arXiv subjects

Fabia Bayer

Publications and source records attributed to Fabia Bayer.

2 recordsLinked to original sources

arXiv2026

Koopman-based stability analysis of differential-algebraic equations with applications to frictional multibody systems

Periodic solutions of differential-algebraic equations (DAEs) and ordinary differential equations (ODEs) can be determined using the harmonic balance method (HBM), which is a frequency-domain approach that approximates the solution by its truncated Fourier series. The Koopman-Hill method, a method to determine the stability of periodic solutions found by HBM, and originally developed for ODEs, is generalized to DAEs in this work. Analogously to the ODE case, the core idea of the proposed Koopman-Hill method for DAEs is to establish a linear time-invariant but high-dimensional DAE which approximately governs the dynamics of small admissible perturbations around the periodic solution. The crucial difference to the ODE case is the fact that the evolution of this linear time-invariant DAE is not simply given by a matrix exponential, but by a more complicated expression involving a Drazin inverse, rendering the resulting monodromy matrix singular. Still, even in the DAE case, this novel relationship between the monodromy matrix and the Hill matrix is essentially given by one single formula, which is the main result of this work. Two academic mechanical systems, a mathematical pendulum formulated as an index-3 DAE and a nonsmooth frictional two-mass oscillator with switching index, demonstrate the applicability of the proposed method and its blindness to the DAE's index.

math.DS↗

arXiv2025

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.

math.NA↗