New approach to the asymptotic theory of nonlinear oscillations and wave-propagation

For a class of nonlinear oscillation problems described by a second order ordinary differential equation containing a small nonnegative perturbation parameter $\varepsilon $ we determine asymptotic solutions which are uniformly valid as $\varepsilon \downarrow 0$ on intervals of the order of magnitude $\varepsilon ^{ - 1} $. These asymptotic solutions are obtained from formal asymptotic solutions subject to a condition in order that they approximate the exact solution asymptotically. These formal asymptotic solutions are constructed in the form of a finite generalized asymptotic power series involving functions of the periodic two variable type.

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Perturbed Nonlinear Oscillations

Initial-value problems for linear ordinary differential equations with a small parameter

Asymptotic solutions of weakly non-linear oscillatory systems are considered in this paper. An algorithm is given for constructing an Nth order asymptotic solution for a class of such systems. The asymptotic solution is almost periodic and has a two-time scale property. Fourier like projections are introduced and used in this paper together with perturbation techniques to arrive at the sought asymptotic solution. The results in this paper extend the results by Hoogstraten and Kaper (1975).

Asymptotic analysis of almost periodic weakly non-linear systems

For a class of weakly non-linear oscillations involving a small parameter e we determine asymptotic solutions as ɛ ↓ 0 which are uniformly valid on some time interval. First, we consider a general initial-value problem in IRn containing a small parameter ɛ. We derive sufficient conditions for asymptotic correctness as ɛ ↓ 0 to be satisfied by formal asymptotic solutions. Next, we consider for the original problem formal asymptotic solutions of a two-variable type. For this type of formal asymptotic solutions the conditions for asymptotic correctness take a form which is very useful in the subsequent development of a construction technique for asymptotic solutions.

H. W. Hoogstraten

Papers

Asymptotic theory for a class of weakly nonlinear oscillations