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Both upper and lower estimates are establishedfor the separatrix splitting of rapidly forced systems with a homoclinic orbit. The general theory is applied to the equation φ + sin φ =S sin(^t-_Є) for illustration. There are two types of results. First,fix η > 0 and let 0 < E ≤ 1 and 0 ≤ S ≤ S_0 where S_0 is sufficiently small. If the separatrices split, they do so by an amount that is no more than C S exp(-^1-є(^π-2-η)) where C = C(S_0) is a constant depending on S_0 but is uniform in E and S . Second, if we replace S by єP S, p ≥ 8, then we have the sharper estimate C_2 є^p S e ^(-π/2є) ≤ splitting distance ≤ C_1є^P S є -^(π/2є) for positive constants C_1 and C_2 depending on So alone. In particular, in this second case, the Melnikov criterion correctly predicts exponentially small splitting and transversal intersection of the separatrices. After developing this theory we discuss some of its applications, concentrating on a 2:I resonance that occurs in a KAM (Kolmogorov, Arnold, and Moser) situation and in the/orced saddle node bifurcation described by X + µX + x^2 + x^3 = Sf(t)

Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations