Frequency analysis for multi-dimensional systems: global dynamics and diffusion

Abstract Frequency analysis is a new method for analyzing the stability of orbits in a conservative dynamical system. It was first devised in order to study the stability of the solar system [J. Laskar, Icarus 88 (1990) 266–291] and then applied to the 2D standard mapping [Laskar et al., Physica D 56 (1992) 253–269]. It is a powerful method for analyzing weakly chaotic motion in Hamiltonian systems or symplectic maps. For regular motions, it yields an analytical representation of the solutions. The analysis of the regularity of the frequency map with respect to the action space and of its variations with respect to time gives rise to two criteria for the regularity of the motion which are valid for multi-dimensional systems. For a 4D symplectic map, plotting the frequency map in the frequency plane provides a clear representation of the global dynamics, and reveals that high order resonances are of great importance in understanding the diffusion of non-regular orbits through the invariant tori. In particular, it appears in several examples that diffusion along the resonance lines (Arnold diffusion), is of less importance than diffusion across the resonances lines, which can lead to large diffusion due to the phenomenon of overlap of higher order resonances chaotic layers. Many fine features of the dynamics are also revealed by frequency analysis, which would require more theoretical study for a better understanding.

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