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

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Bogatov E. M., Mukhin R. R. The relation between the nonlinear analysis, bifurcations and nonlinear dynamics (on the example of voronezh school of nonlinear functional analysis). Izvestiya VUZ. Applied Nonlinear Dynamics, 2015, vol. 23, iss. 6, pp. 74-88. DOI: 10.18500/0869-6632-2015-23-6-74-88

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Language: 
Russian
Heading: 
History of Nonlinear Dynamics. Personalia
Article type: 
Other
UDC: 
51(09)
DOI: 
10.18500/0869-6632-2015-23-6-74-88

The relation between the nonlinear analysis, bifurcations and nonlinear dynamics (on the example of voronezh school of nonlinear functional analysis)

Autors: 
Bogatov Egor Mihajlovich, Stary Oskol technological Institute. A. A. Ugarov (branch) of Federal state educational institution of higher professional education "national research technological University "MISIS" (STI nust Misa)
Mukhin Ravil Rafkatovich, Stary Oskol technological Institute. A. A. Ugarov (branch) of Federal state educational institution of higher professional education "national research technological University "MISIS" (STI nust Misa)
Abstract: 

The paper is devoted to some historical aspects of the rapidly developing field of modern mathematics – nonlinear functional analysis, which is presented as the basis of the mathematical apparatus of nonlinear dynamics. Its methods are demonstrated on the example of bifurcation. The first bifurcations problem – Euler problem on elastic instability rod under longitudinal compressive forces is considered. The formation of Voronezh school of functional analysis and its role in the development of nonlinear analysis in general is also discussed.

Key words: 
Nonlinear functional analysis
Banach space
nonlinear operator
bifurcation
instability
Voronezh school of functional analysis
Soviet mathematics
Krasnosel’skii theorem
Reference: 
  1. Krein M.G., Lyusternik L.A. Functional analysis // Mathematics in the USSR for 30 Years. Moscow–Leningrad: GITTL, 1948 (in Russian).
  2. Istoriya Otechestvennoy Matematiki. V 4 Tomah, 5 Knigah. Kniga 1. Kiev: Naukova dumka, 1970 (in Russian).
  3. Banach S. Theorie des operations lineaires. Warszawa, 1932.  
  4. Karpachev M.D. Voronezh University. Voronezh: Izd-vo VGU, 2003 (in Russian).
  5. Lyusternik L.A., Shnirelman L.G. Proceedings of the Scientific-Research. Inst. Math. and Mech., 1930 (in Russian).
  6. Lyusternik L., Schnirelmann L. Existence des trois lignes geodesiques fermees surla chaque surface de genre 0 // Comр. Ren. 1929. Vol. 188. P. 534.
  7. Lyusternik L., Schnirelmann L. Sur la problem des trois geodesiques fermees sur la chaque surface de genre 0 // Comр. Ren. 1929. Vol. 189. P. 269.
  8. Tikhomirov V.M. Hist. and Mathemat. Studies. 2 Ser., Iss. 5 (40). Moscow: Yanus-K, 2000. P. 112 (in Russian).
  9. Lyusternik L.A. // Uspekhi Mat. Nauk. 1936. Vol. 1. P. 77 (in Russian).
  10. Aleksandrov P.S., Vishik M.I., Ditkin V.A., Kolmogorov A.N., Lavrent’ev M.A., Oleinik O.A. Lazar’ Aronovich Lyusternik (To the 80th Birthday) // Uspekhi Mat. Nauk. 1980. Vol. 35, Iss. 6(216). P. 3 (in Russian).
  11. Bakhtin I.A. Materials to the history of mathematical department of VSU. Voronezh, 1998 (in Russian).
  12. Bogolyubov N.N., Ishlinskii A.Yu., Kantorovich L.V., Sadovskii B.N., Sobolev S.L., Trapeznikov V.A., Bobylev N.A. // Uspekhi Mat. Nauk 1981. Vol. 36, Iss. 2(218). P. 215 (in Russian).
  13. Vershik A.M. The Life and Fate of Functional Analysis in the Twentieth Century. Mathematical Events of the Twentieth Century. Springer-PHASIS, 2006. P. 437.
  14. Mukhin R.R. Essays on the History of Dynamic Chaos. Мoscow: URSS, 2012 (in Russian).
  15. Anosov D.V., Trenogin V.A. Bifurcation // Math. Encyclopedia. Vol. 1. Moscow: Sov. Encyclopedia, 1977. P. 496 (in Russian).
  16. Liapounoff A.M. Sur les figures d’equilibre pen differentes des ellipsoids d’unemasse liquid’de homogene doulee d’un mouvement de rotation. I partie. Etude generale du probleme // St.-Pbg. Imprim. de l’Acad. des Sc.1906. IV+225p.
  17. Andronov A.A., Pontryagin L.S. Rough systems // Dokl. Akad. Nauk SSSR. 1937. Vol. 14, No 5. P. 247 (in Russian).
  18. Krasnosel’skii M.A. Topological Methods in the Theory of Nonlinear Integral Equations. Oxford: Pergamon Press, 1964.
  19. Birkhoff G.D., Kellog O.D. Invariant points in function space // Trans. AMS. 1922. Vol.23. P. 96.
  20. Schauder J. Der Fixpunktsatz in Funktinalraumen // Studia Math. 1930. No 2. S. 171.
  21. Arnold V.I. From Superpositions to the KAM Theory // V.I. Arnold. Selected Works – 60. Moscow: PHASIS, 1997 (in Russian).
  22. M. A. Krasnosel’skii // Uspekhi Mat. Nauk. 1954. Vol. 9, Iss. 3(61). P. 57 (in Russian).
  23. Landau L.D., Lifshitz E.M. Theory of Elasticity: Vol. 7 of a Course of Theoretical Physics. Pergamon Press, 1970.
  24. Timoshenko S.P. The History of the Science of Strength of Materials. Moscow: GITTL, 1957 (in Russian).
  25. A Method for Finding Curves Having the Properties of Maximum or Minimum or Isoperimetric Problem Decision, Taken in the Broadest Sense of Leonhard Euler, a Royal Professor and Member of the St. Petersburg Imperial Academy of Sciences. Moscow-Leningrad: GTTI, 1934 (in Russian).
  26. Krasnosel’skii M. A. // Uspekhi Mat. Nauk. 1957. Vol. 12, Iss. 1(73). P. 203 (in Russian).
  27. Krasnosel’skii M.A. and colleagues. Functional Analysis. Wolters-Noordhoff Publ., Groningen, 1972. 379p.
  28. Curant R., Hilbert D. Methods of Mathematical Physics. Vol. 1. John Wiley & Sons Inc, 1953.
  29. Nonlinear functional analysis and its applications // Proc. of symposia in pure math. California, Berkley, July 11-29, 1983. Providence, Rhode Island, 1986.
  30. To the Memory of M.A. Krasnosel’skii [Online] http://www.aha.ru/ amkr/obitrus.html (accessed 05.11.15) (in Russian).
  31. Krasnosel’skii M.A., Burd V.Sh., Kolesov Yu.S. Nonlinear Almost Periodic Oscillations. New York: Wiley & Sons, 1973.
  32. Borisovitch, Yu.G. M.A. Krasnosel’skii – an Outstanding Scientist and Educator [Online] URL: http://nan.vstu.edu.ru/research-1.htm (accessed 05.11.15) (in Russian).
  33. Ziedler E. Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems. New York: Springer-Verlag, 1986.
  34. Ziedler E. Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Optimization. New York : Springer-Verlag, 1985.
Received: 
25.05.2015
Accepted: 
21.12.2015
Published: 
29.04.2016
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