also satisfy n(t) > 0, v(t) > 0 for t > 0. And the system (1.3) has unique positive equilibrium Then by the linear chain trick technique [4], system(1.3)can be transformed into the following equivalent system: The author obtained when the condition (H) hold, the positive equilibrium (n * , v * ) of (1.3) is linearly asymptotically stable irrespective of the size of the delay τ .XIE(2015) [5] interested in the effect of delay τ on dynamics of system (1.3) when the condition (H) is not satisfied.Taking the delay τ as a parameter, they showed that the stability and a Hopf bifurcation occurs when the delay τ passes through a critical value.We summarize these features of the solution via the existence and stability of a positive equilibrium in following: ) is locally asymptotically stable for 0 ≤ τ < τ 0 and unstable for τ > τ 0 and system (1.3) undergoes Hopf bifurcation at (n * , v * ) when τ = τ n , n=0,1,2,...The organization of this paper is as follows.In Section 2, we study the stability and the Hopf bifurcation of Control system.In the next section, by the normal form method and the center manifold theory introduced by Hassard et al. [2], the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are determined.In addition, the main results illustrated by examples with numerical simulations.

Hopf Bifurcation in Hybrid Control Delay Differential Equation
In this section,we focus on designing a controller to control the Hopf bifurcation in model based on the hybrid control strategy(2015) [7].Apply the hybrid control to system (1.4), we get where 0 < α ≤ 1.The characteristic linear equation (2.1) is We assume λ = iω is a purely imaginary root of (2.2), then we can obtained Separating the real and imaginary parts of (2.3), we obtain (2.4) Since sin 2 ωτ + cos 2 ωτ = 1, we have ) has a solution ω 0 > 0, since equation of (2.4) is Then ±ω 0 is the purely imaginary root of (2.2).
Lemma 2. In the equation of ( 2 Proof.We will calculate dλ dτ using equation (2.2): The sign of the real part of dλ dτ is obtained from Therefore A > 0, and the proof is complete.

Direction and Stability of the Hopf Bifurcation in Discrete Control Model
In Section 2, we obtained conditions for the Hopf bifurcation to occur when In the section we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions, using techniques from normal form and center manifold theory for delay differential equations( 2009) [1].If we consider c 1 = αn * k then system (2.1) can be written as where . By the Riesz representation theorem, there exists a function η(θ, µ) of bounded variation for θ ∈ [−1, 0] such that In fact, we can choose Then system (3.2) is equivalent to where and a bilinear inner product Here η(θ) = η(θ, 0).As < Γ, A(0)φ >=< A * Γ, φ >, obviously A(0) and A * (0) are adjoint operators and ±iω 0 are eigenvalues of A(0) and A * (0).We first need to compute the eigenvector of A(0) and A * (0) corresponding to iω 0 and −iω 0 respectively.Suppose that q(θ) = (1, q 1 ) T e iω 0 θ is the eigenvector of A(0) and q * (s) = D(1, q * 1 )e −iω 0 s is the eigenvector of A * (0) corresponding to iω 0 , −iω 0 respectively.In order to assur < q * , q >= 1, we need to determine the value of the D. From inner product we can obtain To compute the coordinates describing center manifold C 0 at µ = 0. Define z(t) =< q * , x t >, W (t, θ) = x t − 2Rez(t)q(θ).
So W 11 = (W 11 (1) , W 11 ), W 20 = (W 20 (1) , W We define According to the case described above, we can summarize the results in the following theorem.Theorem 3.For the controlled system (3.1), the Hopf bifurcationis determined by the parameters µ 2 , T 2 and β 2 , the conclusions are summarized asfollows: (I) Parameter µ 2 determines the direction of the Hopf bifurcation.if µ 2 > 0, the Hopf bifurcation is supercritical,the bifurcating periodic solutions exist for τ > τ n , if µ 2 < 0 the Hopf bifurcation is subcritical, the bifurcating periodic solutions exist for τ < τ n .

Conclusion
In this paper, the problem of Hopf bifurcation control for an logistic growth model with time delay was studied.In order to control the Hopf bifurcation, a Hybrid controller is applied to the model.This Hybrid controller can successfully delay or advance the onset of an inherent bifurcation.The end theoren helped to improve model.
On the center manifold C 0 we