Asymptotic expansions of periodic solutions of ordinary differential equations with large high-frequency terms

We consider the problem on the periodic solutions of a system of ordinary differential equations of arbitrary order n containing terms oscillating at a frequency ω ? 1 with coefficients of the order of ω ^n/2. For this problem, we construct the averaged (limit) problem and justify the averaging method as well as another efficient algorithm for constructing the complete asymptotics of the solution.     

Critical stability of solutions to linear ordinary differential equations with large high-frequency terms

The stability problem is considered for certain classes of systems of linear ordinary differential equations with almost periodic coefficients. These systems are characterized by the presence of rapidly oscillating terms with large amplitudes. For each class of equations, a procedure for analyzing the critical stability of solutions is constructed on the basis of the Shtokalo-Kolesov method. A verification scheme is described. The theory proposed is illustrated by using a linearized stability problem for the upper equilibrium of a pendulum with a vibrating suspension point.     

First-order partial differential equations with large high-frequency terms

Systems of first-order semilinear partial differential equations with terms that oscillate at a frequency ω ? 1 in a single variable and are proportional to \(\sqrt \omega \) are considered. The Krylov-Bogolyubov-Mitropol’skii averaging method is substantiated for such equations. Based on the two-scale expansion method, an algorithm for constructing complete asymptotics of solutions is proposed and justified.     

Asymptotic integration of some evolution problems with large high-frequency terms

We present results concerning the justification of the averaging method, the construction of a complete justified asymptotics, and the time stability of solutions of semilinear parabolic equations and Navier-Stokes systems with polynomial nonlinearities and large rapidly oscillating terms.     

Asymptotic analogs of the Floquet-Lyapunov theorem for some classes of periodic systems of ordinary differential equations

We prove asymptotic analogs of the Floquet-Lyapunov theorem and some reducibility theorems for various classes of linear and quasilinear systems of ordinary differential equations with periodic matrices with large and small amplitudes. We study such problems with the help of new versions of the splitting method in the theory of regular and singular perturbations, which complements the known results. We also adduce several examples.     

Asymptotic integration of a system of differential equations with high-frequency terms in the critical case

We construct the complete asymptotics of a periodic solution of a linear normal system of differential equations with high-frequency coefficients. We study the Lyapunov stability and instability of that solution. More specifically, we consider the critical case in which the matrix coefficient of the formally averaged stationary system has one eigenvector and one generalized (in the Vishik-Lyusternik sense) associated vector.     

Asymptotic solvers for ordinary differential equations with multiple frequencies

We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms,focusing on the case of multiple,non-commensurate frequencies.We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question.Numerical examples illustrate the effectiveness of the method.     

Asymptotic stability of solutions for some classes of impulsive differential equations with distributed delay

In this paper we show the asymptotic stability of the solutions of some differential equations with delay and subject to impulses. After proving the existence of mild solutions on the half-line, we give a Gronwall–Bellman-type theorem. These results are prodromes of the theorem on the asymptotic stability of the mild solutions to a semilinear differential equation with functional delay and impulses in Banach spaces and of its application to a parametric differential equation driving a population dynamics model.     

Asymptotic behavior of delay differential equations with instantaneously terms

This paper deals with scalar delay differential equation with instantaneously term

     

Asymptotic behavior of solutions of linear ordinary differential equations

We present some conditions which ensure that the solution Y(x) of the ordinary differential equation Y′(x) = A(x) Y(x), Y(x₀) = I, where x₀ ? x < ∞ and A(x), Y(x) are n × n complex matrix-valued functions with A(x) continuous, has a nonsingular limit as x → ∞.     

Asymptotic forms of solutions of certain linear ordinary differential equations

We analyze the asymptotic behavior as x → ∞ of the product integral Π_x0^xe^A(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x₀,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y.     

On two classes of autonomous third-order nonlinear ordinary differential equations

Two autonomous, nonlinear, third-order ordinary differential equations whose dynamics can be represented by second-order nonlinear ordinary differential equations for the first-order derivative of the solution are studied analytically and numerically. The analytical study includes both the obtention of closed-form solutions and the use of an artificial parameter method that provides approximations to both the solution and the frequency of oscillations. It is shown that both the analytical solution and the accuracy of the artificial parameter method depend greatly on the sign of the nonlinearities and the initial value of the first-order derivative.