Möbius transformations in stability theory

1970 ◽  
Vol 68 (1) ◽  
pp. 143-151 ◽  
Author(s):  
Russell A. Smith
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Characteristic Equation ◽  
Stability Theory ◽  
Trivial Solution ◽  
Real Variable ◽  
Continuous Functions ◽  
Asymptotically Stable ◽  
Image Position ◽  
Special Case

1. Introduction: Consider the system of ordinary differential equationswhere the unknown x(t) is a complex m-vector, t is a real variable, D is the operator d/dt and a0, …, an are complex m × m matrices whose elements are continuous functions of t, x, Dx, …, Dn−1x. Furthermore, det a0 ╪ 0. In the special case when a0, …, an are constant matrices the trivial solution x = 0 is asymptotically stable if and only if all the roots of the characteristic equation det f(ζ) = 0 have negative real parts, where

On the asymptotic behaviour of nonlinear systems of ordinary differential equations

1985 ◽  
Vol 26 (2) ◽  
pp. 161-170
Author(s):  
Zhivko S. Athanassov
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Ordinary Differential Equations ◽  
Asymptotically Stable ◽  
Future Time ◽  
Zero Function ◽  
Perturbed Systems ◽  
Approach Zero ◽  
Image Position

In this paper we study the asymptotic behaviour of the following systems of ordinary differential equations:where the identically zero function is a solution of (N) i.e. f(t, 0)=0 for all time t. Suppose one knows that all the solutions of (N) which start near zero remain near zero for all future time or even that they approach zero as time increases. For the perturbed systems (P) and (P1) the above property concerning the solutions near zero may or may not remain true. A more precise formulation of this problem is as follows: if zero is stable or asymptotically stable for (N), and if the functions g(t, x) and h(t, x) are small in some sense, give conditions on f(t, x) so that zero is (eventually) stable or asymptotically stable for (P) and (P1).

Numerical methods for differential algebraic equations

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 141-198 ◽  
Author(s):  
Roswitha März
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Image Position

Differential algebraic equations (DAE) are special implicit ordinary differential equations (ODE)where the partial Jacobian f′y(y, x, t) is singular for all values of its arguments.

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

Optimization and α-Disfocality for Ordinary Differential Equations

1985 ◽  
Vol 37 (2) ◽  
pp. 310-323 ◽  
Author(s):  
M. Essén
Keyword(s):  
Distribution Function ◽  
Continuous Function ◽  
Differential Equations ◽  
Convex Hull ◽  
Ordinary Differential Equations ◽  
Lebesgue Measure ◽  
Fixed Positive Number ◽  
Image Position

For f ∊ L−1(0, T), we define the distribution functionwhere T is a fixed positive number and |·| denotes Lebesgue measure. Let Φ:[0, T] → [0, m] be a nonincreasing, right continuous function. In an earlier paper [3], we discussed the equation(0.1)when the coefficient q was allowed to vary in the classWe were in particular interested in finding the supremum and infimum of y(T) when q was in or in the convex hull Ω(Φ) of (see below).

Some General Instability Criteria for Ordinary Differential Equations

1965 ◽  
Vol 8 (4) ◽  
pp. 453-457
Author(s):  
T. A. Burton
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
System Of Differential Equations ◽  
Instability Criteria ◽  
Image Position

We consider a system of differential equations1where 0 = (o, o) is an isolated singular point. Thus, there exists B > o such that S(0, B) contains only one singular point. Here, S(0, B) denotes a sphere centered at 0 with radius B. We shall denote the boundary of S(0, B) by ∂ S(0, B).

On Nagumo's Condition

1972 ◽  
Vol 15 (4) ◽  
pp. 609-611 ◽  
Author(s):  
Thomas Rogers
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Uniqueness Theorem ◽  
Initial Value Problem ◽  
Initial Value ◽  
Image Position

The classical uniqueness theorem of Nagumo [1] for ordinary differential equations is as follows.Theorem. If f(t, y) is continuous on 0≤t≤1, -∞<y<∞ and ifthen there is at most one solution to the initial value problem y'=f(t, y), y(0)=0.

Stability in terms of two measures for population growth models with impulsive perturbations

2020 ◽  
Vol 13 (06) ◽  
pp. 2050051
Author(s):  
Zhinan Xia ◽  
Qianlian Wu ◽  
Dingjiang Wang
Keyword(s):  
Differential Equations ◽  
Population Growth ◽  
Ordinary Differential Equations ◽  
Trivial Solution ◽  
Growth Models ◽  
Two Measures ◽  
Impulsive Perturbations ◽  
The Stability ◽  
Generalized Ordinary Differential Equations ◽  
Population Growth Models

In this paper, we establish some criteria for the stability of trivial solution of population growth models with impulsive perturbations. The working tools are based on the theory of generalized ordinary differential equations. Here, the conditions concerning the functions are more general than the classical ones.

scholarly journals Stability for a Class of Differential Equations with Nonconstant Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Jin Liang ◽  
Tzon-Tzer Lu ◽  
Yashan Xu
Keyword(s):  
Differential Equations ◽  
Positive Constant ◽  
Zero Solution ◽  
Continuous Functions ◽  
Delay Equation ◽  
Asymptotically Stable ◽  
Uniformly Stable

Stability is investigated for the following differential equations with nonconstant delayx't=qtFxt-ptfxt-τt,wherep:[0,+∞)→[0,+∞),q:[0,+∞)→R,τ:[0,+∞)→[0,r], andFandf:R→Rwithxfx>0   for   x≠0   and   x≤a(ais a positive constant) are continuous functions. A criterion is given for the zero solution of this delay equation being uniformly stable and asymptotically stable.

scholarly journals On a Differential Equation and the Construction of Milner's Lamp

1886 ◽  
Vol 5 ◽  
pp. 99-101 ◽  
Author(s):  
Professor Cayley
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Image Position

What sort of an equation isWriteand start with the equationsThis last givesand the system thus isviz., this is a system of ordinary differential equations between the five variables θ, r, X, Y, Z: the system can therefore be integrated with 4 arbitrary constants, and these may be so determined that for the value β of θ, X, Y, Z shall be each = 0; and r shall have the value r0.

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