Exact Stability Criteria Research Articles

Exact stability criteria for linear differential equations with discrete and distributed delays

This study is devoted to the stability analysis of a scalar linear differential equation with discrete and distributed delays. We establish necessary and sufficient conditions guaranteeing the asymptotic stability of the zero solution composed of explicit delay-dependent criteria. The proofs of our results are obtained using the root analysis of the associated characteristic equation.

Corrections to the General Fourth-Order FDTD Schemes

The sampling weights and approximate stability criteria for two general fourth-order accurate FDTD schemes were previously derived by Smith et al. However, inconsistencies between several of their governing equations and corresponding solutions were recently discovered. Upon rederiving the full 3-D, fourth-order FDTD dispersion relation, two separate errors were identified in the original work. This paper details these errors and provides corrected sampling weights. Exact stability criteria are also derived and stated for the first time.

On the Ledinegg instability in parallel channels: A new and exact criterion

Ledinegg instability has important influence on boiling two-phase flow. A new and exact Ledinegg stability criterion for parallel channels is established by using the matrix analysis method based on the characteristic equation. The new criterion agrees well with the previous experimental results for the parallel three channels, and is proved to be equivalent to the criterion obtained by using Hurwitz theorem for a single channel, parallel two-channels, and parallel three-channels, respectively. The new criterion is found to be independent of the fluid equivalent inertia. And it is also found that, when the slope of pump pressure head versus flow rate curve is negative, there is at most one channel whose slope of the pressure drop versus flow rate curve is less than or equal to zero while the other channels' slopes of the pressure drop versus flow rate curves should be greater than zero, in order to make sure that parallel channels is stable. In addition, the new criterion shows that decreasing the slope of pump pressure head versus flow rate curve and introducing extra restrictor can improve the stability of parallel channels.

Effect of composition gradient on magnetothermal instability modified by shear and rotation

We model the intracluster medium as a weakly collisional plasma that is a binary mixture of the hydrogen and the helium ions, along with free electrons. When, owing to the helium sedimentation, the gradient of the mean molecular weight (or equivalently, composition or helium ions' concentration) of the plasma is not negligible, it can have appreciable influence on the stability criteria of the thermal convective instabilities, e.g., the heat-flux-buoyancy instability and the magnetothermal instability (MTI). These instabilities are consequences of the anisotropic heat conduction occurring preferentially along the magnetic field lines. In this paper, without ignoring the magnetic tension, we first present the mathematical criterion for the onset of composition gradient modified MTI. Subsequently, we relax the commonly adopted equilibrium state in which the plasma is at rest, and assume that the plasma is in a sheared state which may be due to differential rotation. We discuss how the concentration gradient affects the coupling between the Kelvin--Helmholtz instability and the MTI in rendering the plasma unstable or stable. We derive exact stability criterion by working with the sharp boundary case in which the physical variables---temperature, mean molecular weight, density, and magnetic field---change discontinuously from one constant value to another on crossing the boundary. Finally, we perform the linear stability analysis for the case of the differentially rotating plasma that is thermally and compositionally stratified as well. By assuming axisymmetric perturbations, we find the corresponding dispersion relation and the explicit mathematical expression determining the onset of the modified MTI.

Collective behavior of globally coupled Langevin equations with colored noise in the presence of stochastic resonance.

The long-time collective behavior of globally coupled Langevin equations in a dichotomous fluctuating potential driven by a periodic source is investigated. By describing the collective behavior using the moments of the mean field and single-particle displacements, we study stochastic resonance and synchronization using the exact steady-state solutions and related stability criteria. Based on the simulation results and the criterion of the stationary regime, the notable differences between the stationary and nonstationary regimes are demonstrated. For the stationary regime, stochastic resonance with synchronization is discussed, and for the nonstationary regime, the volatility clustering phenomenon is observed.

Influence of Gravity and Taper on the Vibration of a Standing Column

AbstractThe stability and natural vibration of a standing tapered vertical column under its own weight are studied. Exact stability criteria are found for the pointy column and numerical stability boundaries are determined for the blunt tipped column. For vibrations we use an accurate, efficient initial value numerical method for the first three frequencies. Four kinds of columns with linear taper are considered. Both the taper and the cross section shape of the column have large influences on the vibration frequencies. It is found that gravity decreases the frequency while the degree of taper may increase or decrease frequency. Vibrations may occur in two different planes.

Non-smooth stability analysis of the parametrically excited impact oscillator

The aim of this paper is to give a Lyapunov stability analysis of a parametrically excited impact oscillator, i.e. a vertically driven pendulum which can collide with a support. The impact oscillator with parametric excitation is described by Hill's equation with a unilateral constraint. The unilaterally constrained Hill's equation is an archetype of a parametrically excited non-smooth dynamical system with state jumps. The exact stability criteria of the unilaterally constrained Hill's equation are rigorously derived using Lyapunov techniques and are expressed in the properties of the fundamental solutions of the unconstrained Hill's equation. Furthermore, an asymptotic approximation method for the critical restitution coefficient is presented based on Hill's infinite determinant and this approximation can be made arbitrarily accurate. A comparison of numerical and theoretical results is presented for the unilaterally constrained Mathieu equation.

Exact lateral–torsional buckling solutions for cantilevered beams subjected to intermediate and end transverse point loads

This paper presents exact stability criteria for the lateral–torsional buckling of cantilever strip beams under intermediate and end point loads. The two-dimensional stability domain is given in closed-form solutions, in terms of Bessel functions. The limit of the convex stability domain tends towards Dunkerley's line when the intermediate load approaches the tip of the cantilever. However, the Dunkerley's line is no longer appropriate when the distance between both loads is sufficiently large. An approximate set of formulas is proposed for quick estimation of the buckling load.

Equivalence of some stability criteria for linear time-delay systems

For linear systems with constant delay, although some of the exact stability criteria are quite different, their qualitative nature for conservatism is equivalent. This paper theoretically proves the equivalence of some stability criteria in the LMI forms. This work brings about a new insight into the comparison of conservatism among different stability criteria, which are usually formulated as the feasibility problem of certain LMIs.

Exact stability criteria for delay differential and difference equations

For linear delay differential and difference equations with one coefficient matrix A , we give explicit necessary and sufficient conditions for the asymptotic stability of the equations in terms of tr A , det A and the delay parameter. We also summarize some exact stability criteria for certain classes of linear delay differential and difference equations.

Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials

We investigate the nonlinear propagation of electromagnetic waves in left-handed materials. For this purpose, we consider a set of coupled nonlinear Schrödinger (CNLS) equations, which govern the dynamics of coupled electric and magnetic field envelopes. The CNLS equations are used to obtain a nonlinear dispersion, which depicts the modulational stability profile of the coupled plane-wave solutions in left-handed materials. An exact (in)stability criterion for modulational interactions is derived, and analytical expressions for the instability growth rate are obtained.

Stability criteria for rectangular plates subjected to intermediate and end inplane loads

This paper is concerned with the elastic buckling of rectangular plates subjected to intermediate and end uniaxial inplane loads, whose direction is parallel to two simply supported edges. The aforementioned buckling problem is solved by decomposing the plate into two subplates at the location where the intermediate uniaxial load acts. Each subplate buckling problem is solved exactly using the Levy approach and the two solutions brought together by matching the continuity equations at the separated interface. It is worth noting that there are five possible solutions for each subplate and consequently there are 25 combinations of solutions to be considered. For different boundary conditions, the buckling solutions comprise of different combinations. For each boundary condition, the correct solution combination depends on the ratio of the intermediate load to the end load. The exact stability criteria, presented both in tabulated and in graphical forms, should be useful for engineers designing walls or plates that have to support intermediate floors/loads.

Exact Buckling Solutions For Rectangular Plates Under Intermediate and End Uniaxial Loads

This paper is concerned with the elastic buckling of rectangular plates subjected to both intermediate and end uniaxial loads. The rectangular plates have two simply supported opposite edges that are perpendicular to the in-plane load direction, while the other two plate edges can have free, simply supported, or clamped edges. The solution procedure involves the use of the Levy approach, the domain decomposition technique, and the state-space concept. The method furnishes exact stability criteria; samples of which are presented in a graphical form for plates with various boundary conditions. These results will be useful to engineers who design plates (or walls) that support intermediate floors.

Stability Criteria for Euler Columns with Intermediate and End Axial Loads

Presented herein are exact stability criteria for Euler columns under intermediate and end concentrated axial loads. The stability criteria cover all combinations of classical boundary conditions, arbitrary location of the intermediate concentrated load, and ratios of the magnitude of the intermediate load to the end load. Also included is the buckling problem of a new class of Euler columns where one segment is in tension while the other segment in compression.

Buckling of Columns with Intermediate Elastic Restraint

This paper presents a comprehensive set of exact stability criteria for Euler columns with an intermediate elastic restraint. A subset of this class of problem is the buckling problem of columns with an intermediate rigid support where the elastic restraint takes on an infinite stiffness. Also, this study reiterates the existence of a critical elastic restraint stiffness in which the buckled mode switches to a higher-buckling mode of the corresponding column without an intermediate support. It is clear that this critical stiffness value exists only when the restraint is placed at the node of the higher-buckling mode and the buckling load associated with this critical stiffness value is the maximum achievable value that can be attained with an intermediate elastic restraint.

Stability criteria for Timoshenko columns with intermediate and end concentrated axial loads

This paper presents exact stability criteria and buckling loads of Timoshenko columns under intermediate and end concentrated loads. The new buckling solutions show clearly the erosive effect of transverse shear deformation on the buckling capacity of the column under compressive loads, especially in stocky and built-up columns with high values of shear parameter s= EI/( κ s GAL 2) and more so when the columns ends are highly restrained such as fixed-pinned columns and fixed-fixed columns. Furthermore, for columns under an intermediate compressive load acting near the base of the column, the effect of transverse shear deformation becomes significant even when the entire column is slender. This is because the column buckling problem behaves like a stocky column problem when the load is acting over a very short portion of the column.

Stability criteria for a general class of finite difference schemes

The von Neumann stabllity analysis is performed on a generalized class of two-level space-centered finite difference schemes known as Lerat-Peyret schemes. In the present study, these schemes are used to solve the convection-diffusion equation in an effort to obtain a better assessment of the accuracy of such schemes. Exact stability criteria are derived for each of five different schemes applied to the Burgers equation. Exact results for the amplification factor and phase error are plotted for several values of Courant number and diffusion parameter for each scheme. These exact results, which are presented here for the flat time, are extremely valuable in assessing the dissipation and dispersion characteristics of each scheme

Exact analysis of nonlinear instability in a discrete Burgers' equation

A family of explicit nonlinear numerical schemes for Burgers' equation is derived by means of a discrete version of the Hopf-Cole transformation. Exact nonlinear stability conditions for these schemes are then found, and for one particular scheme the exact stability criteria are compared to the conventional linearized stability condition.

Competitive instability in nonlinear Fabry-Perot etalons

The stability of Fabry-Perot etalons with competing mechanisms is investigated. For high-finesse cavities with linear absorption, exact stability criteria, in terms of physically measurable parameters, are deduced. It is found that the existence of unstable outputs, over a certain range of input powers, depends on the strengths and time constants of the competing mechanisms and on the detuning of the etalon. The dynamics of these devices, both stable and unstable, are illustrated by means of a numerical model. The practical application of the theory is illustrated with reference to a typical system. >

SUNS: The Sussex University Nonlinear Control Systems Software

The paper outlines the capabilities of the Sussex University Nonlinear Software. The programs based on two theoretical techniques, the describing function and Tsypkin's method, evaluate limit cycle solutions in both continuous and discrete feedback systems. The software allows the display of solution waveforms at the input to each nonlinearity and an exact orbital stability criterion, for continuous systems, is used for the limit cycles predicted by the Tsynkin method. Most of the software is for fixed structure canonical feedback systems although one of the DF based programs allows analysis of systems of a free structure. The software may also be used for frequency domain design where gain and phase margins of compensation schemes may be calculated by finding the minimum amount of additional gain or time delay required for a system to limit cycle.