Features of the Computational Implementation of the Algorithm for Estimating the Lyapunov Exponents of Systems with Delay
https://doi.org/10.18255/1818-1015-2019-4-572-582
Abstract
We consider the computational implementation of the algorithm for Lyapunov exponents spectrum numerical estimation for delay differential equations. It is known that for such systems, as well as for boundary value problems, it is not possible to prove the well-known Oseledets theorem which allows us to calculate the required parameters very efficiently. Therefore, we can only talk about the estimates of the characteristics in some sense close to the Lyapunov exponents. In this paper, we propose two methods of linearized systems solutions processing. One of them is based on a set of impulse functions, and the other is based on a set of trigonometric functions. We show the usage flexibility of these algorithms in the case of quasi-stable structures when several Lyapunov exponents are close to zero. The developed methods are tested on a logistic equation with a delay, and these tests illustrate the “proximity” of the obtained numerical characteristics and Lyapunov exponents.
Keywords
Supporting agencies: The reported study was funded by RFBR according to the research project № 18-29-10055.
About the Author
Vladimir E. Goryunov
P. G. Demidov Yaroslavl State University
Russian Federation
postgraduate student
Russian Federation
postgraduate student
References
1. Kuptsov P. V., “Computation of Lyapunov Exponents for Spatially Extended Systems: Advantages and Limitations of Various Numerical Methods”, Izv. VUZ. Applied Nonlinear Dynamics, 18:5 (2010), 93–112, (in Russian).
2. Oseledets V. I., “A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems”, Trans. Moscow Math. Soc., 19 (1968), 197–231.
3. Bylov B. F., Vinograd R. E., Grobman D. M., Nemytskiy V. V., Teoriya pokazateley Lyapunova i ee prilozheniya k voprosam ustoychivosti, Nauka, 1966, (in Russian).
4. Balyakin A. A., Ryskin N. M., “Peculiarities of Calculation of the Lyapunov Exponents Set in Distributed Self-Oscillated Systems with Delayed Feedback”, Izv. VUZ. Applied nonlinear dynamics, 15:6 (2007), 3–21, (in Russian).
5. Balyakin A. A., Blokhina E. V., “Peculiarities of Calculation of the Lyapunov Exponents Set in Distributed Self-Oscillated Systems with Delayed Feedback”, Izv. VUZ. Applied Nonlinear Dynamics, 16:2 (2008), 87–110, (in Russian).
6. Koloskova A. D., Moskalenko O. I., Koronovskii A. A., “A Method for Calculating the Spectrum of Lyapunov Exponents for Delay Systems”, Technical Physics Letters, 44:5 (2018), 374–377.
7. Aleshin S. V., “The Numerical Evaluation of Attractors Exponents of Delay Differential Equations System”, Comp. Technologies in Sciences. Methods of Simul. on Supercomputers, 2014, 10–17, (in Russian).
8. Farmer J. D., “Chaotic Attractors of an Infinite-Dimensional Dynamical System”, Physica D: Nonlinear Phenomena, 4:3 (1982), 366–393.
9. Hairer E., Nørsett S. P., Wanner G., Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag Berlin Heidelberg, 2008.
10. Cheney W., Kincaid D., Linear Algebra: Theory and Applications, Sudbury, Mass: Jones and Bartlett Publishers, 2009.
11. Nussbaumer H. J., Fast Fourier Transform and Convolution Algorithms, Springer-Verlag Berlin Heidelberg, 1981.
12. Glyzin D. S., Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “The Dynamic Renormalization Method for Finding the Maximum Lyapunov Exponent of a Chaotic Attractor”, Differ. Equ., 41:2 (2005), 284–289.
13. Aleshin S. V., Glyzin D. S., Glyzin S. D., Goryunov V. E., “Estimation of Lyapunov Exponents for Quasi-Stable Attractors of Dynamical Systems with Time Delay.”, Journal of Physics: Conference Series, 1163 (2019), 012045.
14. Kuptsov P. V., Kuznetsov S. P., “Violation of Hyperbolicity in a Diffusive Medium with Local Hyperbolic Attractor”, Phys. Rev. E., 80 (2009), 01620513.
15. Hutchinson G. E., “Circular Causal Systems in Ecology”, Ann. N.Y. Acad. Sci., 50 (1948), 221–246.
16. Hale J., Theory of Functional Differential Equations, Springer-Verlag New York, 1977.
17. Wright E. M., “A Non-Linear Difference-Differential Equation”, J. Reine Angew. Math., 194 (1955), 66–87.
18. Kakutani S., Markus L., “On the Nonlinear Difference-Differential Equation y'(t) = (A — By(t — t))y(t)”, Contributions to the Theory of Nonlinear Oscillations, 4 (1958), 1-18.
19. Kaschenko S. A., “K voprosu ob otsenke v prostranstve parametrov oblasti global’noy ustoychivosti uravneniya Khatchinsona”, Nelineynye kolebaniya v zadachakh ekologii. Yaroslavl: YarGU, 1985, 55–62, (in Russian).
20. Kaschenko S. A., “Asymptotics of Solutions of the Generalized Hutchinson’s Equation”, Modeling and Analysis of Information Systems, 19:3 (2012), 32–62, (in Russian).
Review
For citations:
Goryunov V.E. Features of the Computational Implementation of the Algorithm for Estimating the Lyapunov Exponents of Systems with Delay. Modeling and Analysis of Information Systems. 2019;26(4):572-582. (In Russ.) https://doi.org/10.18255/1818-1015-2019-4-572-582
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