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Over short time intervals planetary ephemerides have been traditionallyrepresented in analytical form as finite sums of periodic terms or sums ofPoisson terms that are periodic terms with polynomial amplitudes. Nevertheless,this representation is not well adapted for the evolution of the planetaryorbits in the solar system over million of years as they present drifts intheir main frequencies, due to the chaotic nature of their dynamics. The aim ofthe present paper is to develop a numerical algorithm for slowly diffusingsolutions of a perturbed integrable Hamiltonian system that will apply to therepresentation of the chaotic planetary motions with varying frequencies. Bysimple analytical considerations, we first argue that it is possible to recoverexactly a single varying frequency. Then, a function basis involvingtime-dependent fundamental frequencies is formulated in a semi-analytical way.Finally, starting from a numerical solution, a recursive algorithm is used tonumerically decompose the solution on the significant elements of the functionbasis. Simple examples show that this algorithm can be used to give compactrepresentations of different types of slowly diffusing solutions. As a testexample, we show how this algorithm can be successfully applied to obtain avery compact approximation of the La2004 solution of the orbital motion of theEarth over 40 Myr ([-35Myr,5Myr]). This example has been chosen as thissolution is widely used for the reconstruction of the climates of the past.

Frequency analysis and the representation of slowly diffusing planetary solutions