Hopf Bifurcation and Stability Analysis of a Congestion Control Model with Delay in Wireless Access Network

and Applied Analysis 3 (e) each positive rootω(τ) ofF(ω, τ) = 0 is continuous and differentiable in τ whenever it exists. Proof. (a) For τ ∈ [τ1, +∞), P (0, τ) + Q (0, τ) = − a22b11 ̸ = 0. (14) (b) ConsiderP(ωi, τ)+Q(ωi, τ) = −ω2−b11a22+i(−ωa11− ωa12) ̸ = 0. (c) From (13), we get lim |λ|→+∞ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨Q (λ, τ) P (λ, τ) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 = lim |λ|→+∞ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 a12λ − b11a22 λ2 − a11λ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 = 0. (15) Hence, lim sup{|Q(λ, τ)/P(λ, τ)| : |λ| → +∞,Re λ ≥ 0} = 0 < 1. (d) From (13), we get F (ω, τ) = |P(ωi, τ)|2 − |Q(ωi, τ)|2 = ω4 + a2 11ω2 − a2 12ω2 − b2 11a2 22. (16) Hence, (d) holds. (e) F(ω, τ) is continuous for ω and τ and differentiable in ω; hence, implicit function theorem implies (e). This completes the proof of the theorem. Supposing thatD(ωi, τ) = 0 and ω > 0, we get sinωτ = ω (a11a22b11 − a12ω2) a2 22b2 11 + a2 12ω2 , cosωτ = −ω2 (a22b11 − a11a12) a2 22b2 11 + a2 12ω2 . (17) Hence, F (ω, τ) = |P(ωi, τ)|2 − |Q(ωi, τ)|2 = ω4 + a2 11ω2 − a2 12ω2 − b2 11a2 22 = 0. (18) Let z = ω2, and then (18) can be rewritten as z2 + (a2 11 − a2 12) z − b2 11a2 22 = 0. (19) Denote h (z, τ) = z2 + (a2 11 − a2 12) z − b2 11a2 22. (20) Since − b2 11a2 22 < 0, the equation h(z, τ) = 0 has one positive root.We denote that the positive root is z+.Then, (18) has positive real root ω(√z+), where ω = ω (τ) = √ − (a2 11 − a2 12) + √ (a2 11 − a2 12)2 + 4b2 11a2 22 2 . (21) For τ ∈ [τ1, +∞), let θ(τ) ∈ (0, 2π) be defined by sin θ (τ) = ω (a11a22b11 − a12ω2) a2 22b2 11 + a2 12ω2 , cos θ (τ) = ω2 (a22b11 − a11a12) a2 22b2 11 + a2 12ω2 , (22) which combines with (18) and defines the following maps: Sn (τ) = τ − θ (τ) + 2nπ ω (τ) , n ∈ N. (23) According to [18] and the above discussion, we have the following result. Theorem 3. Assume that (H1) is satisfied, and then λ = ±ω(τ0)i, τ0 ∈ (0, τ1), are a pair of simple and conjugate pure imaginary roots of the characteristic equation (11) if and only if S0(τ0) = 0 for some n ∈ N. This pair of simple conjugate pure imaginary roots crosses the imaginary axis from left to right if δ(τ0) > 0 and crosses the imaginary axis from right to left if δ(τ0) < 0, where δ (τ0) := sign{ dRe λ dτ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨λ=ω(τ0)i} = sign{ dSn (τ) dτ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨τ=τ0} . (24) By the the expression of a11, a12, and b11, we know that they have singularity at τ = 0. We can not gain the conclusion that the equilibrium (x∗, p∗) by discussing roots of the characteristic equationD(λ, 0) = 0. To our knowledge, this case is rarely considered by papers. But we can get the stability of the system (6) when τ = τ0/2 by discussing the stability of the following auxiliary system: ̇ y1 (t) = c11y1 (t) + c12y1 (t − r) + d11y2 (t) , ̇ y2 (t) = c22y1 (t − r) , (25) where c11 = a11󵄨󵄨󵄨󵄨 τ=τ0/2 , c12 = a12󵄨󵄨󵄨󵄨 τ=τ0/2 , c22 = a22󵄨󵄨󵄨󵄨 τ=τ0/2 , d11 = b11󵄨󵄨󵄨󵄨 τ=τ0/2 . (26) Then, the characteristic equation of the linearized equation (25) is λ − c11λ − c12λe−λr − c22d11e−λr = 0. (27) Definition 4. For simplicity, let D0 (λ, r) = λ2 − c11λ − c12λe−λr − c22d11e−λr. (28) Lemma 5. The equilibrium (0, 0) of system (25) is locally asymptotically stable when r = 0. 4 Abstract and Applied Analysis Proof. When r = 0, (27) becomes λ2 − (c11 + c12) λ − c22d11 = 0. (29)


Introduction
Recently, the wireless access network has been wildly applied to various fields, especially to the Internment; therefore, it has received significant attention.The congestion control in wireless access network also plays a crucial role in the success of the wireless network technology.
The congestion and avoidance mechanism is a combination of the end-to-end TCP congestion control mechanism [1,2] at the end hosts and the queue management mechanism at the routers.Because the congestion control algorithm is a highly complex dynamical model, many researchers have given much study to its dynamics and stability.In [3][4][5], the local stability in congestion control models is studied.In [6][7][8][9], the existence of Hopf bifurcation is analyzed in congestion control models.
For wired access network, the dynamic of window size is captured by the following equation [10]: where   (),   () =   ()/  ,   , and () denote the TCP window size, TCP rate, round trap time at time  of flow , and probability of packet mark at time , respectively.However, there are seldom works which discuss the dynamical behaviors of the congestion control model in wireless access network such as stability and Hopf bifurcation.The observation provides us with the motivation to investigate the dynamical behaviors of the congestion control model in wireless access network.
In this paper, we consider the wireless access networks of only one bottleneck router and let  TCP flows tracer the router.In the down link communication from the network to the sources, the marking probability is fed back to the sources.During channel fading, the source has failed to receive the marking probability.Therefore, we suppose that the drop probability is   .In this case, the source will use the previous packet marking probability to reduce its window size, and also the window size is decreased by one by convention.Thus, we obtain We assume that the   is a constant and not timevarying and the queuing delay is neglected.So, we obtain the following congestion model in wireless network: q () =  (∑   ( − )) − . ( The paper is organized as follows.In Section 2, the stability of trivial solutions and the existence of Hopf bifurcation are discussed and the delay passes through the critical value, the system loses its stability, and a Hopf bifurcation occurs.In Section 3, based on the normal form theory and the center manifold theorem, we derive the formulas for determining the properties of the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions.In Section 4, numerical simulations are given to justify the theoretical analysis.Finally, the conclusions appear in Section 5.
Since we focus on dynamical behavior analysis of the above model in the wireless access networks, we only need to choose the communication delay as the bifurcation parameter.
It is worth to point out that recent many works have been done for wired access network.For details, we refer to [13][14][15][16].

Stability of the System with Communication Delay
In this section, we assume that   (),  = 1, . . .,  is equal to (), so (5) can be rewritten as follows: Let the equilibrium point of the system (6) be ( * ,  * ), which should satisfy and 0 <  * ≤ 1.
Then, the characteristic equation of the linearized equation ( 9) is Note that the coefficients  11 ,  12 , and  11 depend on time delay , since  * is connected with .In order to apply the geometric criterion of Kuang [17,18], we rewrite (, ) = 0 into where  (, ) = Let  =  2 , and then (18) can be rewritten as Since −  2 11  2 22 < 0, the equation ℎ(, ) = 0 has one positive root.We denote that the positive root is  + .Then, (18) has positive real root ( √  + ), where which combines with (18) and defines the following maps: According to [18] and the above discussion, we have the following result.
Theorem 3. Assume that (H1) is satisfied, and then  = ±( 0 ),  0 ∈ (0,  1 ), are a pair of simple and conjugate pure imaginary roots of the characteristic equation (11) if and only if  0 ( 0 ) = 0 for some  ∈ .This pair of simple conjugate pure imaginary roots crosses the imaginary axis from left to right if ( 0 ) > 0 and crosses the imaginary axis from right to left if ( 0 ) < 0, where By the the expression of  11 ,  12 , and  11 , we know that they have singularity at  = 0. We can not gain the conclusion that the equilibrium ( * ,  * ) by discussing roots of the characteristic equation (, 0) = 0. To our knowledge, this case is rarely considered by papers.But we can get the stability of the system (6) when  =  0 /2 by discussing the stability of the following auxiliary system: where Then, the characteristic equation of the linearized equation (25) is Definition 4. For simplicity, let Lemma 5.The equilibrium (0, 0) of system (25) is locally asymptotically stable when  = 0.
From the above discussion about the system (25), we have the following result.Theorem 9. When  <  0 , the equilibrium point of system (25) is locally asymptotically stable.
Further, if is satisfied, we will get the following lemma.
According to [18] and the above discussion, we have following the result.

Direction and Stability of the Hopf Bifurcation
In this section, we will study the direction of Hopf bifurcation and the stability of bifurcating periodic solution of system ( 6) at  =  0 .The approach employed here is the normal form method and center manifold theorem introduced by Hassard [20].More precisely, we will compute the reduced system on the center manifold with the pair of conjugate complex, purely imaginary solutions of the characteristic equation (11).
By this reduction, we can determine the Hopf bifurcation direction, that is, to answer the question of whether the bifurcation branch of periodic solution exists locally for supercritical bifurcation or subcritical bifurcation.

Numerical Simulation Examples
In this section, we use the formulas obtained in Sections 2 and 3 to verify the existence of the Hopf bifurcation and calculate the Hopf bifurcation value and the direction of the Hopf bifurcation of system (6) with  = 1000,  = 50, and  = 0.001.x(t) q(t)  These calculations prove that the system equilibrium ( * ,  * ) is asymptotically stable when  <  0 by computer simulation (see Figures 1, 2, and 3;  = 0.200).When  passes through the critical value  0 , ( * ,  * ) loses its stability and a Hopf bifurcation occurs (see Figures 4,5,and 6;  = 0.206).

Conclusion
A delayed model of congestion control was analyzed in this paper.Based on our theoretical analysis and numerical   simulation, we can find that there exists a critical value for this delay and the whole system is stable when the delay of the system is less than this critical value.By using the time delay as a bifurcation parameter, we have shown that a Hopf bifurcation occurs when this parameter passes through a critical value, which means that the wireless access system will be congested, even collapsed, when the communication delay becomes large.