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The Trouble with Spurious eigenvalues

Stability analysis of spatially discretized approximations to invariant circle solutions to a noninvertible map is shown to produce seemingly spurious eigenvalues, found to exist for all truncation numbers. Numerical analysis of the spectral Galerkin projection procedure used to compute the solutions and the eigenvalues of the linearization shows that the set of discrete eigenvalues converges in terms of its mean value. Further analysis shows that the spurious eigenvalues — those that are farthest from the mean value — correspond to eigenfunctions which produce the largest residual in the eigenvalue problem used to compute the eigenvalues and eigenfunctions.