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 < ∞.

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.

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 ± ∞.

scholarly journals ON APPROXIMATION OF ALMOST-PERIODIC SOLUTIONS FOR A NON-LINEAR COUNTABLE SYSTEM OF DIFFERENTIAL EQUATIONS BY QUASI-PERIODIC SOLUTIONS FOR SOME LINEAR SYSTEM

2021 ◽  
Vol 9 (2) ◽  
pp. 111-123
Author(s):  
Yu. Teplinsky
Keyword(s):  
Dynamical Systems ◽  
Linear System ◽  
Differential Equations ◽  
Periodic Solutions ◽  
System Of Equations ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
System Of Differential Equations ◽  
Dimensional Torus ◽  
Non Linear

It is well-known that many applied problems in different areas of mathematics, physics, and technology require research into questions of existence of oscillating solutions for differential systems, which are their mathematical models. This is especially true for the problems of celestial mechanics. Novadays, by oscillatory motions in dynamical systems, according to V. V. Nemitsky, we call their recurrent motions. As it is known from Birkhoff theorem, trajectories of such motions contain minimal compact sets of dynamical systems. The class of recurrent motions contains, in particular, both quasi-periodic and almost-periodic motions. There are renowned fundamental theorems by Amerio and Favard related to existence of almost-periodic solutions for linear and non-linear systems. It is also of interest to research the behavior of a dynamical system’s motions in a neighborhood of a recurrent trajectory. It became understood later, that the question of existence of such trajectories is closely related to existence of invariant tori in such systems, and the method of Green-Samoilenko function is useful for constructing such tori. Here we consider a non-linear system of differential equations defined on Cartesian product of the infinite-dimensional torus T∞ and the space of bounded number sequences m. The problem is to find sufficient conditions for the given system of equations to possess a family of almost-periodic in the sense of Bohr solutions, dependent on the parameter ψ ∈ T∞, every one of which can be approximated by a quasi-periodic solution of some linear system of equations defined on a finite-dimensional torus.

Tracking Controllers for Uncertain Systems: Application to a Manutec R3 Robot

1989 ◽  
Vol 111 (4) ◽  
pp. 609-618 ◽  
Author(s):  
Martin Corless
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
General Class ◽  
Uncertain Systems ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Tracking Controllers ◽  
Asymptotic Tracking ◽  
Three Degree Of Freedom

We consider a class of uncertain dynamical systems described by ordinary differential equations and characterized by certain structural conditions and known bounding functions. For a feasible class of desired state motions we present a class of controllers which assure asymptotic tracking to within any desired degree of accuracy. The results are applied to a general class of mechanical systems and are illustrated by a simple example and by application to a three degree-of-freedom model of a Manutec r3 robot.

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.

scholarly journals VII.—On the Adelphic Integral of the Differential Equations of Dynamics

1918 ◽  
Vol 37 ◽  
pp. 95-116 ◽  
Author(s):  
E. T. Whittaker
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Two Degrees Of Freedom ◽  
Image Position ◽  
Kinetic And Potential Energies

§ 1. Ordinary and singular periodic solutions of a dynamical system. — The present paper is concerned with the motion of dynamical systems which possess an integral of energy. To fix ideas, we shall suppose that the system has two degrees of freedom, so that the equations of motion in generalised co-ordinates may be written in Hamilton's formwhere (q1q2) are the generalised co-ordinates, (p1, p2) are the generalised momenta, and where H is a function of (q1, q2, p1, p2) which represents the sum of the kinetic and potential energies.

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