Speaker: Kai Diethelm, Technical University of Applied Sciences Wurzburg-Schweinfurt

Time : 15:00 - 16:00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom : A Zoom link will appear here, an hour before the talk

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: incommensurate fractional differential equation system, asymptotic stability, nonlinear eigenvalue problem

Abstract: For d ∈ {2, 3, 4, . . .} and a constant (d × d) matrix A, we first consider the homogeneous linear differential equation system

[ D^α1 x₁(t) , D^α2 x₂(t) , . . .  D^αd x_d(t) ]^T  =  A [ x₁(t) , x₂(t) , . . .  , x_d(t) ]^T         (1)

with α_k ∈ (0, 1] for all k and assuming that the system is incommensurate, i.e. that not all orders α_k are equal. The differential operators on the left-hand side of eq. (1) are chosen in Caputo's sense. The main goal of the talk is to describe the development of an algorithm [1] that allows to determine whether or not this system is asymptotically stable, i.e. whether or not all solutions x = (x₁, x₂, . . . , x_d)^T of the system have the property lim_t→∞ x(t) = 0, no matter how the initial conditions are chosen. To this end, we recall the concept of nonlinear eigenvalue problems from the field of linear algebra [3] and demonstrate that the question can be reformulated in this language as a special case of such a nonlinear eigenvalue problem that we can construct explicitly. Specifically, asymptotic stability is present if and only if all eigenvalues of this problem have negative real parts. Based on an analysis of the precise structure of the eigenvalue problem and exploiting suitable concepts from linear algebra [4], we then develop an algorithm that determines all its eigenvalues and thus allows us to answer the stability question under the (mild) additional assumption that all quotients α_j /α_k are rational numbers. We also provide some remarks about possible approaches in the case when this assumption is not satisfied, and we discuss how our procedure can be applied to investigate analog stability questions for inhomogeneous linear systems and for nonlinear systems.

An implementation of the algorithm in MATLAB or GNU/Octave is available from the Zenodo repository [2].

The talk is based on joint work with Safoura Hashemisharaki.

Biography: Kai Diethelm is Professor for Mathematics and Applied Computer Science at the Technical University of Applied Sciences Wurzburg-Schweinfurt, Germany, where he leads the Scientific Computing Laboratory. He holds a Diploma in Mathematics from TU Braunschweig and a PhD in Computer Science from the University of Hildesheim. His main research interests are approximation theory and its applications, analytical and numerical aspects of fractional calculus (especially, fractional ordinary differential equations), and high performance computing.

Bibliography

[1] Kai Diethelm and Safoura Hashemishahraki. “A stability testing algorithm for incommensurate fractional differential equation systems”. arXiv:2603.02269, submitted for publication (2026).
[2] Kai Diethelm and Safoura Hashemishahraki. “A MATLAB and GNU/Octave code for checking the asymptotic stability of incommensurate fractional differential equation systems”. Zenodo Record # 18730961. https://zenodo.org/records/18730961 (2026).
[3] Stefan G¨uttel and Fran¸coise Tisseur. “The nonlinear eigenvalue problem.” In: Acta Numerica 26 (2017), pp. 1–94.
[4] D. Steven Mackey, Niloufer Mackey, Christian Mehl and Volker Mehrmann. “Vector spaces of linearizations for matrix polynomials.” In SIAM J. Matrix Anal. Appl. 28 (2006), pp. 971–1004