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

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