During the last two decades, in several branches of science (water
waves, crystal growth, travelling waves in one dimensional lattices,
splitting of separatrices, ...) different problems appeared in which the
key point is the computation of exponentially small terms. This
self-contained monograph gives new and rigorous mathematical tools which
enable a systematic study of such problems. Starting with elementary
illuminating examples, the book contains (i) new asymptotical tools for
obtaining exponentially small equivalents of oscillatory integrals
involving solutions of nonlinear differential equations; (ii)
implementation of these tools for solving old open problems of
bifurcation theory such as existence of homoclinic connections near
resonances in reversible systems