Frequency analysis and representation of slowly diffusing solutions

International audienceGiven numerically a KAM solution of a Hamiltonian system, the frequency analysis algorithm NAFF allows to recover its significant periodic terms, and as thus to provide a compact quasi-periodic representation with high precision. In astronomical practice, however, it is often necessary to take into consideration frequency drifts of chaotic and/or dissipative nature. The aim of the present report is to develop a numerical algorithm of representing slowly diffusing solutions of a perturbed integrable Hamiltonian system. By simple analytical considerations, we first argue that it is possible to recover exactly a single varying frequency. Then, a function basis involving time-dependent fundamental frequencies is formulated in a semi-analytical way. Finally, starting from a numerical solution, a recursive algorithm is designed to numerically decompose the solution on the significant elements of the function basis. Simple examples show that this algorithm can be used to give compact representations of different types of slowly diffusing solutions. As practically useful examples, such representations spanning [-35Myr,5Myr] are obtained for the eccentricity and the inclination of the Earth, which are known to be chaotic