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).

Integrals of Systems of Ordinary Differential Equations

1924 ◽  
Vol 22 (3) ◽  
pp. 273-281 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytic Functions ◽  
Convergent Series ◽  
System Of Differential Equations ◽  
Image Position

Letbe a system of differential equations in which X1, … Xn are analytic functions independent of t, expansible in convergent series of powers of x1, … xn. Suppose further that not all the functions X1, … Xn vanish when x1 = … = xn = 0. It will be shown in §§ 1–3 that these equations possess (n — 1) independent integrals independent of t, expansible in convergent series of powers of x1, … xn; i.e. functionssuch thatidentically. In § 4 it is shown how the series for ϕ1,…ϕn−1 may be most directly constructed, and in § 5 is briefly considered the corresponding problem when the origin is a singular point of the equations, i.e. when the expansions of X1,…Xn all begin with terms of the first or higher degrees.

On Integrals developable about a Singular Point of a Hamiltonian System of Differential Equations

1924 ◽  
Vol 22 (3) ◽  
pp. 325-349 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Singular Point ◽  
Hamiltonian System ◽  
Characteristic Function ◽  
Differential Equations ◽  
Convergent Series ◽  
Hamiltonian Form ◽  
System Of Differential Equations ◽  
Image Position

Letbe a system of differential equations of Hamiltonian form, the characteristic function H being independent of t and expansible in a convergent series of powers of x1, … xn, y1, … yn in which the terms of lowest degree are

scholarly journals Stability on the basis of Orthogonal Trajectories

1965 ◽  
Vol 8 (5) ◽  
pp. 647-658
Author(s):  
T. A. Burton
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Second Order ◽  
System Of Differential Equations ◽  
Partial Derivatives ◽  
Connected Set ◽  
Image Position ◽  
Simply Connected

We consider a system of differential equations of second order given by1(' = d/dt) where P and Q have continuous first partial derivatives with respect to x and y in some open and simply connected set R containing O = (0, 0) which we assume to be the only singular point in R. In fact, let R be the whole plane; for if not then the following discussion can be modified.

Solvability of Ordinary Differential Equations Near Singular Points: An Analytic Case

1967 ◽  
Vol 19 ◽  
pp. 1303-1313
Author(s):  
Homer G. Ellis
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Singular Point ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Singular Points ◽  
Initial Value ◽  
Analytic Case ◽  
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The question of solvability of the differential equation1with x ranging over an interval (0, a], and with the boundary condition ƒ(0+) = 0, can be investigated as an initial-value problem at 0, which may be a singular point for the equation.

Integrals developable about a singular point of a Hamiltonian system of differential equations. Part II

1925 ◽  
Vol 22 (4) ◽  
pp. 510-533 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Singular Point ◽  
Hamiltonian System ◽  
Power Series ◽  
Differential Equations ◽  
Taylor Series ◽  
System Of Differential Equations ◽  
Linearly Independent ◽  
Image Position

This paper completes an investigation, of which the first part has already been published, into the integrals of a Hamiltonian system which are formally developable about a singular point of the system. Letbe a system of differential equations of which the origin is a singular point of the first type, i.e. a point at which H is developable in a convergent Taylor series, but at which its first derivatives all vanish. We suppose that H does not involve t, and we consider only integrals not involving t. Let the exponents of this singular point be ± λ1, ± λ2,…±λn. In Part I, I considered the case in which the constants λ1,…λn are connected by no relation of commensurability, i.e. a relation of the formwhere A1…An are integers (positive, negative or zero) not all zero, and showed that the equations (1) possess n, and only n, integrals not involving t which are formally developable as power series in the xk, yk. In this paper I consider the case in which λ1 … λn are connected by one or more relations of commensur-ability. Suppose that there are p, and only p, such relations linearly independent (p > 0): it will be shown that the equations (1) possess (n − p) independent integrals not involving t, formally developable about the origin and independent of H.

The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces

2006 ◽  
Vol 49 (2) ◽  
pp. 170-184
Author(s):  
Richard Atkins
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Shortest Paths ◽  
Equivalence Problem ◽  
Second Order ◽  
System Of Differential Equations ◽  
Lagrange Equations ◽  
Invariant Functions ◽  
Underlying Geometry ◽  
The Relationship

AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

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.

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