Bounds for the characteristic exponents of linear systems

1965 ◽  
Vol 61 (4) ◽  
pp. 889-894 ◽  
Author(s):  
R. A. Smith
Keyword(s):  
Differential Equations ◽  
Linear Systems ◽  
Characteristic Exponent ◽  
Zero Solution ◽  
System Of Differential Equations ◽  
Characteristic Exponents ◽  
Integrable Functions ◽  
Image Position ◽  
Complex Valued

For an n-vector x = (xi) and n × n matrix A = (aij) with complex elements, let |x|2 = Σi|xi|2,|A|2 = ΣiΣj|aij|2. Also, M(A), m(A) denoteℜλ1,ℜλn, respectively, where λ1,…,λA are the eigenvalues of A arranged so that ℜλ1 ≥ … ≥ ℜλn. Throughout this paper A(t) denotes a matrix whose elements aij(t) are complex valued Lebesgue integrable functions of t in (0, T) for all T > 0. Then M(A(t)), m(A(t)) are also Lebesgue integrable in (0, T) for all T > 0. The characteristic exponent μ of a non-zero solution x(t) of the n × n system of differential equationscan be defined, following Perron ((12)), aswhere ℒ denotes lim sup as t → + ∞. When |A(t)| is bounded in (0,∞), μ is finite; in other cases it could be ± ∞.

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.

Limit Point Criteria for Differential Equations, II

1974 ◽  
Vol 26 (02) ◽  
pp. 340-351 ◽  
Author(s):  
Don Hinton
Keyword(s):  
Differential Equations ◽  
Weight Function ◽  
Limit Point ◽  
Differential Operators ◽  
Linearly Independent ◽  
Integrable Functions ◽  
Image Position ◽  
Continuous Weight ◽  
Finite Singularity

We consider here singular differential operators, and for convenience the finite singularity is taken to be zero. One operator discussed is the operator L defined by where q 0 > 0 and the coefficients q t are real, locally Lebesgue integrable functions defined on an interval (a, b). For a given positive, continuous weight function h, conditions are given on the functions qi for which the number of linearly independent solutions y of L(y) = λhy (Re λ = 0) satisfying.

scholarly journals Linear systems of nonoscillatory differential equations with delayed arguments

1984 ◽  
Vol 30 (2) ◽  
pp. 307-314
Author(s):  
K. Gopalsamy
Keyword(s):  
Differential Equations ◽  
Linear Systems ◽  
Sufficient Conditions ◽  
Delayed Arguments ◽  
Image Position

Sufficient conditions are obtained for a not necessarily scalar system of the formto be nonoscillatory.

On Dynamical Systems With One Degree Of Freedom

1955 ◽  
Vol 7 ◽  
pp. 280-283
Author(s):  
C. R. Putnam
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Degree Of Freedom ◽  
System Of Differential Equations ◽  
Component System ◽  
Image Position ◽  
Set Of Points

1. Introduction. Consider the (vector, n-component) system of differential equations1,where f(x) is of class C1. Let Ω denote a set of points, x, consisting of unrestricted solution paths x(t), so that the x(t) exist and lie in Ω for — ∞ < t < ∞.

Functional Equations, Distributions and Approximate Identities

1990 ◽  
Vol 42 (4) ◽  
pp. 696-708 ◽  
Author(s):  
John A. Baker
Keyword(s):  
Functional Equation ◽  
Lebesgue Measure ◽  
Functional Equations ◽  
Schwartz Distributions ◽  
Integrable Functions ◽  
Approximate Identities ◽  
The Subject ◽  
Image Position ◽  
Complex Valued

The subject of this paper is the use of the theory of Schwartz distributions and approximate identities in studying the functional equationThe aj’s and b are complex-valued functions defined on a neighbourhood, U, of 0 in Rm, hj. U → Rn with hj(0) = 0 and fj, g: Rn → C for 1 ≦ j ≦ N. In most of what follows the aj's and hj's are assumed smooth and may be thought of as given. The fj‘s, b and g may be thought of as the unknowns. Typically we are concerned with locally integrable functions f1, … , fN such that, for each s in U, (1) holds for a.e. (almost every) x ∈ Rn, in the sense of Lebesgue measure.

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.

XXVI.—The Universal Integral Invariants of Hamiltonian Systems and Application to the Theory of Canonical Transformations

1948 ◽  
Vol 62 (3) ◽  
pp. 237-246 ◽  
Author(s):  
Hwa-Chung Lee
Keyword(s):  
Hamiltonian System ◽  
Differential Equations ◽  
Hamiltonian Systems ◽  
Hamiltonian Function ◽  
Integral Invariant ◽  
Canonical Transformations ◽  
System Of Differential Equations ◽  
Integral Invariants ◽  
Image Position ◽  
Universal Integral

I. Introduction.—Consider a Hamiltonian system of differential equationswhere H is a function of the 2n variables qi and pi involving in general also the time t. For each given Hamiltonian function H the system (1.1) possesses infinitely many absolute and relative integral invariants of every order r = 1,…, 2n, which can all be written out when (1.1) is integrated. Our interest now is not in these integral invariants, which are possessed by one Hamiltonian system, but in those which are possessed by all Hamiltonian systems. Such an integral invariant, which is independent of the Hamiltonian H, is said to be universal.

scholarly journals Periodic solutions of quasilinear non-autonomous systems with impulses

1985 ◽  
Vol 31 (2) ◽  
pp. 185-197 ◽  
Author(s):  
S.G. Hristova ◽  
D.D. Bainov
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Small Parameter ◽  
Periodic Solutions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
System Of Differential Equations ◽  
Image Position ◽  
The Given

The paper considers a system of differential equations with impulse perturbations at fixed moments in time of the formwhere x ∈ Rn, ε is a small parameter,Sufficient conditions are found for the existence of the periodic solution of the given system in the critical and non-critical cases.

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