Stability of motion of quasiperiodic systems in critical cases

Abstract

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:

Stability of the motion of linear large-scale oscillating systems

“…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.…”

“…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].…”

Stability of the stationary motion of a spherical pendulum interacting with a string

“…The simplified problem formulation assumes vertical oscillation of the string fixed at the ends, and the refined formulation assumes nonvertical oscillation. We will use an (in)stability criterion [5] for solutions of linear differential equations with quasiperiodic coefficients [7][8][9][10][11][12][13].…”

Stability of the stationary motion of a double pendulum interacting with a string

“…The in-plane vertical motion of a simple pendulum interacting with an elastic string was modeled in [4] using a normal-form system of differential equations of motion. The perturbed equations of motion in the neighborhood of the lower equilibrium position, the method of auxiliary matrix-valued functions [12][13][14][15][16][17][18][19], and asymptotic methods of the theory of nonlinear oscillations were used in [4] to analyze the stationary oscillations of a simple pendulum for stability in linear and nonlinear problem formulations.The present paper sets out to model a double pendulum interacting with a string and to analyze its equilibrium positions for asymptotic stability using a stability criterion for linear differential equations with quasiperiodic coefficients [8,9].1. Problem Formulation.…”