Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equations

TL;DR

This paper advances the theory of asymptotic expansions for dissipative nonlinear differential equations with decaying forcing, allowing for complex combinations of exponential, power, and logarithmic functions to describe long-term dynamics.

Contribution

It extends existing asymptotic expansion methods to include multiple base functions and more general decay structures in nonlinear ODEs.

Findings

Solutions admit asymptotic expansions matching the forcing functions

Expansions can involve multiple base functions beyond polynomials

The theory generalizes previous polynomial-based approaches

Abstract

This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in [11]. We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential equations with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are coherent combinations of exponential, power and iterated logarithmic functions. We prove that any decaying solution admits an asymptotic expansion, as time tends to infinity, corresponding to the asymptotic structure of the forcing function. Moreover, these expansions can be generated by more than two base functions and go beyond the polynomial formulation imposed in previous work.