Computation of Invariant Tori by Newton-Krylov Methods in Large-Scale Dissipative Systems

A method to compute invariant tori in high-dimensional systems, obtained as discretizations of PDEs, by continuation and Newton-Krylov methods is described. Invariant tori are found as fixed points of a generalized Poincaré map without increasing the dimension of the original system. Due to the dissipative nature of the systems considered, the convergence of the linear solvers is extremely fast. The computation of periodic orbits inside the Arnold’s tongues is also considered. Thermal convection of a binary mixture of fluids, in a rectangular cavity, has been used to test the method.