A numerical method for computing initial conditions of Lagrangian invariant tori using the frequency map

• A method to compute initial conditions on Lagrangian invariant tori is proposed.
• Initial condition are found by imposing suitable conditions on the frequency map.
• The basic tool is an averaging-extrapolation strategy to perform frequency analysis.
• The proposed approach performs with high accuracy at a moderate computational cost.

We present a numerical method for computing initial conditions of Lagrangian quasi-periodic invariant tori of Hamiltonian systems and symplectic maps. Such initial conditions are found by solving, using the Newton method, a nonlinear system obtained by imposing suitable conditions on the frequency map. The basic tool is a newly developed methodology to perform the frequency analysis of a discrete quasi-periodic signal, allowing to compute frequencies and their derivatives with respect to parameters. Roughly speaking, this method consists in computing suitable weighted averages of the iterates of the signal and using the Richardson extrapolation method. The proposed approach performs with high accuracy at a moderate computational cost. We illustrate the method by considering a discrete FPU model and the vicinity of the point L4L4 in a RTBP.