COMPUTATION OF INVARIANT TORI BY THE FOURIER METHODS

In this paper, we study systems of functional equations (FEs) and firs t-order partial differential equations (PDEs) suggested in [SIAM J. Sc i. Statist. Comput., 12 (1991), pp. 607-647 and SIAM J. Numer. Anal., 29 (1992), pp. 1741-1768] as approximations for the computation of inv ariant tori. The main new ideas of this paper are, first, to investiga te these systems in the setting of Hilbert spaces rather than in the s etting of Banach spaces and, then, to employ Fourier methods instead o f difference methods for a numerical solution. Based on the setting of Sobolev spaces H-s(T-p), proper conditions for the PDE and FE systems to be dissipative are described, and some regularity results for the FE system are proved. We studied two kinds of Fourier methods, the spe ctral method and the pseudospectral method, in detail under dissipativ ity conditions. Convergence and optimal error estimates are shown theo retically for these Fourier methods in the case of general linear syst ems. Numerical results for three examples provided in the last section indicate that the Fourier method behaves very well not only for smoot h solutions but also for nonsmooth solutions.