"JOURNAL OF RADIO ELECTRONICS" (Zhurnal Radioelektroniki ISSN 1684-1719, N 11, 2019

contents of issue      DOI  10.30898/1684-1719.2019.11.9    full text in Russian (pdf)  

UDC 534.1



A. V. Blank, N. A. Suhareva

 M.V.Lomonosov Moscow State University, Faculty of Physics,  Leninskie Gory, 1-2, Moscow 119991, Russia


The paper is received on November 8, 2019


Abstract. The results of an experimental study of the structure of phase trajectories of the energy center of a collimated wave beam are presented. A significant difference was found between phase portraits and values ​​of the dimension of the embedded space for the vertical and horizontal components of the first spatial moment. A qualitative difference between phase portraits realized in laminar flows and coherent turbulence is considered. A preliminary estimation of the spectra of the Poincaré return times is performed. The reconstruction and analysis of phase portraits for stationary modes and phase trajectories for non-stationary can be the basis for the method of visual control of the dynamic modes of spatio-temporal beam aberrations and the classification of states of non-stationary and nonequilibrium atmospheres.

Key words: collimated beam, turbulent atmosphere, phase portrait, chaos maps, Poincare return time, tilt correction.


1. Sauer T., Yorke J. A., Casdagli M. Embedology. Statistical Physics. 1991. Vol.65. P. 79–616.

2. Ruelle D., Takens F. On the nature of turbulence. Communications in Mathematical Physics. 1971. Vol.20. P.167–192.

3. Hasegawa A., Maclennan C., Kodama Y. Nonlinear behavior and turbulence spectra of drift waves and Rossby waves. The Physics of Fluids. 1979. Vol.22. P.2122–2129.

4. Takens F. Detecting strange attractors in turbulence. Berlin: Springer, 1981. 381 p.

5. Wolf A., Swift J., Swinney H. Determining Lyapunov exponents from a time series.  Physica D: Nonlinear Phenomena. 1985. Vol.16. P.285– 317.

6. Ruelle D. Strange attractors. The Mathematical Intelligencer. 1980. Vol.2. P.126–137.

7. Casdagli M. Nonlinear prediction of chaotic time series. Physica D: Non- linear Phenomena. 1989. Vol.35. P.335–356.

8. Temam R. Navier-Stokes equations and nonlinear functional analysis. Philadelphia: Siam. 1987. 160 p.

9. Sagaut P., Cambon C. Homogeneous turbulence dynamics. Berlin: Springer. 2008. 481 p.

10. Fraedrich K. Estimating the dimensions of weather and climate attractors.  Journal of the Atmospheric Sciences. 1986. Vol.43. P.419–432.

11. Hoover W. Canonical dynamics: equilibrium phase-space distributions. Physical Review A. 1985. Vol.31. P.1–3.

12. Trahan C., Wyatt R. Evolution of classical and quantum phase-space distributions: A new trajectory approach for phase space hydrodynamics.  The Journal of Chemical Physics.  2003.  Vol. 19. P. 017–7029.

13. van der Veen R., Huisman S., Dung O. Exploring the phase space of multiple states in highly turbulent Taylor-Couette flow. Physical Review Fluids. 2016. Vol.1. P.1–14.

14. Berera A., Ho R. Chaotic properties of a turbulent isotropic fluid. Physical Review Letters. 2018. Vol.120. Pp.1–5.

15. Lacorata G., Vulpiani A. Chaotic Lagrangian models for turbulent relative dispersion. Physical Review E. 2017. Vol. 95. P.1–9.

16. Hilborn R. Chaos and nonlinear dynamics: an introduction for scientists and engineers. Oxford University Press on Demand, 2000. 654 p.

17. Thompson J., Stewart H. Nonlinear dynamics and chaos.  NY: John Wiley & Sons, 2002. 458 p.

18. Gutman Y., Qiao Y., Szabo G. The embedding problem in topological dynamics and Takens’ theorem. Nonlinearity. 2018. Vol. 1. P.1–25.

19. Harnack D., Laminski E., Schunemann M. Topological Causality in Dynamical Systems.  Physical Review Letters. 2017. Vol. 119. P.1–5.

20. Jiang H., He H. State space reconstruction from noisy nonlinear time series: An autoencoder-based approach. Neural Networks (IJCNN).  2017. P.3191–3198.

21. Kapranov V.V., Matsak I.S., Blank A.V., et al. Atmospheric turbulence effects on the performance of the laser wireless power transfer system. Free-Space Laser Communication and Atmospheric Propagation XXIX. 2017. Vol.10096. P.1–13.

22. Arsenyan T.I., Grebennikov D. Yu., Sukhareva N.A., et al. Reconstruction of phase trajectories of a laser beam propagated through a turbulent medium. Atmospheric and Oceanic Optics. 2014. Vol.27. P.205–210.

23. Babanin E.A., Kapranov V.V., Suhareva N.A., et al. Space-temporal stochastic characteristics of complex amplitude for the sounding vector optical beam. 2017 Progress In Electromagnetics Research Symposium-Spring (PIERS). 2017. P.902–908.

24. Deyle E. R., Sugihara G. Generalized theorems for nonlinear state space reconstruction. PLoS One. 2011. Vol.6. P.1–8.

25. Babanin E. A., Blank A.V., Nasonov A. A., et al. Profile management of astigmatic energy-carrying collimated beam. Proceedings of SPIE — The International Society for Optical Engineering. 2019. P.1–13.

26. Noakes L. The Takens embedding theorem. International Journal of Bifurcation and Chaos. 1991. Vol.1. P.867–872.

27. Pikulev S., Semenkov V., Chernykh A. Tests of an adaptive optical system on a model atmospheric turbulent path. Optoelectronics, Instrumentation and Data Processing. 2012. Vol.48. P.166–173.

28. Tomashevsky A. I., Kapranov M. V. Fractal properties of chaotic dynamical systems in reverse time and its applications.  International Conference Physics and Control. 2005. P.443–446.

29. Blank A.V., Kapranov V.V., Mikhailov R.V., et al. Non-linear dynamics of positional parameters of the collimated coherent beam at the end of the long atmospheric path. Progress In Electromagnetics Research Symposium - Spring PIERS.  2017. St. Petersburg. P.708–714.

30. Matsak I.S., Kapranov V.V., Tugaenko V. Yu. Super narrow beam shaping system for remote power supply at long atmospheric paths. Laser Resonators, Microresonators, and Beam Control XIX. 2017. Vol.10090. P.1–12.

31. Klyatskin V. I. Stochastic Equations: Theory and Applications in Acoustics, Hydrodynamics, Magnetohydrodynamics, and Radiophysics, Coherent Phenomena in Stochastic Dynamic Systems. Berlin: Springer. 2016. 491 p.



For citation:

Blank A.V., Suhareva N.A. Phase portraits of spatial moments of  a collimated wave beam in controlled and free modes. Zhurnal Radioelektroniki - Journal of Radio Electronics. 2019. No. 11. Available at http://jre.cplire.ru/jre/nov19/9/text.pdf

DOI 10.30898/1684-1719.2019.11.9