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This report contains a proof of the theorem announced in Ref.27. Over the last year I have reconsidered this report, removed some (small) errors, improved some of the arguments and added some figures. The resulting work, "The onset of bifurcations in the forced Duffing equation with damping" appeared in the International Journal of Bifurcations and Chaos (no.28 above).
This report contains a proof that the Duffing equation admits of a unique periodic solution if the damping increases slightly quicker than logarithmically with the forcing.
This is an introduction to the use of boundary layer methods in obtaining periodic solutions of Duffing's equation at large damping and forcing. The report contains many numerical examples.