Evolution of the main provisions of the theory of stability
https://doi.org/10.22405/2226-8383-2022-23-4-327-349
Abstract
The aim of the work is to study the evolution of the concept of stability, which is a structureforming concept in all areas of science and technology, and even beyond them. The stages of this long evolution corresponded to the dominant trends in the mathematics of their time.
By the end of the XIX century. the complexity of the concept of stability was realized, the question arose of a mathematically rigorous approach to the problem. A general theory of motion stability was built on a solid mathematical foundation. This became a milestone not
only in the development of the subject itself, but was one of the foundations for constructing a qualitative theory. Subsequently, the theory of stability was divided into two branches: one - the expansion of the theory in breadth based on old ideas, strengthening the links with applications; the other is stability in the context of the theory of dynamical systems. In the latter case, stable movements are considered in the series of all movements; in the stability-instability dichotomy both poles are equal and meaningful. Instability also turns out to be a complex concept that has a variety of forms. Instability has acquired a constructive meaning; it ensures innovation and development. Typical is the coexistence of stability and instability with a complex topology of such a structure. Diverse types of instability demonstrate the phenomenon of turbulence. The study of this phenomenon at the modern level requires the use of mathematics according to the canons of rigor adopted in mathematics itself. One can raise the question of the limits of applicability of the possibilities of the most qualitative description and the concept of stability.
In this regard, there are first results, new ideas are required.
Supporting agencies: Thе work was supported by the RFBR grant No. 20-011-00402 A.
About the Author
Ravil’ Rafkatovich Mukhin
Ugarov Stary Oskol Technological Institute (branch) National University of Science and Technology «MISiS»
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
References
1. Leine R. L. The historical development of classical stability concepts: Lagrange, Poisson,
2. Lyapunov stability // Nonlinear Dyn. 2010. V. 59. Pp. 173–182.
3. The method of finding curves of lines with the properties of a maximum or a minimum, or
4. the solution of an isoperimetric problem, taken in the broadest sense of Leonard Euler, a royal
5. professor and a member of the Imperial Petersburg Academy of Sciences. Moscow-Leningrad:
6. GTTI, 1934. 603 р. (in Russian).
7. Euler L. Scintia novalis seu tractatus de consruendis ac dirigendis. St. Petersburg, Academiae
8. Scientarum, 1749. 536 p.
9. Lagrange J. L. Sur le principe des vitesse virtuelles // Oeuvres de Lagrange. T. VII. Paris:
10. Gautier-Villars, 1877. Pp. 317–321.
11. M´ecanique analiytique, par J. L. Lagrange. T. Premiere. Paris, 1811. 422 p.
12. Lagrange J. L. Analytical mechanics. T. 1. M.-L.: GITTL, 1950. 594 p. (in Russian).
13. Lejeune-Dirichlet P. G. ¨Uber die Stabilit¨at des Gleichgewichts // CRELLE, J. Reine Angew.
14. Math. 1846. B. 32. Pp. 85–88.
15. Lejeune-Dirichlet P. G. On the Stability of Equilibrium // Lagrange J. Analytical Mechanics.
16. T. 1. Moscow-Leningrad: GITTL, 1950. Pp. 537–540 (in Russian).
17. Lagrange J. L. Sur le movement des noeuds des orbits plan´etaires // M´em. Acad. Sci. 1774. Pp.
18. –307.
19. Lagrange J. L. Th´eorie des variations s´eculaires des ´elements des plan´etes // Oeuvres de
20. Lagrange. T. V. Paris: Gautier-Villars, 1870. Pp. 127–207, 208–344.
21. Laplace P. S. Sur le principe de la gravitation universelle et sur le inegalit´e s´eculaires des plan´etes
22. qui en d´ependent // Oeuvres de Laplace. T. VIII. Paris: Gautier-Villars, 1841. Pp. 201–275.
23. Laplace P. S. Sur le ´equation s´eculaire de la lune // M´em. Acad. Sci. 1788. Pp. 243–271.
24. Poisson S. D. M´emoire sur les inegalit´e s´eculaires des moyens mouvements des plan´etes // J.
25. Ecole Polytech. 1808. T. XV. Pp. 1–56.
26. Einstein A., Infeld L. The Evolution of Physics. Simon and Schuster, 1942. 351 p.
27. Thomson W., Tait P. G. Treatise on Natural Philosophy. Oxford: Clarendon Press, 1867. XXIII.
28. p.
29. Wise N. M., William Thomson and Peter Guthrie Tait, Treatise on Natural Philosophy, First
30. edition (1867) // Landmark Writings in Western Mathematics 1640–1940. I. Grattan-Guinnes
31. (ed.). Amsterdam: Elsivier, 2005. Pp. 521–533.
32. Routh E. G. A treatise of stability of a given state of motion. L.: Macmillan and Com., 1877.
33. p.
34. Poincar´e H. Memoire sur les courbes d´efinies par une ´equations differentielle // J. math. pures
35. et appl. S´er. 3. 1881. V. 7. Pp. 375–422; 1882. V. 8. Pp. 251–296; S´er. 4. 1885. V. 1. Pp. 167–244;
36. V. 2. P. 151–217.
37. Poincar´e H. Sur les courbes d´efinies par des ´equations diff´erentielles. Moscow: GITTL, 1947.
38. p. (in Russian).
39. Poincar´e H. Les m´ethodes nouvelles de la m´ecanique c´eleste. Tome 3. Paris: Gauthier-Villars,
40. 410 p.
41. Zhukovsky N. E. On the Strength of Motion // Scholar notes of Moscow University. Department
42. of Physics and Mathematics. sciences. 1882. V. 4. Pp. 10–21 (in Russian).
43. Leonov G. A., Burkin I. M., Shepeljavyi A. I. Frequently methods in oscillation theory.
44. Mathematics and its applications. V. 357. Dordrecht: Kluwer Academic, 1996. 403 p.
45. Lyapunov A. M. The general problem of the stability of motion // Lyapunov A.M. Select. works:
46. works on the stability of motion. Moscow Nauka, 2007. Pp. 27–298 (in Russian).
47. Hurvitz, W. A. The Chicago Colloquium // Bull. AMS. 1920. V. 27. Pp. 65–71.
48. Birkhoff G. D. Dynamical Systems. Providence, Rhod Island: AMS, 1927. IX + 295 p.
49. Birkhoff G. D. Surface transformations and their dynamical applications // George David
50. Birkhoff. Coll. Math. Papers. V. II. N.Y.: AMS, 1950. Pp. 111–229.
51. Levi-Civita T. Sopra alcuni criteri di instabilita // Annali di Math. 1901. Ser. III. V. 5. Pp.
52. –308.
53. Moiseyev N. D. Essais on the developement of the theory of stability. Moscow-Leningrad:
54. GITTL, 1949. 664 p. (in Russian).
55. Mathematics in the USSR for forty years 1917-1957. T. 1. Ed. A.G. Kurosh. M.: State. Publ.
56. House of Phys.-Math. Lit., 1959. 1000 p. (in Russian).
57. Grigoryan A. T., Pogrebyssky I. B. History of mechanics from the end of the 18th century to
58. the middle of the 20th century. M.: Nauka, 1972. 417 p. (in Russian).
59. Chetaev N. G. One theorem on instability // Rep. Acad. Sci. USSR. 1934. T. 1. Pp. 529–530
60. (in Russian).
61. Chetaev N. G. Movement stability. Works on analytical mechanics. M.: Publ. House Acad. Sci.
62. USSR, 1962. 535 p. (in Russian).
63. Poincar´e H. Sur l’´equilibre d’un masse fluide anim´ee d’un mouvement de rotation // Acta.
64. Math. 1885. T. 7. Pp. 259–380 ; Oeuvres de Henri Poincar´e. T. VII. Paris: Gautier-Villars,
65. Pp. 40–140.
66. Andronov A. A., Pontryagin L. S. Rough systems // Rep. Acad. Sci. USSR. 1937. V. 14, Pp.
67. –250. (in Russian).
68. Peixoto M. Structural stability on two-dimensional manifolds // Topology. 1962. V. 1. № 2. Pр.
69. –120.
70. Andronov A. A. Mathematical problems in self-oscillation theory // A. A. Andronov. Coll.
71. Works, M.: Acad. Sci. USSR, 1956. Pp. 32–71 (in Russian).
72. Smale S. A structurally stable differential homomorphysm with an infinite number of periodic
73. points // Proc. Int. Congress on non-linear oscillations. Кiev 1961. Кiev: Ukranian Acad. Sci.,
74. Pp. 365–366.
75. Bylov B. F., Vinograd R. E., Grobman D. N., Nemytsky V. V. The theory of Lyapunov exponents
76. and its applications to stability problems. M.: Nauka, 1966. 576 p. (in Russian).
77. Perron O. Die Ordnungszahlen linearer Differentialgleichungssysteme // Mathem. Zeitschr.
78. B. 31. Pp. 748–766.
79. Oseledets V. I. The multiplicative ergodic theorem. Characteristic Lyapunov exponents of
80. dynamical systems // Proc. Moskow mat. soc. Т. 19. Pp. 179–210 (in Russian).
81. Millionshchikov M. D. A stability criterion for the probability spectrum of linear systems of
82. differential equations with recurrent coefficients and a criterion for almost reducibility of systems
83. with almost periodic coefficients // Mat. collection 1969. T. 78. № 2. Pp. 179-202 (in Russian).
84. Arnold V. I. On the instability of dynamical systems with many degrees of freedom // Rep.
85. Acad. Sci. USSR. 1964. V. 156. № . 1. Pp. 9–12 (in Russian).
86. Zaslavsky G. M., Zakharov M.Yu., Sagdeev R. Z., Usikov D. A., Chernikov A. A. Generation
87. of ordered structures with an axis of symmetry from Hamiltonian dynamics // JETP Letters.
88. T. 144. V. 7. Pp. 349–353 (in Russian).
89. Zaslavsky G. M., Zakharov M.Yu., Sagdeev R. Z., Usikov D. A., Chernikov A. A. Stochastic
90. web and diffusion of particles in a magnetic field // JETP. 1986. T.91. V. 5. Pp. 500–516 (in
91. Russian).
92. Zaslavsky G. M., Sagdeev R. Z., Usikov D. A., Chernikov A. A. Weak chaos and quasi-regular
93. structures. M.: Nauka, 1991. 236 p. (in Russian).
94. Stephenson A. On a class of forced oscillations // Quart. J. Pure and Appl. Math. 1906. V. 37.
95. № 148. Pp. 353–360.
96. Bogolyubov N. N. Perturbation theory in nonlinear mechanics // Proc. Institute Struct.
97. Mechanics Acad. Sci. Ukrainian SSR. 1950. T. 14., Рр. 9–34 (in Russian).
98. Kapitsa P. L. Dynamic stability of a pendulum with an oscillating suspension point // JETP..
99. V. 21. Issue 5. Рp. 588–597 (in Russian).
100. Bogatov E. M., Mukhin R. R. Averaging method, pendulum with vibrating suspension:
101. N. N. Bogolyubov, A. Stephenson, P. L. Kapitsa and others // Proc. of universities “PND”.
102. V. 25. № . 5. Pp. 69–87 (in Russian).
103. Chelomey V. N. On the Possibility of Increasing the Stability of Elastic Systems Using
104. Vibrations// Rep. Acad. Sci. USSR. 1956. V. 110. No. 3. Pp. 345–347 (in Russian).
105. Chelomey V. N. Paradoxes in mechanics caused by vibrations // Rep. Acad. Sci. USSR. 1983.
106. V. 270. No. 1. Рр. 62–67 (in Russian).
107. Landau L. D., Lifshitz E. M. Hydrodynamics. M.: Nauka, 1986. 736 p. (in Russian).
108. Goldstein S. Fluid mechanics in the first half of this century // Annu. Rev. Fluid Mech. 1969.
109. V. 1. Pp. 1–29.
110. Marsden J. Relationship between the Navier-Stokes equations and turbulence // Strange
111. attractors. M.: Mir, 1981. Рр. 7–20 (in Russian).
112. Sommerfeld A. Ein Beitrag zur hydrodyhamischen Erkl¨arung der turbulenten Flussigkeitsbewegungen
113. / / Proc. 4th Int. Congr. Rome.1908. Pр. 116–124.
114. Orr W. The stability or instability of the steady motions of a liquid // Proc.Roy. Irish Acad.
115. A 27. V. 27. Pp. 9–68, 69–138.
116. Heisenberg W. ¨Uber Stabilit¨at und Turbulenz von Flissigkeitsstr¨omen // Ann. der Phys. 1924.
117. Bd.74. No. 15. Pp. 577–624.
118. Lyapunov A. M. On one problem of Chebyshev // Proc. Akad. Sciences in Phys.-Math.
119. department 1905. 8 ser. T. 17. No. 3. Pp. 1–32; A.M. Lyapunov. Coll. Works. T. 3. M.-L.:
120. Publ. House Acad. Sci. USSR, 1959. Рр. 207–236 (in Russian).
121. Leray J. Etude de diverses ´equations int´egrals non lin´earais et de quelques probl`emes que pose
122. l’hydrodynamique // J. Math. Pures Appl. 1933. V. 12. Pp. 1–82.
123. Leray J. Essai sur le mouvements plans d’un liquide visqueux que limitent des parios // J.
124. Math. Pures Appl. 1934. V. 13. Pp. 341–418.
125. Leray J. Sur le mouvements d’un liquide visqueux emplissant l’espace // Acta Math. 1934. V.
126. Pp. 193–248.
127. Oseen C. W. Neuere Methoden und Ergebnisse in der Hydrodynamik. Leipzig : Academie Verlag,
128. 337 S.
129. Bitsadze A. V. Some classes of partial differential equations. M.: Nauka, 1981. 448 p. (in
130. Russian).
131. Hopf E. ¨Uber die Anfangswertaufgabe f¨ur die hydrodynamischen Gundgleichungen // Math.
132. Nachrich. 1951. B. 4. S. 213–231.
133. Ladyzhenskaya O. A. Attractors for Semigroups and Evolution Equations. Lincei Lectures.
134. Cambridge: CUP, 1991. 73 p.
135. Scheffer V. (1976), Turbulence and Hausdorff dimension. // Turbulence and Navier-Stokes
136. equations. Proc. Conf. Univ. Paris-Sud, Orsay, 1975. Berlin: Springer Verlag, 1976. Pp. 174—
137.
138. Caffarelli L., Kohn R., Nirenberg L. Partial regularity of suitable weak solutions of the Navier-
139. Stokes equations // Comm. Pure Appl. Math. 1982. V. 35(6). Pp. 771-–831.
140. Monin A. S., Yaglom A. M. Statistical fluid mechanics : mechanics of turbulence. Boston: MIT
141. Press, 1971. 782 p.
142. Ruelle D., Takens F. On the Nature of Turbulence // Comm. Math. Phys. 1971. V. 20. Pp.
143. –192 / Рус. пер. в кн.: Странные аттракторы. М.: Мир, 1981. Pp. 117–151.
144. Arnold V. I. Kolmogorov’s hydrodynamic attractors // Proc. Roy. Soc. London. Ser. A. 1991.
145. V. 434. Pp. 19-22.
146. Arnold V., Khesin B. Topogical methods in hydrodynamics. N.Y.: Springer, 1998. 374 p.
147. Obukhov A. M. On integral invariants in systems of hydrodynamic type // Rep. Acad. Sci.
148. USSR. 1969. V. 184. № . 2. Pp. 309–312 (in Russian).
149. Nonlinear systems of hydrodynamic type. Ed. A.M. Obukhov. M.: Nauka, 1974. 160 p. (in
150. Russian).
151. Afraimovich V. A., Shilnikov L.P. On strange attractors and quasiattractors // Nonlinear
152. dynamics and turbulence. Boston-London-Melbourn: Pitman, 1983. Pр. 1-–34.
153. Zaslavsky G. M. Chaotic Dynamics and the Origin of Statistical Laws // Physics Today. 1999.
154. V. 52. Pр. 39—45.
155. Gonchenko S. V. et al. On Newhouse domains of two-dimensional diffeomorphisms that are
156. close to diffeomorphism with a structurally unstable heteroclinic contour // Proc. o MIAN.
157. Moscow: Nauka, 1997. V. 216. Pp. 76—125 (in Russian).
158. Gonchenko S. V. et al. Mathematical theory of dynamical chaos and its applications: review //
159. Proc. Universities “PND”. 2017. V. 25. Issue 2. Pp. 4—36 (in Russian).
160. Holmes P., Lumley J., Bercooz G., Rowley C. Turbulence, Coherent Structures, Dynamical
161. Systems and Symmetry. Cambridge: CUP, 2012. 386 p.
162. Fomin N. N., Chechetkin V. M. Coherent hydrodynamic structures and vortex dynamics. IPM
163. M.V. Keldysh. Preprint No. 1 for 2015. (in Russian).
164. Shilnikov L.P. Homoclinic trajectories from Poincare to the Present // Mathematical events of
165. the Twentieth Century. Berlin: Springer-Verlag, 2006. Pp. 347—370.
166. Gonchenko S. V., Shilnikov L.P., Turaev D. V. Homoclinic Tangencies of an Arbitrary Order in
167. Newhouse Domains // J. Math. Sci., 2001. V. 105. Issue 1. Pp. 1738-–1778.
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Mukhin R.R. Evolution of the main provisions of the theory of stability. Chebyshevskii Sbornik. 2022;23(4):327-349. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-4-327-349
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