scholarly journals A stability theory for perturbed differential equations

1979 ◽  
Vol 2 (2) ◽  
pp. 283-297
Author(s):  
Sheldon P. Gordon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Direct Method ◽  
Mathematical Technique ◽  
General Differential Equation ◽  
Stability And Instability ◽  
The Difference ◽  
Perturbed Differential Equation ◽  
Eventual Stability ◽  
Liapunov's Direct Method

The problem of determining the behavior of the solutions of a perturbed differential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered isX′=f(t,X)and the associated perturbed differential equation isY′=f(t,Y)+g(t,Y).The approach used is to examine the difference between the respective solutionsF(t,t0,x0)andG(t,t0,y0)of these two differential equations. Definitions paralleling the usual concepts of stability, asymptotic stability, eventual stability, exponential stability and instability are introduced for the differenceG(t,t0,y0)−F(t,t0,x0)in the case where the initial valuesy0andx0are sufficiently close. The principal mathematical technique employed is a new modification of Liapunov's Direct Method which is applied to the difference of the two solutions. Each of the various stabillty-type properties considered is then shown to be guaranteed by the existence of a Liapunov-type function with appropriate properties.

A reduction theory of second order meromorphic differential equations, I

1987 ◽  
Vol 101 (2) ◽  
pp. 323-342
Author(s):  
W. B. Jurkat ◽  
H. J. Zwiesler
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Standard Form ◽  
Reduction Theory ◽  
Fundamental Solution Matrix ◽  
Equivalent Equation ◽  
The Difference ◽  
Solution Matrix ◽  
Meromorphic Differential Equation ◽  
Meromorphic Differential

In this article we investigate the meromorphic differential equation X′(z) = A(z) X(z), often abbreviated by [A], where A(z) is a matrix (all matrices we consider have dimensions 2 × 2) meromorphic at infinity, i.e. holomorphic in a punctured neighbourhood of infinity with at most a pole there. Moreover, X(z) denotes a fundamental solution matrix. Given a matrix T(z) which together with its inverse is meromorphic at infinity (a meromorphic transformation), then the function Y(z) = T−1(z) X(z) solves the differential equation [B] with B = T−1AT − T−1T [1,5]. This introduces an equivalence relation among meromorphic differential equations and leads to the question of finding a simple representative for each equivalence class, which, for example, is of importance for further function-theoretic examinations of the solutions. The first major achievement in this direction is marked by Birkhoff's reduction which shows that it is always possible to obtain an equivalent equation [B] where B(z) is holomorphic in ℂ ¬ {0} (throughout this article A ¬ B denotes the difference of these sets) with at most a singularity of the first kind at 0 [1, 2, 5, 6]. We call this the standard form. The question of how many further simplifications can be made will be answered in the framework of our reduction theory. For this purpose we introduce the notion of a normalized standard equation [A] (NSE) which is defined by the following conditions:(i) , where r ∈ ℕ and Ak are constant matrices, (notation: )(ii) A(z) has trace tr for some c ∈ ℂ,(iii) Ar−1 has different eigenvalues,(iv) the eigenvalues of A−1 are either incongruent modulo 1 or equal,(v) if A−1 = μI, then Ar−1 is diagonal,(vi) Ar−1 and A−1 are triangular in opposite ways,(vii) a12(z) is monic (leading coefficient equals 1) unless a12 ≡ 0; furthermore a21(z) is monic in case that a12 ≡ 0 but a21 ≢ 0.

Existence of conjugate points for self-adjoint linear differential equations

1989 ◽  
Vol 113 (1-2) ◽  
pp. 73-85 ◽  
Author(s):  
Ondřej Došlý
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equations ◽  
Conjugate Points ◽  
General Differential Equation ◽  
Linear Differential

SynopsisThe conjecture of Muller-Pfeiffer [4] concerning the oscillation behaviour of the differential equation (–l)n(p(x)y(n))(n) + q(x)y = 0 is proved, and a similar conjecture concerning the more general differential equation ∑nk=0(−l)k(Pk(x)y(k)(k + q(x)y= 0 is formulated.

Asymptotic stability criteria for delay-differential equations

1988 ◽  
Vol 110 (1-2) ◽  
pp. 31-44
Author(s):  
L.C. Becker ◽  
T.A. Burton
Keyword(s):  
Differential Equations ◽  
Direct Method ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
State Variables ◽  
Stability Criteria ◽  
Bounded State ◽  
Functional Differential ◽  
Delay Differential ◽  
Liapunov's Direct Method

SynopsisThis paper is concerned with the problem of showing uniform stability and equiasymptotic stability of thezero solution of functional differential equations with either finite or infinite delay. The investigations are based on Liapunov's direct method and attention is focused on those equations whose right-hand sides are unbounded for bounded state variables.

Stability Conditions for a Class of Distributed-Parameter Systems

1966 ◽  
Vol 88 (2) ◽  
pp. 475-479 ◽  
Author(s):  
R. E. Blodgett
Keyword(s):  
Equilibrium State ◽  
Local Stability ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Chemical Reactor ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Stability Conditions ◽  
Stability And Instability ◽  
Liapunov's Direct Method

The purpose of this paper is to obtain stability conditions for a class of nonlinear distributed-parameter systems by using a generalization of Liapunov’s direct method. Sufficient conditions for local stability and instability of the equilibrium state are derived. An application is given in which conditions are obtained for stability of a chemical-reactor process.

A Procedure for Investigating the Liapunov Stability of Nonautonomous Linear Second-Order Systems

1973 ◽  
Vol 40 (4) ◽  
pp. 1103-1106 ◽  
Author(s):  
S. E. Jones ◽  
T. R. Robe
Keyword(s):  
Differential Equation ◽  
Ad Hoc ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Liapunov Function ◽  
Null Solution ◽  
Liapunov Stability ◽  
The Stability ◽  
Second Order Systems ◽  
Liapunov's Direct Method

Utilizing Liapunov’s direct method a procedure is presented which generates sufficient conditions for the stability of the null solution of the nonautonomous differential equation x¨ + b(t)x˙ + a(t)x = 0. This procedure systematically leads to the construction of a Liapunov function for a given differential equation and thus eliminates the normally ad hoc nature of the direct method. Four examples illustrating the procedure are discussed.

scholarly journals The fundamental importance of differential equations with three singularities in Mathematical Statistics

1985 ◽  
Vol 4 (1) ◽  
pp. 18-24
Author(s):  
H. S. Steyn
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Probability Distributions ◽  
Mathematical Physics ◽  
Second Order ◽  
Discrete Distributions ◽  
Mathematical Statistics ◽  
Multivariate Distributions ◽  
The Difference ◽  
Linear Differential

It is well-known that the solution of a second order linear differential equation with at most five singularities plays a fundamental role in Mathematical Physics. In this paper it is shown that this statement also applies to Mathematical Statistics but with the difference that an equation with three singularities will suffice. Two wide classes of probability distributions are defined as solutions of such a differential equation, one for continuous distributions and one for discrete distributions. These two classes contain as members all the distributions which are normally considered as of importance in Mathematical Statistics. In the continuous case the probability functions are solutions of the relevant second order equation, while in the discrete case the probability generating functions are solutions there-of. By defining appropriate multidimentional extensions corresponding differential equations are obtained for continuous and discrete multivariate distributions.

scholarly journals Methods for stability and instability of the solution of one non-linear differential equation

1996 ◽  
Vol 18 (3) ◽  
pp. 1-7
Author(s):  
Nguyen Dang Bich ◽  
Nguyen Vo Thong
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Analytical Solutions ◽  
Flexible Structures ◽  
Stability Conditions ◽  
Stability And Instability ◽  
Non Linear ◽  
Linear Differential ◽  
The Stability

When research on instability states of tall and flexible structures is carried out, the solutions of non-linear differential equations have to be investigated. Although the fully analytical solutions of the moving rule of the structures cannot be found, but based on the conditions applied to the parameters of the moving differential equations the authors have studied the characteristics of the solutions when t→∞. Then the instability of the structures may be investigated, and the stability conditions can be concluded. This is the content which this paper would like to present. 

Climate: A Chain of Identical Reservoirs

1991 ◽  
Author(s):  
James C. G. Walker
Keyword(s):  
Differential Equation ◽  
Energy Balance ◽  
Differential Equations ◽  
Climate Model ◽  
Diffusion Problem ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
The Difference ◽  
A Chain ◽  
The One

One class of important problems involves diffusion in a single spatial dimension, for example, height profiles of reactive constituents in a turbulently mixing atmosphere, profiles of concentration as a function of depth in the ocean or other body of water, diffusion and diagenesis within sediments, and calculation of temperatures as a function of depth or position in a variety of media. The one-dimensional diffusion problem typically yields a chain of interacting reservoirs that exchange the species of interest only with the immediately adjacent reservoirs. In the mathematical formulation of the problem, each differential equation is coupled only to adjacent differential equations and not to more distant ones. Substantial economies of computation can therefore be achieved, making it possible to deal with a larger number of reservoirs and corresponding differential equations. In this chapter I shall explain how to solve a one-dimensional diffusion problem efficiently, performing only the necessary calculations. The example I shall use is the calculation of the zonally averaged temperature of the surface of the Earth (that is, the temperature averaged over all longitudes as a function of latitude). I first present an energy balance climate model that calculates zonally averaged temperatures as a function of latitude in terms of the absorption of solar energy, which is a function of latitude, the emission of long-wave planetary radiation to space, which is a function of temperature, and the transport of heat from one latitude to another. This heat transport is represented as a diffusive process, dependent on the temperature gradient or the difference between temperatures in adjacent latitude bands. I use the energy balance climate model first to calculate annual average temperature as a function of latitude, comparing the calculated results with observed values and tuning the simulation by adjusting the diffusion parameter that describes the transport of energy between latitudes. I then show that most of the elements of the sleq array for this problem are zero. Nonzero elements are present only on the diagonal and immediately adjacent to the diagonal. The array has this property because each differential equation for temperature in a latitude band is coupled only to temperatures in the adjacent latitude bands.

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