Dobrov Institute for Scientific and Technological Potential and Science History Studies of the NAS of Ukraine

Nauka naukozn. 2021, 2(111): 137-153

Section: Science and technology history
Language: Ukrainian
Abstract:   Recently, the concept of turbulence has gone beyond the natural sciences and is widely used in social sciences, economics, world politics, history of science and technology. Turbulent processes are the processes of self-organization of a large system of particles into stable dissipative structures of physical, biological, social, economic nature. Therefore, it is very important and relevant to carry out a more complete than before analysis of the evolution of the concept of turbulence and methods of its study.

The purpose of study is to present the history of discoveries in the theory of turbulence, which have influenced the formation of nonlinear dynamics. The research was performed using methods of logical analysis, generalization, classification, and systematization. The historicalscientific method was used to evaluate the scientific results of scientists.

The article considers the contribution of A.M. Kolmogorov to development of the theory of turbulence; emerging of the theory of turbulence of Landau — Hopf (1944—1948) and its influence on the formation of concepts, ideas and models of nonlinear dynamics. The shift to the turbulence of Landau-Hopf influenced on the inception of the theory of turbulent flows (the theory of Ruelle — Takens, 1971). The difference between these theories have been established. The article discusses the further development of turbulence research in the 60—80s of the twentieth century; the study of plasma turbulence (there appear highly nonlinear problems) — in particular, the study of isolated waves in plasma (N. Zabuski and M. Kruskal, 1965), the theory of Feigenbaum (1978—1979), the theory of turbulent motion of Pomeau — Manneville (1980). It is concluded that the theory of turbulence of A.M. Kolmogorov (1941), the origin and development of the theory of Landau — Hopf (1944—1948) and the theory of Ruelle — Takens (1971), and also the further studies of turbulence influenced on the formation of nonlinear dynamics and its concepts (formation of the concept of a strange attractor, a constructive concept of self-similarity).

It is concluded that the development of the theory of turbulence influenced the formation of nonlinear dynamics of its concepts (the concept of a strange attractor, the constructive concept of self-similarity). In 1971, D. Ruel and F. Tuckens, criticizing the Landau — Hopf theory, were the first to point to the existence of a strange attractor that demonstrates sensitivity to initial conditions (the strange attractor — one of the basic concepts of nonlinear dynamics).

The research of J. McLaughlin and P. Martin (1974) and J. Gollab and H. Swinney (1975) also contributed to the formation of the concept of a strange attractor and the development of the Ruel — Takens turbulence. Based on the turbulence of Landau — Hopf, in clarifying the question of how turbulence arose, in 1978, G.M. Zaslavsky proposed a simpler model of a strange attractor, which establishes a connection between two types of chaotic motion. In 1983, V.S. Afrai novich and L.P. Shilnikov introduced the concept of quasi-attractors.

In 1996, the Ukrainian scientist A.N. Sharkovsky proposed the concept of “ideal turbulence” — a new mathematical phenomenon in deterministic systems.

The theory of turbulence is developing rapidly in Ukraine and abroad, gradually spreading to various fields of science and technology.

Keywords: turbulence, nonlinear dynamics, dynamical systems, attractor.


  1. Chaplygin, S.A. (1894). On some cases of motion of a rigid body in a liquid. Works of physics science department at the society of natural science lovers, vol. 6, issue 2, 20—42 [in Russian].
  2. Zhukovsky, N.Ye. (1935). Complete works. Vol. 2. Hydrodynamics. Moscow: ONTI NKTP USSR, 1935, 357 p.; Vol. 3. Hydrodynamics. Moscow: ONTI NKTP USSR, 1936, 486 p.; Vol. 7. Hydrodynamics. Moscow: ONTI NKTP USSR, 1937, 410 p. [in Russian].
  3. Arnold, V.I. (1993). About A.N. Kolmogorov. Kolmogorov in memories. Moscow: Nauka [in Russian].
  4. Mukhin, R.R. (2003). A. N. Kolmogorov and the statistical theory of turbulence. IIFM. Moscow: Nauka [in Russian].
  5. Kolmogorov, A.N. (1985). Introduction to works on turbulence. Selected Works. Book 1, Mat hematics and mechanics. Moscow: Nauka [in Russian].
  6. Klimontovich, Yu.L. (2005). Recollectionsof people and his personal notes about people. V.S. Anischenko, V. Ebeling, & Yu.M. Roanovsky (Eds.). Saratov: Publishing house “College” [in Russian].
  7. Betyaev, S.K. (1995). Hydrodynamics: Problems and Paradoxes. UFN, 165, issue 3, 299—330. https://doi.org/10.1070/PU1995v038n03ABEH000076 [in Russian].
  8. Monin, A.S. (1978). The nature of turbulence. UFN, 125, issue 1, 97—122. https://doi.org/10.3367/UFNr.0125.197805f.0097 [in Russian].
  9. Mukhin, R.R. (2018). Essays on the history of dynamic chaos: Research in the USSR in the 1950—1980s, issue 63, 320 p. [in Russian].
  10. Kilochytska, T.V. (2019). Development of world research in plasma physics and turbulence (1960—1970). Proceedings from: The 18th All-Ukrainian Scientific Conference of Young Historians of Science, Technology, Education and Specialists. (pp. 81—84). Kyiv [in Ukraine].
  11. Kilochytska, T.V. (2019). History of research on hydrodynamic turbulence and nonlinear dynamics (global context). Proceedings from: The 18th All-Ukrainian Conference “Current Issues in the History of Science and Technology”. (pp. 128—131). Kyiv [in Ukraine].
  12. Swinney, H., & Gollab, J. (Eds.) (1984). Hydrodynamic instabilities and transition to turbulence. Moscow: Mir [in Russian].
  13. Kolmogorov, A.N. (1941). Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers. DAN USSR, vol. 30, issue 4, 299—303 [in Russian].
  14. Kolmogorov, A.N. (1941) The degeneration of isotropic turbulence in an incompressible viscous fluid. DAN USSR, vol. 31, issue 6, 538—541 [in Russian].
  15. Kolmogorov, A.N. (1941). Energy dissipation at locally isotropic turbulence. DAN USSR, vol. 32, issue 1, 19—21 [in Russian].
  16. Landau, L.D. (1944). The problem of turbulence. DAN SSSR, 44, 339 [in Russian].
  17. Millionschikov, M.D. (1941). The theory of homogeneous isotropic turbulence. Reports of the USSR Academy of Sciences, vol. 32, issue 9, 611—614 [in Russian].
  18. Wilson, K., & Kohut, J. (1975). Renormalization group and e-expansion. Moscow: Mir [in Russian].
  19. Kuznetsov, S.P. (1993). Renormalization chaos in systems exhibiting period doubling. Nonlinear waves. Physics and Astrophysics. Moscow: Nauka, 286—299 [in Russian].
  20. Zaslavsky, G.V. (1978). The simplest case of a strange attractor. Phys. Lett., vol. 15. 240—243. https://doi.org/10.1016/0375-9601(78)90195-0
  21. Ruelle, D., & Takens, F. (1971). The Nature of Turbulence. Comm. Math. Phys., 20, 167—192. https://doi.org/10.1007/BF01646553
  22. Smale, S. (1967). Differentiable dynamical systems. Bull. AMS, 73, 747—817. https://doi.org/10.1090/S0002-9904-1967-11798-1
  23. Mukhin, R.R. (2012). Essays on the history of dynamic chaos: Research in the USSR in the 1950—1980s. Moscow: Book House “LIBROKOM” [in Russian].
  24. Academician L.I. Mandelstam (1979). To the 100th anniversary of his birth. Moscow: Nauka, 312 p. [in Russian].
  25. McLaughlin, J.B., & Martin, P.C. (1974). Transition to turbulence of a statistically stressed fluid. Phys. Rev. Lett., 33, 1189—1192. https://doi.org/10.1103/PhysRevLett.33.1189
  26. Gollab, J.P., & Swinney, H.L. (1975). Onset of turbulence in a rotating fluid. Phys. Rev. Lett., 35, 927—930. https://doi.org/10.1103/PhysRevLett.35.927
  27. Li, T.-Y., & Yorke, J.A. (1975). Period Three Implies Chaos. Amer. Math. Monthly, 82, 982— 985. https://doi.org/10.1080/00029890.1975.11994008
  28. Lorenz, E. (1963). Deterministic Nonperiodic Flow. J. Atmosph. Sci., 20, 130—141. https://doi.org/10.1175/1520-0469(1963)0202.0.CO;2
  29. Newhouse, S.E., Ruelle, D., & Takens, F. (1978). Occurrence of Strange Axiom A Attractors Near Quasi Periodic Flows on Tm (m = 3 or more). Comm. Math. Phys., 64, 35—40. https://doi.org/10.1007/BF01940759
  30. Eckmann, J.-P. (1981). Roads to turbulence in Dissipative Dynamical Systems.Rev. Mod. Phys., vol. 53, issue 4, part 1, 643—654. https://doi.org/10.1103/RevModPhys.53.643
  31. Bogolyubov, N.N. (1964). On quasiperiodic solutions in problems of nonlinear mechanics. Proceedings of the first summer mat. school, 1, 11—101 [in Ukraine]. https://doi.org/10.1016/B978-1-4832-0078-1.50007-9
  32. Klimontovich, Yu.L. (1964). Statistical theory of nonequilibrium processes in plasma. Moscow State University [in Russian].
  33. Vedenov, A.A., Velikhov, E.P., & Sagdeev, R.Z. (1961). Nonlinear oscillations of a rarefied plasma. Nuclear fusion, vol. 1, issue 1, 82—105 [in Russian]. https://doi.org/10.1088/0029-5515/1/2/003
  34. Drummond, W.E., & Pines, D. (1962). Nonlinear stabilization of plasma oscillations. Nucl. Fusion Supp., 3, 1049.
  35. Mitropolskiy, Yu.A. (1964). The study of an integral manifold for a system of nonlinear equations close to equations with variable coefficients in a Hilbert space. Ukrainian mathematical journal, vol. 16, issue 3, 334—338 [in Ukraine].
  36. Vedenov, A.A., & Rudakov, L.I. (1964). The interaction of waves in continuous media. DAN USSR, vol.159, issue 4, 767—770 [in Russian].
  37. Zakharov, V.E. (1972). Collapse of Langmuir waves. Journal of Electro-Technic Phys., vol. 62, issue 5, 1745—1759 [in Russian].
  38. Kadomtsev, B.B. (1964). Plasma turbulence. Issues of Plasma Theory, 4, 188—339 [in Russian].
  39. Sitenko, A.G. (1965). Electromagnetic fluctuations in plasma. Kharkiv: Publishing house of Kharkiv State University, 185 p. [in Ukraine].
  40. Dolzhansky, F.V., Klyatskin, V.I., Obukhov, A.M., & Chusov, M.A. (1974). Nonlinear systems of hydrodynamic type. Moscow: Nauka [in Russian].
  41. Sitenko, A.G. (1977). Fluctuations and nonlinear interaction of waves in plasma. Kyiv: Naukova dumka [in Ukrainian]. https://doi.org/10.1007/978-1-4757-1571-2_27
  42. Feigenbaum, M.J. (1978). Quantitative universality for a class on nonlinear transformations. J. Stat. Phys., 19(1), 25—52. https://doi.org/10.1007/BF01020332
  43. Feigenbaum, M.J. (1979). The universal metric properties of nonlinear transformations. J. Stat. Phys., vol. 21, issue 6, 669—706. https://doi.org/10.1007/BF01107909
  44. Pomeau, Y., & Manneville, P. (1980). Intermittent Transition to Turbulence in Dissipative Dynamical Systems. Comm. Math. Phys., 74, 189—197. https://doi.org/10.1007/BF01197757
  45. Manneville, P., & Pomeau, Y. (1980). Different ways to turbulence in dissipative dynamical systems. Physica ID, 219—226. https://doi.org/10.1016/0167-2789(80)90013-5
  46. Romanenko, O.Y., & Sharkovsky, O.M. (1996). From one-dimensional to infinite-dimensional dynamical systems: ideal turbulence. Ukrainian Mathematical Journal, vol. 48, issue 12, 1604—1627. https://doi.org/10.1007/BF02375370
  47. Landa, P.S. (1995). Hydrodynamic turbulence and coherent structures. Proceedings of universities. Applied nonlinear dynamics, vol. 3, issue 2, 4—5 [in Russian].
  48. Sinai, Ya.G. (2001). How mathematicians study chaos. Mathematical education, 3rd series, vol. 5, 32—46 [in Russian].
  49. Dobrocheev, O.V. (2019). The mechanics of very large systems, life and mind. Moscow: TEIS, 144 p. [in Russian].

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