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DCDS-B

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.

DCDS

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.

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