Dynamical System Modeling and Stability Investigation�DSMSI-2025
May 08-10, 2025, Kyiv, Ukraine
APPROXIMATION OF SYSTEMS WITH DELAY AND THEIR APPLICATION
Igor Cherevko, Svitlana Ilika,
Oleksandr Krasnokutskyi, Oleksandr Matviy,
Yuriy Fedkovych Chernivtsi National University�
Introduction
Delay differential systems are widely used to model various real-world phenomena in fields such as automatic control and regulation systems, as well as chemical, biological, technical, economic, and other processes, where the evolution depends on their history. Specifically, in the modeling of processes in semiconductor laser devices with optical feedback, delays arise due to the time required for the signal to travel from the laser to the mirror. In biological systems, evolution is related to long-term processes such as reproduction, development, and extinction, which do not occur instantaneously but involve a certain delay. With the help of such equations, it was possible to identify and describe new effects and phenomena in physics, biology, technology [1-4].
1. Fathalla A. Rihan. Delay Differential Equations and Applications to Biology. Singapore : Springer Verlag, 2021. 303 p.
2. Domoshnitsky A., Rasin A., Padhi S. Functional Differential Equations and Applications: Fdea-2019, Ariel, Israel, September 22-27. Springer Proceedings in Mathematics & Statistics, 379. 2022.
3. Schiesser W.E. Time Delay ODE/PDE Models. Applications in Biomedical Science and Engineering. Boca Raton : CRC Press, 2019. 250 p.
4. Rodrıguez F., Carlos J., Lopez C., Castro M. Models of Delay Differential Equations . – MDPI: Basel , 2021. – 248 p.
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Introduction
One of the most important problems in the theory of delay differential equations is studying the stability of their solutions. It is well known [5, 6] that even a small delay can lead to a loss of stability. Thus, it is essential to consider the impact of delay on solution stability. There is also significant interest in determining the critical delay values at which the stability of the solutions is preserved.
Stability in linear delay differential systems is the most extensively studied topic in this field [5-9]. Numerous approaches in scientific research that determine stability conditions for linear differential equations with delay can be divided into two main categories.
One approach is based on the direct Lyapunov method. For systems with delay, this method uses Lyapunov-Krasovskii functionals.
5. Kolmanovskii V.B., Myshkis A.D. Introduction to the theory and applications of functional differential equations. Kluwer Acad. Publ. 1999. 664 p.
6. Gopalsamy K. Stability and Oscillation in Delay Differential Equations of Population Dynamics. Dordrecht: Kluwer Academic Publishers, 1992. Vol. 74. 501 p.
7. Hale J. K. , Verduyn Lunel S. M. Introduction to Functional Differential Equations . New York : Springer Verlag, 1993. 458 p.
8. Berezansky l., Braverman E. On stability of some linear and nonlinear delay differential equations. J. Math. Anal. Appl. 2006. Vol. 314, № 2. P. 391-411.
9. Khusainov D.Ya., Shatyrko A.V. The Lyapunov function method in studying the stability of differential functional systems. Kyiv: Kyiv University Publishing House, 1987. 236 p.
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Introduction
The second approach is based on studying the distribution of roots of the corresponding characteristic equations, which are referred to as quasipolynomials in this case. In this case, it is necessary to establish that all roots of the characteristic quasipolynomial lie in the left half-plane. Such approaches generally yield more accurate results but often require complex computations [10-11].
This paper discusses a scheme for investigating the stability of solutions to linear delay differential equations based on their approximation by sequences of ordinary differential equations [12-14].
10. Scholl T. H., Groll L. Stability criteria for time-delay systems from an insightful perspective on the characteristic equation. IEEE Trans. Automat. Contr., 2023. Vol. 68. P. 2352–2359.
11. Olgac N., Sipahi R. An exact method for the stability analysis of time-delayed linear time invariant (LTI) systems. IEEE Transactions on Automatic Control, 2002. Vol. 47. P.793-797.
12. Matviy O.V., Cherevko I.M. About approximation of system with delay and them stability. Nonlinear oscilations, 2004. Vol.7, №2. P. 208-216.
13. Tuzyk I., Cherevko I. Algorithms for studying the stability of linear systems with many delay. 12th International Conference on Advanced Computer Information Technologies, 26-28 September 2022, Spisska Kapitula, Slovakia. P. 164-167.
14. Petryk M., Cherevko I., Ilika S. Approximation of Systems with Delay and their Application. CEUR Workshop Proceedings, 2023, 3687. P. 107–114.
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Approximation schemes
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Dynamical System Modeling and Stability Investigation, DSMSI-2025
15. Matviy O.V., Cherevko I.M. About approximation of system with delay and them stability. Nonlinear oscilations, 2004. Vol.7, №2. P. 208-216.
16. Ilika S.A., Piddubna L.A., Tuzyk I. I., Cherevko I.M. Approximation of linear differential-difference equations and their application// Bukovinian Mathematical Journal, 2018. Vol. 6, № 3-4. P. 80-83.
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Investigation of the stability of systems with a delay
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Example
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Conclusions
To The purpose of this work is to apply approximation schemes for linear delay differential systems using a sequence of specially constructed ordinary differential equation (ODE) systems in order to analyze the stability of linear delay-difference equations.
The stability analysis of linear systems with delay is reduced to verifying whether all roots of the corresponding quasi-polynomials have negative real parts. The study establishes that for a sufficiently high-dimensional approximating ODE system, this verification can be replaced by analyzing the location of the roots of the corresponding characteristic polynomials. The roots are computed using the NumPy library in the Python programming language.
For a model example involving a third-order linear delay system, explicit formulas for the coefficients of the characteristic polynomial of the approximating ODE system are derived. Based on numerical simulations, two regions of the delay parameter were identified within which the system is exponentially stable. The obtained bounds refine the estimates produced using the generalized D-decomposition method presented in [11].
Dynamical System Modeling and Stability Investigation, DSMSI-2025
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