Transitions from strongly to weakly nonlinear motions of damped nonlinear oscillators

We construct analytical approximations for the transition from strongly non linear, early-time oscillations to weakly nonlinear, late-time motions of s ingle degree of freedom, damped, nonlinear oscillators. Two methods are dev eloped. The first relies on (a) derivation of an analytic solution for the initial value problem of an exactly integrable damped system, (b) developme nt of separate early- and late-time approximations to the damped motion usi ng the integrable solution, and (c) patching of the two approximations in t he time domain by imposing continuity conditions on the composite solution at the point of matching. The second approach relies on a multiple-scales a pplication of the method of nonsmooth transformations first developed by Pi lipchuck, but complemented with a corrected frequency-amplitude relation. T his improved relation is obtained by developing two separate frequency-ampl itude asymptotic expansions in the frequency-amplitude plane, that are vali d for large and small amplitudes, respectively, and then matching them usin g two-point diagonal Pade approximants. Comparisons between analytical appr oximations and numerical results validate the two approaches developed.