Bifurcations in an Internet congestion control system with distributed delay

Applied Mathematics and Computation 347 (2019) 54–63

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Bifurcations in an Internet congestion control system with distributed delay Yang Cao Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China

a r t i c l e

i n f o

Keywords: Congestion control Bifurcation Distributed delay

a b s t r a c t The problem of bifurcation of an Internet congestion control system with distributed delay is fully considered. Some criteria of stability and bifurcation conditions for the positive equilibrium of the proposed system are precisely established with the help of the Routh– Hurwitz criterion. Our results extremely extend previous ones, which enhance the stability performance of Internet congestion control system averts network collapse. Finally, numerical simulations are provided to verify the theoretical analysis. © 2018 Elsevier Inc. All rights reserved.

1. Introduction The Internet has grown into one of the grandness creations and has quickly altered the way of communication[1,2], life [3,4], entertainment [5,6] and education [7–9]. The stability is a momentous issue for Internet congestion control system, which has attracted considerable attention from scholars [10–13]. The global stability of a delayed Internet congestion control model with a single link and single source was considered [11]. Although the Internet has continued to expand over the past few decades, the size constraints still exist. Consequently, the issue of how to effectively process resources or information becomes more important. In the past decade, congestion control already becomes one of the main issues of current Internet owing to the continuously exploding increase of users, which can destabilize the network and even collapse it. The core target of the network congestion control algorithm is to regulate the transmission traffic of the terminal host to avoid congestion of the network link. Hence, in order to ameliorate the stability performance of Internet congestion control system and avoid collapsing, it is crucial and urgent to explore the dynamics of the Internet congestion control system for polishing up the performance of the network. There is an excellent instrument on Hopf bifurcation handling the dynamics of nonlinear systems for achieving desirable dynamics. [14–17]. A lot of effort has been made in Hopf bifurcation analysis of congestion control system in the past decades [18–22]. A hybrid controller was carefully developed to control the Hopf bifurcation of a twofold model of Internet congestion control system [19], which revealed that by adjusting the appropriate control parameters, Hopf bifurcation can be delayed without altering the initial poise point of the system. In [21], the authors discussed the problem of bifurcation control of an Internet congestion model with a single link and two sources via nonlinear state feedback controller, it suggested that the linear term of the controller can delay the start of the Hopf bifurcation and can adjust the bifurcation dynamics by choosing the appropriate higher term to obtain some desirable behaviors. To better depict the dynamical behaviors of some complicated phenomena, the distributed time delay has been introduced into many practical systems. It was shown that the distributed time delay has a noticeable impact on the dynamical

E-mail address: [email protected] https://doi.org/10.1016/j.amc.2018.10.093 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

Y. Cao / Applied Mathematics and Computation 347 (2019) 54–63

55

properties of complex systems [23–25]. With the vigorous expansion of fractional calculus, some investigators are surprised to find that fractional calculus can be integrated into the Internet congestion control models while fractional versions are constructed,which can more accurately reflect dynamic attributes due to their infinite memory attributes. Some valuable results of fractional-order Internet congestion control models have been reported [16,26–28]. It is a great pity that most existing results on Internet congestion models rarely discussing the issue of bifurcation of Internet congestion control models with distributed delays [30]. Motivated by this fact, we shall deal with this challenging problem in this article. The fabric of the thesis is highlighted as follows: In Section 2, the mathematical models are raised. In Section 3, the conditions of Hopf bifurcation are established by separating three cases. Numerical examples are performed to confirm the efficiency of the proposed theory in Section 4. Conclusions are finally drawn. 2. Mathematical model Ding et all considered the following first-order delay differential equation in [19,20]

p˙ (t ) = kp(t )[ f ( p(t − τ )) − c],

(1)

where p(t) denotes the price of the bottleneck link, f(p(t)) represents a non-negative continuous, strictly reduced requirement function with at the lowest a third order continuous derivative. c > 0 indicates the ability of the bottleneck link, k denotes a gain parameter, and τ is the round trip time comprises propagation delay and queuing delay. Let p∗ be the non-zero poise point of system (1). In order to reach the problem of bifurcation for system (1) that the communication delay is used as a bifurcation parameter [19]. As is known to all, the dynamical systems with distributed delays are more general than those with discrete delay. In this paper, we propose to extend model (1) by replacing the discrete time delay τ with a distributed delay as follows

 

p˙ (t ) = kp(t ) f



t

p(r )g(t − r )dr

−∞



−c ,

(2)

where g( · ) is a gamma distribution, i.e.

g( a ) =

 m m am−1 e− mT a ( m − 1 )!

T

,

(3)

with m a positive integer and T ≥ 0 a parameter associated with the mean time delay of the distribution. There are two special cases should be considered in (3): m = 0 and m = 1 are called the weak delay kernel and the strong delay kernel, respectively. It is well known that dynamical systems with distributed delays are more general than those with discrete delay. As T → 0 notice that the distribution function approaches the Dirac distribution, and, thus, one recovers the discrete delay case. 3. Main results To investigate the dynamical properties of Eq. (2) around the equilibrium point p∗ , we consider its linearized version. Letting x = p − p∗ , then linearize the system of (2) at the origin as

x˙ (t ) = kp∗ f  ( p∗ )



t

−∞

x(r )g(t − r )dr.

(4)

Substituting x(t ) = α eλt into Eq. (4), the related characteristic equation of the linearized system can be obtained as

λ − kp∗ f  ( p∗ )



t

−∞

e−λ(t−r ) g(t − r )dr = 0.

By using



t

e−λ(t−r ) g(t − r )dr

−∞



= =

+∞

0

 m m 

=

e−λv g(v )dv

T

1+

1 ( m − 1 )!

λT m

−m

,



+∞ 0

vm−1 e−( T +λ )v dv m

(5)

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Y. Cao / Applied Mathematics and Computation 347 (2019) 54–63

the characteristic Eq. (5) can be transformed into the following form

 m λT λ 1+ − kp∗ f  ( p∗ ) = 0,

(6)

m

which is a polynomial equation of degree M = m + 1 of the form

λM + c1 λM−1 + · · · + cM−1 λ + cM = 0, where the coefficients cj , j = 1, . . . , M, are real constants. By applying the Routh–Hurwitz criterion, we can decide whether or not are negative parts of a polynomial. Let M be the following M-dimensional square matrix

⎡

c1 ⎢c 3 ⎢c 5 M = ⎢ ⎢. ⎣ .. 0

1 c2 c4 .. . 0

0 c1 c3 .. . 0

0 1 c2 .. . 0

··· ··· ··· ··· ···

⎤

0 0⎥ ⎥ 0 ⎥, .. ⎥ .⎦ cN

(7)

where c j = 0 if j > M. It is clear that all the roots of (6) are negative or have negative real parts if and only if the determinant of all matrices M is positive. Thus, a direct consequence is that all coefficients cj are positive. It is not hard to see that the criterion can be simplified as: (i) when M = 2, c1 > 0 and c2 > 0; (ii) when M = 3, c1 > 0, c3 > 0 and c1 c2 > c3 ; (iii) when M = 3, c1 > 0, c3 > 0, c4 > 0 and c1 c2 c3 > c32 + c12 c4 . So as to analytically consider local stability of our poise we will consider the following special cases: m = 1, m = 2 and m = 3. Case m = 1 Substituting m = 1 in (6) reveals that the characteristic equation reduces to a second order algebraic equation in λ,

λ2 + c1 (T )λ + c2 (T ) = 0,

(8)

where

1 , T kp∗ f  ( p∗ ) c2 ( T ) = − . T c1 ( T ) =

It is apparent that c1 (T) > 0, c2 (T) > 0. With the help of the Routh–Hurwitz criterion, the equilibrium p∗ of (2) is locally asymptotically stable for all T > 0. Case m = 2 The characteristic Eq. (6) can be acquired with respect to λ as

λ3 + c1 (T )λ2 + c2 (T )λ + c3 (T ) = 0,

(9)

where

4 , T 4 c2 ( T ) = 2 , T 4kp∗ f  ( p∗ ) c3 ( T ) = − . T2 c1 ( T ) =

It can be seen that that c1 (T) > 0, c2 (T) > 0 and c3 (T) > 0. Hence, if c1 (T)c2 (T) > c3 (T), in view of the Routh–Hurwitz criterion, the equilibrium p∗ of (2) is locally asymptotically stable, i.e. if

T <−

4 = T∗ . kp∗ f  ( p∗ )

(10)

Y. Cao / Applied Mathematics and Computation 347 (2019) 54–63

57

On the basis of the Hopf bifurcation theorem, If the characteristic Eq. (9) has a pair of pure virtual roots and there are without else roots have zero real parts, then the real part of these roots will change the sign with the bifurcation parameter, which may lead to the limit cycle of T = T∗ . The curve T = T∗ partitions the parameter space into stable and unstable parts. When T = T∗ , one has c1 (T∗ )c2 (T∗ ) = c3 (T∗ ). So that the characteristic Eq. (9) can be factored as



[λ + c1 (T∗ )]

 λ2 + c2 (T∗ ) = 0.

It is then obvious that (9) has a pair of purely imaginary roots and a root with non-zero real parts. More precisely, we have

 λ1,2 = ±i c2 (T∗ ) = ±iω∗

and

λ3 = −c1 (T∗ ). Next, we choose T as the bifurcation parameter and treat the characteristic Eq. (9) as a continuous function of T. A differentiation of (9) with respect to T gives



3λ2 + 2c1 (T )λ + c2 (T )

 dλ

dT   = − c1 (T )λ2 + c2 (T )λ + c3 (T ) ,

(11)

where

4 , T2 8 c2 (T ) = − 3 , T 8kp∗ f  ( p∗ ) c3 (T ) = . T3 c1 (T ) = −

Hence,

dλ = dT

φ1 ( T ) , φ2 ( T )

where

φ1 (T ) = −[c1 (T )λ2 + c2 (T )λ + c3 (T )], φ2 (T ) = 3λ2 + 2c1 (T )λ + c2 (T ). Producing the real form of the derivative of λ about T in the following form



dλ Re dT

  

λ=iω∗

=

ψ1 (T∗ ) , ψ2 (T∗ )

where

ψ1 (T∗ ) = −c1 (T∗ )c2 (T∗ ) − c1 (T∗ )c2 (T∗ ) + c3 (T∗ ),   ψ2 (T∗ ) = 2 c2 (T∗ ) + c12 (T∗ ) . By calculation, we have



dλ Re dT

  

λ=iω∗

=

1 [kP ∗ f  P ∗ ] 16 . 2[ T42 + T4 2 ]

( ) ( )

It is not hard to obtain that the transversality condition



Re

dλ dT

  

λ=iω∗

> 0.

This signifies that as T increases, the root crossing the imaginary axis from left to right at λ = iω∗ . Summarising the above discussion we get the following result. Theorem 1. Let T∗ = −4/[kp∗ f  ( p∗ )]. Then (i) The equilibrium p∗ of (2) is locally asymptotically stable for T < T∗ and unstable for T > T∗ . (ii) Eq. (2) undergoes a Hopf bifurcation at the equilibrium p∗ when T = T∗ .

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Y. Cao / Applied Mathematics and Computation 347 (2019) 54–63

Case m = 3 In this case the characteristic Eq. (6) is

λ4 + c1 (T )λ3 + c2 (T )λ2 + c3 (T )λ + c4 (T ) = 0,

(12)

where

9 , T 27 c2 ( T ) = 2 , T 27 c3 ( T ) = 3 , T 27kp∗ f  ( p∗ ) c4 ( T ) = − . T3 c1 ( T ) =

We can derive that c1 (T) > 0, c2 (T) > 0, c3 (T), c4 (T) > 0. It is clear that the equilibrium p∗ of (2) is locally asymptotically stable by means of the Routh–Hurwitz criterion if

ϕ (T ) = c1 (T )c2 (T )c3 (T ) − c32 (T ) − c12 (T )c4 (T ) > 0, obviously,

T <−

8 = T∗ . 3kp∗ f  ( p∗ )

Noticing that ϕ (T∗ ) = 0. When T = T∗ , the characteristic Eq. (12) can be formulated as







c1 (T∗ )λ2 + c3 (T∗ ) c1 (T∗ )λ2 + c12 (T∗ )λ + c1 (T∗ )c2 (T∗ ) − c3 (T∗ ) = 0,

which indicates that two roots are purely imaginary



λ1,2 = ±i

c3 (T∗ ) = ±iω∗ , c1 (T∗ )

and the other two are

λ3,4 =

ϕ1 (T∗ ) . ϕ2 (T∗ )

where

ϕ1 (T∗ ) = −c12 (T∗ ) ± ϕ2 (T∗ ) = 2c1 (T∗ ).



c14 (T∗ ) − 4c1 (T∗ )[c1 (T∗ )c2 (T∗ ) − c3 (T∗ )],

It can be observed that λ3 + λ4 = −c1 (T∗ ) < 0 and λ3 λ4 = [c1 (T∗ )c2 (T∗ ) − c3 (T∗ )]/c1 (T∗ ) > 0. Therefore, λ3 and λ4 have real parts different from zero. In the following, the rate of change of the real part of λ about T. can be calculated. It follows from (12) that



4λ3 + 3c1 (T )λ2 + 2c2 (T )λ + c3 (T )

 dλ

dT   = − c1 (T )λ3 + c2 (T )λ2 + c3 (T )λ + c4 (T ) , i.e.

c (T )λ3 + c2 (T )λ2 + c3 (T )λ + c4 (T ) dλ =− 1 3 , dT 4λ + 3c1 (T )λ2 + 2c2 (T )λ + c3 (T ) where

9 , T2 54 c2 (T ) = − 3 , T 81  c3 ( T ) = − 5 , T 81 kp∗ f  ( p∗ ) c4 (T ) = . T5 c1 (T ) = −

Y. Cao / Applied Mathematics and Computation 347 (2019) 54–63

Then,



dλ Re dT

  

λ=iω∗

=−

59

ψ1 (T∗ )ψ2 (T∗ ) , ψ3 (T∗ )

where

ψ1 (T∗ ) = c1 (T∗ ), ψ2 (T∗ ) = c1 (T∗ )c2 (T∗ )c3 (T∗ ) + c1 (T∗ )c2 (T∗ )c3 (T∗ ) + c1 (T∗ )c2 (T∗ )c3 (T∗ ) − 2c3 (T∗ )c3 (T∗ ) − 2c1 (T∗ )c1 (T∗ )c4 (T∗ ) − c12 (T∗ )c4 (T∗ ),   ψ3 (T∗ ) = 2 c13 (T∗ )c3 (T∗ ) + [c1 (T∗ )c2 (T∗ ) − 2c3 (T∗ )]2 . In terms of direct calculations, we have ϕ  (T∗ ) < 0. Thus,



dλ Re dT

  

λ=iω∗

> 0.

Hence, as T increases that the root is passing the imaginary axis from right to left. The above discussions lead to the following results. Theorem 2. Let T∗ = −8/[3kp∗ f  ( p∗ )]. Then (i) The equilibrium p∗ of (2) is locally asymptotically stable for T < T∗ and unstable for T > T∗ . (ii) A Hopf bifurcation occurs at the equilibrium point p∗ as T passes through T∗ . 4. Numerical simulations In this section, numerical example are presented to demonstrate the efficiency and application of the derived new results. To numerically examine the behaviour of Eq. (2), we can apply the linear chain trick technique [29], which consists in replacing Eq. (1) by an equivalent system of (m + 1 ) ordinary differential equations. In our simulations, the same parameter are taken from Ding et al. [19]: f ( p(t )) = p(1t ) , c = 50, k = 0.01. By direct calculation, the equilibrium point can be acquired as p∗ = 0.02. Example 1. In this example, we consider the case that m = 1. Hence, Eq. (2) can be converted into

 

p˙ (t ) = kp(t ) f

t −∞

p( r )

1 T



e

− T1 (t−r )

dr



−c .

Introducing the following new variable

u(t ) =



t −∞

p( r )

1 T

e− T (t−r ) dr. 1

Eq. (2) can be written as



p˙ (t ) = kp(t )[ f (u(t )) − c], 1 u˙ (t ) = [ p(t ) − u(t )]. T

(13)

By utilizing Eq. (13), the analysis can now be performed. It is obvious that

a1 ( T ) > 0, and

a2 ( T ) = −

kp∗ f  ( p∗ ) 1 = > 0. T 2T

By employing the Routh–Hurwitz criterion, p∗ is locally asymptotically stable for all T > 0. We arbitrarily choose T = 0.6, 3.5, 9.2, simulation results are displayed in Fig. 1. Example 2. In what follows, the case m = 2 shall be studied. Hence Eq. (2) can be transformed into

 

p˙ (t ) = kp(t ) f

t −∞

p( r )

 2 2 T



(t − r )e

− T2 (t−r )

dr



−c .

60

Y. Cao / Applied Mathematics and Computation 347 (2019) 54–63

Fig. 1. Waveform plot of the system (2) with m = 1, and p∗ is locally asymptotically stable for T = 0.6, 3.5, 9.2.

Introducing the new variables

⎧  ⎪ ⎨u(t ) =

t

 2 2

(t − r )e− T (t−r ) dr, T −∞  t 2 2 ⎪ ⎩v(t ) = p( r ) e− T (t−r ) dr. T −∞ p( r )

2

Eq. (2) rewrites as

⎧ p˙ (t ) = kp(t )[ f (u(t )) − c] ⎪ ⎪ ⎪ ⎪ ⎨ 2 u˙ (t ) = [v(t ) − u(t )], T ⎪ ⎪ ⎪ 2 ⎪ ⎩ v˙ (t ) = [ p(t ) − v(t )].

(14)

T

Based on the system (14), the analysis can now be carried out. By calculation, we conclude that T ∗ = 8. In terms of Theorem 1, the equilibrium p∗ of is locally asymptotically stable for T = 7.2 < T∗ ,as shown in Fig. 2, and unstable for T = 8.5 > T∗ generates a Hopf bifurcation at the equilibrium p∗ , see Fig. 3. Case m = 3 Eq. (2) becomes

 

p˙ (t ) = kp(t ) f

t −∞

p( r )

 3 3 (t − r )2 e− T3 (t−r )  T

2

Introducing the new variables

⎧  t  3 3 (t − r )2 e− T3 (t−r ) ⎪ ⎪ p( r ) dr, ⎪u(t ) = ⎪ T 2 ⎪ −∞ ⎪ ⎨  t  3 2 (t − r )e− T3 (t−r ) v(t ) = p( r ) dr, ⎪ T 2 −∞ ⎪ ⎪  3  3  e− T (t−r ) ⎪ t ⎪ ⎪ ⎩z(t ) = p( r ) dr. −∞

T

2

dr

 −c ,

Y. Cao / Applied Mathematics and Computation 347 (2019) 54–63

Fig. 2. Waveform plot of the system (2) with m = 2, and p∗ is locally asymptotically stable for T = 7.2 < T ∗ = 8.

Fig. 3. Waveform plot of the system (2) with m = 2, and Hopf bifurcation occurs from p∗ for T = 8.5 > T ∗ = 8.

61

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Y. Cao / Applied Mathematics and Computation 347 (2019) 54–63

Fig. 4. Waveform plot of the system (2) with m = 3, and p∗ is locally asymptotically stable for T = 4.5 < T ∗ = 5.3333.

Fig. 5. Waveform plot of the system (2) with m = 3, and Hopf bifurcation occurs from p∗ for T = 5.6 > T ∗ = 5.3333.

Y. Cao / Applied Mathematics and Computation 347 (2019) 54–63

63

Eq. (2) rewrites as

⎧ p˙ (t ) = kp(t )[ f (u(t )) − c] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪u˙ (t ) = 3 [v(t ) − u(t )], ⎨ ⎪ v˙ (t ) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z˙ (t ) =

T 3 [z(t ) − v(t )], T 3 [ p(t ) − z(t )]. T

(15)

Thus, the analysis of this system can now be done. By some computation, we get that T ∗ = 5.3333. With the help of Theorem 1, it can see that the poise p∗ of is locally asymptotically stable for T = 4.5 < T∗ , which is depicted in Fig. 4, and unstable for T = 5.6 > T∗ a Hopf bifurcation bifurcates from the equilibrium p∗ , which is illustrated in Fig. 5. 5. Conclusion In this paper, the novel research subject of avoiding Internet congestion involving distributed delay has been carefully discussed. The core problem of the Hopf bifurcation of an Internet congestion control system with distributed delay is investigated. The bifurcation point of the positive equilibrium of such system is accurately figured out with the help of the Routh–Hurwitz criterion. It is believed that our studies are meritorious for enhancing the performance of Internet congestion control system and avoiding network collapse. The effectiveness is ultimately confirmed by providing three simulation examples. References [1] N. Miteku, Li-Fi over Wi-Fi in Internet data communication, Int. J. Innovative Res. Electr. Electron Instrum. Control Eng. 3 (12) (2016) 153–159. [2] E.V. Brodovskaya, A.Y. Dombrovskaya, V.D. Nechaev, A.V. Sinyakov, National profiles of internet-communication results of cross-national cluster analysis, Eur. J. Sci. Theol. 11 (3) (2015) 125–130. [3] I. Prigogine, M. Adamidou, Life and the internet, Int. Hist. Rev. 23 (4) (2015) 757–766. [4] H. Boz, S.E. Karatas, A review on internet use and quality of life of the elderly, Cypriot J. Educ. Sci. 10 (3) (2015) 182–191. [5] S. Pantea, B. Martens, The value of the internet as entertainment in five European countries, J. Med. Econ. 29 (1) (2016) 16–30. [6] J.C. Wu, Expanding civic engagement in China: Super girl and entertainment-based online community, Inf. Commun. Soc. 17 (1) (2014) 105–120. [7] Y.C. Kuo, A.E. Walker, K.E.E. Schroder, B.R. Belland, Interaction, Internet self-efficacy, and self-regulated learning as predictors of student satisfaction in online education courses, Int. Higher Educ. 20 (1) (2014) 35–50. [8] R. Kim, L. Olfman, T. Ryan, E. Eryilmaz, Leveraging a personalized system to improve self-directed learning in online educational environments, Comput. Educ. 70 (1) (2014) 150–160. [9] H.G. Peach Jr, J.P. Bieber, Faculty and online education as a mechanism of power, Distance Educ. 36 (1) (2015) 26–40. [10] Y. Lei, G.E. Dullerud, R. Srikant, Global stability of internet congestion controllers with heterogeneous delays, IEEE/ACM Trans. Netw. 14 (3) (2006) 579–591. [11] M. Peet, S. Lall, Global stability analysis of a nonlinear model of internet congestion control with delay, IEEE Trans. Autom. Control 52 (3) (2007) 553–559. [12] J.Y. Choi, Global stability analysis scheme for a class of nonlinear time delay systems, Automatica 45 (10) (2009) 2462–2466. [13] D.W. Ding, X.Y. Zhang, D. Xue, Stability analysis of internet congestion control model with compound TCP under RED, Open Autom. Control Syst. J. 7 (1) (2015) 1986–1992. [14] C.D. Huang, J.D. Cao, Impact of leakage delay on bifurcation in high-order fractional BAM neural networks, Neural Netw. 98 (2018) 223–235. [15] C.D. Huang, J.D. Cao, M. Xiao, A. Alsaedi, T. Hayat, Bifurcations in a delayed fractional complex-valued neural network, Appl. Math. Comput. 292 (2017) 210–227. [16] M. Xiao, W.X. Zheng, G.P. Jiang, J.D. Cao, Stability and bifurcation of delayed fractional-order dual congestion control algorithms, IEEE Trans. Autom. Control 62 (9) (2017) 4819–4826. [17] C.D. Huang, J.D. Cao, M. Xiao, A. Alsaedi, T. Hayat, Effects of time delays on stability and Hopf bifurcation in a fractional ring-structured network with arbitrary neurons, Commun. Nonlinear Sci. Numer. Simulat. 57 (2018) 1–13. [18] M.J. Motlagh, S. Khorsandi, M. Analoui, Hopf bifurcation analysis on an internet congestion control system of arbitrary dimension with communication delay, Nonlinear Anal. Real World Appl. 11 (5) (2010) 3842–3857. [19] D.W. Ding, J. Zhu, X.S. Luo, Y.L. Liu, Delay induced Hopf bifurcation in a dual model of Internet congestion control algorithm, Nonlinear Anal. Real World Appl. 10 (2009) 2873–2883. [20] D.W. Ding, X.M. Qin, N. Wang, T.T. Wu, D. Liang, Hybrid control of Hopf bifurcation in a dual model of internet congestion control system, Nonlinear Dyn. 76 (2) (2014) 1041–1050. [21] W.Y. Xu, J.D. Cao, M. Xiao, Bifurcation analysis and control in exponential RED algorithm, Neurocomputing 129 (2014) 232–245. [22] S.T. Guo, X.F. Liao, Q. Liu, H.X. Wu, Linear stability and Hopf bifurcation analysis for exponential RED algorithm with heterogeneous delays, Nonlinear Anal. Real World Appl. 10 (4) (2009) 2225–2245. [23] M. Adimy, F. Crauste, S. Ruan, Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics, Nonlinear Anal. Real World Appl. 6 (2005) 651–670. [24] H.Y. Zhao, L. Wang, Hopf bifurcation in Cohen-Grossberg neural network with distributed delays, Nonlinear Anal. Real World Appl. 873–89. [25] K. Balachandran, Y. Zhou, J. Kokila, Relative controllability of fractional dynamical systems with delays in control., Comput. Math. Appl. 17 (2012) 3508–3520. [26] M. Xiao, W.X. Zheng, J.X. Lin, G.P. Jiang, L.D. Zhao, J.D. Cao, Fractional-order PD control at Hopf bifurcations in delayed fractional-order small-world networks, J. Frankl. Inst. 354 (17) (2017) 7643–7667. [27] Y.H. Tang, M. Xiao, G.P. Jiang, J.X. Lin, J.D. Cao, W.X. Zheng, Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system, Nonlinear Dyn. 90 (3) (2017) 2185–2198. [28] H. Hamidian, M.T.H. Beheshti, A robust fractional-order PID controller design based on active queue management for TCP network, Int. J. Syst. Sci. 49 (1) (2018) 211–216. [29] N. MacDonald, Time Lags in Biological Systems, Springer, New York, 1978. [30] M. Krstic, Delay Compensation for Nonlinear, Adaptive, and PDE Systems, Springer, 2009.