Hill-Type Stability Analysis of Periodic Solutions of Fractional-Order Differential Equations
Paul-Erik Haacker, Remco I. Leine, Renu Chaudhary, Kai Diethelm, André Schmidt, Safoura Hashemishahraki
TL;DR
The paper tackles stability analysis of periodic solutions in fractional-order differential equations by extending Floquet/Hill theory to Liouville-Weyl-type FODEs that admit exact periodic solutions. Through linearization around a periodic orbit, it derives a linear time-periodic fractional system and introduces the fractional Hill matrix, yielding a nonlinear eigenvalue problem whose solutions identify exponentially growing perturbations. A key finding is that the extended Floquet framework cannot access algebraically decaying modes, revealing a fundamental limitation compared to classical Floquet theory and illustrating this with linear-time-invariant analyses. The authors propose a truncated fractional Hill matrix for practical computation and demonstrate numerical results in scalar and low-dimensional systems, highlighting open questions about monodromy operators and full stability characterization in LTP-FODEs. This work advances a structured approach for stability analysis of memory-bearing fractional systems with periodic forcing or self-excitation.
Abstract
This 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.