scholarly journals Note on a Resolution of Linear Differential Systems

1934 ◽  
Vol 4 (1) ◽  
pp. 36-40
Author(s):  
I. S. ◽  
E. S. Sokolnikoff
Keyword(s):  
Differential System ◽  
Lower Order ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Differential System ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Equivalent Systems ◽  
Linear Differential ◽  
Necessary And Sufficient

In a recent paper J. M. Whittaker considered the problem of resolving a linear differential system into a product of two or more systems of lower order. This note is a contribution to this problem and furnishes the necessary and sufficient conditions for the resolution of the system into two equivalent systems of lower order of the type considered by Whittaker.

scholarly journals On Transformation and Oscillation of Linear Differential System

1977 ◽  
Vol 29 (2) ◽  
pp. 392-399 ◽  
Author(s):  
Donald F. St. Mary
Keyword(s):  
Differential System ◽  
Comparison Theorem ◽  
Linear Case ◽  
Linear Differential System ◽  
Oscillation Theorem ◽  
Differential Systems ◽  
Order Linear ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
The Relationship

In this paper we study second order linear differential systems. We examine the relationship between oscillation of n-dimensional systems and certain associated n-dimensional systems, where m ≧ n. Several theorems are presented which unify and encompass in the linear case a number of results from the literature. In particular, we present a transformation which extends an oscillation theorem due to Allegretto and Erbe [1], and a comparison theorem due to Kreith [9], and explains some work of Howard [7].

Beurling–Borg Type Theorem for Two-Dimensional Linear Differential Systems

2008 ◽  
Vol 15 (4) ◽  
pp. 677-682
Author(s):  
Ivan Kiguradze
Keyword(s):  
Differential System ◽  
Trivial Solution ◽  
Type Theorem ◽  
Linear Differential System ◽  
Two Dimensional ◽  
Differential Systems ◽  
Number Of Zeros ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
Upper Estimate

Abstract For the two-dimensional linear differential system with Lebesgue integrable coefficients 𝑝𝑖𝑘 : [𝑎, 𝑏] → ℝ (𝑖 = 1, 2), a Beurling–Borg type theorem is proved on an upper estimate of the number of zeros of an arbitrary non-trivial solution.

scholarly journals Variational control problems for linear differential systems with Stieltjes boundary conditions

1978 ◽  
Vol 20 (4) ◽  
pp. 434-445 ◽  
Author(s):  
W. L. Chan ◽  
S. K. Ng
Keyword(s):  
Boundary Conditions ◽  
Bounded Variation ◽  
Sufficient Conditions ◽  
Control Problems ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Sufficient Conditions For Optimality ◽  
Quadratic Performance ◽  
Linear Differential ◽  
Necessary And Sufficient

Necessary and sufficient conditions for optimality in the control of linear differential systems ẋ = Ax + Bu with Stieltjes boundary conditions , where ν is an r × n matrix valued measure of bounded variation, are obtained, Feedback-like control is given in the case of quadratic performance.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

Center Problems and Limit Cycle Bifurcations in a Class of Quasi-Homogeneous Systems

2015 ◽  
Vol 25 (10) ◽  
pp. 1550135 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han ◽  
Yong Wang
Keyword(s):  
Limit Cycle ◽  
Differential System ◽  
Homogeneous Polynomial ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Perturbed System ◽  
Generalized Polynomial ◽  
Necessary And Sufficient

In this paper, we first classify all centers of a class of quasi-homogeneous polynomial differential systems of degree 5. Then we extend this kind of systems to a generalized polynomial differential system and provide the necessary and sufficient conditions to have a center at the origin. Furthermore, we study the Poincaré bifurcation for its perturbed system as it has a center at the origin, find the Poincaré cyclicity up to first order of ε.

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