Czechoslovak Mathematical Journal, Vol. 75, No. 1, pp. 289-296, 2025
Characterization of shadowing for linear autonomous delay differential equations
Mihály Pituk, John Ioannis Stavroulakis
Received April 22, 2023. Published online March 20, 2024. OPEN ACCESS
Abstract: A well-known shadowing theorem for ordinary differential equations is generalized to delay differential equations. It is shown that a linear autonomous delay differential equation is shadowable if and only if its characteristic equation has no root on the imaginary axis. The proof is based on the decomposition theory of linear delay differential equations.
Keywords: delay differential equation; linear autonomous equation; shadowing
References: [1] L. Backes, D. Dragičević: Shadowing for infinite dimensional dynamics and exponential trichotomies. Proc. R. Soc. Edinb., Sect. A, Math. 151 (2021), 863-884. DOI 10.1017/prm.2020.42 | MR 4259329 | Zbl 1470.37028
[2] L. Backes, D. Dragičević, M. Pituk, L. Singh: Weighted shadowing for delay differential equations. Arch. Math. 119 (2022), 539-552. DOI 10.1007/s00013-022-01769-3 | MR 4496984 | Zbl 1515.34064
[3] J. Brzdek, D. Popa, I. Raşa, B. Xu: Ulam Stability of Operators. Mathematical Analysis and its Applications. Academic Press, London (2018). DOI 10.1016/c2015-0-06292-x | MR 3753562 | Zbl 1393.39001
[4] C. Buse, O. Saierli, A. Tabassum: Spectral characterizations for Hyers-Ulam stability. Electron. J. Qual. Theory Differ. Equ. 2014 (2014), Article ID 30, 14 pages. DOI 10.14232/ejqtde.2014.1.30 | MR 3218777 | Zbl 1324.34022
[5] J. K. Hale, S. M. Verduyn Lunel: Introduction to Functional Differential Equations. Applied Mathematical Sciences 99. Springer, New York (1993). DOI 10.1007/978-1-4612-4342-7 | MR 1243878 | Zbl 0787.34002
[6] K. Palmer: Shadowing in Dynamical Systems: Theory and Applications. Mathematics and its Applications (Dordrecht) 501. Kluwer, Dordrecht (2000). DOI 10.1007/978-1-4757-3210-8 | MR 1885537 | Zbl 0997.37001
Affiliations: Mihály Pituk (corresponding author), Department of Mathematics, University of Pannonia, Egyetem út 10, 8200 Veszprém, Hungary and HUN-REN- ELTE Numerical Analysis and Large Networks Research Group, Budapest Hungary, e-mail: pituk.mihaly@mik.uni-pannon.hu; John Ioannis Stavroulakis, Department of Mathematics, Ariel University, Ariel, Israel, e-mail: Ioannis.Stavroul@gmail.com, Ioannis.Stavroul@msmail.ariel.ac.il
