On Whitham's averaging method

Landau Institute of Theoretical Physics, Russian Academy of Sciences. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 1, pp. 7–24, January–March, 1995.     

On one generalization of the averaging method

We consider one case where it is possible to establish sufficient conditions for the convergence and analyticity of matrix series used for the construction of a system of moment equations. Kiev National Economic University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 906–911, July, 1997.     

On the existence and geometry of invariant tori in the method of averaging

Summary In this paper some results on the Method of Averaging are presented. We give new conditions for the existence of attractive invariant tori together with some information on their domain of attraction. The results are based on invariant manifold theory and error estimation. An application to the rotational motion of Mercury is given.Dedicated to Professor Hans-Wilhelm Knobloch on the Occasion of his 60th Anniversary     

On the method of averaging in the theory of stability of pulse systems

We consider the functional-differential equation with constant coefficients where

On the averaging method for rod equations with quadratic non-linearity

The averaging method is used to approximate solutions of systems of linearly coupled, (quadratic) non-linear dispersive wave equations, which describe extensional–torsional dynamics of a rod. Existence and uniqueness results are established. Error estimates confirm the asymptotic validity of the approximation method on a long time-scale. The linear couplings between the equations imply that resonance can occur inside a single mode of the solution, but energy can also be transferred to other modes.     

On a modification of the averaging method for seeking higher approximations

Systems in the N.M. Bogolyubov standard form as well as systems with rapid phases are considered. It is proposed to seek the solution in the form of an asymptotic series in a small parameter with coefficients representable in the form of the sum of two functions. The first depends on slow time and is found as the solution of a simpler equation in a finite segment. The second is a trigonometric polynomial of the time (or the angular displacements) with coefficients which depend on the slow time (it is found in an explicit manner). It is convenient to use the results in solving certain problems in celestial mechanics.     

On the asymptotic stability of discontinuous systems analysed via the averaging method

The averaging method is one of the most powerful methods used to analyse differential equations appearing in the study of nonlinear problems. The idea behind the averaging method is to replace the original equation by an averaged equation with simple structure and close solutions. A large number of practical problems lead to differential equations with discontinuous right-hand sides. In a rigorous theory of such systems, developed by Filippov, solutions of a differential equation with discontinuous right-hand side are regarded as being solutions to a special differential inclusion with upper semi-continuous right-hand side. The averaging method was studied for such inclusions by many authors using different and rather restrictive conditions on the regularity of the averaged inclusion. In this paper we prove natural extensions of Bogolyubov’s first theorem and the Samoilenko-Stanzhitskii theorem to differential inclusions with an upper semi-continuous right-hand side. We prove that the solution set of the original differential inclusion is contained in a neighbourhood of the solution set of the averaged one. The extension of Bogolyubov’s theorem concerns finite time intervals, while the extension of the Samoilenko-Stanzhitskii theorem deals with solutions defined on the infinite interval. The averaged inclusion is defined as a special upper limit and no additional condition on its regularity is required.     

Melnikov method and detection of chaos for non-smooth systems

We extend the Melnikov method to non-smooth dynamical systems to study the global behavior near a non-smooth homoclinic orbit under small time-periodic perturbations. The definition and an explicit expression for the extended Melnikov function are given and applied to determine the appearance of transversal homoclinic orbits and chaos. In addition to the standard integral part, the extended Melnikov function contains an extra term which reflects the change of the vector field at the discontinuity. An example is discussed to illustrate the results.     

The averaging method and the persistence of attractors

The theorems that are presented in this paper, are a contribution to the foundations of the averaging method for ordinary differential equations. They involve the study of the persistent features of vector fields, under non autonomous perturbations of mean value zero. The problem of obtain ing qualitative information from the study of the averaged equation is considered and theorems that give new conditions to guarantee the uniform validitv of the approximation over the time interval [ 0.∞), are proved. A general icsult on the persistence of attractors is presented. The analysis uses in a fundamental way, a generalization of the notion of a solution stable under persistent disturbances. The proofs do not require special behavior of the linearized system and the results obtained are not only local, but give relevant information about the persistence of domains of attraction.     

The averaging method and the asymptotic behavior of solutions to differential inclusions

We prove one version of the first Bogolyubov theorem for differential inclusions with multivalued mappings that satisfy certain one-sided constraints. We study the dependence of solutions to differential inclusions on the parameters.     

A geometric averaging procedure for constructing supertransitive approximation to binary comparison matrices

The usefulness of encoding the fuzzy evaluations of alternatives and the importance weights of criteria, in a multiple objective decision problem through binary comparison matrices (or pairwise judgment matrices) is receiving considerable attention. The methodology for identifying the best alternative in a given decision problem involves the computation of the principal eigenvectors of the binary comparison matrices. The eigenvectors transform the fuzzy evaluations of the importance of the criteria and the ratings of the alternatives into a ratio scale. A difficulty that is often experienced in using this approach in practice, is the inconsistency of the binary evaluations. This paper proposes a simple averaging procedure to construct a supertransitive approximation to a binary comparison matrix, where inconsistency is a problem. It is further suggested that such an adjustment might be necessary to more closely reflect the inherent fuzziness of the evaluations contained in a binary comparison matrix. The procedure is illustrated by means of examples.     

An averaging method applied to a duffing equation

We approximate a Duffing equation by an averaged system. We solve the system explicitely and draw bifurcation diagrams in dependence of the forcing term. We discuss the goodness of the averaging method, relative to the behavior of the solutions in dependence of involved parameters, by comparing with results obtained in [6,10] for the original Duffing equation.     

The method of coordinate averaging

In this paper a perturbation technique for nearly linear oscillatory systems is developed and a set of second order averaged equations is obtained. The standard form treated is $\text{dx}\text{dt}\text{=}\text{F}\text{(}\text{x, t}\text{; ?), |?| ? 1}$, and a well known example is considered in detail so as to show how the asymptotic approximations of other methods can be obtained.     

Assessment of errors in the averaging method as applied to linear viscoelastic dynamics

Errors of the averaging method are assessed as applied to the oscillations of a undimensional, linear viscoelastic oscillator.M. V. Lomonosov State University, Moscow. Translated from Mekhanika Polimerov, No. 5, pp. 943–944, September–October, 1972.     

Periodic solutions of parabolic inclusions and the averaging method

We obtain an analog of the second Bogolyubov theorem for differential inclusions with multimappings acting in Sobolev spaces and satisfying some monotonicity and compactness conditions. As a consequence, we obtain criteria for the existence of periodic solutions of secondorder parabolic inclusions.     

Tensor approach to averaging Jacobi fields along geodesicswith random curvature

The paper considers the Jacobi field along a geodesic on a Riemannian manifold on which the curvature is a stochastic process. The author introduces the concept of a linearizing tensor forming the base for obtaining equations for the 2-, 3- and 4-order moments. A theorem on the general form of the moment equation is proved.     

On one approach to the discretization of the lavrent’ev method

We propose a new scheme of discretization of the Lavrent’ev method for operator equations of the first kind with self-adjoint nonnegative operators of certain “smoothness.” This scheme is more economical in the sense of the amount of used discrete information as compared with traditional approaches.     

On the convergence of averaging Hermite interpolators

We investigate two sequences of polynomial operators, H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x), of degrees 2n − 2 and 2n − 3, respectively, defined by interpolatory conditions similar to those of the classical Hermite-Féjer interpolators H_{2n − 1}(f, x). If H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x) are based on the zeros of the jacobi polynomials P_n^(α,β)(x), their convergence behaviour is similar to that of H_{2n − 1}(f;, x). If they are based on the zeros of (1 − x²)T_{n − 2}(x), their convergence behaviour is better, in some sense, than that of H_{2n − 1}(f, x).