General Stability Theory Research Articles

Gully Erosion, Northwestern Colorado: A Threshold Phenomenon

The widespread occurrence of discontinuous gullies in the oil-shale region of northwestern Colorado is of particular concern because of the resulting progressive destruction of the valley floors. Furthermore, the integration of a semi-arid drainage network can cause a rapid increase in the sediment yield of the basin, with subsequent harmful effects downstream. Field work in the Piceance Creek and Yellow Creek drainage basins indicates that these discontinuous gullies developed on oversteepened segments of the valley floors. Although the critical slope of entrenchment is probably related to magnitude of run-off, discharge measurements are not available; therefore, drainage-basin area was selected as the most representative measure of discharge. An inverse relation between drainage-basin area and critical slope of entrenchment applies, and the lower limit of scatter of the data establishes a critical slope-area relation, which can be used to identify potentially unstable valley floors. This relation can help the land manager determine areas of instability where preventive measures can most economically and successfully be undertaken. It is stressed that this particular quantitative relation is applicable only to the Piceance Creek and Yellow Creek drainage basins. In more heterogeneous basins, other variables will need to be included in the analysis; however, the general theory of valley stability will remain applicable.

Actual Order of Convergence of Runge–Kutta Methods on Differential Equations with Discontinuities

This paper examines the order of convergence of Runge–Kutta methods for the ordinary differential equation $y'(t) = f(t,y(t))$ in Banach space, where the function f is badly behaved off the solution trajectory, and the solution y has derivatives only of low order. Under the assumption that f is everywhere Lipschitz continuous in its second argument, we get orders of convergence for the case when f has no derivatives at all, and for when it has only directional derivatives along the solution trajectory $y(t)$. Utilization of the authors’general theory of convergence, stability and consistency enables us to obtain results which apply to actual computations in the presence of rounding error. Some numerical experiments which support the conjecture that these convergence results are the best possible are briefly described.

Grundlagen einer nichtlinearen Stabilitätstheorie der stationären Zustände isothermer elektrischer Netzwerke

To approach recent stability problems in biophysics and -chemistry by the general stability theory of Schlogl, a thermodynamic test function for the stationary states of nonlinear isothermal electrical networks has been given previously for arbitrary network topology. To examine the range of this theory several topologically simple, characteristic applications (most of them containing nonlinear resistances) are treated here. Furthermore it is shown that the test function and its time derivative can be used to obtain additional stability statements which do not follow from the criterion proper.

On the general theory of stability for elastic bodies

This paper deals with the stability of loaded equilibrium configuration of elastic bodies, within the framework of nonlinear thermoelasticity. Although most of the results are obtained in the context of thermo-mechanical stability, the parallel developments for the purely mechanical stability are also considered. By use of an appropriate thermodynamic restriction, several theorems are proved concerning a sufficient condition for stability of the equilibrium configuration. These theorems hold under a general class of motions and their validity is not limited to small motions and small temperature changes from a loaded equilibrium state. Various generalizations of our main results are discussed and the reasonableness of the proposed stability criterion is demonstrated.

Continuum and finite element branching studies of the circular plate

The general theory of elastic stability for discrete conservative systems is applied to the post-buckling of a circular plate using the kinematically-admissible finite-element procedure and a corresponding continuum perturbation analysis is written down for comparison. The energy functional for moderately large deflections is first established, with a careful discussion of the precise mode of compressive loading. The calculus of variations is then used to generate the differential equations of equilibrium, and a continuum perturbation scheme employing the two basic displacement functions is developed as far as the second load derivative. The finite-element study is then made on the basis of the transformed energy functional, and the results are observed to converge rapidly to those of the exact continuum analysis, twelve elements being necessary for accurate results. The perturbation patterns of the finite-element and continuum schemes are seen to be essentially identical, and it is felt that the finite-element perturbation approach is a viable method for more complex practical problems of elastic post-buckling.

Singular critical points in the general theory of elastic stability

Some of the concepts introduced in an earlier work with regard to one-degree-of-freedom, non-linear structural systems are extended to systems with “N” degrees of freedom and “two” independent loading parameters.

Stability and convergence in fluid flow problems

The general convergence and stability theory for finite-difference approximations to fluid flow problems is described. The methods of analysis are outlined and applied to a typical problem. Simple models are then used to consider the behaviour of common difference schemes under practical conditions of finite mesh intervals. Practical stability limits, mode propagation and conservation properties, and nonlinear instabilities are particular characteristics considered. The leapfrog method when used with appropriate safeguards emerges as the most generally suitable basic method.

Linkage disequilibrium and homozygosity of chromosome segments in finite populations

In considering populations of finite size, there are two approaches to studying the correlation of genotype frequencies at two linked loci. The first is based on disequilibrium parameters, and the second on identity-by-descent methods. The basic connection between the two approaches may be stated in the following way. The expected square of the correlation of gene frequencies at two loci r 2 = D 2 p A p a p B p b , is approximately equal to Q, the probability that, given two genes at one locus which are identical by descent, then the two genes at the second locus will be identical by descent through the same pathways. Using this relationship it is shown that for a monoecious population of effective size N e with no selection, the expected value of r 2 will tend to approximately 1 (1 + 4N ec) , where c is the recombination frequency, the rate of approach being given by (1 − 1 2N e )(1 − c) 2 . Computer simulation has shown that this formula holds with reasonable accuracy, and that it is also quite accurate in populations with heterozygote advantage stabilizing gene frequencies. It is suggested that the measurement of linkage disequilibrium in natural populations might thus be used to give information about the effective population size. The identity-by-descent approach is extended to derive the distribution of the length of homozygous chromosome segment surrounding a locus which is identical by descent. The mean such length is approximately 1 2N e ( log N e − 1) . It is suggested that a general theory of stability of the genotype in small populations might be based on parameters such as this.

Fundamental principles in the buckling of structures under combined loading

The basic concepts of the general theory of elastic stability are developed with reference to combined loading, only discrete conservative structural systems being considered.It is observed that the two well known critical points, the limit and bifurcation points, are not quite adequate to describe the buckling behaviour of structures under combined loading. A more appropriate classification which describes the nature of buckling more aptly is given. Thus, in the elastic stability of structures under combined loading, mainly two types of critical point are involved, the “general” and “special” critical points. The conditions giving rise to these two distinct phenomena and their relation to the limit and bifurcation points are examined in detail. The connexion between the shape of the equilibrium surface and the nature of buckling is demonstrated analytically.

ON PARTIAL DIFFERENCE EQUATIONS IN MATHEMATICAL PHYSICS

Abstract A rather general theory of nonlinear computational stability is reported. Instability is caused by both spatial and temporal high frequencies that, if not present initially, will appear from nonlinear interactions. It appears that through simple remedies relative stability, if not perfect stability, can be achieved.

An improved two-layer scheme for sets of parabolic equations with constant coefficients

DIFFERENCE schemes of improved accuracy have been proposed for various types of partial differential equations; see [1, 2] for the references. An improved two-layer scheme was described in [2, 3] for sets of parabolic equations with constant coefficients and no mixed derivatives. Here, to obtain a priori estimates, fairly rigid conditions were imposed on the matrix of space operator coef- ficients. It is easy to construct a set of parabolic equations with constant coefficients which does not satisfy these conditions. We show in the present paper, by using Samarskii's general theory of stability of difference schemes [4, 5], that absolute stability, a priori estimates and convergence theorems can be proved for the improved schemes of [2, 3] without supplementary conditions on the space operator. It should be mentioned that the set of algebraic equations obtained may be solved by the matrix pivotal condensation method. In [6], Samarskii's method of regularisation was used to construct absolutely stable three-layer difference schemes of improved accuracy for sets of parabolic equations with constant coefficients and no mixed derìvatives, and their convergence in a uniform metric was proved in the case of a p-dimensional parallelepiped ( p = 2, 3). As distinct from the two-layer scheme of [2, 3], the sets of algebraic equations obtained in [6] may be solved by means of a uniform pivotal condensation algorithm.

Mechanism for the anti‐redeposition action of sodium carboxy‐methyl cellulose with cotton. II. colloid‐stability theory applied to the fibre‐soil system

AbstractThe deposition of particulate soil on to a fabric in a wash system has been treated as a particular case of the general colloid stability theory of Derjaguin, Landau, Verwey & Overbeek. The zeta potentials on the fabric and the soil in detergent solutions have been calculated from electrophoretic mobility measurements. The colloid stability of the systems was determined by a sedimentation method. In particular, it is shown that the anti‐redeposition action of sodium carboxy‐methyl cellulose may be adequately explained by repulsive electrostatic and attractive van der Waals forces.

Detonation Stability for Disturbances of Small Transverse Wavelength

The stability of one-dimensional, steady detonations to periodic disturbances transverse to the flow is examined in the limit of small wavelength, 2π/ε → 0. The asymptotic criterion for stability is found to depend largely on the steady-state profile of c02η (where c0 is the frozen sound speed, η is the sonic parameter 1 − u2/c02, and u is the mass velocity relative to the von Neumann shock) as a function of distance from the shock. Detonations for which c02η decreases monotonically are found to be stable in the ε → ∞ limit but stability in cases in which this quantity increases either monotonically or up to a maximum is determined through simple integral functions of the steady-flow variables. In contrast to the labor involved with application of the general theory of detonation stability, the asymptotic result can be applied straightforwardly to any detonation, irrespective of the equation of state and the complexity of the chemical kinetics. The results for an idealized, one-reaction, A → B, system with an Arrhenius rate constant are detailed.

Discrete branching points in the general theory of elastic stability

The loss of stability of a general conservative structural system described by n generalized coordinates, a loading parameter, and an imperfection parameter, is studied. Attention is restricted to discrete critical points of the ‘perfect’ system and three branching points which might be described as asymmetric, stable-symmetric and unstable-symmetric are examined in detail. For each point explicit expressions for the derivatives of the post-buckling path of the ‘perfect’ system and for the derivatives relating the critical load of an ‘imperfect’ system to the magnitude of the imperfection parameter are derived intrinsically, without resort to power-series expansions. The general theory is finally applied to three ‘models’ designed to illustrate the salient features of the three branching points under consideration.

On Hydrodynamic Instability of Gas-Lubricated Journal Bearings

Paper starts with a study of static stability response of gas-lubricated bearing, followed by a general small perturbations theory of the dynamic stability of journal bearings. Then the pressure equation for bearings subjected to variable forces and velocities is analyzed, by pointing out the existence of a limiting solution which can occur both for high speeds or for high frequency of the bearing eccentricity. At the same time the squeeze effect can be strongly altered by the lubricant compressibility so that, for motions with high tangential speeds or with high frequencies, the pressures depend only on the thickness h and not on the derivative with respect to time h˙ as is the case of incompressible films. Finally, the analysis of the stability conditions reveals that bearings operating at low numbers H are unstable according to the small perturbations theory. The same situation occurs to the bearings operating with small eccentricity ratios, for any number H. The frequency of undamped oscillations is proportional to the shaft angular speed ω for low numbers H but tends to a bounded value ω0* for high number H. Quasi-resonant conditions may also occur when the number H is increasing, a fact which allows the deduction of a simple half-empirical stability condition.

Some static and dynamic implications of of the general theory of elastic stability

Some static and dynamic implications of of the general theory of elastic stability

Stability of Idealized One-Reaction Detonations

The hydrodynamic stability of steady, one-dimensional detonations in an ideal-gas medium, undergoing the irreversible, unimolecular reaction A → B, having Arrhenius rate constant, is obtained in numerical form by application of the general theory of detonation stability. Stability results, as well as the pressure and progress variable profiles for the steady detonations, are presented for a (constant) heat capacity ratio γ of 1.2, a heat of reaction Q, relative to thermal energy in the cold reactants, of 50, activation energies Q‡ (in the same units) of 10 and 50, and at several detonation velocities, D. The neutral stability curves, consisting of the values of the disturbance wavelength transverse to the steady flow for which the steady detonation of given velocity changes stability (as functions of detonation velocity), are investigated. One-dimensional disturbances are found to be unstable at low detonation velocities (i.e., near the CJ point), for the larger activation energy (but not the smaller) but are stabilized by sufficient overdrive. At fixed detonation velocity, a range of wave lengths are unstable and for Q‡ = 50, this instability appears to persist for all detonation velocities. For decreasing heat of reaction (at fixed Q‡, γ, and D/DCJ) the detonation is found to become stable although for Q‡ = 50 the instability persists to very small values of the heat of reaction. Thus, the system Q = 0.3, Q‡ = 50, γ = 1.2 is found to be unstable when f = (D/DCJ)2 ≤ 1.1. The phenomenon of one-dimensional instability and its dependence on Q is also investigated.

The Form of a North Borneo Nativistic Behavior*

IN THE past forty years, systematic studies of micro-temporal cultural change processes of innovation, acculturation, and assimilation have led to formulation of a series of models, on the basis of which dynamic alterations in regularities in customary behavior may be recognized and made a basis for prediction and empirical test, thereby contributing to a general theory of cultural stability and change (Keesing 1953). Only in the last two decades, however, has detailed attention been accorded to sometimes abrupt, often violent, fluctuations in cultural behavior which occur under a variety of conditions. Such short term, turbulent modifications in custom generally have been termed nativistic movements and have been defined as adaptive or reformulative responses in internal organization which some cultural systems appear to undergo as a consequence of conditions of disequilibrium arising in cultural contact situations. In this discussion I shall describe the form of a nativistic, or revitalization, movement among the Murut of North Borneo. I shall offer also an analysis of these data with reference to current theory of this type of micro-temporal cultural change. These data are derived from ethnological studies conducted in 1959-60 and 1962-64 as part of research among the Dusun and Murut peoples of North Borneo. The Murut comprise a population of some 22,343 persons, concentrated in the southern portion of the North Borneo Interior Residency.1 Descriptions of some forms of Murut customary behavior are found in the works of Roth (1896), Haddon (1901), and Rutter (1922, 1929). Landgraf's report (1956) of his brief survey of Murut population is the initial study of the group by an ethnographer. I have published accounts recently of Murut customary behavior (1960, 1961). Murut peoples generally subsist today in an economy based on an annual crop of swidden rice. The rice diet is supplemented by root crops grown in small mountainside gardens. The basic subsistence activities are extended through a variety of hunting and gathering techniques. The Murut population long has been in contact with agents of differing cultural traditions. There are evidences, in contemporary material cultural

On flows and fronts in a stratified fluid

The instantaneous state of a fluid system is supposed to be defined only by a representative set of numbers, so that the transition from one state to another is represented by a transformation from one set to another, this being effected by the intercession of an operator. By using an operator based on properties of the characteristic lines, a theory of stratified fluid flow may then be developed in such away as to lead to results that are easily translated into program s for machine computation. The equations of motion appropriate to the assumption of a hydrostatic pressure distribution and the equations of continuity are expressed, together with their corresponding equations of variation, in the form of a matrix relation. The condition of indeterminacy leading to the characteristics and the Riemann differential equation is applied to this relation, and, after following certain matrix operations, the conditions are reduced to simplified forms. Using these forms we derive a complete solution for a two-layer system. This derivation then introduces the study of front formation and transition within the system , two distinct types of fluid front being identified. The first, named a 'St Venant front', is associated with a coincidence of characteristics, while the second, named a ‘wave front’, occurs when one of the characteristics becomes time stationary. The conditions leading to these fronts are discussed and the associated criteria are expressed graphically. The results are then related to certain front observations, and to the general theory of hydrodynamic stability. The extension of the front criteria to two-dimensional flows and the possible influence of large vertical motions are also discussed. A program for the solution of the general problem is described. It is shown that the characteristic directions are the latent roots of an easily generated matrix and that an operator may be generated in the machine following the same general procedure as in the two-layer case. The front conditions and transitions are also shown to occur as in the case of two layers.

Basic principles in the general theory of elastic stability

T he basic concepts of elastic stability are developed systematically for a structural system described by n generalized co-ordinates and a variable loading parameter. The expansion of the total potential energy function as a power series in a locally principal set of co-ordinates allows a critical equilibrium state and the neighbouring equilibrium paths to be studied analytically. The critical path configurations associated with the distinct phenomena of ‘snapping’ and ‘buckling’ are examined in detail. The snapping condition, in which an equilibrium path loses its stability on yielding a local extremum of the loading parameter, is the more general, and arises when a single stability coefficient vanishes. The buckling condition, in which a path loses its stability at a point of intersection (bifurcation), arises when a single stability coefficient and a second energy coefficient vanish simultaneously.