Dynamical systems on networks

Limit Cycles ◽

Spectral Properties ◽

Stability Conditions ◽

Short Introduction ◽

An introduction to the theory of dynamical systems on networks. This chapter starts with a short introduction to classical (non-network) dynamical systems theory, including linear stability analysis, fixed points, and limit cycles. Dynamical systems on networks are introduced, focusing initially on systems with only one variable per node and progressing to multi-variable systems. Linear stability analysis is developed in detail, leading to master stability conditions and the connection between stability and the spectral properties of networks. The chapter ends with a discussion of synchronization phenomena, the stability of limit cycles, and master stability conditions for synchronization.

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

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

10.1177/0954406211413520 ◽

2011 ◽

Vol 225 (12) ◽

pp. 2804-2818 ◽

Cited By ~ 5

Author(s):

A Amamou ◽

M Chouchane

Keyword(s):

Stability Analysis ◽

Hopf Bifurcation ◽

Linear Stability ◽

Limit Cycles ◽

Critical Speed ◽

Design Parameters ◽

Stable Limit ◽

Non Linear ◽

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.

Stability conditions for synchronization of networks with mixed couplings by linear stability analysis

Journal of Shanghai University (English Edition) ◽

10.1007/s11741-007-0311-3 ◽

2007 ◽

Vol 11 (3) ◽

pp. 245-250

Author(s):

Rui-xin Wu ◽

Hai-feng Zhang ◽

Xin-chu Fu

Keyword(s):

Stability Analysis ◽

Linear Stability ◽

Stability Conditions

A Study of the Lateral Stability of Railroad Vehicles Using a Nonlinear Constrained Multibody Formulation

10.1115/imece2001/rtd-25704 ◽

2001 ◽

Author(s):

Davide Valtorta ◽

Khaled E. Zaazaa ◽

Ahmed A. Shabana ◽

Jalil R. Sany

Keyword(s):

Stability Analysis ◽

Linear Stability ◽

Linear Theory ◽

Numerical Study ◽

Operating Conditions ◽

Lateral Stability ◽

Railroad Vehicles ◽

Contact Formulation ◽

Abstract The lateral stability of railroad vehicles travelling on tangent tracks is one of the important problems that has been the subject of extensive research since the nineteenth century. Early detailed studies of this problem in the twentieth century are the work of Carter and Rocard on the stability of locomotives. The linear theory for the lateral stability analysis has been extensively used in the past and can give good results under certain operating conditions. In this paper, the results obtained using a linear stability analysis are compared with the results obtained using a general nonlinear multibody methodology. In the linear stability analysis, the sources of the instability are investigated using Liapunov’s linear theory and the eigenvalue analysis for a simple wheelset model on a tangent track. The effects of the stiffness of the primary and secondary suspensions on the stability results are investigated. The results obtained for the simple model using the linear approach are compared with the results obtained using a new nonlinear multibody based constrained wheel/rail contact formulation. This comparative numerical study can be used to validate the use of the constrained wheel/rail contact formulation in the study of lateral stability. Similar studies can be used in the future to define the limitations of the linear theory under general operating conditions.

Linear stability of confined flow around a 180-degree sharp bend

10.1017/jfm.2017.266 ◽

2017 ◽

Vol 822 ◽

pp. 813-847 ◽

Cited By ~ 4

Author(s):

Azan M. Sapardi ◽

Wisam K. Hussam ◽

Alban Pothérat ◽

Gregory J. Sheard

Keyword(s):

Reynolds Number ◽

Stability Analysis ◽

Linear Stability ◽

Critical Reynolds Number ◽

Three Dimensional ◽

Base Flow ◽

Two Dimensional ◽

Two Dimensional Flow ◽

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

Non-parallel linear stability analysis of the vertical boundary layer in a differentially heated cavity

10.1017/s0022112097007258 ◽

1997 ◽

Vol 352 ◽

pp. 265-281 ◽

Cited By ~ 11

Author(s):

A. M. H. BROOKER ◽

J. C. PATTERSON ◽

S. W. ARMFIELD

Keyword(s):

Boundary Layer ◽

Stability Analysis ◽

Linear Stability ◽

Navier Stokes ◽

Parabolized Stability Equations ◽

The Stability ◽

Flow Quantity ◽

Energy Equations ◽

Made In

A non-parallel linear stability analysis which utilizes the assumptions made in the parabolized stability equations is applied to the buoyancy-driven flow in a differentially heated cavity. Numerical integration of the complete Navier–Stokes and energy equations is used to validate the non-parallel theory by introducing an oscillatory heat input at the upstream end of the boundary layer. In this way the stability properties are obtained by analysing the evolution of the resulting disturbances. The solutions show that the spatial growth rate and wavenumber are highly dependent on the transverse location and the disturbance flow quantity under consideration. The local solution to the parabolized stability equations accurately predicts the wave properties observed in the direct simulation whereas conventional parallel stability analysis overpredicts the spatial amplification and the wavenumber.

Rayleigh–Bénard instability generated by a diffusion flame

10.1017/jfm.2013.549 ◽

2013 ◽

Vol 736 ◽

pp. 464-494 ◽

Cited By ~ 5

Author(s):

P. Pearce ◽

J. Daou

Keyword(s):

Stability Analysis ◽

Rayleigh Number ◽

Linear Stability ◽

Diffusion Flame ◽

Damkohler Number ◽

Benard Convection ◽

The Stability ◽

Rayleigh Benard Convection ◽

Rayleigh Benard

AbstractWe investigate the Rayleigh–Bénard convection problem within the context of a diffusion flame formed in a horizontal channel where the fuel and oxidizer concentrations are prescribed at the porous walls. This problem seems to have received no attention in the literature. When formulated in the low-Mach-number approximation the model depends on two main non-dimensional parameters, the Rayleigh number and the Damköhler number, which govern gravitational strength and reaction speed respectively. In the steady state the system admits a planar diffusion flame solution; the aim is to find the critical Rayleigh number at which this solution becomes unstable to infinitesimal perturbations. In the Boussinesq approximation, a linear stability analysis reduces the system to a matrix equation with a solution comparable to that of the well-studied non-reactive case of Rayleigh–Bénard convection with a hot lower boundary. The planar Burke–Schumann diffusion flame, which has been previously considered unconditionally stable in studies disregarding gravity, is shown to become unstable when the Rayleigh number exceeds a critical value. A numerical treatment is performed to test the effects of compressibility and finite chemistry on the stability of the system. For weak values of the thermal expansion coefficient $\alpha $, the numerical results show strong agreement with those of the linear stability analysis. It is found that as $\alpha $ increases to a more realistic value the system becomes considerably more stable, and also exhibits hysteresis at the onset of instability. Finally, a reduction in the Damköhler number is found to decrease the stability of the system.

The effect of interruption probability in lattice model of two-lane traffic flow with passing

International Journal of Modern Physics C ◽

10.1142/s0129183116500509 ◽

2016 ◽

Vol 27 (05) ◽

pp. 1650050 ◽

Cited By ~ 3

Author(s):

Guanghan Peng

Keyword(s):

Numerical Simulation ◽

Stability Analysis ◽

Traffic Flow ◽

Linear Stability ◽

Stability Condition ◽

Lattice Model ◽

Mkdv Equation ◽

Reaction Coefficient ◽

A new lattice model is proposed by taking into account the interruption probability with passing for two-lane freeway. The effect of interruption probability with passing is investigated about the linear stability condition and the mKdV equation through linear stability analysis and nonlinear analysis, respectively. Furthermore, numerical simulation is carried out to study traffic phenomena resulted from the interruption probability with passing in two-lane system. The results show that the interruption probability with passing can improve the stability of traffic flow for low reaction coefficient while the interruption probability with passing can destroy the stability of traffic flow for high reaction coefficient on two-lane highway.

On the porosity influence on stability of flow over porous medium

Vestnik Udmurtskogo Universiteta Matematika Mekhanika Komp yuternye Nauki ◽

10.35634/vm200110 ◽

2020 ◽

Vol 30 (1) ◽

pp. 134-144

Author(s):

K.B. Tsiberkin

Keyword(s):

Porous Medium ◽

Stability Analysis ◽

Linear Stability ◽

Short Wave ◽

Wave Instability ◽

Plane Parallel ◽

Long Wave ◽

The Stability ◽

Critical Wavenumber

The stability of incompressible fluid plane-parallel flow over a layer of a saturated porous medium is studied. The results of a linear stability analysis are described at different porosity values. The considered system is bounded by solid wall from the porous layer bottom. Top fluid surface is free and rigid. A linear stability analysis of plane-parallel stationary flow is presented. It is realized for parameter area where the neutral stability curves are bimodal. The porosity variation effect on flow stability is considered. It is shown that there is a transition between two main instability modes: long-wave and short-wave. The long-wave instability mechanism is determined by inflection points within the velocity profile. The short-wave instability is due to the large transverse gradient of flow velocity near the interface between liquid and porous medium. Porosity decrease stabilizes the long wave perturbations without significant shift of the critical wavenumber. Simultaneously, the short-wave perturbations destabilize, and their critical wavenumber changes in wide range. When the porosity is less than 0.7, the inertial terms in filtration equation and magnitude of the viscous stress near the interface increase to such an extent that the Kelvin-Helmholtz analogue of instability becomes the dominant mechanism for instability development. The stability band realizes in narrow porosity area. It separates the two branches of the neutral curve.

Competition between the centrifugal and strato-rotational instabilities in the stratified Taylor–Couette flow

10.1017/jfm.2018.15 ◽

2018 ◽

Vol 840 ◽

pp. 5-24 ◽

Cited By ~ 6

Author(s):

Junho Park ◽

Paul Billant ◽

Jong-Jin Baik ◽

Jaemyeong Mango Seo

Keyword(s):

Stability Analysis ◽

Linear Stability ◽

Couette Flow ◽

Reynolds Numbers ◽

Axisymmetric Mode ◽

Stability Threshold ◽

Taylor Couette ◽

The Stability ◽

Taylor Couette Flow

The stably stratified Taylor–Couette flow is investigated experimentally and numerically through linear stability analysis. In the experiments, the stability threshold and flow regimes have been mapped over the ranges of outer and inner Reynolds numbers: $-2000<Re_{o}<2000$ and $0<Re_{i}<3000$, for the radius ratio $r_{i}/r_{o}=0.9$ and the Brunt–Väisälä frequency $N\approx 3.2~\text{rad}~\text{s}^{-1}$. The corresponding Froude numbers $F_{o}$ and $F_{i}$ are always much smaller than unity. Depending on $Re_{o}$ (or equivalently on the angular velocity ratio $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}_{o}/\unicode[STIX]{x1D6FA}_{i}$), three different regimes have been identified above instability onset: a weakly non-axisymmetric mode with low azimuthal wavenumber $m=O(1)$ is observed for $Re_{o}<0$ ($\unicode[STIX]{x1D707}<0$), a highly non-axisymmetric mode with $m\sim 12$ occurs for $Re_{o}>840$ ($\unicode[STIX]{x1D707}>0.57$) while both modes are present simultaneously in the lower and upper parts of the flow for $0\leqslant Re_{o}\leqslant 840$ ($0\leqslant \unicode[STIX]{x1D707}\leqslant 0.57$). The destabilization of these primary modes and the transition to turbulence as $Re_{i}$ increases have been also studied. The linear stability analysis proves that the weakly non-axisymmetric mode is due to the centrifugal instability while the highly non-axisymmetric mode comes from the strato-rotational instability. These two instabilities can be clearly distinguished because of their distinct dominant azimuthal wavenumber and frequency, in agreement with the recent results of Park et al. (J. Fluid Mech., vol. 822, 2017, pp. 80–108). The stability threshold and the characteristics of the primary modes observed in the experiments are in very good agreement with the numerical predictions. Moreover, we show that the centrifugal and strato-rotational instabilities are observed simultaneously for $0\leqslant Re_{o}\leqslant 840$ in the lower and upper parts of the flow, respectively, because of the variations of the local Reynolds numbers along the vertical due to the salinity gradient.

Linear stability analysis of non-Newtonian blood flow with magnetic nanoparticles: application to controlled drug delivery

International Journal of Numerical Methods for Heat &amp Fluid Flow ◽

10.1108/hff-03-2021-0161 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Pascalin Tiam Kapen ◽

Cédric Gervais Njingang Ketchate ◽

Didier Fokwa ◽

Ghislain Tchuen

Keyword(s):

Magnetic Field ◽

Drug Delivery ◽

Stability Analysis ◽

Magnetic Nanoparticles ◽

Linear Stability ◽

Controlled Drug Delivery ◽

Darcy Number ◽

Content Type ◽

Purpose For this purpose, a linear stability analysis based on the Navier–Stokes and Maxwell equations is made leading to an eigenvalue differential equation of the modified Orr–Sommerfeld type which is solved numerically by the spectral collocation method based on Chebyshev polynomials. Unlike previous studies, blood is considered as a non-Newtonian fluid. The effects of various parameters such as volume fraction of nanoparticles, Casson parameter, Darcy number, Hartmann number on flow stability were examined and presented. This paper aims to investigate a linear stability analysis of non-Newtonian blood flow with magnetic nanoparticles with an application to controlled drug delivery. Design/methodology/approach Targeted delivery of therapeutic agents such as stem cells and drugs using magnetic nanoparticles with the help of external magnetic fields is an emerging treatment modality for many diseases. To this end, controlling the movement of nanoparticles in the human body is of great importance. This study investigates controlled drug delivery by using magnetic nanoparticles in a porous artery under the influence of a magnetic field. Findings It was found the following: the Casson parameter affects the stability of the flow by amplifying the amplitude of the disturbance which reflects its destabilizing effect. It emerges from this study that the taking into account of the non-Newtonian character is essential in the modeling of such a system, and that the results can be very different from those obtained by supposing that the blood is a Newtonian fluid. The presence of iron oxide nanoparticles in the blood increases the inertia of the fluid, which dampens the disturbances. The Strouhal number has a stabilizing effect on the flow which makes it possible to say that the oscillating circulation mechanisms dampen the disturbances. The Darcy number affects the stability of the flow and has a stabilizing effect, which makes it possible to increase the contact surface between the nanoparticles and the fluid allowing very high heat transfer rates to be obtained. It also emerges from this study that the presence of the porosity prevents the sedimentation of the nanoparticles. By studying the effect of the magnetic field on the stability of the flow, it is observed that the Hartmann number keeps the flow completely stable. This allows saying that the magnetic field makes the dissipations very important because the kinetic energy of the electrically conductive ferrofluid is absorbed by the Lorentz force. Originality/value The originality of this paper resides on the application of the linear stability analysis for controlled drug delivery.