Hamiltonian Mechanics pp 13-34 | Cite as

# Averaging under Fast Quasiperiodic Forcing

## Abstract

We consider a non autonomous system of ordinary differential equations. Assume that the time dependence is quasiperiodic with large basic frequencies, *ω/ε* and that the *ω* 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(−*cε* ^{−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.

## Keywords

Invariant Manifold Homoclinic Orbit Invariant Torus Moderate Time Interval Melnikov Function## Preview

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