Computation of invariant tori by orthogonal collocation

A partial differential equation formulation has been used recently for the problem of computing invariant tori for systems of ordinary differential equations. We present an O(h⁴) collocation method for solving these resulting nonlinear first order partial differential equations with periodic boundary conditions. A convergence analysis is given, and numerical results for the method are contrasted with those of some previously tested methods. We also introduce an adaptive grid refinement scheme and use it to study the torus breakdown.