Computation of invariant tori by the method of characteristics

In this paper we present a technique for the numerical approximation of a branch of invariant tori of finite-dimensional ordinary differential equations systems. Our approach is a discrete version of the graph transform technique used in analytical work by Fenichel [Indiana Univ. Math. J., 21 (1971), pp. 193–226]. In contrast to our previous work [L. Dieci, J. Lorenz, and R. D. Russell, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 607–647], the method presented here does not require a priori knowledge of a suitable coordinate system for the branch of invariant tori, but determines and updates such a coordinate system during a continuation process. We give general convergence results for the method and present its algorithmic description. We also show how the method performs on two physically important nonlinear problems, a system of two coupled oscillators and the forced van der Pol oscillator. In the latter case, we discuss some modifications needed to approximate an invariant curve for the Poincare map.