Discrete Systems

Abstract

The classical method of perturbation is based on the assumption that the influence of the nonlinear part of the considered differential equations is small in comparison to the influence of the linear part of the equations, or that the oscillation amplitude is small. The perturbation technique can also be used even if the deviations from the true (sought) solution are not small, but are localized in a small space. This is emphasized by the formal or natural introduction of the “small” perturbation parameter ε to the differential equation. The solution of the equations are sought in the form of power series because of the parameter ε (for ε = 0 the fundamental solution — the first term of the required series — is the solution to the linear differential equation). The next solution components, standing by the successive powers of ε, are obtained from the recurrent sequence of linear differential equations with constant coefficients.