Averaging under Fast Quasiperiodic Forcing
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.
KeywordsInvariant Manifold Homoclinic Orbit Invariant Torus Moderate Time Interval Melnikov Function
Unable to display preview. Download preview PDF.
- 1.Abramowitz M. and Stegun I.: “Handbook of Mathematical Functions”, Dover, 1972.Google Scholar
- 2.Bogoljubov N.N, Mitropolski Ju. A. and Samoilenko A.M.: “Methods of Accelerated Convergence in Nonlinear Mechanics”, Springer, 1976.Google Scholar
- 3.Broer H., Roussarie R. and Simó C.: Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms, in preparation.Google Scholar
- 7.Gelfreich V.G.: Splitting of separatrices for the rapidly forced pendulum, preprint, 1990.Google Scholar
- 9.Jorba A. and Simó C.: On quasiperiodic perturbations of elliptic equilibrium points, preprint, 1993.Google Scholar
- 10.Martínez R. and Pinyol C.: Parabolic orbits in the elliptic restricted three-body problem, to appear in J. of Diff. Eq.Google Scholar
- 11.Martínez R. and Simó C: A note on the existence of heteroclinic orbits in the planar three-body problem, in V.F. Lazutkin, editor, Proceed. S. Petersburg Workshop on Hamiltonian Dynamical Systems, to appear.Google Scholar
- 13.Simó C.: Averaging and splitting for Hamiltonians with fast quasiperiodic forcing, in preparation.Google Scholar