On Numerical Computation of the Spectrum of Monodromy Operators Based on Higher-Order Hold Approximation

Abstract

A numerical computation method of the spectrum of the monodromy operator based on fast-sample/hold approximation is investigated. Through the numerical examples of the previous work with the zero-th, 1st and 3rd order polynomial hold functions, it is observed that the computational efficiency improves as the approximation order becomes higher. However, it is reasonable to expect that such a monotonous tendency will hit the ceiling as the approximation order grows. Motivated by this observation, our primary objective in this paper is to investigate the computational efficiency of higher order approximations. Since the order reflects the smoothness of the domain where the monodromy operator is considered, one must justify the approximation procedure for each function space (inductively) before developing the numerical algorithms. Due to the lack of scalability in the proofs of earlier results, this is the first challenge. Then the matrix formula for general order approximation is derived. After checking the error convergence property of the proposed method, we examine the computational efficiency of higher-order approximations through numerical examples.