01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date:
5.11.1934
Keywords:
dynamical systems; Hamiltonian systems; limit cycles; Integrability of systems.
Subject:
For two-dimensional autonomous analytic dynamical systems it is proved the existence of "canonical" and "quasicanonical" integrals, the important special case of which are Darbouxian integrals and their generalizations. Influence of limit sets on singularity of the first integrals and the integrating factors is studied. Erugin's problem of the existence polynomial vector fields with center and limit cycles is solved. The counterexample to K. S. Sibirsky's hypothesis on everywhere dense set of algebraic equations with Darbouxian integral in the set of equations with the center is constructed.
Biography
Graduated from Faculty of Mechanics and Mathematics of NI Lobachevsky Gorky State University in 1962 (department of differential equations and mathematical analysis). Ph.D. thesis was defended in 1967. D.Sci. thesis was defended in 1984. A list of my works contains more than 100 titles. I am a vice-chairman of Doctoral Counsil D212.166.06 since 1995 and sit on a board of Nizhny Novgorod Mathematical Society since 1996.
Main publications:
Dolov M. V. Ob algebraicheskikh predelnykh tsiklakh polinomialnykh vektornykh polei na ploskosti // Differentsialnye uravneniya. 2001. T. 37. # 9. C. 1155–1160.
Dolov M. V., Chistyakova S. A. O strukture obschego resheniya i integriruyuschego mnozhitelya v okrestnosti prostoi osoboi tochki // Differentsialnye uravneniya. 2001. T. 37. # 5. C. 710–713.
Dolov M. V. Integriruyuschii mnozhitel v okrestnosti uzla // Differentsialnye uravneniya. 1997. T. 33. # 2. C. 158–160.
Dolov M. V., Chistyakova S. A. Algebraicheskie differentsialnye uravneniya s integriruyuschim mnozhitelem tipa Darbu // Differentsialnye uravneniya. 1997. T. 33. # 5. C. 618–622.
Dolov M. V., Kruglov E. V. O chisle polualgebraicheskikh chastnykh integralov odnogo klassa dinamicheskikh sistem s tsilindricheskim fazovym prostranstvom // Differentsialnye uravneniya. 1995. T. 31. # 6. C. 949–954.
M. V. Dolov, E. V. Kruglov, “On the number of linear particular integrals of
polynomial vector fields”, Zhurnal SVMO, 18:1 (2016), 27–30
2009
2.
M. V. Dolov, Chistyakova S.A., “On limit cycles of dynamical systems with nonlinearities of fourth order and invariant curves of elliptic type”, Trudy SVMO, 11:1 (2009), 63–71
2004
3.
M. V. Dolov, “On the Number of Algebraic Invariant Curves of Polynomial Vector Fields”, Differ. Uravn., 40:6 (2004), 838–839; Differ. Equ., 40:6 (2004), 896–897
M. V. Dolov, Yu. V. Pavlyuk, “Invariant Algebraic Curves of Polynomial Dynamical Systems”, Differ. Uravn., 39:8 (2003), 1038–1043; Differ. Equ., 39:8 (2003), 1091–1097
2002
5.
M. V. Dolov, Yu. V. Pavlyuk, “Elliptic Limit Cycles of Two-Dimensional Autonomous Differential Systems”, Differ. Uravn., 38:10 (2002), 1303–1309; Differ. Equ., 38:10 (2002), 1385–1392
2001
6.
M. V. Dolov, “Algebraical Limit Cycles of Polynomial Vector Fields on the Plane”, Differ. Uravn., 37:9 (2001), 1155–1160; Differ. Equ., 37:9 (2001), 1211–1216
7.
M. V. Dolov, S. A. Chistyakova, “The Structure of the General Solution and Integrating Factors in a Neighborhood of a Simple Singular Point”, Differ. Uravn., 37:5 (2001), 710–713; Differ. Equ., 37:5 (2001), 747–750
1997
8.
M. V. Dolov, S. A. Chistyakova, “Algebraic differential equations with an integrating factor of Darboux type”, Differ. Uravn., 33:5 (1997), 618–622; Differ. Equ., 33:5 (1997), 621–625
9.
M. V. Dolov, “An integrating factor in a neighborhood of a node”, Differ. Uravn., 33:2 (1997), 158–160; Differ. Equ., 33:2 (1997), 158–160
1995
10.
M. V. Dolov, E. V. Kruglov, “On the number of semi-algebraic partial integrals of a class of dynamical systems with a cylindrical phase space”, Differ. Uravn., 31:6 (1995), 949–954; Differ. Equ., 31:6 (1995), 888–893
11.
M. V. Dolov, E. V. Kruglov, “On limit cycles for a class of dynamical systems with a cylindrical phase space”, Differ. Uravn., 31:5 (1995), 752–758; Differ. Equ., 31:5 (1995), 695–701
12.
M. V. Dolov, E. V. Kruglov, “Singularities of Darboux type of integrals of semialgebraic differential equations”, Matem. Mod., 7:5 (1995), 31
1994
13.
M. V. Dolov, R. V. Kuz'min, “On limit cycles of systems with a given partial integral”, Differ. Uravn., 30:7 (1994), 1125–1132; Differ. Equ., 30:7 (1994), 1044–1050
M. V. Dolov, A. A. Alekseev, “On the absence of limit cycles in dynamical systems with an integrating factor of a special type”, Differ. Uravn., 30:6 (1994), 947–954; Differ. Equ., 30:6 (1994), 876–883
1993
15.
M. V. Dolov, R. V. Kuz'min, “On the limit cycles of a class of systems”, Differ. Uravn., 29:9 (1993), 1481–1485; Differ. Equ., 29:9 (1993), 1282–1285
M. V. Dolov, V. V. Kosarev, “Darboux integrals and the analytic structure of solutions of differential equations”, Differ. Uravn., 19:4 (1983), 697–700
1981
17.
M. V. Dolov, “The structure of a single-valued integrating factor near a cycle”, Differ. Uravn., 17:8 (1981), 1490–1492
18.
M. V. Dolov, “Quasicanonical integrals”, Differ. Uravn., 17:3 (1981), 411–422
1978
19.
M. V. Dolov, “Differential equations that have Darboux integrals”, Differ. Uravn., 14:10 (1978), 1765–1774
20.
M. V. Dolov, “Darboux integrals in the case of a focus”, Differ. Uravn., 14:7 (1978), 1173–1178
1977
21.
M. V. Dolov, “Limit cycles and Darboux integrals in the case of a node”, Differ. Uravn., 13:3 (1977), 406–415
1976
22.
M. V. Dolov, “The canonical integral in the neighborhood of a focus”, Differ. Uravn., 12:11 (1976), 1946–1953
M. V. Dolov, “Limit cycles and algebraic integrals in the case of a center”, Differ. Uravn., 11:11 (1975), 1935–1941
1974
24.
M. V. Dolov, “The order of an algebraic curve that is a solution of a differential equation”, Differ. Uravn., 10:3 (1974), 544–546
1973
25.
M. V. Dolov, “The domains of existence of canonical integrals”, Izv. Vyssh. Uchebn. Zaved. Mat., 1973, no. 10, 29–37
1972
26.
M. V. Dolov, “Limit cycles in the case of the center”, Differ. Uravn., 8:9 (1972), 1691–1692
1970
27.
M. V. Dolov, “Canonical integrals and singular cycles”, Differ. Uravn., 6:8 (1970), 1378–1383
1969
28.
M. V. Dolov, “Limit Cycles and analytic integrals”, Differ. Uravn., 5:11 (1969), 2026–2033
29.
M. V. Dolov, “The limit cycles of a certain class of differential equations”, Differ. Uravn., 5:7 (1969), 1330–1334
1968
30.
M. V. Dolov, “A certain method of investigation of limit cycles”, Differ. Uravn., 4:5 (1968), 812–820
1967
31.
M. V. Dolov, “Differential equations $yy'=Q_4(x,y)$, having an ellipse as limit cycle”, Differ. Uravn., 3:6 (1967), 898–904
1966
32.
M. V. Dolov, “Algebraic limit cycles of a differential equation”, Differ. Uravn., 2:11 (1966), 1469–1473
1991
33.
N. I. Avdonin, V. N. Belykh, L. N. Belyustina, M. V. Dolov, Yu. I. Neimark, S. F. Morozov, “Nikolaǐ Fedorovich Otrokov”, Differ. Uravn., 27:5 (1991), 912–913; Differ. Equ., 27:5 (1991), 637–638
1986
34.
D. A. Gudkov, M. V. Dolov, A. F. Leont'ev, V. V. Napalkov, Yu. I. Neimark, N. F. Otrokov, M. A. Soldatov, “Anatolii Alekseevich Mirolyubov (obituary)”, Uspekhi Mat. Nauk, 41:4(250) (1986), 141–142; Russian Math. Surveys, 41:4 (1986), 171–172
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