Geodesic
Construction of the planar vector fields with~nonsimple critical point of~prescribed topological structure
Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 68 (2022) no. 4, pp. 575-595.

Voir la notice de l'article provenant de la source Math-Net.Ru

The problem of constructing $n$-linear ($n\geq 2$) plane vector fields with isolated critical point and given separatrices of prescribed types is considered. Such constructions are based on the use of vector algebra, the qualitative theory of second-order dynamic systems and classical methods for investigating their critical points. This problem is essentially an inverse problem of the qualitative theory of ordinary differential equations, and its solution can be used to synthesize mathematical models of controlled dynamical systems of various physical nature.
Keywords: vector field, topological structure, critical point, separatrix, inverse problem of qualitative theory of ODE, mathematical model, programmed motion, controlled particle.
Mots-clés : ODE, phase portrait 
@article{CMFD_2022_68_4_a2,
     author = {S. V. Volkov},
     title = {Construction of the planar vector fields with~nonsimple critical point of~prescribed topological structure},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {575--595},
     publisher = {mathdoc},
     volume = {68},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2022_68_4_a2/}
}
TY  - JOUR
AU  - S. V. Volkov
TI  - Construction of the planar vector fields with~nonsimple critical point of~prescribed topological structure
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2022
SP  - 575
EP  - 595
VL  - 68
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2022_68_4_a2/
LA  - ru
ID  - CMFD_2022_68_4_a2
ER  - 
%0 Journal Article
%A S. V. Volkov
%T Construction of the planar vector fields with~nonsimple critical point of~prescribed topological structure
%J Contemporary Mathematics. Fundamental Directions
%D 2022
%P 575-595
%V 68
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2022_68_4_a2/
%G ru
%F CMFD_2022_68_4_a2
S. V. Volkov. Construction of the planar vector fields with~nonsimple critical point of~prescribed topological structure. Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 68 (2022) no. 4, pp. 575-595. http://geodesic.mathdoc.fr/item/CMFD_2022_68_4_a2/

[1] Almukhamedov M. I., “Obratnaya zadacha kachestvennoi teorii differentsialnykh uravnenii”, Izv. vuzov. Ser. Mat., 1963, no. 4, 3–6 | MR

[2] Almukhamedov M. I., “O konstruirovanii differentsialnogo uravneniya, imeyuschego svoimi predelnymi tsiklami zadannye krivye”, Izv. vuzov. Ser. Mat., 1965, no. 1, 12–16 | MR

[3] Volkov S. V., “Upravlenie kvazilineinymi dinamicheskimi sistemami s dvumya stepenyami svobody”, Dokl. RAN, 384:1 (2002), 43–46 | MR

[4] Galiullin A. S., Obratnye zadachi dinamiki, Mir, M., 1984

[5] Galiullin A. S., Metody resheniya obratnykh zadach dinamiki, Nauka, M., 1986

[6] Erugin N. P., “Postroenie vsego mnozhestva sistem differentsialnykh uravnenii, imeyuschikh zadannuyu integralnuyu krivuyu”, Prikl. mat. mekh., 16:6 (1952), 659–670

[7] Mescherskii I. V., Raboty po mekhanike tel peremennoi massy, Gostekhizdat, M.–L., 1948 | MR

[8] Suslov G. K., O silovoi funktsii, dopuskayuschei dannye chastnye integraly, Dokt. diss., Kievskii un-t, Kiev, 1890

[9] Andronov A. A., Leontovich E. A., Gordon I. I., Maier A. G., Qualitative theory of second-order dynamic systems on a plane, John Wiley Sons, New York, 1973 | MR

[10] Argémi J., “Sur les points singuliers muptiples de systèms dynamiques dans $\mathbb{R}^2$”, Ann. Mat. Pura Appl. (4), 79 (1968), 35–69 | DOI | MR

[11] Bertrand M. J., “Thèoréme relatif au mouvement d'un point attiré vers un centre fixe”, Comp. Rend. Acad. Sci., 77 (1873), 849–853

[12] Frommer M., “Die integralkurven einer gervonlichen differentialgleichung erster ordnung in der umgebung rationaler unbestimmtheitsstellen”, Math. Ann., 99 (1928), 222–272 | DOI | MR

[13] Jaumes G., “Synthèse d'un systéme dynamique correspondant a un portrait topologique donné”, Int. J. Nonlinear Mech., 7:6 (1972), 597–608 | DOI

[14] Newton I., Phylosophiae Naturalis Principia Mathematica, London, 1687 | MR

[15] Schecter S., Singer M. F., “Separatrices at singular points of planar vector fields”, Acta Math., 145 (1980), 47–78 | DOI | MR

[16] Shaub H., Junkins L. J., Analytical mechanics and aerospace systems, 2002 www.fisica.net/mecanicaclassica/Analytical

[17] Sverdlove R., “Inverse problems for dynamical systems”, J. Differ. Equ., 42:1 (1981), 72–105 | DOI | MR

[18] Volkov S. V., “Modelling of the controlled motion of a point on a plane”, J. Appl. Math. Mech., 69 (2005), 173–182 | DOI | MR