Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Constructive Method of Integration and Qualitative Study of Plane Dynamical Systems

Author(s):

Elbert Naziyev

Ukraine, 03056, Kyiv, Peremogy Avenue, 37,
National technical University of Ukraine «KPI»,
Department of higher mathematics,
associate professor, PhD in mathematics

gik8@yandex.ru

Abstract:

For 2nd order dynamical systems the tasks of integration in quadratures and qualitative research are considered. The paper describes the methods of the solution of these problems which are based on the use of the first order differential equation adjoint to the system and its integrating multipliers. The functions inverse to the multipliers (integrating functions) are introduced, and the differential equation for which the functions are solutions is considered. All the main results are illustrated on the examples of specific dynamical systems.

References:

1. Ovsyannikov L. V. Group analysis of differential equations. М. : «Nauka», 1978. 399 p
2. Naziyev E. H. On the study of the behavior of the trajectories in the vicinity of equilibrium States of dynamic systems of the second order. Kiev, 1988. – 28 p. – Dep. in UkrNIINTI 27. 10. 88, №2750-Uk88
3. Naziyev E. H. Phase portraits of homogeneous dynamical systems of the second order. Kiev, 1991. – 34p. – Dep. in UkrNIINTI 30. 10. 91, №1429-Uk91
4. Andronov А. А. , Leontovich Е. А., Gordon I. I., Mayer А. G. Qualitative theory of dynamical systems of the second order. М. : Nauka, 1956. 568 p
5. Bautin N. N., Leontovich Е. А. Methods and techniques of the qualitative study of dynamical systems on the plane. М. : Nauka, 1990. 486 p
6. Andreev А. F. Introduction to the local qualitative theory of differential equations. The training manual. - SPB. : Publishing house St. Petersburg state University, 2001. – 106 p
7. Basov V. V. Method of normal forms in the local area of the qualitative theory of differential equations. The training manual. - SPB. : Publishing house St. Petersburg state University, 2001. – 47 p
8. Ushho D. S. Introduction to the qualitative theory of dynamic systems of the second order. Maikop, 2003г. – 104 p
9. Bogdanov R. I. Nonlinear dynamic systems on a plane and their applications. М. «Vuzovskaya kniga», 2003. 373 p
10. Bogdanov R. I. Phase portraits of dynamic systems on a plane and their invariants. М. «Vuzovskaya kniga», 2008. 427 p
11. Stepanov V. V. A course of differential equations. М. : GIFML, 1958. 468 p
12. Elsgolc S. E. Differential equations and calculus of variations. М. : Nauka, 1965. 271 p
13. Kartan E. Integral invariants. М. -L. : GITTL, 1940. 216 p
14. Eisenhart L. P. Continuous groups of transformations. М. : GIIL, 1947. 359 p
15. Lefshec S. Geometric theory of differential equations. М. : GIIL, 1961. 387 p
16. Nemickiy V. V., Stepanov V. V. Qualitative theory of differential equations. М. -L. : GITTL, 1947. 448 p
17. Golovina L. I. Linear algebra and some of its applications. М. : «Nauka», 1975. 407 p
18. Malkin I. G. The theory of stability of motion. М. : «Nauka», 1966. 530 p
19. Naziyev E. H., Naziyev А. H., Keleinikova G. I. On homogeneous linear differential systems with constant coefficients and of the full problem of eigenvalues (Part 2). Ryazan, 2012. – 40 p. – Journal of Ryazan State University named S.. Esenin №1/34
20. Kamenkov G. V. Selected works. v. II. М. : «Nauka», 1972. 214 p

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