Linear stability and Hopf bifurcation in an exponential RED algorithm model

2009 ◽  
Vol 59 (3) ◽  
pp. 463-475 ◽  
Author(s):  
Haijun Hu ◽  
Lihong Huang
Keyword(s):  
Hopf Bifurcation ◽  
Linear Stability ◽  
Exponential Red

Subcritical bifurcation and bistability in thermoacoustic systems

2013 ◽  
Vol 715 ◽  
pp. 210-238 ◽  
Author(s):  
Priya Subramanian ◽  
R. I. Sujith ◽  
P. Wahi
Keyword(s):  
Hopf Bifurcation ◽  
Linear Stability ◽  
Multiple Scales ◽  
Landau Equation ◽  
Linear Instability ◽  
Numerical Continuation ◽  
Rijke Tube ◽  
Subcritical Hopf Bifurcation ◽  
Fold Bifurcation ◽  
Parameter Values

AbstractThis paper analyses subcritical transition to instability, also known as triggering in thermoacoustic systems, with an example of a Rijke tube model with an explicit time delay. Linear stability analysis of the thermoacoustic system is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation. We then use the method of multiple scales to recast the model of a general thermoacoustic system near the Hopf point into the Stuart–Landau equation. From the Stuart–Landau equation, the relation between the nonlinearity in the model and the criticality of the ensuing bifurcation is derived. The specific example of a model for a horizontal Rijke tube is shown to lose stability through a subcritical Hopf bifurcation as a consequence of the nonlinearity in the model for the unsteady heat release rate. Analytical estimates are obtained for the triggering amplitudes close to the critical values of the bifurcation parameter corresponding to loss of linear stability. The unstable limit cycles born from the subcritical Hopf bifurcation undergo a fold bifurcation to become stable and create a region of bistability or hysteresis. Estimates are obtained for the region of bistability by locating the fold points from a fully nonlinear analysis using the method of harmonic balance. These analytical estimates help to identify parameter regions where triggering is possible. Results obtained from analytical methods compare reasonably well with results obtained from both experiments and numerical continuation.

Complex Dynamics and Periodic Oscillation Mechanism in Two Novel Gene Expression Models with State-Dependent Delays

2021 ◽  
Vol 31 (01) ◽  
pp. 2150002
Author(s):  
Shuo Wang ◽  
Lijun Pei
Keyword(s):  
Gene Expression ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Linear Stability ◽  
Periodic Oscillation ◽  
Degradation Process ◽  
The Real ◽  
State Dependent ◽  
Transcription Process ◽  
Gene Regulatory

The reaction time delay in the transcription process depends on the concentration of the protein because the transportation of mRNA from the nucleus to the cytoplasm becomes saturated. Thus the gene regulatory network is a state-dependent delayed model. This study aims to provide some mathematical explanations for the dynamics of the system, such as the linear stability and periodic oscillation, using mathematical techniques, such as formal linearization, linear stability analysis, the method of multiple scale (MMS), and the normal form. First, Hopf bifurcation of the state-dependent delayed gene regulatory networks model in the gene expression is analyzed by the method of multiple scales (MMS). Mechanism of periodic oscillations is obtained by Hopf bifurcation. The findings show that when degradation effects of the mRNA and protein are very strong, the oscillatory gene expression disappears. Then, a more realistic version of the aforementioned model with both constant and state-dependent time delays is established due to the existence of the constant time delay in the protein degradation process. Its nonresonant double Hopf bifurcation is found and analyzed using MMS. Interesting complex dynamic phenomena, such as periodic, quasi-periodic, and global period-[Formula: see text] solutions, are also discovered. These observations indicate that both state-dependent delay and constant delay could induce richer dynamics of the system, and the modified model may potentially describe the real dynamical mechanism (both the transcription process and the degradation process) more accurately in the gene expression. The findings may provide important guidance or hints to understand the real dynamic mechanism of the gene expression process.

Linear stability and Hopf bifurcation in a three-unit neural network with two delays

2006 ◽  
Vol 70 (1-3) ◽  
pp. 219-228 ◽  
Author(s):  
Shaofen Zou ◽  
Lihong Huang ◽  
Yuming Chen
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Linear Stability ◽  
Two Delays

Non-linear stability analysis of floating ring bearings using Hopf bifurcation theory

2011 ◽  
Vol 225 (12) ◽  
pp. 2804-2818 ◽  
Author(s):  
A Amamou ◽  
M Chouchane
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Limit Cycles ◽  
Critical Speed ◽  
Design Parameters ◽  
Stable Limit ◽  
Non Linear ◽  
The Stability

Floating ring bearings are used to support and guide rotors in several high-speed rotating machinery applications. They are usually credited for lower heat generation and higher vibration suppressing ability. Similar to conventional hydrodynamic bearings, floating ring bearings may exhibit unstable behaviour above a certain stability critical speed. Linear stability analysis is usually applied to predict the stability threshold speed. Non-linear stability analysis, however, is needed to predict the presence and the size of stable limit cycles above the stability threshold speed or unstable limit cycles below the stability critical speed. The prediction of limit cycles is an important step in bearing stability analysis. In this article, a non-linear dynamic model is derived and used to investigate the stability of a perfectly balanced symmetric rigid rotor supported by two identical floating ring bearings near the critical stability boundaries. The fluid film hydrodynamic reactions of the floating ring bearings are modelled by applying the short bearing theory and the half Sommerfeld solution. Hopf bifurcation theory is then utilized to determine the existence and the approximate size of stable and unstable limit cycles in the neighbourhood of the stability critical speed depending on the bearing design parameters. Numerical integration of the non-linear equations of motion is then carried out in order to compare the trajectories obtained by numerical integration to those obtained analytically using Hopf bifurcation analysis. Stability boundary curves for typical bearing design parameters have been decomposed into boundaries with supercritical stable limit cycles and boundaries with subcritical unstable limit cycles. The shape and size of the limit cycles for selected bearing parameters are presented using both analytical and numerical approaches. This article shows that floating ring stability boundaries may exhibit either stable supercritical limit cycles or unstable subcritical limit cycles predictable by Hopf bifurcation.

EFFECTS DUE TO INDUCED AZIMUTHAL EDDY CURRENTS IN A SELF-EXCITING FARADAY DISK HOMOPOLAR DYNAMO WITH A NONLINEAR SERIES MOTOR II: THE GENERAL CASE

2000 ◽  
Vol 10 (12) ◽  
pp. 2701-2716 ◽  
Author(s):  
IRENE M. MOROZ ◽  
RAYMOND HIDE
Keyword(s):  
Hopf Bifurcation ◽  
Linear Stability ◽  
Eddy Currents ◽  
Quadratic Function ◽  
Oscillatory Behavior ◽  
Equilibrium Solutions ◽  
Double Zero ◽  
Oscillatory Solutions ◽  
Special Cases ◽  
In Series

This paper forms the second part of a two-part study into the effects of azimuthal eddy currents in the Faraday disk self-exciting homopolar dynamo, connected in series with the coil when the applied couple driving the disk is steady. The Lorentz couple driving the armature of the motor is a general quadratic function I(1 - ε + εSI) of the current I(t), where 0 ≤ ε ≤ 1. Here we investigate how cases with 0 < ε < 1 relate to the two special cases of ε = 0 and ε = 1, considered in Part I of our study [Hide & Moroz, 1999]. One key difference is that the lack of reflectional symmetry in the general ε problem means that the linear stability curves for the onset of both steady and oscillatory behavior for both the trivial and the nontrivial equilibrium solutions no longer coincide. This results in distinct Takens–Bogdanov double-zero bifurcations for these states, as well as multiple branches to the Hopf bifurcation curves, associated with bifurcations from the nontrivial equilibrium states. Multiple bifurcations involving simultaneous steady and nondegenerate oscillatory solutions are also possible.

Unicellular natural circulation in a shallow horizontal porous layer heated from below by a constant flux

1995 ◽  
Vol 294 ◽  
pp. 231-257 ◽  
Author(s):  
S. Kimura ◽  
M. Vynnycky ◽  
F. Alavyoon
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Rayleigh Number ◽  
Aspect Ratio ◽  
Linear Stability ◽  
Porous Layer ◽  
Linear Stability Analysis ◽  
Natural Circulation ◽  
Stable Mode ◽  
Constant Flux

Natural convection in a saturated horizontal porous layer heated from below and cooled at the top with a constant flux is studied both analytically and numerically. Linear stability analysis indicates that unicellular recirculation remains a stable mode of flow as the aspect ratio (A) of the layer is increased, in contrast to the situation for an isothermally heated and cooled layer. An analytical solution is presented for fully developed counterflow in the infinite-aspect-ratio limit; this flow is found to be linearly stable to transverse disturbances for Rayleigh number (Ra) as high as 506, at which point a Hopf bifurcation sets in; however, further analysis indicates that an exchange of stability due to longitudinal disturbances will occur much sooner at Ra ≈ 311.53. The velocity and temperature profiles of the counterflow solution, whilst not strictly speaking valid in the extreme end regions of the layer, otherwise agree very well with full numerical computations conducted for the ranges 25 [les ] Ra [les ] 1050, 2 [les ] A [les ] 10. However, for sufficiently high Rayleigh number (Ra between 630 and 650 for A = 8 and Ra between 730 and 750 for A = 4, for example), the computations indicate transition from steady unicellular to oscillatory flow, in line with the Hopf bifurcation predicted by the linear stability analysis for infinite aspect ratio.

Cops-on-the-dots: The linear stability of crime hotspots for a 1-D reaction-diffusion model of urban crime

2019 ◽  
Vol 31 (5) ◽  
pp. 871-917 ◽  
Author(s):  
ANDREAS BUTTENSCHOEN ◽  
THEODORE KOLOKOLNIKOV ◽  
MICHAEL J. WARD ◽  
JUNCHENG WEI
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Population Density ◽  
Phase Diagrams ◽  
Linear Stability ◽  
Reaction Diffusion ◽  
Urban Crime ◽  
Steady States ◽  
Three Component Reaction ◽  
Policing Strategy

In a singularly perturbed limit, we analyse the existence and linear stability of steady-state hotspot solutions for an extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Jones et. al. [Math. Models. Meth. Appl. Sci., 20, Suppl., (2010)], which models urban crime with police intervention. In our extended RD model, the field variables are the attractiveness field for burglary, the criminal population density and the police population density. Our model includes a scalar parameter that determines the strength of the police drift towards maxima of the attractiveness field. For a special choice of this parameter, we recover the ‘cops-on-the-dots’ policing strategy of Jones et. al., where the police mimic the drift of the criminals towards maxima of the attractiveness field. For our extended model, the method of matched asymptotic expansions is used to construct 1-D steady-state hotspot patterns as well as to derive nonlocal eigenvalue problems (NLEPs) that characterise the linear stability of these hotspot steady states to ${\cal O}$(1) timescale instabilities. For a cops-on-the-dots policing strategy, we prove that a multi-hotspot steady state is linearly stable to synchronous perturbations of the hotspot amplitudes. Alternatively, for asynchronous perturbations of the hotspot amplitudes, a hybrid analytical–numerical method is used to construct linear stability phase diagrams in the police vs. criminal diffusivity parameter space. In one particular region of these phase diagrams, the hotspot steady states are shown to be unstable to asynchronous oscillatory instabilities in the hotspot amplitudes that arise from a Hopf bifurcation. Within the context of our model, this provides a parameter range where the effect of a cops-on-the-dots policing strategy is to only displace crime temporally between neighbouring spatial regions. Our hybrid approach to study the NLEPs combines rigorous spectral results with a numerical parameterisation of any Hopf bifurcation threshold. For the cops-on-the-dots policing strategy, our linear stability predictions for steady-state hotspot patterns are confirmed from full numerical PDE simulations of the three-component RD system.

Stability and performance analysis of Compound TCP with the Exponential-RED and the Drop-Tail queue policies

2017 ◽  
Vol 36 (2) ◽  
pp. 399-421
Author(s):  
Sai Prasad ◽  
Gaurav Raina
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Performance Metrics ◽  
Orbital Stability ◽  
Feedback Delay ◽  
Queuing Delay ◽  
Dimensional Parameter ◽  
Current Form ◽  
Exponential Red ◽  
The Stability

Abstract The analysis of transport protocols, along with queue management policies, forms an important aspect of performance evaluation for the Internet. In this article, we study Compound TCP (C-TCP), the default TCP in the Windows operating system, along with the Exponential-RED (E-RED) queue policy and the widely used Drop-Tail queue policy. We consider queuing delay, link utilization and the stability of the queue size as the performance metrics. We first analyse the stability properties of a nonlinear model for C-TCP coupled with the E-RED queue policy. We observe that this system, in its current form, may be difficult to stabilize as the feedback delay gets large. Further, using an exogenous and non-dimensional parameter, we show that the system loses local stability via a Hopf bifurcation, which gives rise to limit cycles. Employing Poincaré normal forms and the center manifold theory, we outline an analytical framework to characterize the type of the Hopf bifurcation and to determine the orbital stability of the emerging limit cycles. Numerical examples, stability charts and bifurcation diagrams complement our analysis. We also conduct packet-level simulations, with E-RED and Drop-Tail, in small and large buffer-sizing regimes. With large buffers, E-RED can achieve small queue sizes compared with Drop-Tail. However, it is difficult to maintain the stability of the E-RED policy as the feedback delay gets large. On the other hand, with small buffers, E-RED offers no clear advantage over the simple Drop-Tail queue policy. Our work can offer insights for the design of queue policies that can ensure low latency and stability.

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