Time-varying delayed feedback control for an internet congestion control model
Shu Zhang Jian Xu
Discrete & Continuous Dynamical Systems - B 2011, 16(2): 653-668 doi: 10.3934/dcdsb.2011.16.653
A proportionally-fair controller with time delay is considered to control Internet congestion. The time delay is chosen to be a controllable parameter. To represent the relation between the delay and congestion analytically, the method of multiple scales is employed to obtain the periodic solution arising from the Hopf bifurcation in the congestion control model. A new control method is proposed by perturbing the delay periodically. The strength of the perturbation is predicted analytically in order that the oscillation may disappear gradually. It implies that the proved control scheme may decrease the possibility of the congestion derived from the oscillation. The proposed control scheme is verified by the numerical simulation.
keywords: delayed differential equation time-varying delay. Internet congestion control method of multiple scales Hopf bifurcation
Localized Birkhoff average in beta dynamical systems
Bo Tan Bao-Wei Wang Jun Wu Jian Xu
Discrete & Continuous Dynamical Systems - A 2013, 33(6): 2547-2564 doi: 10.3934/dcds.2013.33.2547
In this note, we investigate the localized multifractal spectrum of Birkhoff average in the beta-dynamical system $([0,1], T_{\beta})$ for general $\beta>1$, namely the dimension of the following level sets $$ \Big\{x\in [0,1]: \lim_{n\to \infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi(T^jx)=f(x)\Big\}, $$ where $f$ and $\psi$ are two continuous functions defined on the unit interval $[0,1]$. Instead of a constant function in the classical multifractal cases, the function $f$ here varies with $x$. The method adopted in the proof indicates that the multifractal analysis of Birkhoff average in a general $\beta$-dynamical system can be achieved by approximating the system by its subsystems.
keywords: $\beta$-expansion Hausdorff dimension. localized Birkhoff average

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