94-41 Simo, Carles

Averaging under Fast Quasiperiodic Forcing (615K, PostScript) Mar 2, 94

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Abstract. We consider a non autonomous system of ordinary differential equations. Assume that the time dependence is quasiperiodic with large basic frequencies, $\omega /\varepsilon$ and that the $\omega$ vector satisfies a diophantine condition. Under suitable hypothesis of analyticity, there exists an analytic (time depending) change of coordinates, such that the new vector field is the sum of an autonomous part and a time dependent remainder. The remainder has an exponentially small bound of the form $\exp(-c\varepsilon^{-a})$, where $c$ and $a$ are positive constants. The proof is obtained by iteration of an averaging process. An application is made to the splitting of the separatrices of a two-dimensional normally hyperbolic torus, including several aspects: formal approximation of the torus and their invariant manifolds, numerical computations of the splitting, a first order analysis using a Melnikov approach and the bifurcations of the set of homoclinic orbits. (This is the text of a paper delivered at the NATO-ASI "Integrability and chaos in Hamiltonian systems" held in Torun, Poland 1993)

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