A new form of asymptotic expansion for non-smooth differential equations with time-decaying forcing functions

TL;DR

The paper introduces a novel asymptotic expansion method for solutions of non-smooth differential equations with complex, decaying forcing functions, using complexification to explicitly construct the expansion.

Contribution

It presents a new type of asymptotic expansion for non-smooth ODEs with decaying forcing, including a new variable and a closed-form expression.

Findings

Expansion exists for solutions with decaying forcing functions

Explicit construction of the expansion using complexification

New variable enhances the expansion's form without loss of accuracy

Abstract

This article is focused on the asymptotic expansions, as time tends to infinity, of solutions of a system of ordinary differential equations with non-smooth nonlinear terms. The forcing function decays to zero in a very complicated but coherent way. We prove that every decaying solution admits an asymptotic expansion of a new type. This expansion contains a new variable that allows it to be established in a closed-form, but does not affect the meaning and precision of the expansion. Moreover, the expansion is constructed explicitly with the use of the complexification method.