Publications

The chair works on nonautonomous dynamical systems, that is, on systems whose rules change over time. The main lines are spectral and dichotomy theory, the control of time-varying systems, dynamics on finite time horizons, and fractional differential equations, together with the question of how this theory bears on biology, networks and electrodynamics.

Selected publications

Spectral theory and dichotomies of nonautonomous systems

Which notions of spectrum survive when a system is not autonomous? Dichotomy spectrum, Bohl dichotomy, nonuniform spectra.

• A. Czornik, K. Kitzing, S. Siegmund. The new notion of Bohl dichotomy for non-autonomous difference equations and its relation to exponential dichotomy. Journal of Difference Equations and Applications 30 (2024), 626–658. [DOI] [arXiv]
• A. Czornik, K. Kitzing, S. Siegmund. Spectra based on Bohl exponents and Bohl dichotomy for nonautonomous difference equations. Journal of Dynamics and Differential Equations (2023). [DOI] [arXiv]
• A. Czornik, K. Kitzing, S. Siegmund. Dichotomies uniform on subspaces and formulas for dichotomy spectra. Preprint (2024). [arXiv]

Control theory and stabilisation of time-varying systems

How much of the spectrum can feedback prescribe in a time-varying system, and when does local information suffice?

• A. Babiarz, A. Czornik, S. Siegmund. On stabilization of discrete time-varying systems. SIAM Journal on Control and Optimization 59 (2021), 242–266. [DOI]
• P.T. Anh, A. Czornik, T.S. Doan, S. Siegmund. Proportional local assignability of dichotomy spectrum of one-sided continuous time-varying linear systems. Journal of Differential Equations 309 (2022), 176–195. [DOI]

Bifurcation theory and nonautonomous normal forms

Normal forms, linearisation and smoothing when the passage of time is itself part of the problem.

• L.V. Cuong, T.S. Doan, S. Siegmund. A Sternberg theorem for nonautonomous differential equations. Journal of Dynamics and Differential Equations 31 (2019), 1279–1299. [DOI]
• P. Bonckaert, P. De Maesschalck, T.S. Doan, S. Siegmund. Partial linearization for planar nonautonomous differential equations. Journal of Differential Equations 258 (2015), 1618–1652. [DOI]
• S. Siegmund. Normal forms for nonautonomous differential equations. Journal of Differential Equations 178 (2002), 541–573. [DOI]

Finite-time dynamics and Lyapunov methods

Hyperbolicity, entropy and spectrum without the limit t → ∞: dynamics on the time window one actually observes.

• L.H. Duc, S. Siegmund. A concept of local metric entropy for finite-time nonautonomous dynamical systems. Journal of Difference Equations and Applications 24 (2018), 165–179. [DOI]
• M. Budišić, S. Siegmund, T.S. Doan, I. Mezić. Mesochronic classification of trajectories in incompressible 3D vector fields over finite times. Discrete and Continuous Dynamical Systems – Series S 9 (2016), 923–958. [DOI] [arXiv]
• T.S. Doan, D. Karrasch, T.Y. Nguyen, S. Siegmund. A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents. Journal of Differential Equations 252 (2012), 5535–5554. [DOI]

Fractional differential equations and Hilbert space methods

Equations with memory, cast functional-analytically: stability, invariant manifolds, well-posedness in Hilbert space.

• K. Diethelm, K. Kitzing, R. Picard, S. Siegmund, S. Trostorff, M. Waurick. A Hilbert space approach to fractional differential equations. Journal of Dynamics and Differential Equations (2022). [DOI] [arXiv]
• H.T. Tuan, S. Siegmund. Stability of scalar nonlinear fractional differential equations with linearly dominated delay. Fractional Calculus and Applied Analysis 23 (2020), 250–267. [DOI] [arXiv]

Applications: biology, networks, electrodynamics

Where the basic research reaches into concrete models: epidemics and pest control, networks of coupled systems, storm protection, electrodynamics.

• S. Siegmund. Protecting against hurricane damage with mathematics. Notices of the American Mathematical Society 65:8 (2018), 967–968. [DOI]
• J. Páez Chávez, T. Götz, S. Siegmund, K.P. Wijaya. An SIR-Dengue transmission model with seasonal effects and impulsive control. Mathematical Biosciences (2017). [DOI]
• A.L. Do, S. Boccaletti, J. Epperlein, S. Siegmund, T. Gross. Topological stability criteria for networking dynamical systems with Hermitian Jacobian. European Journal of Applied Mathematics 27 (2016), 888–903. [DOI]
• M. Geyer, J. Hausmann, K. Kitzing, M. Senkyr, S. Siegmund. Maxwell’s equations revisited – mental imagery and mathematical symbols. Archivum Mathematicum (Brno) 59 (2023), 47–68. [DOI] [arXiv]

Editorial board memberships

Active

• Journal of Nonlinear Evolution Equations and Applications (JNEEA), since 2011

Former

• Discrete and Continuous Dynamical Systems – Series S (2008–2010)
• Journal of Difference Equations and Applications (2008–2019)
• Abstract and Applied Analysis (2009–2019)
• Differential Equations and Dynamical Systems (2008–2018)
• Advances in Difference Equations, now Advances in Continuous and Discrete Models (2015–2018)
• International Journal of Pure Mathematics (2013–2015)
• Journal of Advances in Mathematics (2013–2015)

All publications

152 Entries

2001

• The dichotomy spectrum for noninvertible systems of linear difference equationsAulbach, B. & Siegmund, S., 2001, In: Journal of difference equations and applications. 7, 6, p. 895-913, 19 p.Electronic (full-text) versionResearch output: Contribution to journal > Research article

2000

• A spectral theory for nonautonomous difference equationsAulbach, B. & Siegmund, S., 2000, p. 45-55, 11 p.Research output: Contribution to conferences > Paper

This information is provided by the FIS.

Publication sources, continuously updated: FIS · arXiv · ORCID · Google Scholar · MathSciNet · zbMATH Open · DBLP · IEEE Xplore