01.01.02 (Differential equations, dynamical systems, and optimal control)
Keywords:
linear and nonlinear functional analysis; nonlinear equations; branching theory of solutions of nonlinear equations; stability; symmetry in nonlinear analysis; applications to mathematical physics and natural sciences.
Subject:
In 1964–1969 — works on spectral theory of linear operators (perturbations of discrete spectrum, numerical range and spectrum, von Neumann spectral sets, generalizations of normal operators). From 1969 — works on branching theory of solutions of nonlinear equations, mainly under group symmetry conditions. They were proved theorems about symmetry of basic nonlinear equation inheritance by branching equations (BEq). On their base group analysis of differential equations methods were applied to construction of BEq general form on allowing by its group symmetry both in stationary and nonstationary branching. Role of generalized Jordan structure of the linearized operator-function of bifurcational parameter in branching theory, recently in application to the construction and investigation of resolving systems in branching theory problems, in particular to BEq in the root subspaces. Applications to mathematical physics: phase transitions in statistical crystal theory; capillary-gravity surface waves in fluid layers; von Karman plate in supersonic gas flow; pseudoperturbation methods in spectral theory.
Biography
Graduated from Faculty of Mathematics and Mechanics of V. I. Lenin Tashkent State University (TashSU) in 1961 (chair of Mathematical Analysis);
Ph.D. thesis was defended in 1965 (V. I. Romanovsky Mathematical Institute Acad. Sci. UzbekSSR);
1965–1966 — assistent of "Math. Analysis" Chair of Tashkent State University, UzbekSSR;
1966–1985 — senior sci. contributor of "Differential Equations" Dept.;
1985–1993 — Head of "Applied Mathematics" Dept. V. I. Romanovsky Mathematical Institute, Academy of Sciences of Uzbekistan;
1993 — by the present time Professor of "Higher Mathematics" Chair, Ulyanovsk state technical University. A list of my works contains more than 200 titles.
Member-correspondent "Natural Sci" (1995); academician of "Nonlinear Sciences", International Academy (1997); member of GAMM (Gesellschaft fur Angewandte Mathematik and Mechanik) (1996); American Mathematical Society (1997); Middle-Volga Math. Society (Mordova, Saransk, 1996); Reviewer of RZ Matematika, Moscow, Russia (from 1972), Mathematical Reviews (from 1972), Zentralblatt fur Mathematik (from 1975); order "Sign of Honour" N 457134 (1986); supervisor of 13 kandidate phys.-math. sci. (Ph.D.).
Main publications:
Loginov B. V. Determination of the bifurcation equation by its group symmetry — Andronov–Hopf bifurcation // Nonlinear Analysis, TMA, 28(1997), 12, 2033–2047.
L. R. Kim-Tyan, B. V. Loginov, Yu. B. Rousak, “Normal forms of the degenerate autonomous differential equations with the maximal Jordan chain and simple applications”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 10:3 (2017), 5–15
2016
2.
B. V. Loginov, Yu. B. Rousak, L. R. Kim-Tyan, “Differential equations with degenerate, depending on the unknown function operator at the derivative”, CMFD, 59 (2016), 119–147
P. A. Shamanaev, B. V. Loginov, “The branching of periodic solutions of inhomogeneous linear differential equations with a the perturbation in the form of small linear term with delay”, Zhurnal SVMO, 18:3 (2016), 61–69
A. A. kjashkin, B. V. Loginov, P. A. Shamanaev, “The branching of periodic solutions of inhomogeneous linear differential equations with degenerate or identity operator in the derivative and the disturbance in the form of small linear term”, Zhurnal SVMO, 18:1 (2016), 45–53
B. V. Loginov, Yu. B. Rousak, L. R. Kim-Tyan, “Normal forms of the degenerate differential autonomous and non autonomous equations with the maximal Jordan chain of length two and three”, Bulletin of Irkutsk State University. Series Mathematics, 12 (2015), 58–71
T. E. Badokina, B. V. Loginov, “Models of multiparameter bifurcations in boundary value problems for ODEs of the fourth order on divergence of elongated plate in supersonic gas flow”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8:2 (2015), 24–35
2014
7.
A. A. kjashkin, B. V. Loginov, P. A. Shamanaev, “Comments to the problem on branching of periodical solutions at Andronov-Hopf bifurcation in differential equations with degenerated operator before the derivative”, Zhurnal SVMO, 16:4 (2014), 33–40
A. V. Gerasimov, B. V. Loginov, N. N. Yuldashev, “Eigenvalue problem for the Laplace operator in $s$-dimensional unit ball $\Omega\subset\R^{s+1}$ with displacements in derivatives II”, Zhurnal SVMO, 16:4 (2014), 7–22
9.
A. N. Kuvshinova, B. V. Loginov, “Cayley-Hamilton theorem for two variants of matrix spectral problems on E.Shmidt and development of characteristic polynomial”, Zhurnal SVMO, 16:3 (2014), 7–20
10.
I. V. Konopleva, B. V. Loginov, “Group Symmetry for Dynamica Bifurcation Problems with Schmidt Spectrum in Linearization”, Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 14:4 (2014), 50–63
11.
A. V. Gerasimov, B. V. Loginov, N. N. Yuldashev, “Eigenvalue problem for the Laplace operator with displacement in derivatives”, Vestnik Samarskogo Gosudarstvennogo Universiteta. Estestvenno-Nauchnaya Seriya, 2014, no. 3(114), 41–45
2013
12.
B. V. Loginov, N. N. Yuldashev, A. V. Gerasimov, “The eigenvalue problem for the Laplace operator in $s$-dimensional unit ball $\Omega$ with displacement in derivatives”, Zhurnal SVMO, 15:4 (2013), 136–147
13.
A. A. kjashkin, B. V. Loginov, P. A. Shamanaev, “Comments to the problems of small perturbations of linear
equations and linear term of the spectral characteristics of a Fredholm
operator”, Zhurnal SVMO, 15:3 (2013), 100–107
B. V. Loginov, Yu. B. Rusak, L. R. Kim-Tyan, “Degenerated differential equations with variable degeneration operator”, Zhurnal SVMO, 15:3 (2013), 8–20
2012
15.
T. E. Badokina, B. V. Loginov, Yu. B. Rusak, “Construction of the asymptotics of solutions of nonlinear boundary value problems for fourth order differential equation with two bifurcation parameters”, Bulletin of Irkutsk State University. Series Mathematics, 5:1 (2012), 2–12
16.
B. V. Loginov, I. V. Konopleva, L. V. Mironova, “Dynamic bifurcation problems with E.Schmidt spectrum in the linearization under group symmetry conditions”, Zhurnal SVMO, 14:1 (2012), 25–35
17.
B. V. Loginov, L. R. Kim-Tyan, “Group symmetry of potential type bifurcation equations in dynamic branching”, Tr. Inst. Mat., 20:2 (2012), 75–83
18.
V. A. Trenogin, B. V. Loginov, L. R. Kim-Tyan, “Stability of solutions to the Cauchy problem with respect to linear approximation, and branching equation in the root subspace”, Trudy Mat. Inst. Steklova, 278 (2012), 260–268; Proc. Steklov Inst. Math., 278 (2012), 251–259
2011
19.
B. V. Loginov, I. V. Konopleva, Y. B. Rousak, “Implicit operator theorems under group symmetry conditions”, Bulletin of Irkutsk State University. Series Mathematics, 4:1 (2011), 31–43
20.
B. V. Loginov, V. E. Pospeev, “Regularization of the problem about perturbations of linear equation by small linear terms.}”, Zhurnal SVMO, 13:3 (2011), 146–147
21.
B. V. Loginov, I. V. Konopleva, L. V. Mironova, “Stationary bifurcation problems with E. Schmidt spectrum in linearization under group symmetry conditions.”, Zhurnal SVMO, 13:3 (2011), 15–30
2010
22.
V. A. Trenogin, B. V. Loginov, L. R. Kim-Tyan, “Solutions stability on linear approximation to Cauchy problem , branching equation in the root subspace, symmetry”, Zhurnal SVMO, 12:3 (2010), 8–17
2009
23.
B. V. Loginov, O. V. Makeev, “Bifurcation theory methods in problem about crystallization of liquid phase state in statistical crystal theory”, Trudy SVMO, 11:2 (2009), 194–196
24.
B. V. Loginov, O. V. Makeeva, “Pseudoperturbation operator in generalized eigenvalue
problem under group symmetry”, Trudy SVMO, 11:2 (2009), 43–48
25.
B. V. Loginov, O. V. Makeev, “Bifurcation theory methods in problem about crystallization of liquid phase state in statistical crystal theory”, Trudy SVMO, 11:1 (2009), 10–28
2008
26.
B. V. Loginov, O. V. Makeeva, “On application of pseudoperturbation method to the solving of algebraic equations”, Trudy SVMO, 10:1 (2008), 315–320
27.
I. V. Konopleva, B. V. Loginov, O. V. Makeev, Yu. B. Rusak, “Symmetry of $SO(2)$ and $SH(2)$ in Poincaré-Andronov-Hopf bifurcation problems with potential branching equations”, Trudy SVMO, 10:1 (2008), 106–112
B. V. Loginov, I. V. Konopleva, Yu. B. Rusak, “Symmetry and potentiality in a general problem in bifurcation theory”, Izv. Vyssh. Uchebn. Zaved. Mat., 2006, no. 4, 30–40; Russian Math. (Iz. VUZ), 50:4 (2006), 28–38
B. V. Loginov, O. V. Makeev, “Solutions with subgroup symmetry for singular equations in bifurcation theory”, Lobachevskii J. Math., 20 (2005), 91–101
30.
B. Karasözen, I. V. Konopleva, B. V. Loginov, “Differential-algebraic equations in the theory of invariant manifolds for singular equations”, Lobachevskii J. Math., 20 (2005), 77–89
2004
31.
B. Karasozen, B. V. Loginov, “Invariant Reduction of Partially Potential Branching Equations and Iterative Methods in the Problem on a Bifurcation Point with a Symmetry”, Differ. Uravn., 40:3 (2004), 380–388; Differ. Equ., 40:3 (2004), 410–419
2001
32.
B. V. Loginov, “Bifurcation and symmetry in problems of capillary-gravity waves”, Sibirsk. Mat. Zh., 42:4 (2001), 868–887; Siberian Math. J., 42:4 (2001), 731–747
B. V. Loginov, V. A. Trenogin, “A problem with a free boundary for the Laplace equation, capillary-gravitational waves in space, bifurcation and symmetry”, Differ. Uravn., 34:8 (1998), 1100–1106; Differ. Equ., 34:8 (1998), 1104–1110
1997
34.
B. V. Loginov, N. A. Sidorov, Yu. B. Rusak, “Теоремы существования бифуркации в присутствии одной обобщенной жордановой цепочки нечетной длины”, Matem. Mod., 9:10 (1997), 30–31
1995
35.
P. A. Vel'misov, B. V. Loginov, “Group analysis in twopoint boundary value problems describing the rod bending”, Matem. Mod., 7:5 (1995), 37–38
B. V. Loginov, “Group analysis in problems with symmetry violatation”, Matem. Mod., 7:5 (1995), 36
1993
37.
B. V. Loginov, “On the determination of a bifurcation equation by its symmetry
group”, Dokl. Akad. Nauk, 331:6 (1993), 677–680; Dokl. Math., 48:1 (1994), 200–205
38.
B. V. Loginov, L. R. Kim-Tyan, Yu. B. Rusak, “Modification of the Lyapunov–Schmidt method and the stability of
solutions of differential equations with a degenerate operator of finite
index multiplying the derivative”, Dokl. Akad. Nauk, 330:6 (1993), 687–690; Dokl. Math., 47:3 (1993), 599–603
B. V. Loginov, N. A. Sidorov, “Group symmetry of the Lyapunov–Schmidt branching equation and iterative methods in the problem of a bifurcation point”, Mat. Sb., 182:5 (1991), 681–691; Math. USSR-Sb., 73:1 (1992), 67–77
V. A. Trenogin, N. A. Sidorov, B. V. Loginov, “The bifurcation equation: potentiality, bifurcation, symmetry”, Dokl. Akad. Nauk SSSR, 309:2 (1989), 286–289; Dokl. Math., 40:3 (1990), 517–520
A. O. Kuznetsov, B. V. Loginov, “Construction of the asymptotic behavior of small solutions of a
problem of rolling waves”, Dokl. Akad. Nauk SSSR, 304:6 (1989), 1320–1323; Dokl. Math., 34:2 (1989), 107–108
1988
42.
B. V. Loginov, A. M. Nagornyi, “On the spectrum of a Bitsadze–Samarskiǐ problem”, Differ. Uravn., 24:11 (1988), 2012–2016
B. V. Loginov, L. R. Kim-Tyan, “On the stability of bifurcating solutions”, Differ. Uravn., 24:4 (1988), 695–698
1981
44.
B. V. Loginov, “On the use of vector invariants for determining the general form of a bifurcation equation under group invariance conditions”, Dokl. Akad. Nauk SSSR, 259:5 (1981), 1045–1050
45.
B. V. Loginov, D. G. Rakhimov, “On the connection between the method of false perturbations, analytical perturbations, and differential equations with degeneration”, Differ. Uravn., 17:11 (1981), 2090–2093
1979
46.
B. V. Loginov, “Construction of periodic solutions of the three-dimensional problem of waves above a level bottom”, Dokl. Akad. Nauk SSSR, 247:2 (1979), 324–328
B. V. Loginov, N. A. Sidorov, “Calculation of eigenvalues and eigenvectors of bounded operators by the false-perturbation method”, Mat. Zametki, 19:1 (1976), 105–108; Math. Notes, 19:1 (1976), 62–64
B. V. Loginov, V. A. Trenogin, “The use of group properties to determine milti-parameter families of solutions of nonlinear equations”, Mat. Sb. (N.S.), 85(127):3(7) (1971), 440–454; Math. USSR-Sb., 14:3 (1971), 438–452
E. N. Artem'eva, I. V. Boykov, M. A. Borisov, D. I. Boyarkin, P. A. Vel'misov, V. K. Gorbunov, T. A. Gorshunova, V. Z. Grines, Yu. N. Deryugin, E. V. Desyaev, D. K. Egorova, R. V. Zhalnin, O. E. Kaledin, V. N. Krizsky, E. B. Kuznetsov, B. V. Loginov, T. Ph. Mamedova, S. I. Martynov, N. D. Morozkin, S. M. Muryumin, I. P. Nikitin, O. V. Pochinka, D. V. Pashutkin, A. Yu. Pavlov, E. E. Peskova, I. P. Ryazantseva, V. I. Safonkin, G. A. Smolkin, S. I. Spivak, L. A. Sukharev, A. O. Syromyasov, M. T. Terekhin, V. F. Tishkin, S. A. Firsova, E. A. Chernoivanova, I. I. Chuchaev, P. A. Shamanaev, O. S. Yazovtseva, Z. Ya. Yakupov, “On the 80th anniversary of professor E.V. Voskresensky's birthday”, Zhurnal SVMO, 19:4 (2017), 95–99
2015
56.
B. V. Loginov, V. V. Pukhnachev, G. A. Sviridyuk, A. V. Keller, S. A. Zagrebina, O. A. Romanova, D. N. Sidorov, A. Dreglea, R. Yu. Leontiev, “Nikolay Aleksandrovich Sidorov (to the 75th Anniversary)”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8:2 (2015), 143–148
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