Universität Stuttgart
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Item Open Access Hill-type stability analysis of periodic solutions of fractional-order differential equations(2026) Haacker, Paul-Erik; Leine, Remco I.; Chaudhary, Renu; Diethelm, Kai; Schmidt, André; Hashemishahraki, SafouraThis paper explores stability properties of periodic solutions of (nonlinear) fractional-order differential equations (FODEs). As classical Caputo-type FODEs do not admit exactly periodic solutions, we propose a framework of Liouville-Weyl-type FODEs, which do admit exactly periodic solutions and are an extension of Caputo-type FODEs. Local linearization around a periodic solution results in perturbation dynamics governed by a linear time-periodic differential equation. In the classical integer-order case, the perturbation dynamics is therefore described by Floquet theory, i.e. the exponential growth or decay of perturbations is expressed by Floquet exponents which can be assessed using the Hill matrix approach. For fractional-order systems, however, a rigorous Floquet theory is lacking. Here, we explore the limitations when trying to extend Floquet theory and the Hill matrix method to linear time-periodic fractional-order differential equations (LTP-FODEs) as local linearization of nonlinear fractional-order systems. A key result of the paper is that such an extended Floquet theory can only assess exponentially growing solutions of LTP-FODEs. Moreover, we provide an analysis of linear time-invariant fractional-order systems (LTI-FODEs) with algebraically decaying solutions and show that the inaccessibility of decaying solutions through Floquet theory is already present in the time-invariant case.Item Open Access Model reduction of a periodically forced slow-fast continuous piecewise linear system(2023) Karoui, A. Yassine; Leine, Remco I.In this paper, singular perturbation theory is exploited to obtain a reduced-order model of a slow-fast piecewise linear 2-DOF oscillator subjected to harmonic excitation. The nonsmooth nonlinearity of piecewise linear nature is studied in the case of bilinear damping as well as with bilinear stiffness characteristics. We propose a continuous matching of the locally invariant slow manifolds obtained in each subregion of the state space, which yields a reduced-order model of the same nature as the full dynamics. The frequency-response curves obtained from the full system and the reduced-order models suggest that the proposed reduction method can capture nonlinear behaviors such as super- and subharmonic resonances.Item Open Access State observers for the time discretization of a class of impulsive mechanical systems(2022) Preiswerk, Pascal V.; Leine, Remco I.In this work, we investigate the state observer problem for linear mechanical systems with a single unilateral constraint, for which neither the impact time instants nor the contact distance is explicitly measured. We propose to attack the observer problem by transforming and approximating the original continuous‐time system by a discrete linear complementarity system (LCS) through the use of the Schatzman-Paoli scheme. From there, we derive a deadbeat observer in the form of a linear complementarity problem. Sufficient conditions guaranteeing the uniqueness of its solution then serve as observability conditions. In addition, the discrete adaptation of an existing passivity‐based observer design for LCSs can be applied. A key point in using a time discretization is that the discretization acts as a regularization, that is, the impacts take place over multiple time steps (here two time steps). This makes it possible to render the estimation error dynamics asymptotically stable. Furthermore, the so‐called peaking phenomenon appears as singularity within the time discretization approach, posing a challenge for robust observer design.Item Open Access Sorting-free Hill-based stability analysis of periodic solutions through Koopman analysis(2023) Bayer, Fabia; Leine, Remco I.In this paper, we aim to study nonlinear time-periodic systems using the Koopman operator, which provides a way to approximate the dynamics of a nonlinear system by a linear time-invariant system of higher order. We propose for the considered system class a specific choice of Koopman basis functions combining the Taylor and Fourier bases. This basis allows to recover all equations necessary to perform the harmonic balance method as well as the Hill analysis directly from the linear lifted dynamics. The key idea of this paper is using this lifted dynamics to formulate a new method to obtain stability information from the Hill matrix. The error-prone and computationally intense task known by sorting , which means identifying the best subset of approximate Floquet exponents from all available candidates, is circumvented in the proposed method. The Mathieu equation and an n -DOF generalization are used to exemplify these findings.Item Open Access Tuning and optimal performance of an asymmetric vibro-impact nonlinear energy sink with dry friction(2026) Youssef, Balkis; Leine, Remco I.This paper presents a design-oriented framework for asymmetric vibro-impact nonlinear energy sinks (VI-NES) with dry friction. In this context, asymmetry arises from directional differences in friction and restitution properties. The proposed approach builds on analytical results from the Multiple Scales Method (MSM) and an impact map formulation developed to fully characterize near-resonant dynamics in symmetric frictional configurations. The absence of closed-form solutions for asymmetric systems motivates a pragmatic design hypothesis that bounds their response using symmetric reference systems. Based on this insight, a performance-driven design strategy is established, enabling efficient evaluation and optimization without exhaustive simulations. The cavity length, identified as the dominant tuning parameter, governs activation, dissipation efficiency, and regime transitions. Analytical, numerical, and experimental results confirm the predicted tunability of the asymmetric VI-NES through geometric adjustment and demonstrate the robustness and applicability of the proposed strategy for passive vibration mitigation in realistic mechanical systems.Item Open Access Explicit error bounds and guaranteed convergence of the Koopman-Hill projection stability method for linear time-periodic dynamics(2026) Bayer, Fabia; Leine, Remco I.The Koopman-Hill projection method offers an efficient, numerically validated approach for stability analysis of linear time-periodic systems and thereby also for the Floquet stability analysis of periodic solutions of nonlinear systems. However, it has previously only been motivated via the Koopman framework, involving the optimistic truncation of an ill-posed bi-infinite initial value problem, therefore lacking any rigorous theoretical guarantees. We close this gap for a class of dynamical systems with exponentially decaying Fourier coefficients by presenting a closed-form bound for the difference between the fundamental solution matrix and its Koopman-Hill approximation. The bound converges to zero as the truncation order goes to infinity. It is derived using constructive series expansions for fixed truncation order, making the results fully independent from the ill-posed bi-infinite Koopman lift. The bound is not sharp, but nevertheless provides a solid theoretical foundation for the Koopman-Hill projection method. In addition, it enables conservative but reliable inference of Floquet multipliers and associated stability properties. The same methodology applied to a subharmonic Koopman-Hill formulation yields a bound with improved convergence rate. Numerical examples, including the Mathieu equation and the Duffing oscillator, illustrate the practical relevance of the bound and illuminate how it can be used in practice to assess the accuracy of computed Floquet multipliers quantitatively.