scholarly journals Periodic, Quasi-Periodic and Chaotic Oscillations in Two Heterogeneous AIMD/RED Network Congestion Models with State-Dependent Round-Trip Delays

2021 ◽  
Vol 31 (06) ◽  
pp. 2150124
Author(s):  
Lijun Pei ◽  
Fanxin Wu ◽  
Shuo Wang
Keyword(s):  
Chaotic Oscillation ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Network Congestion ◽  
Round Trip ◽  
Periodic Oscillations ◽  
Analytical Approximations ◽  
State Dependent ◽  
Period 2 ◽  
Improved Model

AIMD and RED are two dominant algorithms for controlling Internet congestion. So this paper explores the periodic solutions and complex dynamical phenomena in the state-dependent round-trip delayed AIMD/RED network congestion model with heterogeneous flows, and its improved model. We first use the semi-analytical and semi-numerical method, known as the harmonic balance method with alternating frequency/time (HB-AFT) domain technique, to derive the analytical approximations of periodic oscillations of the system. The obtained results are compared with the numerical results by WinPP, and they show good consistency. At the same time, this suggests that the method used in this paper is correct and valid. Then for the sake of making the system more realistic, we improve the model by using the hyperbolic tangent function. We obtain the approximate solutions, and find some rich dynamical behaviors of this delayed heterogeneous system, including Period-1 to torus, Period-1 to Period-2 to Period-3 motions and two kinds of mechanisms of chaos, i.e. the windows of Period-2 and Period-3 orbits to chaos, where to the best knowledge of the authors, the former route has never been reported. The periodic oscillations may induce synchronization and further congestion, where chaotic oscillation usually means that the system is unstable and may even collapse. Hence, we need to avoid these abundant dynamics discovered in this paper because they are undesirable and harmful. The derived results can help researchers better understand the performance of the AIMD/RED system, and they can be a guide for choosing parameters in a suitable range in order to maintain the network stability and optimize system performance.

Hopf Bifurcation of the Wireless Network Congestion Model with State-Dependent Round Trip Delay

2018 ◽  
Vol 28 (09) ◽  
pp. 1850110 ◽  
Author(s):  
Lijun Pei ◽  
Yangyang Wu
Keyword(s):  
Hopf Bifurcation ◽  
Wireless Network ◽  
Work Conditions ◽  
Simulation Software ◽  
Network Congestion ◽  
Round Trip ◽  
Nonlinear Simulation ◽  
State Dependent ◽  
And Performance ◽  
Round Trip Delay

In this paper, the bifurcation analysis software, DDE-BIFTOOL, is employed to analyze the Hopf bifurcation of the wireless network congestion model with state-dependent round trip delay. Hopf bifurcations are investigated for the four typical work conditions. The corresponding stable and unstable bifurcating periodic solutions are quantitatively and qualitatively verified by nonlinear simulation software, WinPP, respectively, which agree with those of DDE-BIFTOOL very well. The results imply that the channel loss probabilities [Formula: see text] and [Formula: see text] can play a more important role than the speed of the network, i.e. the related link bandwidth [Formula: see text]. For larger [Formula: see text] and [Formula: see text] in Cases 3 and 4, the smaller [Formula: see text] and [Formula: see text] can induce Hopf bifurcation. This will result in the loss of stability and performance degradation. So [Formula: see text] and [Formula: see text] should be set smaller to avoid congestion, providing a sound theoretical basis and instructions for the congestion control of the wireless network.

On the Perturbation Methods for Vibration Analysis of Linear Time-Varying Systems

2016 ◽  
Vol 08 (03) ◽  
pp. 1650035 ◽  
Author(s):  
Xiao-Dong Yang ◽  
Ming Liu ◽  
Wei Zhang ◽  
Ying-Jing Qian ◽  
Roderick V. N. Melnik
Keyword(s):  
Perturbation Methods ◽  
Multiple Scales ◽  
Linear Time ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Beam Model ◽  
Time Varying ◽  
Analytical Approximations ◽  
Time Varying Systems ◽  
New Perspective

Some perturbation methods in the studying vibrations of the linear time-varying (LTV) system are discussed. Three classical perturbation methods, namely, averaging method, harmonic balance method, and multiple scales method with linear scales, have been used from a new perspective based on analytical approximations to the corresponding LTV ordinary differential equations. The deploying beam model has been taken as an example to validate the explicit approximate solutions obtained by these perturbation methods. It is demonstrated that such approximate solutions have good agreement with numerical and exact solutions, excluding the vicinity of the turning point.

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

scholarly journals The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Y. H. Qian ◽  
J. L. Pan ◽  
S. P. Chen ◽  
M. H. Yao
Keyword(s):  
Exact Solutions ◽  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Nonlinear Dynamical Systems ◽  
Harmonic Balance Method ◽  
Time History ◽  
Approximate Solutions ◽  
Balance Method ◽  
Lumped Mass ◽  
Strongly Nonlinear

The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems.

The Residue Harmonic Balance Method for Duffing-Van Der Pol Oscillator with Fractional Derivative

2013 ◽  
Vol 774-776 ◽  
pp. 103-106
Author(s):  
Xin Xue ◽  
Lian Zhong Li ◽  
Dan Sun
Keyword(s):  
Fractional Derivative ◽  
Harmonic Balance ◽  
Numerical Solutions ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Solution Procedure ◽  
Balance Method ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Fractional Orders

Duffing-van der Pol oscillator with fractional derivative was constructed in this paper. The solution procedure was proposed with the residue harmonic balance method. The effect of different fractional orders on resonance responses of the system in steady state were analyzed for an example without parameters. The approximate solutions were contrasted with numerical solutions. The results show that the residue harmonic balance method to Duffing-van der Pol differential equation with fractional derivative is very valid.

ANALYTICAL APPROXIMATE SOLUTIONS TO LARGE-AMPLITUDE FREE VIBRATIONS OF UNIFORM BEAMS ON PASTERNAK FOUNDATION

2014 ◽  
Vol 06 (06) ◽  
pp. 1450075 ◽  
Author(s):  
YONGPING YU ◽  
BAISHENG WU
Keyword(s):  
Large Amplitude ◽  
Approximate Solutions ◽  
Free Vibrations ◽  
Integration Scheme ◽  
Nonlinear Frequency ◽  
Pasternak Foundation ◽  
Analytical Approximations ◽  
Numerical Integration Scheme ◽  
Amplitude Vibration ◽  
Large Amplitude Vibration

This paper is concerned with the large-amplitude vibration behavior of simply supported and clamped uniform beams, with axially immovable ends, on Pasternak foundation. The combination of Newton's method and harmonic balance one is used to deal with these vibrations. Explicit and brief analytical approximations to nonlinear frequency and periodic solution of the beams for various values of the two stiffness parameters of the Pasternak foundation, small as well as large amplitudes of oscillation are presented. The analytical approximate results show excellent agreement with those from numerical integration scheme. Due to brevity of expressions, the present analytical approximate solutions are convenient to investigate effects of various parameters on the large-amplitude vibration response of the beams.

Analytical Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Numerical Simulations ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Balance Method ◽  
Periodic Motions ◽  
Conditions Of Stability ◽  
Generalized Harmonic Balance Method

The analytical solutions of the period-1 motions for a hardening Duffing oscillator are presented through the generalized harmonic balance method. The conditions of stability and bifurcation of the approximate solutions in the oscillator are discussed. Numerical simulations for period-1 motions for the damped Duffing oscillator are carried out.

scholarly journals Harmonic Balance Solution Of Coupled Nonlinear Non-Conservative Differential Equation

2013 ◽  
Vol 32 ◽  
pp. 1-14
Author(s):  
M Saifur Rahman ◽  
M Majedur Rahman ◽  
M Sajedur Rahaman ◽  
M Shamsul Alam
Keyword(s):  
Differential Equation ◽  
Numerical Solution ◽  
Limit Cycle ◽  
Nonlinear Differential Equation ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Balance Method ◽  
Balance Solution ◽  
Good Agreement

A modified harmonic balance method is employed to determine the second approximate solutions to a coupled nonlinear differential equation near the limit cycle. The solution shows a good agreement with the numerical solution. DOI: http://dx.doi.org/10.3329/ganit.v32i0.13640 GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 32 (2012) 1 – 14

Closed-Form Approximations to the Solution of V-Belt Force and Slip Equations

1985 ◽  
Vol 107 (2) ◽  
pp. 292-300 ◽  
Author(s):  
J. P. Dolan ◽  
W. S. Worley
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Design Practice ◽  
Tension Distribution ◽  
Analytical Approximations ◽  
Current Design

A method for generating accurate numerical solutions of the exact differential equations describing tension distribution and radial penetration of a flexible V-belt on driveN and driveR sheaves is presented and results are compared with approximate solutions reported in the literature. Analytical approximations for these solutions of higher accuracy than any previously published have been found and are presented. They suggest important modifications of current design practice for belt tensioning and life appraisal.

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