On stability of some solutions for equations of locked lasing of optically coupled lasers with periodic pumping

Lila, D. M.; Martynyuk, À. À.

doi:10.1007/s11072-010-0089-x

Cited by 5 publications

(5 citation statements)

References 6 publications

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“…To this end, the method of matrix-valued functions [3,18,19,21] is generalized. The unified approach to setting up Lyapunov functions in the form of quadratic forms [9,10,[15][16][17] becomes more specific because of the introduction of an additional cubic form in the coordinates of the phase vector with variable coefficients. …”

Section: Introductionmentioning

confidence: 99%

Stability of motion of quasiperiodic systems in critical cases

Lila

2010

Int Appl Mech

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A method of setting up a matrix-valued Lyapunov function for a system of differential equations with quasiperiodic coefficients is proposed. This function is used to establish asymptotic-stability conditions for some class of mechanical systems described by nonlinear systems of equations. The stability of motion of these systems in critical cases is analyzed Keywords: matrix-valued Lyapunov function, differential equations with quasiperiodic coefficients, asymptotic-stability conditions Introduction.In solving stability problems for mechanical systems in critical cases, it is not sufficient to analyze the linear approximation of the perturbed equations of motion [4,6,11,12]. Critical cases occur if and only if the characteristic equation of the first-order system does not have roots with positive real parts and has roots with zero real parts. Developing general methods for the stability analysis of mechanical systems in critical cases is of importance because it is these cases where stability may often be observed. On the other hand, many mechanical systems [20,23] are such that the characteristic equation of the first-order system has critical roots [12,13,24]. Since in analyzing unsteady motions in critical cases that necessitate nonlinear terms in the corresponding equations, nonlinearity can lead to either stability or instability, it is important to be able to find Lyapunov functions.The present paper sets out to outline a new method of setting up a Lyapunov function for nonlinear systems with quasiperiodic coefficients [14,22]. To this end, the method of matrix-valued functions [3,18,19,21] is generalized. The unified approach to setting up Lyapunov functions in the form of quadratic forms [9, 10, 15-17] becomes more specific because of the introduction of an additional cubic form in the coordinates of the phase vector with variable coefficients.2. Setting up the Lyapunov Function. Consider a system of nonlinear equations of perturbed motion with quasiperiodic coefficients:

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Section: Introductionmentioning

confidence: 99%

Stability of motion of quasiperiodic systems in critical cases

Lila

2010

Int Appl Mech

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“…On the basis of the mentioned references, the theoretical objective is to find an asymptotically stable periodic solution of system (1). To this purpose, the very recent paper [4] presents a numerical-analytic scheme of investigation of periodic solutions of system (1) by means of a technique of construction of matrix-valued Lyapunov functions [5]. The aim of this paper is to contribute to the literature from a different point of view, in such a way that we are able to identify explicit regions of parameters where there exists a unique periodic solution which is asymptotically stable.…”

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confidence: 99%

Stabilization of optically coupled lasers with periodic pumping

Torres

2012

Nonlinear Oscill

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We study a periodically forced system modeling the synchronization of two optically coupled lasers pumped by an alternating current. A necessary and sufficient condition for existence of a periodic solution is given, as well as a sufficient condition for uniqueness and asymptotic stability.Вивчаються перiодично збурюванi системи, що моделюють синхронiзацiю двох оптично зв'язаних лазерiв, що накачуються за допомогою змiнного струму. Наведено необхiднi i достатнi умови iснування перiодичного розв'язку, а також достатню умову для його єдиностi та асимптотичної стiйкостi.

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“…New sufficient conditions for the stability of large-scale periodic systems decomposed into an even number of subsystems are formulated Keywords: large-scale periodic system, linear system with periodic coefficients, block-diagonal matrix-valued function, time-periodic Lyapunov function, asymptotic stability conditions Introduction. In natural sciences and engineering, motions repeating in equal and approximately equal time intervals are attributed to oscillatory phenomena [10,14,21]. Only feasible motions of this kind are of practical interest in mechanics.…”

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confidence: 99%

Stability of the motion of linear large-scale oscillating systems

Lila

2011

Int Appl Mech

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A method for setting up a time-periodic Lyapunov function for a linear system with periodic coefficients based on an auxiliary matrix-valued function is proposed and developed. New sufficient conditions for the stability of large-scale periodic systems decomposed into an even number of subsystems are formulated Keywords: large-scale periodic system, linear system with periodic coefficients, block-diagonal matrix-valued function, time-periodic Lyapunov function, asymptotic stability conditions Introduction. In natural sciences and engineering, motions repeating in equal and approximately equal time intervals are attributed to oscillatory phenomena [10,14,21]. Only feasible motions of this kind are of practical interest in mechanics. The principle of choice (from the set of all theoretically possible solutions of the corresponding equations of motion) tells us that such motions correspond to stable solutions.The positive experience of using the direct Lyapunov method in developing reliable approaches to the stability analysis of large-scale systems and systems with an infinite number of degrees of freedom and the justified universality of the Lyapunov function method [1,6,8,12,22] with its generalizations and modifications [3,5,7,9,11], which are currently used to develop the method of multicomponent functions [4,19,20] and establish new stability conditions for some classes of systems [13,15,16,18], give importance to the problem of setting up an approximating V-function for systems of linear differential equations with periodic coefficients [2].We will employ the result obtained in [2,17] as to a method of setting up a Lyapunov function based on the general theorem of asymptotic stability of periodic system (based on a matrix-valued auxiliary function). Generalizing this result, we will formulate sufficient conditions for the uniform asymptotic stability of large-scale periodic systems. We will also analyze the general case of a system of arbitrary order that has matrix trigonometric polynomials as coefficients and consists of an even number of subsystems. The idea of forming the off-diagonal elements of a block-diagonal matrix-valued function by solving Fredholm equations of the second kind will be applied to the system under consideration. The derivation of diagonal elements is reduced to quadratures.1. Lyapunov function. Stability Conditions. Consider a system

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On stability of some solutions for equations of locked lasing of optically coupled lasers with periodic pumping

Abstract: Based on a numerical-analytic scheme of investigation of periodic solutions, we propose a method for the construction of a matrix-valued Lyapunov function for the linear approximation of a system of equations of locked lasing of two optically coupled lasers with periodic pumping.

Cited by 5 publications

References 6 publications

Stability of motion of quasiperiodic systems in critical cases

Stability of motion of quasiperiodic systems in critical cases

Stabilization of optically coupled lasers with periodic pumping

Stability of the motion of linear large-scale oscillating systems

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