Behavior of the integral curves of a system of differential equations near a singular point in a space of more than one dimension

1961 ◽  
pp. 173-186 ◽  
Author(s):  
L. È. Reĭzin′
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
One Dimension ◽  
System Of Differential Equations ◽  
Integral Curves

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

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).

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.

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.

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