Construction of Whiskers for the Quasiperiodically Forced Pendulum

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations. We extend analytically the solutions of the equations of motion, order by order in the perturbation parameter, to a uniform neighbourhood of the time axis. 1. Main Concepts and Results 1.1. Background and history. A quasiperiodic motion of a mechanical system is composed of incommensurable periodic motions; the trajectory in phase space winds around on a torus filling its surface densely. An integrable Hamiltonian system has a great profusion of quasiperiodic motions: if one picks an initial phase point according to a uniform distribution, the trajectory will be quasiperiodic with probability one. The remaining trajectories are periodic. KAM theory deals with the stability of quasiperiodic motions, or persistence of invariant tori, under small perturbations. Poincaré [Poi93a] called this the general problem of dynamics. In 1954, Kolmogorov [Kol54] outlined a result, made rigorous by Arnold in 1963 [Arn63], that quasiperiodic motions are typical also for nearly integrable analytic Hamiltonians under suitable nondegeneracy conditions. Thus, only a small fraction of the tori would be destroyed by the perturbation. Moser managed to prove the same for twist maps [Mos62] in 1962, and later for Hamiltonians [Mos66b, Mos66a], in the smooth (non-analytic) setting (see also [Mos67]). The difficult problem to overcome is the following. Suppose that the Hamiltonian reads H = H0 + λH1, where H0 is integrable and λ is considered small. Then one can formally represent a solution to the equations of motion by a power series in λ, known as the Lindstedt series in this context, conditioned to agree for λ = 0 with a quasiperiodic solution obtained in the integrable case. When one computes the coefficients of the Lindstedt series, however, one encounters expressions containing arbitrarily small denominators. The latter seem to imply that the kth coefficient grows like k! with a large power α. Thus, there is little hope of being able to sum the series and obtain a true solution, unless a miracle occurs. The proofs mentioned above relied on a rapidly convergent Newton-type iteration scheme, which is interesting in its own right, and yields solutions analytic in λ. On the other hand, one is then left to wonder why the Lindstedt series does converge. 2000 Mathematics Subject Classification. 70K43; Secondary 37J40, 37D10, 70H08, 70K44. 1