Frequency map analysis and quasiperiodic decompositions

Frequency Map Analysis is a numerical method based on refined Fourier techniques which provides a clear representation of the global dynamics of many multi-dimensional systems, and which is particularly adapted for systems of 3-degrees of freedom and more. This method relies heavily on the possibility of making accurate quasiperiodic approximations of of quasiperiodic signal given in a numerical way. In the present paper, we will describe the basis of the frequency analysis method, focussing on the quasi periodic approximation techniques. Application of these methods for the study of the global dynamics and chaotic diffusion of Hamiltonian systems and symplectic maps in different domains can be found in (Laskar, 1988, 1990, Laskar and Robutel, 1993, Robutel and Laskar, 2001, Nesvorny and Ferraz-Mello, 1997) for solar system dynamics, and in (Papaphilippou and Laskar, 1996, 1998, Laskar, 2000, Wachlin and Ferraz-Mello, 1998, Valluri and Merritt, 1998, Merritt and Valluri, 1999) for galactic dynamics. The method has been particularly successful for its application in particle accelerators (Dumas and Laskar, 1993, Laskar and Robin, 1996, Robin et al., 2000, Comunian et al., 2001, Papaphilippou and Zimmermann, 2002, Steier et al., 2002), and was also used for the understanding of atomic physics (Milczewski et al., 1997), or more general dynamical system issues (Laskar et al., 1992, Laskar, 1993, 1999, Chandre et al., 2001).