Abstract :
We present two new spectral implementations for computing invariant tori. The underlying nonlinear partial differential equation (Dieci et al., 1991), although hyperbolic by nature, has periodic boundary conditions in both space and time. Our first approach uses a spatial spectral discretization, and finds the solution via a shooting method. The second one employs a full two-dimensional Fourier spectral discretization, and uses Newtonʹs method. This leads to very large, sparse, unsymmetric systems, although with highly structured matrices. A modified conjugate gradient type iterative solver was found to perform best when the dimensions get too large for direct solvers. The two methods are implemented for the van der Pol oscillator, and compared to previous algorithms.
