ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Ponomarenko V. I., Prokhorov M. D. Reconstruction of self-oscillating systems with delay time modulation. Izvestiya VUZ. Applied Nonlinear Dynamics, 2025, vol. 33, iss. 1, pp. 27-37. DOI: 10.18500/0869-6632-003131, EDN: HSXIUA

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
537.86
DOI: 
10.18500/0869-6632-003131
EDN: 
HSXIUA

Reconstruction of self-oscillating systems with delay time modulation

Autors: 
Ponomarenko Vladimir Ivanovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Prokhorov Mihail Dmitrievich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

The aim of our research is to study the possibility of reconstruction from time series the self-oscillating systems with variable time delay, demonstrating regimes of turbulent and laminar chaos.

Methods. The object of study is self-oscillating systems described by delay-differential equations, in which the delay time is modulated by an external periodic signal. The possibility of estimating the parameters of systems with delay time modulation from their time series is considered using the known method for reconstructing systems with constant delay time, which is based on statistical analysis of time intervals between all possible pairs of extrema in time series. A new method for estimating the parameters of systems with variable delay time is proposed, based on statistical analysis of time intervals between two successive extrema in time series.

Results. It is shown that in some cases the known methods for reconstructing systems with constant delay time are also effective for reconstructing systems with varying delay time. With their help, one can estimate the mean delay time and recove the nonlinear function of the system. The proposed method, aimed at application to time-delay systems with delay time modulation, allows one to estimate the frequency and amplitude of delay time modulation.

Conclusion. The obtained results are of interest to various scientific disciplines that study systems with variable delay times based on their time series.
 

Key words: 
systems with delay time modulation
laminar chaos
reconstruction of systems from time series
statistics of extrema
Acknowledgments: 
The work was carried out within the framework of the state task of Saratov Branch of the Institute of Radio Engineering and Electronics of Russian Academy of Sciences.
Reference: 
  1. Erneux T. Applied Delay Differential Equations. New York: Springer-Verlag; 2009. 204 p. DOI: 10.1007/978-0-387-74372-1.
  2. Kuang Y. Delay Differential Equations with Applications in Population Dynamics. Boston: Academic Press; 1993. 398 p.
  3. Farmer J. Chaotic attractors of an infinite-dimensional dynamical system. Physica D: Nonlinear Phenomena. 19824;4(3):366—393. DOI: 10.1016/0167-2789(82)90042-2.
  4. Senthilkumar DV, Lakshmanan M. Delay time modulation induced oscillating synchronization and intermittent anticipatory/lag and complete synchronizations in time-delay nonlinear dynamical systems. Chaos. 2007;17(1):013112. DOI: 10.1063/1.2437651.
  5. Lazarus L., Davidow M., Rand R. Dynamics of an oscillator with delay parametric excitation. Int. J. Nonlinear Mech. 2016;78:66-71. DOI: 10.1016/j.ijnonlinmec.2015.10.005.
  6. Grigorieva EV, Kashchenko SA. Quasi-periodic and chaotic relaxation oscillations in a laser model with variable delayed optoelectronic feedback. Doklady mathematics. 2017;95(3):282-286. DOI: 10.1134/S1064562417030073.
  7. Muller D, Otto A, Radons G. Laminar chaos. Phys. Rev. Lett. 2018;120:084102. DOI: 10.1103/PhysRevLett.120.084102.
  8. Kul’minskii DD, Ponomarenko VI, Prokhorov MD. Laminar chaos in a delayed-feedback generator. Technical Physics Letters. 2020;46(5):423-426. DOI: 10.1134/S1063785020050090.
  9. Muller-Bender D, Otto A, Radons G. Resonant Doppler effect in systems with variable delay. Phil. Trans. R. Soc. A. 2019;377(2153):20180119. DOI: 10.1098/rsta.2018.0119. 
  10. Muller-Bender D, Radons G. Laminar chaos in systems with quasiperiodic delay. Physical Review E. 2023;107(1):014205. DOI: 10.1103/PhysRevE.107.014205.
  11. Hart JD, Roy R, Muller-Bender D, Otto A, Radons G. Laminar chaos in experiments: Nonlinear systems with time-varying delays and noise. Physical Review Letters. 2019;123(15):154101. DOI: 10.1103/PhysRevLett.123.154101.
  12. Jungling T, Stemler T, Small M. Laminar chaos in nonlinear electronic circuits with delay clock  modulation. Phys. Rev. E. 2020;101(1):012215. DOI: 10.1103/PhysRevE.101.012215.
  13. Kul’minskii DD, Ponomarenko VI, Prokhorov MD. Laminar chaos in coupled time-delay systems. Technical Physics Letters. 2022;48(2):53-56. DOI: 10.21883/PJTF.2022.04.52077.19044.
  14. Ponomarenko VI, Lapsheva EE, Kurbako AV, Prokhorov MD. Laminar chaos in an experimental system with quasiperiodic delay time modulation. Technical Physics Letters. 2024;50(11):34-37 (in Russian).
  15. Bunner MJ, Ciofini M, Giaquinta A, Hegger R, Kantz H, Meucci R, Politi A. Reconstruction of systems with delayed feedback: II. Application. Eur. Phys. J. D. 2000;10:177–187. DOI: 10.1007/s100530050539.
  16. Udaltsov VS, Goedgebuer J-P, Larger L, Cuenot J-B, Levy P, Rhodes WT. Cracking chaosbased encryption systems ruled by nonlinear time delay differential equations. Phys. Lett. A. 2003;308(1):54–60. DOI: 10.1016/S0375-9601(02)01776-0.
  17. Prokhorov MD, Ponomarenko VI, Karavaev AS, Bezruchko BP. Reconstruction of time-delayed feedback systems from time series. Physica D. 2005;203(3–4):209–223. DOI: 10.1016/j.physd.2005.03.013.
  18. Bezruchko BP, Karavaev AS, Ponomarenko VI, Prokhorov MD. Reconstruction of time-delay systems from chaotic time series. Physical Review E. 2001;64(5):056216. DOI: 10.1103/PhysRevE.64.056216.
  19. Muller-Bender D, Otto A, Radons G, Hart JD, Roy R. Laminar chaos in experiments and nonlinear delayed Langevin equations: A time series analysis toolbox for the detection of laminar chaos. Physical Review E. 2020;101(3):032213. DOI: 10.1103/PhysRevE.101.032213.
Received: 
07.07.2024
Accepted: 
18.07.2024
Available online: 
22.10.2024
Published: 
31.01.2025
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