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Oscillation deduced from the NeimarkSacker bifurcation in a discrete dual Internet congestion control algorithm
Advances in Difference Equations volume 2013, Article number: 136 (2013)
Abstract
In this paper, we consider a dual Internet congestion control algorithm applied to a nonstandard finitedifference scheme, which responds to congestion signals from the network. By choosing delay as a bifurcation parameter, the local asymptotic stability of the positive equilibrium and the existence of NeimarkSacker bifurcations are analyzed. Then the explicit algorithm for determining the direction of NeimarkSacker bifurcations and the stability of invariant closed curves are derived. In addition, we give specific examples to illustrate the phenomenon that coincides with our theoretical results.
MSC:34K18, 34K20.
1 Introduction
Congestion control algorithms and active queue management (AQM) for Internet have been the focus of intense research since the seminal work of Kelly et al. [1]. Congestion control schemes can be divided into three classes: primal algorithms, dual algorithms and primaldual algorithms [2]. In primal algorithms, the users adapt the source rates dynamically based on the route prices (the congestion signal generated by the link), and the links select a static law to determine the link prices directly from the arrival rates at the links. However, in dual algorithms, the links adapt the link prices dynamically based on the link rates, and the users select a static law to determine the source rates directly from the route prices and the source parameters. Primaldual algorithms combine these two schemes and dynamically compute both user rates and link prices. In [3], a stability condition was provided for a single proportionally fair congestion controller with delayed feedback. Since then, this result was extended to networks in [4, 5]. In [4], the author studied the case of a network with users having arbitrary propagation delays to and from any links in the network. In [5], the authors studied a more general network with different roundtrip delays amongst different TCP connections.
In this paper, we consider a dual congestion control algorithm, which can be formulated as a congestion control system with feedback delay. The model is described by [7, 8]
where $f(\cdot )$ is a nonnegative continuous, strictly decreasing demand function and has at least thirdorder continuous derivatives. The scalar c is the capacity of the bottleneck link and the variable p is the price at the link. k is a positive gain parameter, τ is the sum of forward and return delays. A great deal of research has been devoted to the global stability, periodic solutions and other properties of this model, for which we refer to [7, 8].
Considering the need of scientific computation and realtime simulation, our interest is focused on the behaviors of a discrete dynamics system corresponding to (1.1). In this paper, we use a nonstandard finitedifference scheme [11, 12] to make the discretization for system (1.1). Firstly, we consider the autonomous delay differential equations
Then we get the following:
with the function Φ such that
where $h=\frac{1}{m}$ ($m\in {\mathbb{Z}}_{+}$) stands for stepsize, and ${u}_{k}$ denotes the approximate value to $u(kh)$. This method can be seen as a modified forward Euler method.
The NeimarkSacker bifurcation is the discrete analogue of the Hopf bifurcation that occurs in continuous systems. The Hopf bifurcation is extremely important in the continuous dual congestion control algorithm [7, 8]. Similarly, the NeimarkSacker bifurcation is also highly relevant to the discrete dual congestion control algorithm. The purpose of this paper is to discuss this version as a discrete dynamical system by using NeimarkSacker bifurcation theory of discrete systems.
The paper is organized as follows. In Section 2, we analyze the distribution of the characteristic equation associated with the discrete model and obtain the existence of the local NeimarkSacker bifurcation. In Section 3, the direction and stability of a closed invariant curve from the NeimarkSacker bifurcation of the discrete delay model are determined by using the theories of discrete systems in [13]. In Section 4, some computer simulations are performed to illustrate the theoretical results. Conclusions are given in the final section.
2 NeimarkSacker bifurcation analysis
Let $u(t)=p(\tau t)$. Then (1.1) can be rewritten as
Assume Eq. (2.1) has a positive equilibrium point ${u}^{\ast}$. Then ${u}^{\ast}$ satisfies
If we employ the nonstandard finitedifference scheme (1.2) to Eq. (2.1) and choose the function Φ as
it yields the delay difference equation
Note that ${u}^{\ast}$ also is the unique positive equilibrium of (2.2). Set ${y}_{n}={u}_{n}{u}^{\ast}$, then it follows that
By introducing a new variable ${Y}_{n}={({y}_{n},{y}_{n1},\dots ,{y}_{nm})}^{T}$, we can rewrite (2.3) in the form
where $F={({F}_{0},{F}_{1},\dots ,{F}_{m})}^{T}$, and
Clearly the origin is an equilibrium of (2.4), and the linear part of (2.4) is
where
The characteristic equation of A is given by
It is well known that the stability of the zero equilibrium solution of (2.4) depends on the distribution of zeros of the roots of (2.6). In this paper, we employ the results from Zhang et al. [9] and He et al. [10] to analyze the distribution of zeros of characteristic Eq. (2.6). In order to prove the existence of the local NeimarkSacker bifurcation at equilibrium, we need some lemmas as follows.
Lemma 2.1 There exists a $\overline{\tau}>0$ such that for $0<\tau <\overline{\tau}$ all roots of (2.6) have modulus less than one.
Proof When $\tau =0$, (2.6) becomes
The equation has, at $\tau =0$, an mfold root $\lambda =0$ and a simple root $\lambda =1$.
Consider the root $\lambda (\tau )$ such that $\lambda (0)=1$. This root depends continuously on τ and is a differential function of τ. From (2.6), we have
and
Noticing $f(\cdot )$ is a nonnegative continuous, strictly decreasing function, we have
So, with the increase of $\tau >0$, λ cannot cross $\lambda =1$. Consequently, all roots of Eq. (2.6) lie in the unit circle for sufficiently small positive $\tau >0$, and the existence of the $\overline{\tau}$ follows. □
A NeimarkSacker bifurcation occurs when a complex conjugate pair of eigenvalues of A cross the unit circle as τ varies. We have to find values of τ such that there are roots on the unit circle. Denote the roots on the unit circle by ${e}^{i{\omega}^{\ast}}$, ${\omega}^{\ast}\in (\pi ,\pi )$. Since we are dealing with complex roots of a real polynomial, we only need to look for ${\omega}^{\ast}\in (0,\pi )$.
Lemma 2.2 There exists an increasing sequence of values of the time delay parameter $\tau ={\tau}_{j}$, $j=0,1,2,\dots ,[\frac{m1}{2}]$ satisfying
where ${\omega}_{j}\in (\frac{2j\pi}{m},\frac{(2j+1)\pi}{m})$, $j=0,1,2,\dots ,[\frac{m1}{2}]$.
Proof Denote the roots of Eq. (2.6) on the unit circle by ${e}^{i{\omega}^{\ast}}$, ${\omega}^{\ast}\in (0,\pi )$. Then
Separating the real part and the imaginary part from Eq. (2.10), there are
and
So,
Then the roots ${e}^{i{\omega}^{\ast}}$ of (2.6) satisfy Eqs. (2.10)(2.13). From (2.12) we get
Substituting (2.14) into (2.11), we have
Then Eq. (2.15) has a unique solution ${\omega}^{\ast}$ in every interval $(\frac{2j\pi}{m},\frac{(2j+1)\pi}{m})$, $j=0,1,2,\dots ,[\frac{m1}{2}]$, we set
From (2.14), we have
This completes the proof. □
Lemma 2.3 Let ${\lambda}_{j}(\tau )={r}_{j}(\tau ){e}^{i{\omega}_{j}(\tau )}$ be a root of (2.6) near $\tau ={\tau}_{j}$ satisfying ${r}_{j}({\tau}_{j})=1$ and ${\omega}_{j}({\tau}_{j})={\omega}_{j}$. Then
Proof From (2.11) and (2.12), we obtain that
It is easy to see that
From (2.7), (2.8) and using (2.18)(2.20), we have
This completes the proof. □
Lemmas 2.12.3 immediately lead to the stability of the zero equilibrium of Eq. (2.3), and equivalently, of the positive equilibrium ${u}^{\ast}$ of Eq. (2.2). So, we have the following results on stability and bifurcation in system (2.2).
Theorem 2.1 There exists a sequence of values of the timedelay parameter $0<{\tau}_{0}<{\tau}_{1}<\cdots <{\tau}_{[\frac{m1}{2}]}$ such that the positive equilibrium ${u}^{\ast}$ of Eq. (2.2) is asymptotically stable for $\tau \in [0,{\tau}_{0})$ and unstable for $\tau >{\tau}_{0}$. Equation (2.2) undergoes a NeimarkSacker bifurcation at the positive equilibrium ${u}^{\ast}$ when $\tau ={\tau}_{j}$, $j=0,1,2,\dots ,[\frac{m1}{2}]$, where ${\tau}_{j}$ satisfies (2.13).
3 Direction and stability of the NeimarkSacker bifurcation
In the previous section, we have obtained the conditions under which a family of invariant closed curves bifurcates from the steady state at the critical value $\tau ={\tau}_{j}$, $j=0,1,2,\dots ,[\frac{m1}{2}]$. Without loss of generality, denote the critical value $\tau ={\tau}_{j}$ by ${\tau}^{\ast}$. In this section, following the idea of Hassard et al. [14], we study the direction and stability of the invariant closed curve when $\tau ={\tau}_{0}$ in the discrete Internet congestion control model. The method we use is based on the theories of a discrete system by Kuznetsov [13].
Rewrite Eq. (2.3) as
So, system (2.3) is turned into
where
and
Let $q=q({\tau}_{0})\in {\mathbb{C}}^{m+1}$ be an eigenvector of A corresponding to ${e}^{i{\omega}_{0}}$, then
We also introduce an adjoint eigenvector ${q}^{\ast}={q}^{\ast}(\tau )\in {\mathbb{C}}^{m+1}$ having the properties
and satisfying the normalization $\u3008{q}^{\ast},q\u3009=1$, where $\u3008{q}^{\ast},q\u3009={\sum}_{i=0}^{m}{\overline{q}}_{i}^{\ast}{q}_{i}$.
Lemma 3.1 (See [15])
Define a vectorvalued function $q:\mathbb{C}\to {\mathbb{C}}^{m+1}$ by
If ξ is an eigenvalue of A, then $Ap(\xi )=\xi p(\xi )$.
In view of Lemma 3.1, we have
Lemma 3.2 Suppose ${q}^{\ast}={({q}_{0}^{\ast},{q}_{1}^{\ast},\dots ,{q}_{m}^{\ast})}^{T}$ is the eigenvector of ${A}^{T}$ corresponding to the eigenvalue ${e}^{i{\omega}_{0}}$, and $\u3008{q}^{\ast},q\u3009=1$. Then
where $\alpha =(1{e}^{\tau kh}){u}^{\ast}{f}^{\prime}({u}^{\ast})$ and
Proof Assign ${q}^{\ast}$ satisfies ${A}^{T}{q}^{\ast}=\overline{z}{q}^{\ast}$ with $\overline{z}={e}^{i{\omega}_{0}}$, then the following identities hold:
Let ${q}_{m}^{\ast}=\alpha {e}^{i{\omega}_{0}}\overline{D}$, then
From normalization $\u3008{q}^{\ast},q\u3009=1$ and computation, Eq. (3.7) holds. □
Let $a(\lambda )$ be a characteristic polynomial of A and ${\lambda}_{0}={e}^{i{\omega}_{0}}$. Following the algorithms in [13] and using a computation process similar to that in [15], we can compute an expression for the critical coefficient ${c}_{1}({\tau}_{0})$,
where
and
By (3.3), (3.5) and Lemma 3.2, we get
Substituting these into (3.9), we can obtain ${c}_{1}({\tau}_{0})$.
Lemma 3.3 (See [6])
Given the map (2.4), assume

(1)
$\lambda (\tau )=r(\tau ){e}^{i\omega (\tau )}$, where $r({\tau}^{\ast})=1$, ${r}^{\prime}({\tau}^{\ast})\ne 0$ and $\omega ({\tau}^{\ast})={\omega}^{\ast}$;

(2)
${e}^{ik{\omega}^{\ast}}\ne 1$ for $k=1,2,3,4$;

(3)
$Re[{e}^{i{\omega}^{\ast}}{c}_{1}({\tau}^{\ast})]\ne 0$.
Then an invariant closed curve, topologically equivalent to a circle, for map (2.4) exists for τ in a one side neighborhood of ${\tau}^{\ast}$. The radius of the invariant curve grows like $O(\sqrt{\tau {\tau}^{\ast}})$. One of the four cases below applies.

(1)
${r}^{\prime}({\tau}^{\ast})>0$, $Re[{e}^{i{\omega}^{\ast}}{c}_{1}({\tau}^{\ast})]<0$. The origin is asymptotically stable for $\tau <{\tau}^{\ast}$ and unstable for $\tau >{\tau}^{\ast}$. An attracting invariant closed curve exists for $\tau >{\tau}^{\ast}$.

(2)
${r}^{\prime}({\tau}^{\ast})>0$, $Re[{e}^{i{\omega}^{\ast}}{c}_{1}({\tau}^{\ast})]>0$. The origin is asymptotically stable for $\tau <{\tau}^{\ast}$ and unstable for $\tau >{\tau}^{\ast}$. A repelling invariant closed curve exists for $\tau <{\tau}^{\ast}$.

(3)
${r}^{\prime}({\tau}^{\ast})<0$, $Re[{e}^{i{\omega}^{\ast}}{c}_{1}({\tau}^{\ast})]<0$. The origin is asymptotically stable for $\tau >{\tau}^{\ast}$ and unstable for $\tau <{\tau}^{\ast}$. An attracting invariant closed curve exists for $\tau <{\tau}^{\ast}$.

(4)
${r}^{\prime}({\tau}^{\ast})<0$, $Re[{e}^{i{\omega}^{\ast}}{c}_{1}({\tau}^{\ast})]>0$. The origin is asymptotically stable for $\tau >{\tau}^{\ast}$ and unstable for $\tau <{\tau}^{\ast}$. A repelling invariant closed curve exists for $\tau >{\tau}^{\ast}$.
From the discussion in Section 2, we know that ${r}^{\prime}({\tau}^{\ast})>0$; therefore, by Lemma 3.3, we have the following result.
Theorem 3.1 For Eq. (2.2), the positive equilibrium ${u}^{\ast}$ is asymptotically stable for $\tau \in [0,{\tau}_{0})$ and unstable for $\tau >{\tau}_{0}$. An attracting (repelling) invariant closed curve exists for $\tau >{\tau}_{0}$ if $Re[{e}^{i{\omega}_{0}}{c}_{1}({\tau}_{0})]<0$ (>0).
4 Computer simulation
In this section, we confirm our theoretical analysis by numerical simulation. We choose $f(u)=\frac{1}{u}$ as in [8] and set $k=0.01$, $m=20$, $c=50$. Then ${\tau}_{0}=3.0$ is the NeimarkSacker bifurcation value.
Figures 14 are about delay difference Eq. (2.2) when step size $h=0.05$.
In Figures 1 and 2, we show the waveform plot and phase plot for (2.2) with initial values ${u}_{j}=0.05$ ($j=0,1,\dots ,20$) for $\tau =2.9<{\tau}_{0}=3.0$. The equilibrium ${u}^{\ast}=0.02$ of Eq. (2.2) is asymptotically stable. In Figures 3 and 4 we show the waveform plot and phase plot for (2.2) with initial values ${u}_{j}=0.05$ ($j=0,1,\dots ,20$). The equilibrium ${u}^{\ast}=0.02$ of (2.2) is unstable for $\tau =3.1>{\tau}_{0}=3.0$. When τ varies and passes through ${\tau}_{0}=3.0$, the equilibrium loses its stability and an invariant closed curve bifurcates from the equilibrium for $\tau =3.1>{\tau}_{0}=3.0$. That is the delay difference Eq. (2.2) which has a NeimarkSacker bifurcation at ${\tau}_{0}$.
5 Conclusion
In this paper, we focus our study on the NeimarkSacker bifurcation in a fair dual algorithm of an Internet congestion control system. By using communication delay as a bifurcation parameter, we have shown that when the communication delay of the system passes through a critical value, a family of periodic orbits bifurcates from the equilibrium point. Furthermore, we have analyzed the stability and direction of the bifurcating periodic solutions. The results of numerical simulations agree with the theoretical results very well and verify the theoretical analysis.
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Acknowledgements
The author is grateful to anonymous referees for their excellent suggestions, which greatly improved the paper. Also, this work was supported by the Foundation of Fujian Education Bureau (JB12046).
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Li, Y. Oscillation deduced from the NeimarkSacker bifurcation in a discrete dual Internet congestion control algorithm. Adv Differ Equ 2013, 136 (2013). https://doi.org/10.1186/168718472013136
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Keywords
 dual congestion control
 NeimarkSacker bifurcation
 stability
 time delay
 discrete dynamic system