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This article is focused on the asymptotic expansions, as time tends toinfinity, of decaying solutions of a system of nonlinear ordinary differentialequations. The nonlinear terms in the system may not be smooth in a smallneighborhood of the origin. The forcing function decays to zero in a verycomplicated but coherent way. We prove that, under suitable conditions on theforcing function, every decaying solution admits the same asymptotic expansionof a new type. This expansion contains a new variable that allows it to beestablished in a closed-form, but does not affect the meaning and precision ofthe expansion. The new variable is introduced to overcome the unsuitability ofcertain terms resulted from the scaling and shifting method, which is used todeal with the lack of smoothness of the nonlinear terms, and the general formof the forcing function. Moreover, the expansion is constructed explicitly withthe use of the complexification method.

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