Explicit error bounds and guaranteed convergence of the Koopman-Hill projection stability method for linear time-periodic dynamics

The Koopman-Hill projection method offers an efficient approach for stability analysis of linear time-periodic systems, and thereby also for the Floquet stability analysis of periodic solutions of nonlinear systems. However, its accuracy has previously been supported only by numerical evidence, lacking rigorous theoretical guarantees. This paper presents the first explicit error bound for the truncation error of the Koopman-Hill projection method, establishing a solid theoretical foundation for its application. The bound applies to linear time-periodic systems whose Fourier coefficients decay exponentially with a sufficient rate, and is derived using constructive series expansions. The bound quantifies the difference between the true and approximated fundamental solution matrices, clarifies conditions for guaranteed convergence, and enables conservative but reliable inference of Floquet multipliers and stability properties. Additionally, the same methodology applied to a subharmonic formulation demonstrates improved convergence rates of the latter. Numerical examples, including the Mathieu equation and the Duffing oscillator, illustrate the practical relevance of the bound and underscore its importance as the first rigorous theoretical justification for the Koopman-Hill projection method.