  We study the precise asymptotic behavior of a non-trivial solution that
converges to zero, as time tends to infinity, of dissipative systems of
nonlinear ordinary differential equations. The nonlinear term of the equations
may not possess a Taylor series expansion about the origin. This absence
technically cripples previous proofs in establishing an asymptotic expansion,
as an infinite series, for such a decaying solution. In the current paper, we
overcome this limitation and obtain an infinite series asymptotic expansion, as
time goes to infinity. This series expansion provides large time approximations
for the solution with the errors decaying exponentially at any given rates. The
main idea is to shift the center of the Taylor expansions for the nonlinear
term to a non-zero point. Such a point turns out to come from the non-trivial
asymptotic behavior of the solution, which we prove by a new and simple method.
Our result applies to different classes of non-linear equations that have not
been dealt with previously.
