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


Cite this article as:

Bogatov E. M. On the development of qualitative methods for solving nonlinear equations and some consequences. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 1, pp. 96-114. DOI: https://doi.org/10.18500/0869-6632-2019-27-1-96-114

Published online: 
28.02.2019
Language: 
Russian
UDC: 
51 (09)

On the development of qualitative methods for solving nonlinear equations and some consequences

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)
Abstract: 

Aim. The aim of the paper is investigation of the development of the fixed-point method and mapping degree theory associated with the names of P. Bohl, L. Brouwer, K. Borsuk, S. Ulam and others and its application to study of the trajectories of dynamical systems behavior and stable states of ordered media. Method. The study is based on an analysis of the fundamental works of the mentioned mathematicians 1900–1930’s, as well as later results of N. Levinson, G. Volovik, V. Mineev, J. Toland and H. Hofer of an applied nature. Results. Brouwer made an essential contribution to the solvability theory of nonlinear equations of the form f(x) = x in a finite-dimensional statement. This was preceded by the study of singular points of vector fields undertaken by H. Poincare, as well as the proof of Bohl theorem on the impossibility  of mapping a disk onto its boundary. The first mathematician who used the fixed point method in the study of systems of differential equations was Bohl. This theme was continued 40 years later in the works of Levinson, who showed the existence at least one periodic solution in deterministic dissipative dynamical systems. The fundamental concept of the mapping degree (deg f) introduced by Brouwer «began to play» in the most unexpected situations. Investigations of Volovik and Mineev revealed a direct dependence of ordered media defects on the topological invariant deg f, characterizing the transformation f of a neighborhood of a singular point onto the sphere. Another non-standard application of the mapping degree was discovered by Toland and Hofer in the study of some Hamiltonian systems. Calculating deg f for mappings of a special kind helped them to prove the existence of periodic, homoclinic, and heteroclinic trajectories of these systems. Discussion. The fixed point method and mapping degree are the basic tools of qualitative methods for solving nonlinear equations. They proved to be in demand not only within the framework of mathematics, but also in applications, and this trend, apparently, will persist even in the transition to the infinite-dimensional case.   Acknowledgements. The author thanks Professor R. R. Mukhin (Stary Oskol Technological Institute of «MISIS») for posing the problem and useful discussions, Professor Yu. E. Gliklikh ( Voronezh State University) for advice on topological methods of analysis and familiarity with the manuscript, as well as V.P. Bogatova for help with access to primary sources and translation from German.  

DOI: 
10.18500/0869-6632-2019-27-1-96-114
References: 

1. Bogatov E.M. On the history of the application of qualitative methods for solving nonlinear integral equations. Science and technology: Questions of history and theory. Materials of the XXXVII intern. annual conf. St. Petersburg Dep. Rus. Nat. Comm. Hist. Philos. Science and Techn. RAS, 2016, November, 21–25,. Issue XXXII, SPb, 2016, pp. 102–104 (in Russian). 2. Poincare, Н. Sur l’Analysis Situs. C.R., 1892, 11, pp. 633–636.  ? 3. Poincare, Н. Analyse des travaux scientifiques de Henri Poincar  ? e faite par lui-m  ? eme.  ? Acta math., 1921, vol. 38, pp. 1–135. 4. Gombrich E.H. The Story of Art (14th ed.). Englewood Cliffs, New Jersey: Prentice-Hall, Inc. 1984. 5. Volovik G.E. and Mineev V.P. Investigation of singularities in superfluid He3 and liquid crystals by homotopic topology methods. Soviet Phys. JETP, 1977, vol. 45, pp. 1186–1196. 6. Bogatov E.M., Mukhin R.R The relation between the nonlinear analysis, bifurcations and non-linear dynamics: On example of Voronezh school of nonlinear functional analysis. Izv. VUZ. Applied Nonlinear Dynamics, 2015, vol. 23, iss. 6, pp. 74–88 (in Russian). 7. Bogatov E.M., Mukhin R.R. About the history of nonlinear integral equations. Izv. VUZ. Applied Nonlinear Dynamics, 2016, vol. 24, iss. 2, pp. 77–114 (in Russian). 8. Poincare, Н. M  ? emoire sur les courbes d  ? efini  ? es par une  ? equation differentielle I.  ? J. de Math., 1881, 7, pp. 375–422. 9. Poincare, Н. On curves defined by differential equations. Moscow; Leningrad, OGIZ, 1947 (in  ? Russian). 10. Kronecker L., Uber Systeme von Funktionen mehrerer Variabeln I. Monatsber. Berlin Akad.,  1869, pp. 159–193. 11. Chetaev N. G. Stability of Motion. Works on Analytical Mechanics. USSR Acad. Sci. Publishing, Moscow, 1962 (in Russian). 12. Siegberg, H. Some historical remarks concerning degree theory. Amer. Math. Monthly, 1981, vol. 88, iss. 2, pp. 125–139. 13. Mawhin J. Poincare’s early use of Analysis situs in nonlinear differential equations: Variations around the theme of Kronecker’s integral. Philosophia Scientiae. 2000, vol. 4, pр. 103–143. 14. Poincare, Н. M  ? emoire sur les courbes d  ? efini  ? es par une  ? equation differentielle IV.  ? J. Math. Pure Appl., 1885, vol. 1, pp. 167–244. 15. Bohl P. Sur certains equations differentielles d’un type general utilisables en m  ? ecanique.  ? Bulletin de la Societ ? e math  ? ematique de France  ? , 1910, vol. 38, pp. 5–138. 16. Myshkis A.D., Rabinovich I.M. Mathematician Piers Bohl from Riga. Riga, Zinatne, 1965 (in Russian). 17. Bohl Piers. Collection of sci. works. Transl. I.M. Rabinovich; ed. L.E. Reizihn; Introd. article and comments L.E. Reizihn and I.A. Khenihn; Acad. Sci. Latvian SSR, Institute of Physics, Riga: Zinatne, 1974. 18. Bohl Piers Georgievich, 1865–1921. Selected works; introductory article by A.D. Myshkis and I.M. Rabinovich. Acad. Sci. Latv. SU, Astrophysics lab. 1961 (in Russian). 19. Borsuk K. Sur les retractes.  ? Fund. Math. 1931, vol. 17, no. 1, s. 152–170. 20. Bohl P. Uber die Bewegung eines mechanischen Systems in der N  ? ahe einer Gleichgewichtslage.  ? Journal fur die reine und angewandte Mathematik  ? , 1904, vol. 127, s. 179–276. 21. Birkhoff G.D. & Kellogg O.D. Invariant points in function space. Trans. Amer. Math. Soc., 1922, vol. 23, pp. 95–115. 22. van Dalen D. Luitzen Egbertus Jan Brouwer. History of Topology, ed. I.M. James, North Holland, 1999, pp. 947–964. 23. Johnson Dale M. The Problem of the Invariance of Dimension in the Growth of Modern Topology, Part II. Arch. Hist. Ex. Sci, 1981. pp. 85–267. 24. Brouwer L.E.J. On continuous vector distributions on surfaces. KNAW Proc., 1909, vol. 11, pp. 850–858. 25. Brouwer L.E.J. On continuous vector distributions on surfaces, II. KNAW Proc., 1910,vol. 12, pp. 716–734. 26. Brouwer L.E.J. On continuous vector distributions on surfaces, III. KNAW Proc., 1910, vol. 12, pp. 171–186. 27. Freudenthal H. (ed.) L.E.J. Brouwer Collected Works, Vol. 2, Geometry, Analysis, Topology and Mechanics. North–Holland, Amsterdam, 1976. 28. Brouwer L.E.J. Potentiaaltheorie en Vectoranalyse. Exercise book (unpublished manuscript), 1910. 29. Aleksandrov P.S. Poincare and topology.  ? Russian Math. Surveys, 1972, vol. 27, no. 1, pp. 157–168. 30. Boss V. Lekcii po matematike: Topologiya. T. 13, Izd. 3-e, LENAND, Moscow, 2014 (in Russian). 31. Brouwer L.E.J. Uber Abbildungvon Mannigfaltigkeiten.  ? Math. Annal, 1912, vol. 71, s. 97–115. 32. Brouwer L.E.J. Beweis der Invarianz der Dimensionenzahl. Math. Annal, 1911, vol. 70, s. 161–165. 33. Matousek J. Using the Borsuk–Ulam theorem: Lectures on topological methods in combinatorics ? and geometry. Springer Science & Business Media, 2008. 34. Borsuk K. Drei Satze  ? uber die  ? n-dimensionale euklidische Sphare.  ? Fund. Math. 1933, vol. 20, no. 1, s. 177–190. 35. Krein M.G., Nudelman A.A. Kvant, 1983, no. 8, pp. 20–25 (in Russian). 36. Lyusternik L.A., Shnirelman L.G. Topological methods in variational problems. Mathematics and Mechanics Research Institute at 1st MSU, Moscow, 1930 (in Russian). 37. Volovik G.E., Mineev V.P. Physics and topology. Moscow: Znanie, 1980 (in Russian). 38. Mineev V.P. Topological objects in nematic liquid crystals, In: V.G. Boltyansky, V.A. Efremovich, Visual topology, pp. 148–158 (Bibliotechka Kvant, issue 21. Moscow, Nauka, 1982) (in Russian). 39. Volovik G.E. Superfluid properties of 3He-A. Sov. Phys. Usp., 1984, vol. 27, pp. 363–384. 40. Poincare, Н. Sur un th  ? eor  ? eme en g  ? eom ? etrie.  ? Rendiconti del Circolo matematico di Palermo, 1912, vol. 33, pp. 375–407. http://henripoincarepapers.univ-lorraine.fr/bibliohp/ajax.php?bibkey==hp... 41. Poincare Н. Selected Works. Vol. 2. New methods of celestial mechanics. Topology. Theory of  ? numbers, Ed. N.N. Bogolyubov, V.I. Arnold, I.B. Pogrebyskii. Moscow: Nauka, 1972 (in Russian). 42. Levinson N. Transformation theory of non-linear differential equations of the second order. Annals Math., Second Series, 1944, vol. 45, no. 4, pp. 723–737. 43. Bogatov E.M., Mukhin R.R. The averaging method, a pendulum with a vibrating suspension: N.N. Bogolyubov, A. Stephenson, P.L. Kapitza and others. Izv. VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, no. 5, pp. 69–87 (in Russian). 44. Coddington E.A., Levinson N. Theory of ordinary differential equations. Tata McGraw-Hill Education, 1955. 45. Hofer H., Toland J. Homoclinic, heteroclinic, and periodic orbits for a class of indefinite Hamiltonian systems. Math. Annalen, 1984, vol. 268, no. 3, pp. 387–403. 46. Toland J.F. Solitary wave solutions for a model of the two-way propagation of water waves in a channel. Math. Proc. Camb. Phil. Soc., 1981, vol. 90, pp. 343–360. 47. Peletier L.A., and Troy W.C. Spatial patterns: higher order models in physics and mechanics. Vol. 45. Springer Science & Business Media, 2012. 48. Nash C. Topology and physics – a historical essay. History of Topology, I.M. James ed., North Holland, 1999, pp. 359–416. 49. Ugi I., Dugundij J., Kopp R., & Marquarding D. Perspectives in theoretical stereochemistry. Lecture note series, Vol. 36. Springer, Heidelberg, 1984. 50. Dolbilin N.P. Criterion of a crystal and locally antipodal sets of Delaunay. Vestnik of the ChelSU, 2015, no. 17, pp. 6–17 (in Russian). 51. Dieudonne J. A History of Algebraic and Differential Topology, 1900–1960. Modern Birkh  ? auser,  ? Boston, 1989. 52. Park S. Ninety years of the Brouwer fixed point theorem. Vietnam J. Math., 1999, vol. 27, no. 3, pp. 187–222. 53. Mawhin J. In Memoriam Jean Leray (1906–1998). Topol. Meth. Nonlin. Anal., 1998, vol. 12. 14, pp. 199–206. 54. Mawhin J. Juliusz Schauder, topology of functional spaces and partial differential equations. Wiadomosci matematyczne  ? , 2012, vol. 48, no. 2, pp. 173–183. 55. Bogatov E.M. Key moments of the mutual influence of the Polish and Soviet schools of nonlinear functional analysis in the 1920’s–1950’s. Antiq. Math., 2017, vol. 11, pp. 131–156. 56. Krasnoselskii M.A., Zabreiko P.P. Geometrical Methods of Nonlinear Analysis. Berlin–Heidelberg–New York–Tokyo, Springer-Verlag, 1984. 57. Ladyzhenskaya O.A. Mathematical problems in the dynamics of a viscous incompressible flow. Gordon & Breach, New York, 1963. 58. Beckert, H. Existenzbeweis fur permanente Kapillarwellen einer schweren Fl  ? ussigkeit.  ? Arch. Rat. Mech. Anal., 1963, vol. 13, pp. 15–45. 59. Krasnoselskii M.A., Burd V.Sh., Kolesov Yu.S. Nonlinear Almost Periodic Oscillations. Wiley, New York, 1973. 60. Vorovich I.I. On the existence of solutions in the nonlinear theory of shells. Izv. Akad. Nauk SSSR Ser. Mat., 1955, vol. 19, no. 4, pp. 173–186 (in Russian). 61. Hutson V.C.L., Pym J.S. Applications of Functional Analysis and Operator Theory. Academic Press, 1980. 62. Nonlinear functional analysis and its applications, Part 2. (Proceedings of Symposia in Pure Mathematics, Vol. 45.) F.E. Browdered. AMS, Providence. Rhode Island, 1986.

Short text (in English):
(downloads: 55)
Full text:
(downloads: 122)