  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.
