Abstract In this article, we consider the problem of Hopf bifurcation control in a second-order Westwood + TCP flow control model. First, we propose a dynamic time-delayed state feedback controller to delay the onset of undesirable Hopf bifurcation. Moreover, we show that the Westwood + TCP model with time-delayed state feedback can maintain a stationary state on a certain domain in the delay parameter space. In particular, by the dynamic time-delayed feedback controller, a suitable Hopf bifurcation is created at a desired location with preferred properties. Furthermore, we analyse the dynamic behaviour of the rightmost characteristic root of the proposed controlled time-delayed systems via Lambert W function. In addition, by the Hopf bifurcation analysis, we show that with the proposed controller, one can increase the allowable critical value of the communication delay, and postpone the onset of the undesired Hopf bifurcation by successfully controlling unstable steady states or periodic orbits. Finally, we justify by numerical results the validity of the time-delayed state feedback controller in Hopf bifurcation control. 1. Introduction In recent years, due to rapid progress in wireless communication technology, wireless communication network has been widely deployed and applied in various fields. However, network flow problem is still one of the main concerns which can bring social and economic problems such as network instability and network performance degrading. Thus, network flow control has attracted significant attention from both industry and academia (see Guo et al., 2008a,b, 2009, 2010a,b; Ding et al., 2009a,b; Liu et al., 2011, 2012; Pei et al., 2011; Rezaie et al., 2011; Dong et al., 2013; Yu et al., 2016). In order to deal with flow congestion, Jacobson (1998) first proposed a classic transmission control protocol (TCP), which is based on an additive increase multiplicative decease algorithm in per round trip time (RTT). The congestion and avoidance mechanism is a combination of the end-to-end TCP flow control mechanism at the end and the active queue management (AQM) mechanism at the routers. So far, many AQM algorithms, such as random early detection (RED) (see Floyd & Jacobson, 1993) and random early marking (see Athuraliya et al., 2001), have been proposed. The classic TCP model has become a standard to ensure end-to-end reliable communication and has been used by a large variety of applications. However, the classic TCP is originally designed for wired networks with low-error rate, and it assumes that all packet losses are caused by network congestion. Unlike wired networks, in wireless networks, packet losses are mainly caused by noisy and fading radio channel. However, this kind of packet losses is often misinterpreted as a symptom of congestion by the classic TCP scheme and an unnecessary window reduction is caused. Therefore, a low throughput is induced because of packet losses but not network congestion, for example in the case of radio link (see Barakat & Altman, 2002). In order to overcome such drawbacks, a new bandwidth estimation algorithm, i.e. Westwood + TCP, was proposed in Grieco & Mascolo (2005). The main idea of Westwood + TCP is to exploit the stream of return acknowledgment packets to estimate the available bandwidth by using the additive increase adaptive decrease (AIADD) paradigm. It is extremely effective for throughput improvement in mixed (wired and wireless) communication networks. In general, a flow control system can be considered as a complex nonlinear feedback system with delay by reducing the mathematical model of a flow control system to a nonlinear delay differential equation. Therefore, in recent years, nonlinear dynamical behaviours, such as stability analysis and Hopf bifurcation, of flow control systems in communication networks with delay have attracted many researchers' attention (see Guo et al., 2008a,b, 2009, 2010a,b; Ding et al., 2009a,b; Liu et al., 2011, 2012; Pei et al., 2011; Rezaie et al., 2011; Dong et al., 2013; Yu et al., 2016). In Guo et al. (2008b), several necessary and sufficient conditions that can ensure Hopf bifurcation to occur for exponential RED algorithm with communication delay were derived. In Liu et al. (2012), the Hopf bifurcation of eXplicit Control Protocol (XCP) for a congestion control system was investigated. More recently, dynamics of a flow control model for wireless access networks was studied in Dong et al. (2013). It is worth noting that Hopf bifurcation control in nonlinear time delayed systems is difficult to be tackled. Fortunately, the seminal work by Ott et al. (1990), which is beyond the classic control theory, opened the field of chaos control which has become an interesting field in nonlinear science. Since then, many studies have applied the idea of chaos control in bifurcation control, especially in the Hopf bifurcation control. Therefore, bifurcation control becomes an increasing interest for many researchers. The key idea of bifurcation control is to make the bifurcation characteristics of systems undergoing bifurcation achieve some desirable dynamical behaviours, such as stabilizing an unstable bifurcation solution and delaying the onset of bifurcation, by designing an appropriate controller. Moreover, a very powerful control scheme, named time-delayed feedback control, was proposed by Pyragas (1992), through constructing a control force which utilizes the difference between the present state and its delayed value, i.e. s(t)−s(t−τ). If the stabilized state is reached, the control force will vanish by a proper choice of the time delay τ (see Hövel, 2010). The main advantage of the Pyragas scheme is that it does not need to use a reference system since it generates the control force from information of the system itself. Besides, the Pyragas scheme is easy to be implemented in a practical system. Therefore, in recent years, many researchers have become very interested in Hopf bifurcation control for congestion control models over communication network, especially for those with communication delay (see Ott et al., 1990; Pyragas, 1992; Xiao et al., 2013; Xiao & Cao 2007; Xiao et al., 2014; Guo et al., 2008c; Hövel, 2010; Xu et al., 2014, 2016; Redhu & Gupta (2015), Li et al., 2015; Chen et al., 2000). In Redhu & Gupta (2015) and Li et al. (2015), a time-delayed feedback control method for a lattice hydrodynamic traffic flow model was investigated. Various theories, methods and applications of bifurcation control were surveyed in Chen et al. (2000). Recently, a time-delay feedback control method has been used to control an exponential RED algorithm model in Guo et al. (2008c) and Xiao et al.(2014). Nevertheless, to the best of our knowledge, there are few papers to discuss the bifurcation control of Westwood + TCP flow control model, especially when communication delay is considered. Motivated by the above discussion, in this article, we study Hopf bifurcation control of Westwood + TCP model with communication delay by using the time-delayed feedback control method. Firstly, we develop a time-delayed feedback controller for postponing the occurrence of Hopf bifurcation for Westwood + TCP model with delay. Secondly, we analyse the nonlinear dynamical behaviour of Westwood + TCP model with time-delayed feedback control. Then, we study the effects of control gain parameters on nonlinear dynamical behaviours and the conditions for bifurcation occurrence. Finally, we present numerical results to verify the effectiveness of the proposed control strategy. The rest of the article is organized as follows. Section 2 presents the Westwood + TCP flow control model with time-delayed feedback control. Section 3 gives linear stability and Hopf bifurcation analysis of the Westwood + TCP model with time-delayed feedback control. Section 4 presents calculation of the rightmost characteristic root via Lambert W function. Section 5 provides simulation results. Finally, Section 6 concludes this article. 2. Westwood + TCP flow control model with time-delayed feedback control The Westwood + TCP flow control model proposed by Grieco & Mascolo (2005) is based on an AIADD flow control algorithm and filters the stream of returning acknowledgements. The Westwood + TCP flow control model is given by { dr(t)dt=r(t)[1−pτR2r(t)+p(B(t)τRminτR−r(t))],dB(t)dt=−B(t)T0+1−pT0r(t), (2.1) where the first equation characterizes the evolution of the source sending rate at time t and the second one reflects the evolution of bandwidth demand. p is the drop probability of a segment, τR denotes the mean RTT, T0 is a time constant, r(t) denotes the transmission rate of source at time t, B(t) specifies the bandwidth estimation from the impulse response of a first-order low-pass filter and τRmin stands for the minimum RTT, i.e. slow start threshold to a value of bandwidth estimation times. It is shown in Grieco & Mascolo (2005) that the flow control model is able to improve the steady-state throughput and the tracking of available bandwidth of classic TCP. Although Grieco & Mascolo (2005) analysed the dynamics of Westwood + TCP control model and provided the locally asymptotically stable conditions around equilibrium points, they did not consider communication delay in the Westwood + TCP model. For tackling the above mentioned problem, we propose a Westwood + TCP flow control model with communication delay (see Yu et al., 2016), which is given by { dr(t)dt=r(t−τ)[1−pτR2r(t)+p(B(t)τRminτR−r(t))],dB(t)dt=−B(t−τ)T0+1−pT0r(t−τ), (2.2) where τ denotes communication delay including wired communication delay and wireless delay. We consider communication delay as an important factor that cannot be ignored, because it plays an imperative role in improving network stability, fair bandwidth allocation and resource utilization of high speed wired and wireless communication networks. We study the nonlinear dynamic behaviours, i.e. linear stability and Hopf bifurcation. Taking communication delay as the bifurcation parameter, we derive the linear stability criteria which depend on communication delay. It is observed that Hopf bifurcation analysis is very useful for communication networks with delay, since network is always stable if communication delay is not beyond critical values. In this article, we will design a time-delayed state feedback controller to control the Hopf bifurcation for the Westwood + TCP flow control model (2.2). Similar to Pyragas's method in Pyragas (1992), by adding a time-delayed control force to model (2.2), we can obtain the time-delayed feedback control system as follows { dr(t)dt=r(t−τ)[1−pτR2r(t)+p(B(t)τRminτR−r(t))],dB(t)dt=−B(t−τ)T0+1−pT0r(t−τ)−K[B(t)−B(t−τ)], (2.3) where K≥0 denotes the control gain. It is worth noting that the control force vanishes for an unstable periodic orbit B(t) with period T by choosing the time delay τ=T, i.e. B(t)=B(t−T)=B(t−τ). This is an advantage and important feature of time-delayed feedback control, that is to say, only a minimum knowledge of the system about B(t) is required to achieve target state. Thus, it can be shown that Hopf bifurcation control scheme is easy to be designed and implemented in practical applications. It is clear that for K=0, the model (2.3) is the same as the model (2.2). The main goal of time delayed feedback control is to achieve some desirable dynamical behaviours, such as stabilizing an unstable bifurcation solution and delaying the onset of bifurcation by designing an appropriate controller. Remark 2.1 Stability of ur proposed flow control algorithm is derived by modifying the bandwidth demand estimation function. Therefore, the dynamic delayed feedback controller can be applied to the bandwidth estimation function on the router to achieve the stable sending rate. This is justifiable since it can adjust the evolution of the bandwidth estimation function, and can thus influence the sending rate. To elaborate, the feedback is implemented by measuring the rate r(t) of data packet at the time t and the rate r(t−τ) at the delayed time t−τ on the router, respectively, and taking the difference r(t)−r(t−τ) as the input of the controller. We then use the controller output as the feedback to control the variance of the bandwidth estimation function. Therefore, the evolution of the bandwidth estimation function depends on the current bandwidth estimation B(t), the rate r(t−τ) at the delayed time t−τ and the output of the controller, which can be implemented by the software embedded in the router. 3. Linear stability and Hopf bifurcation analysis with time-delayed feedback control In this section, we first briefly review the results obtained in Yu et al. (2016) about local stability and Hopf bifurcation of model (2.2). Throughout this article, we assume that τ,τRmin,τR≥0 and p are constants. Let (r*,B*) be the non-zero equilibrium point of system (2.2), then we can obtain { r*=1τR[τR−(1−p)τRmin](1−p)p,B*=(1−p)r*. (3.1) Let x(t)=r(t)−r* and y(t)=B(t)−B*. Linearizing system (2.2) around the equilibrium point, by using equation (3.1), we can obtain { dxdt=ax(t)+by(t),dydt=cx(t−τ)+dy(t−τ), (3.2) where a=r*p[(1−p)τRminτR−2], b=r*pτRminτR, c=1−pT0, d=−1T0. Then the characteristic equation of equation (3.2) can be given by D(λ,τ)=λ2−(a+de−λτ)λ+(ad−bc)e−λτ=0. (3.3) Lemma 3.1 (see Yu et al., 2016) When τ=τ0*, equation (3.3) has a simple pair of purely imaginary roots ±iω0*, where ω0*=(d2−a2)+(d2−a2)2+4(ad−bc)22 (3.4) and τ0*=1ω0*{π+arctan[ω0*2d+a(ad−bc)bcω0*]} . (3.5) Theorem 3.1 (see Yu et al. 2016) For system (2.2), the following results hold: (i) When τ<τ0*, the equilibrium point of system (2.2) is locally asymptotically stable. (ii) When τ>τ0*, the equilibrium point of system (2.2) is unstable. (iii) When τ=τ0*, system (2.2) undergoes a Hopf bifurcation at the equilibrium point. In Theorem 3.1, τ0* denotes the critical value of Hopf bifurcation occurrence. In other words, if communication delay is not beyond this value, then system (2.2) is always stable. Otherwise, system (2.2) loses its stability and undergoes a Hopf bifurcation at the equilibrium point. The detailed derivation of the above formulas and theorem can be found in section III in Yu et al. (2016). In the following, by studying the time-delayed control system model (2.3), we will consider how to control the Hopf bifurcation for the Westwood + TCP flow control model (2.2). It is very easy to verify that model (2.2) and model (2.3) have the same equilibrium point( r*,B*), which is shown in (3.1). It is shown that time-delayed feedback control does not need to change the initial state of control system. Linearizing system (2.3) around the equilibrium point, by using equation(3.1), we can obtain { dxdt=ax(t)+by(t),dydt=cx(t−τ)+(d+K)y(t−τ)−Ky(t), (3.6) where a=r*p[(1−p)τRminτR−2], b=r*pτRminτR, c=1−pT0, d=−1T0. Then the characteristic equation of equation (3.6) can be given by D(λ,τ)=[λ2+(K−a)λ−aK]+[−λd+aK+(ad−bc)]e−λτ=0. (3.7) In the following, we will study the existence of Hopf bifurcation of equation(3.6). Then we can obtain the following lemmas. Lemma 3.2 Consider D(λ,τ)=[λ2+(K−a)λ−aK]+[−λd+aK+(ad−bc)]e−λτ=0. (3.8) If τ=0, then all zeros of D(λ,τ) have negative real parts. Proof When τ=0, we have D(λ,0)=λ2−(a+d−K)λ+(ad−bc). Since r*>0,0<p<1,0<τRmin<τR,T0>0, then 0<1−p<1,0<τRminτR<1. We can obtain a=r*p[(1−p)τRminτR−2]<0, then a+d−K=r*p[(1−p)τRminτR−2]+(−1T0)−K<0 and ad−bc=r∗p[(1−p)τRminτR−2](−1T0)−r∗pτRminτR(1−pT0)=2r∗p[τR−τRmin(1−p)]τRT0. By equation (3.1) in Section 3, we know τR−τRmin(1−p)>0. Then, we can obtain ad−bc>0. We also consider △=(a+d−K)2−4(ad−bc)=K2−2K(a+d)+(a−d)2+4bc>0. Then it follows that all zeros of D(λ,0) have negative real parts. This completes the proof. □ Lemma 3.3. When τ=τ0 and 0≤K≤ad−bc−2a, equation (3.7) has a simple pair of purely imaginary roots ±iω0, where ω0=M+M2+4N2, (3.9) where M=d2−a2−K2, N=(ad−bc+aK)2−a2K2 and τ0=1ω0{π+arctan[PQ]}, (3.10) where τ0 denotes the critical value of Hop bifurcation occurrence in (2.3), P=−ω03d+(K−a)ω0(ad−bc)+aω0K(K−a−d), Q=−bcω02+aK(ad−bc+aK+ω02). Proof We know that iω0, ω0>0, is a root of equation (3.7) if and only if iω0 satisfies −ω02+(K−a)iω0−aK+[aK−diω0+(ad−bc)]e−iω0τ=0. Equating the real and imaginary parts of both sides, we get { (ad−bc+aK)cos(ω0τ)−dω0sin(ω0τ)=ω02+aK,dω0cos(ω0τ)+(ad−bc+aK)sin(ω0τ)=(K−a)ω0. (3.11) From equation (3.11), it follows that ω04+[K2+a2−d2]ω02+[a2K2−(ad−bc+aK)2]=0. (3.12) It is clear that equation (3.12) has a unique positive root ω02 when 0≤K≤ad−bc−2a. Therefore, we have ω0=M+M2+4N2, where M=d2−a2−K2 , N=(ad−bc+aK)2−a2K2. From equation (3.11), we also have sinω0τ=−ω03d+(K−a)ω0(ad−bc)+aω0K(K−a−d)ω02d2+(ad−bc+aK)2,cosω0τ=−bcω02+aK(ad−bc+aK+ω02)ω02d2+(ad−bc+aK)2. Then, it follows that τ0=1ω0{π+arctan[PQ]}, where P=−ω03d+(K−a)ω0(ad−bc)+aω0K(K−a−d) , Q=−bcω02+aK(ad−bc+aK+ω02). This completes the proof. □ Next we show that λ=±iω0 are simple roots of equation (3.7) when τ=τ0. Differentiating the function D(λ,τ)=λ2+(K−a)λ−aK+[−dλ+aK+(ad−bc)]e−λτ with respect to λ, we can get dD(λ,τ)dλ=2λ+K−a−de−λτ−τ[−dλ+aK+(ad−bc)]e−λτ. (3.13) Substituting λ=iω0 into equation (3.13) and applying equation (3.11), we can obtain dD(iω0,τ)dλ=A+Bi≠0 (3.14) where A=K−a−dcosω0τ−τ(ω02+aK) , B=2ω0+dsinω0τ+τ(K−a)ω0 when a−K<d. Similarly, we can get dD(−iω0,τ0)dλ≠0. (3.15) Hence, λ=±iω0 are simple roots of equation (3.7) when τ=τ0. Furthermore, according to our previous work (see Yu et al., 2016), we have the following Lemmas. Lemma 3.4 When τ<τ0, all the roots of equation (3.7) have strictly negative real parts. Lemma 3.5 When τ=τ0, except for the pair of purely imaginary roots ±iω0, all other roots of equation (3.7) have strict negative real parts. Lemma 3.6 Let λ(τ)=α(τ)+iω(τ) be the root of equation (3.7) satisfying α(τ0)=0, ω(τ0)=ω0. The following transversally condition holds: dRe(λ(τ))dτ|τ=τ0>0. (3.16) Proof By equation (3.7) with respect to τ and applying the implicit function theorem, we get dλ(τ)dτ=−λ[dλ−(aK+ad−bc)]e−λτ2λ+K−a+[τ(dλ−(aK+ad−bc))−d]e−λτ, then (dλ(τ)dτ)−1=[2λ+K−a]eλτ+τ[dλ−(aK+ad−bc)]−d−λ[dλ−(aK+ad−bc)]. Since λ(τ0)=iω0, hence, by equation (3.11), we can get (dReλ(τ)dτ)−1|τ=τ0=2ω02+a2−d2+K2d2ω02+(aK+ad−bc)2. (3.17) Since 2ω02>d2−a2−K2, then we have (dReλ(τ)dτ)−1|τ=τ0>0. This completes the proof. □ Lemma 3.7 When τ>τ0, equation (3.7) has at least one root with strictly positive real parts. Proof From Lemmas 3.5 and 3.6, by use of thee lemma in Cooke & Grossman (1982), we can see that if τ>τ0, equation (3.7) has at least one root with strictly positive real parts. □ Theorem 3.2 For system (2.3), when 0≤K≤ad−bc−2a, then 1ω0*≤1ω0 holds. Proof By equation (3.9) and (3.4), we know that ω0*=(d2−a2)+(d2−a2)2+4(ad−bc)22 and ω0=M+M2+4N2, where M=d2−a2−K2 and N=(ad−bc+aK)2−a2K2. For 0≤K≤ad−bc−2a, we have ω0*≥ω0, i.e. 1ω0*≤1ω0. This completes the proof. □ In the following, we give the condition that Hopf bifurcation is delayed to occur. Theorem 3.3 For system (2.3), when 0≤K≤ad−bc−2a, then Hopf bifurcation occurrence is postponed via time-delayed feedback, i.e. τ0*≤τ0 is equivalent to f(K)≥0, where f(K) is given by f(K)= [a2K2+amK−Q0]{(a2P0+Q0aω02)K2+ [aP0m−Q0ω0(a2+bc)]K+Q0P1−P0Q1}, where P0=−ω0*3d−aω0*(ad−bc),Q0=−bcω0*2 , P1=−ω03d−aω0(ad−bc),Q1=−bcω02 , m=ad−bc+ω02. Proof Using equations (3.5), (3.10) and Theorem 3.2, if we have τ0*≤τ0, then the following equivalent condition is necessarily satisfied P0Q0≤P1+Kω0(ad−bc)+aω0K(K−a−d)Q1+aK(ad−bc+aK+ω02), where P0=−ω0*3d−aω0*(ad−bc), Q0=−bcω0*2, P1=−ω03d−aω0(ad−bc), Q1=−bcω02. After a simple calculation, we obtain the following result of inequality on control gain K. [a2K2+amK−Q0]{(a2P0+Q0aω02)K2+ [aP0m−Q0ω0(a2+bc)]K+Q0P1−P0Q1}≥0, where m=ad−bc+ω02. In order to facilitate the description, let f(K)= [a2K2+amK−Q0]{(a2P0+Q0aω02)K2+ [aP0m−Q0ω0(a2+bc)]K+Q0P1−P0Q1}. Then τ0*≤τ0 is equivalent to f(K)≥0. This completes the proof. □ It is worth noting that it is very difficult to obtain a closed-form solution of control gain K. However, we can determine K such that τ0*≤τ0 for all K∈A, where A={K|0≤K≤ad−bc−2a}∩{K|f(K)≥0}. To this end, the following procedures are performed: First, we need to determine the relationship between control gain K and equivalent condition function f(K). Figures 1–3 depict the relationships between K and f(K) and illustrate the existence of control gain K when p=0.2,0.3,0.5, respectively. We can observe from Figs 1–3 that there exist some K satisfying f(K)≥0. Second, we need to explore the relationship between control gain K and bifurcation critical value τ0. We can see from Fig. 4 that the amplitude of bifurcation critical value increases with control gain K∈A. Therefore, the time-delayed feedback control plays an important role in hopf bifurcation control, specially at the initial time stage of bifurcation control. Moreover, we have τ0=τ0* for K=0, which implies that the model (2.2) is a special case of the model(2.3). Remark 3.1 Theorem 3.3 reveals the proposed time-delayed feedback controller can delay the onset the Hopf bifurcation when we choose a proper K which satisfies K∈A, that is, there exists one control parameter K which can modulate the threshold of the creation of Hopf bifurcation. Figure 1. View largeDownload slide Relationship between K and f(K) with p=0.2 and τR=1,τRmin=0.3,T0=2. Figure 1. View largeDownload slide Relationship between K and f(K) with p=0.2 and τR=1,τRmin=0.3,T0=2. Figure 2. View largeDownload slide Relationship between K and f(K) with p=0.3 and τR=1,τRmin=0.3,T0=2. Figure 2. View largeDownload slide Relationship between K and f(K) with p=0.3 and τR=1,τRmin=0.3,T0=2. Figure 3. View largeDownload slide Relationship between K and f(K) with p=0.5 and τR=1,τRmin=0.3,T0=2. Figure 3. View largeDownload slide Relationship between K and f(K) with p=0.5 and τR=1,τRmin=0.3,T0=2. Figure 4. View largeDownload slide Relationship between K and τ0, where p=0.2, 0.3 and 0.5, respectively. Figure 4. View largeDownload slide Relationship between K and τ0, where p=0.2, 0.3 and 0.5, respectively. Based on Lemmas 3.2–3.7 and Theorems 3.2 and 3.3, we can obtain the following theorem about local stability and hopf bifurcation of system (2.3) by applying Hopf bifurcation theory for delay differential equation (see Hale, 1977; Hassard et al., 1981) and time-delayed feedback control method (see Pyragas, 1992; Hövel, 2010). Theorem 3.4 For system (2.3), the following results hold: (i) For K∈A, there exists τ0≥τ0* and when τ<τ0, the equilibrium point of system (2.3) is locally asymptotically stable. (ii) For K∈A, there exists τ0≥τ0* and when τ>τ0, the equilibrium point of system (2.3) is unstable. (iii) For K∈A, there exists τ0≥τ0* and when τ=τ0, system (2.3) undergoes a Hopf bifurcation at the equilibrium point. 4. Calculation of the rightmost characteristic root via Lambert W function In this section, we will investigate the behaviour of the real part of the rightmost root and the delay stability region. In fact, when studying the asymptotic stability of the system (3.6), the main goal is to determine necessary and sufficient conditions for stability in either the delay parameter space or the controller-parameter space. It is well known that for a given set of delays, system (3.6) is asymptotically stable if and only if all of the roots of (3.6) lie in the open left-half complex plane C−. More specifically, the equilibrium of system (3.6) is asymptotically stable if only if α0<0, where α0 is given by α0=max{Re(λ):D(λ,τ)=0}, (4.1) where Re(λ) denotes the real part of λ. In other words, the computation of the rightmost characteristic root of time delay system, especially time delay feedback system, becomes an important issue. However, it is difficult to find a closed-form expression of the characteristic roots for general time-delayed systems. Similar to Wang et al. (2008) and Sipahi et al. (2011), we obtain an iteration method, which is based on the Lambert W function, for the calculation of the rightmost root so that the stability of time delay system (3.6) is determined. The Lambert W function W(z) is defined as the solution of a complex transcendental equation W(z)exp(W(z))=z,z∈C. (4.2) Moreover, we assume that λ is a root of the characteristic equation D(λ,τ)=0. Then (a−K)λexp((a−K)λ)=((a−K)λ−D(λ,τ))exp((a−K)λ). (4.3) Due to the property of the Lambert W function, an auxiliary function is constructed as follows. F(λ)=(a−K)λ−W0{((a−K)λ−D(λ,τ))exp((a−K)λ)}, (4.4) where W0 denotes the principal branch of the Lambert W function. And then the rightmost root of D(λ,τ) can be obtained from the following iteration scheme, i.e. Newton–Raphson's scheme. λj+1=λj−F(λj)F'(λj),j=1,2,3,… (4.5) where the derivative can be computed by the property of Lambert W function W′0(z)=W0(z)z+zW0(z). (4.6) Finally, the iteration is stopped if |λj+1−λj|<ε, where ε is a given tolerance value and λ0 denotes the initial value. The standard algorithm in solving nonlinear equation related to the Lambert W function can be implemented in the software Matlab. Numerical simulation about the real part of the rightmost roots of the characteristic equation D(λ,τ)=0 is presented in Section 5.5. 5. Numerical simulation In this section, we do some numerical simulations to verify the effect of the proposed time-delayed feedback controller on the Hopf bifurcation control of Westwood + TCP flow control model with communication delay for wireless communication networks. We first demonstrate the relationships between control gain K and condition function f(K), and between control gain K and bifurcation critical value τ0. We then compare the dynamics of Westwood + TCP flow control model in the following cases: control gain K=0, i.e. without time-delayed feedback control, and control gain K>0, i.e. with time-delayed feedback control. 5.1. Relationship between control gain K and condition function f(K) In this subsection, we will show the existence of control gain K as well as the correctness of Theorem 3.3 by observing the phase portraits of K−f(K) which are plotted in Figs 1–3 when p=0.2,0.3,0.5, respectively. Figures 1–3 depict the evolution of f(K)≥0 for different control gain K∈A. We can see that control gain K indeed exists by Theorem 3.3, that is, there exist some K satisfying f(K)≥0. Also, we can observe that if K=0, then f(K)=0, which means that model (2.2) is a special case of model (2.3). It is clear to observe from Figs 1–3 that if K∈A, then f(K)≥0, which verifies the correctness of Theorem 3.3. In addition, we can find that not all values of K can satisfy the conditions in Theorem 3.3, such as K=0.3 for p=0.2,0.3. 5.2. Relationship between control gain K and bifurcation critical value τ0 In this subsection, we present numerical results to verify the analytical prediction obtained in previous section. We show the function phase of K−τ0 in Fig. 4, where τR=1,τRmin=0.3,T0=2 and p=0.2,0.3,0.5, respectively. It is easy to find from Fig. 4 that when control gain K=0, we have τ0=τ0*=3.6435. Besides, when we choose 0<K<0.2 satisfying K∈A, we have that τ0 increases with control gain K and τ0>τ0*. Thus we can choose an appropriate value of K to control the Hopf bifurcation by delaying the onset of the Hopf bifurcation. Moreover, we can observe that as K increases such that it does not belong to the set A, the controller cannot postpone the occurrence of the Hopf bifurcation. 5.3. The case: control gain K=0 In this subsection, we use the results obtained in Sections 2 and 3 to verify the existence of the Hopf bifurcation and calculate the Hopf bifurcation value for two scenarios with p=0.2,τR=1,τRmin=0.3 and T0=2. By substituting control gain K=0 in equation (3.1), we can obtain that non-zero equilibrium point of system (r*,B*)=(2.2942,1.8353) and a=−0.8075 , b=0.1376, c=0.4 and d=−0.5. Moreover, based on equation (3.4) and (3.5), we have ω0*=0.4488, τ0*=3.6435. These results show that the system equilibrium (r*,B*) is asymptotically stable when τ<τ0* (see Figs 5–7, τ=3.6<3.6435). When τ passes through the critical value τ0*=3.6435, (r*,B*) loses its stability and a Hopf bifurcation occurs. In this situation, we can observe from Figs 8–10 that a family of periodic solution bifurcated out from (r*,B*). Figure 5. View largeDownload slide Waveform plot of t−r(t) with τ=3.6 and K=0. Figure 5. View largeDownload slide Waveform plot of t−r(t) with τ=3.6 and K=0. Figure 6. View largeDownload slide Waveform plot of t−B(t) with τ=3.6 and K=0. Figure 6. View largeDownload slide Waveform plot of t−B(t) with τ=3.6 and K=0. Figure 7. View largeDownload slide Phase portrait of r(t)−B(t) with τ=3.6 and K=0. Figure 7. View largeDownload slide Phase portrait of r(t)−B(t) with τ=3.6 and K=0. Figure 8. View largeDownload slide Waveform plot of t−r(t) with τ=3.645 and K=0. Figure 8. View largeDownload slide Waveform plot of t−r(t) with τ=3.645 and K=0. Figure 9. View largeDownload slide Waveform plot of t−B(t) with τ=3.645 and K=0.} Figure 9. View largeDownload slide Waveform plot of t−B(t) with τ=3.645 and K=0.} Figure 10. View largeDownload slide Phase portrait of r(t)−B(t) with τ=3.645 and K=0.} Figure 10. View largeDownload slide Phase portrait of r(t)−B(t) with τ=3.645 and K=0.} 5.4. The case: control gain K>0 By choosing K=0.1 and applying equation (3.1), we can obtain that non-zero equilibrium point of system (r*,B*)=(2.2942,1.8353) and τ0=5.6180>τ0*=3.6435. It is worth noting that the proposed flow control model (2.3) has the same equilibrium point as that of the original flow control model (2.2), but the bifurcation critical value τ0 increases from 3.6435 to 5.6180. That is to say, the onset of Hopf bifurcation is delayed. We choose K=0.1, τ=5.6>3.6435. Hence, according to Theorem x, the controlled model (2.3) converges to the equilibrium point (r*,B*) rather than undergoes a Hopf bifurcation, as shown in Figs 11–13. When τ=5.92 passes through the critical value τ0=5.6180, (r*,B*) loses its stability and a Hopf bifurcation occurs. In other words, a family of periodic solution bifurcates out from (r*,B*), as shown in Figs 14–16. Figure 11. View largeDownload slide Waveform plot of t−r(t) with τ=5.6 and K=0.1. Figure 11. View largeDownload slide Waveform plot of t−r(t) with τ=5.6 and K=0.1. Figure 12. View largeDownload slide Waveform plot of t−B(t) with τ=5.6 and K=0.1. Figure 12. View largeDownload slide Waveform plot of t−B(t) with τ=5.6 and K=0.1. Figure 13. View largeDownload slide Phase portrait of r(t)−B(t) with τ=5.6 and K=0.1. Figure 13. View largeDownload slide Phase portrait of r(t)−B(t) with τ=5.6 and K=0.1. Figure 14. View largeDownload slide Waveform plot of t−r(t) with τ=5.92 and K=0.1. Figure 14. View largeDownload slide Waveform plot of t−r(t) with τ=5.92 and K=0.1. Figure 15. View largeDownload slide Waveform plot of t−B(t) with τ=5.92 and K=0.1. Figure 15. View largeDownload slide Waveform plot of t−B(t) with τ=5.92 and K=0.1. Figure 16. View largeDownload slide Phase portrait of r(t)−B(t) with τ=5.92 and K=0.1. Figure 16. View largeDownload slide Phase portrait of r(t)−B(t) with τ=5.92 and K=0.1. Therefore, from theoretical and simulation results, we can conclude that the proposed time-delayed feedback control plays an important role in Hopf bifurcation control for the Westwood + TCP model with communication delay. If we choose a proper control gain K, the flow control model may not undergo a Hopf bifurcation even for large τ. This verifies and indicates that the time-delayed feedback controller can delay the onset of Hopf bifurcation so as to guarantee a stationary state for a larger communication delay in wireless network. 5.5. Behaviour of the real part of the rightmost characteristic root In this subsection, we investigate behaviour of the real part of the rightmost root by using an iteration method with Lambert W function discussed in Section 4. Moreover, we present rightmost characteristic roots on complex plane to show the location of the rightmost roots of system (3.6) by designing different delay parameter or control gain parameter. In Fig. 17, this plot depicts how the real part of the rightmost characteristic roots behaves with respect to delay parameter τ. with τR=1,τRmin=0.3,T0=2 and p=0.2,λ0=−5+2i. The sign change of the real part indicates that system (3.6) switches from stability to instability. It is observed from Fig. 17 that the real part of the rightmost root of the system (3.6) for K=0 is always negative when τ∈[0,τ0*), where τ0*=3.6435. The system becomes unstable when τ crosses the delay margin τ0* of the system (3.6) for K=0. On the other hand, we choose control gain K=0.1 and observe from Fig. 17 that the real part of the rightmost root of the system (3.6) is always negative when τ∈[0,τ0), where τ0=5.6180 and τ0>τ0*. The system becomes unstable when τ crosses the delay margin τ0 of the system (3.6) for K=0.1. Figure 17. View largeDownload slide Behaviour of the real part of the rightmost root with respect to delay parameter τ∈[0,8] with p=0.2 and control parameter K=0, 0.1. Figure 17. View largeDownload slide Behaviour of the real part of the rightmost root with respect to delay parameter τ∈[0,8] with p=0.2 and control parameter K=0, 0.1. We study the behaviour of the rightmost root of system (3.6) as delay value is increased from zero. As shown in Fig. 17, we see that the real part of the rightmost root changes it signs as delay parameter τ varies along the delay axis. Moreover, it is shown that with proposed controller, one can increase the allowable delay critical value of the communication delay, and postpone the onset of the undesired Hopf bifurcation. 6. Conclusion In this article, for wireless network, we have addressed the problem of how to control Hopf bifurcation for Westwood + TCP flow control model with communication delay. In order to stabilize the flow control system with delay, we propose a time-delayed state feedback control method to control Hopf bifurcation by choosing proper control parameters. Furthermore, we analyse the local stability of equilibrium and the Hopf bifurcation of Westwood + TCP model. Numerical simulations verify the validity of this control method. Up to now, there is little work to investigate the problem of bifurcation control for Westwood + TCP flow control systems, especially with communication delay. Funding The work described in this paper was partially supported by the National Natural Science Foundation of China (No. 61373179, 61373178, 61402381, 61503309), Natural Science Key Foundation of Chongqing (cstc2015jcyjBX0094), the Fundamental Research Funds for the Central Universities (XDJK2015C010, XDJK2015D023, XDJK2016A011, XDJK2016D047), Natural Science Foundation of Chongqing (CSTC2016JCYJA0449), China Postdoctoral Science Foundation (2016M592619), Chongqing Postdoctoral Science Foundation (XM2016002), The Doctoral Research Funds of Southwest University (No. SWU113020) and Educational Science Research Programming of Hubei Province (NO.B20132508). 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IMA Journal of Mathematical Control and Information – Oxford University Press
Published: Sep 21, 2018
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