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Bistability 2
Published in James E. Ferrell, Systems Biology of Cell Signaling, 2021
To understand why the off-state and on-state are stable and to get a better understanding of what the saddle represents, we turn to what is called linear stability analysis, which tells us whether trajectories that start out very close to a steady-state will converge to it or not. The basic idea behind linear stability analysis is to (1) choose a steady state; (2) perturb the system away from that steady state by an infinitesimal amount; (3) calculate the rate at which the system returns to or is repelled from the steady state; and (4) repeat for the rest of the steady states. Linear stability analysis is easier to do for a one-variable system than a two-variable system, so let us start by returning to our one-variable Mos model from Chapter 8.
Bifurcation Analysis of Spatial Xenon Oscillations in Large Pressurized Heavy Water Reactors Using Multipoint Reactor Kinetics with Thermal-Hydraulic Feedback
Published in Nuclear Science and Engineering, 2021
Abhishek Chakraborty, Suneet Singh, M. P. S. Fernando
Hopf bifurcation is a codimension-1 bifurcation where there is only one free parameter for a bifurcation to occur. Changes in the parameter value of the system result in the change of its eigenvalues. These eigenvalues help determine the linear stability of a system in steady state. The positive real part of any one of the eigenvalues makes the system unstable whereas when real parts of all the eigenvalues are negative, the system is stable. Hopf bifurcation occurs at the point at which a pair of complex eigenvalues crosses the imaginary axis. Linear stability of the system is lost at this critical parameter value, and a family of limit cycles bifurcates at that point, which is a nonlinear phenomenon. Consider an autonomous system of ODEs:
Assessing Numerical Aspects of Transitional Flow Simulations Using the RANS Equations
Published in International Journal of Computational Fluid Dynamics, 2021
Rui Lopes, Luís Eça, Guilherme Vaz, Maarten Kerkvliet
The most time-tested alternative aimed at accurately predicting transition is the method, developed independently by van Ingen (1956) and Smith and Gamberoni (1956). It is based on linear-stability theory, in which the Navier-Stokes equations are linearised in a process that leads to the Orr-Sommerfeld equation. The method consists in finding the local amplification rate for disturbances in the flow, through the Orr-Sommerfeld equation. The local amplification rate is then integrated along the streamlines in order to find a global amplification factor. Upon reaching a critical value, transition is considered to start. Despite its theory-based formulation, the method is a semi-empirical method (Sousa and Silva 2004): the choice of the critical value that signifies the onset of transition is an empirical choice (a value of 9 is commonly used). Another disadvantage is that the method only provides the start of transition. Additionally, the coupling with Reynolds-Averaged Navier-Stokes calculations, which are the bulk of the simulations performed in an industrial setting, is troublesome since it requires well-resolved velocity profiles. Despite the inherent difficulties, some work on this approach (Probst, Radespiel, and Rist 2012) has been performed.
Bedload transport: a walk between randomness and determinism. Part 1. The state of the art
Published in Journal of Hydraulic Research, 2020
Bedforms such as dunes and anti-dunes have often been considered the signatures of streambed instability. Linear stability analysis is the standard tool for studying the conditions necessary for the development of instabilities. The simplest set of equations enabling the implementation of this tool is a combination of the shallow water (or Saint-Venant) equations for the mass and momentum balance equation of the water stream: and the Exner equation for the mass balance of the bed (when the suspended load is neglected): in which denotes the flow depth, is the bed elevation, is the depth-averaged velocity, x is the downstream position, t is time, ρ is the water density, is the bottom shear stress, ζ is the bed porosity, D and E represent the deposition and entrainment rates, respectively, and ν is the eddy viscosity. The bed slope is defined as . The governing equations are closed by empirical relationships for the flow resistance and bedload transport rate , which are assumed to depend on the flow variables and h and other additional parameters (e.g. bed roughness and slope).