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Bruno, Alexander Dmitrievich

Statistics Math-Net.Ru
Total publications: 168
Scientific articles: 166
Presentations: 9

Number of views:
This page:3873
Abstract pages:20891
Full texts:8485
References:1264
Professor
Doctor of physico-mathematical sciences (1969)
Speciality: 01.01.02 (Differential equations, dynamical systems, and optimal control)
Phone: +7 (499) 250 78 84
Fax: +7 (499) 972 07 37
E-mail:
Website: http://brunoa.name
Keywords: number theory, complex analysis, differential equations, algebraic equations, analytical mechanics, stability of motion, celestial mechanics, hydrodynamics.
UDC: 511.36, 513, 514.172, 517, 517.52, 517.9, 517.91, 517.913, 517.925, 517.93, 519, 521.1, 521.41, 529.7, 531.31, 531.38
MSC: 11, 30, 34, 35, 40, 41, 52, 70, 76, 85, 34C20

Subject:

(I) The new calculus "Power Geometry" was created for nonlinear equations and systems of equations of any type (algebraic, ordinary differential and partial differential). It gives the general algorithms for:

  1. the isolation of their first approximations by means of the Newton polyhedrons and their analogous;
  2. simplification of the first approximations by means of the power and logarithmic transformations;
  3. finding self-similar solutions to quasihomogeneous systems (to which belong all first approximations);
  4. finding asymptotic forms of their solutions and
  5. the computation of the asymptotic expansions of their solutions.
It allows to study any singularities (including the singular perturbations) in the mentioned equations and systems. For the autonomous ODE system in a neighborhood of the stationary solution (and also near the periodic solution or the invariant torus), there were proven: (a) the existence of the formal invertible change of coordinates transforming the system to the resonant normal form, (b) which can be reduced to a system of lower order (equal to the multiplicity of the resonance) by means of the power transformation. (c) There were found the conditions $\omega$ on eigenvalues and A on the normal form that are necessary and sufficient for the analyticity of the normalizing transformation. (d) If the condition A is violated, there are sets ${\cal A}$ (if small divisors are absent) and ${\cal B}$ (if they are present) on which the normalizing transformation is analytic. The sets are computed via the normal form, they contain all invariant tori found by means of the KAM theory and allow to simplify the study of bifurcations of the periodic solutions and of the invariant tori. (e) The further simplifications of the resonant normal form were considered. In particular, for systems with the one-fold resonance, there was given the polynomial normal form, all coefficients of which are the formal invariants. (f) Similar results were found for the resonant Hamiltonian normal form of the Hamiltonian system. In particular, the theory of the Hamiltonian normal form for the linear Hamiltonian systems with constant or periodic coefficients was finished. (g) It was shown that the normal form is very convenient for the study of stability. In particular, it was shown that the proof of the stability of the stationary point in the Hamiltonian system with two degrees of freedom, given by V. I. Arnol'd in 1963, contains the wrong statement. (h) The Power Geometry and the normal forms were applied in problems of Mechanics (in particular, all power expansions of motions of the rigid body were calculated for the generic case with $y_0=z_0=0$ and a lot of the new integrable cases was found), of Celestial Mechanics (the families of periodic solutions in the planar restricted three-body problem and in the Beletsky equation, describing the planar motions of a satellite around its masscenter, were studied) and of Hydrodynamics (on a needle the boundary layer was given and the surface waves on the water were studied). (i) For the ordinary differential equation of any order I proposed an algorithm of computing asymptotic expansions of its solutions near a singularity. I have find new types of such expansions: power-logarithmic, complicated, exotic and power-periodic. I obtained conditions of their convergency. All that was made for solutions, for which order of derivative differs from the order of the solution by $-1$ as well as for solution, for which that difference is arbitrary. Finally, by these methods we calculated all asymptotic expansions of solutions of all six Painlev\'e equations. (k) For algebraic equations of $n$ variables, I proposed new methods of computation of approximate values of roots for $n=1$ and of approximate uniformizations its solutions, i. e. algebraic curves and surfaces, for $n>1$. These methods are based on the Hadamar open polygon and polyhedron. I also developed an algorithm of computation of asymptotic expansions of its solutions near singularity (including infinity).

(II) In Number Theory it was shown that the continued fractions of the cubic irrationalities have the same structure as the continued fractions for the almost all numbers. There were attempts to find the multidimensional generalizations of the continued fractions, based on the Klein polyhedra. In particular, the quality of the matrix algorithms of Euler, Jacobi, Poincare, Brun, Parusnikov and Bruno was compared. It appears that the Poincare algorithm is the worst. For the multidimensional generalization of the continued fraction, I proposed a modular polyhedron instead of the Klein polyhedron (that name was given by me instead of the name «Arnold polyhedron»). Preimages of vertices of the modular polyhedron give the best Diophantine approximations. The modular polyhedron can be computed by means of a standard program for computing convex hulls. It gives a solution of the problem, which majority of main mathematicians of XIX century tried to solve. In the algebraic case, using the modular polyhedron it is possible to find all fundamental units of some rings. Using them it is possible to compute all periods of the generalized continued fraction and to compute all solutions to Diophantine equations of some class. This approach gives also simultaneous Diophantine approximations.

Biography

Graduated from Faculty of Mathematics and Mechanics of the M. V. Lomonosov Moscow State University (MSU) in 1962 (department of differential equations). Ph.D. thesis was defended in 1966. D.Sci. thesis was defended in 1969. Professor since 2007. The list of my publications contains more than 380 titles.

In 1956 and 1957 I received the 3rd and the 1st prizes at the Moscow mathematical olympiades for pupiles. In 1960 and 1961 I received the 2nd prizes at the competition of the students works in the Faculty of Mechanics and Mathematics. Since 1965 I am a member of the Moscow Mathematical Soc., since 1993 of the American Math. Soc. and since 1996 I am a member of the Academy of Nonlinear Sciences. My biographical data were published in Who's Who in the World, Marquis, 12th ed., 1995, p. 178; 16th ed., 1999, p. 222. Outstanding People of the 20th Century, Intern. Biogr. Centre, Cambridge, 1st ed., 1999, p. XXXIV–XXXV, 82. Five Hundred Leaders of Influence, ABI, 8th ed., 1999, p. 44; 2000 Outstanding Scholars of the 20th Century, IBC, 2000, p. 46–47; 2000 Outstanding Intellectuals of the 20th Century, IBC, 2000, p. 44; The First Five Hundred in the New Millennium, IBC, 2000, p. 52–53.

   
Main publications:
  • Brjuno A. D. Analytical form of differential equation // Transaction of Moscow Mathematical Society, 1971, 25, 131–288; 1972, 26, 199–239.
  • Bruno A. D. Local Methods in Nonlinear Differential Equations. Berlin: Springer-Verlag, 1989.
  • Bruno A. D. The Restricted 3-Body Problem. Berlin: Walter de Gruyter, 1994.
  • Bruno A. D. Power Geometry in Algebraic and Differential Equations. Amsterdam: Elsevier Science, 2000.
  • Bruno A. D., Shadrina T. V. Axisymmetric boundary layer on a needle // Transactions of Moscow Math. Soc. 68 (2007) 201--259

http://www.mathnet.ru/eng/person9101
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List of publications on ZentralBlatt
https://mathscinet.ams.org/mathscinet/MRAuthorID/198723
http://elibrary.ru/author_items.asp?spin=8698-1667
http://orcid.org/0000-0002-7465-1258
http://www.scopus.com/authid/detail.url?authorId=7102246723
https://www.researchgate.net/profile/A_Bruno

Publications in Math-Net.Ru
2020
1. A. D. Bruno, “On types of stability in Hamiltonian systems”, Keldysh Institute preprints, 2020, 021, 24 pp.  mathnet
2019
2. A. D. Bruno, “Orbital stability of the periodic solution of a Hamiltonian system”, Keldysh Institute preprints, 2019, 120, 16 pp.  mathnet
3. A. D. Bruno, “The newest methods of celestial mechanics”, Keldysh Institute preprints, 2019, 079, 18 pp.  mathnet  elib
4. A. D. Bruno, “Normalization of the periodic Hamiltonian system”, Keldysh Institute preprints, 2019, 064, 18 pp.  mathnet  elib
5. A. D. Bruno, “Normal form of a Hamiltonian system with a periodic perturbation”, Keldysh Institute preprints, 2019, 057, 27 pp.  mathnet  elib
6. A. D. Bruno, “On the Parametrization of an Algebraic Curve”, Mat. Zametki, 106:6 (2019),  837–847  mathnet; Math. Notes, 106:6 (2019), 885–893  isi  scopus
2018
7. A. D. Bruno, “Normal form of the periodic Hamiltonian system with $n$ degrees of freedom”, Keldysh Institute preprints, 2018, 223, 15 pp.  mathnet  elib
8. A. D. Bruno, “Complicated and exotic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2018, 118, 44 pp.  mathnet
9. A. D. Bruno, “Expansion of ODE solutions into transseries”, Keldysh Institute preprints, 2018, 117, 19 pp.  mathnet  elib
10. A. D. Bruno, “Asymptotic solution of some nonlinear problems”, Keldysh Institute preprints, 2018, 035, 24 pp.  mathnet
11. A. D. Bruno, “Power geometry and expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2018, 021, 15 pp.  mathnet
12. A. D. Bruno, “On complicated expansions of solutions to ODES”, Zh. Vychisl. Mat. Mat. Fiz., 58:3 (2018),  346–364  mathnet  elib; Comput. Math. Math. Phys., 58:3 (2018), 328–347  isi  scopus
2017
13. A. D. Bruno, “Complicated and exotic expansions of solutions to the fifth Painlevé equation”, Keldysh Institute preprints, 2017, 107, 18 pp.  mathnet
14. A. D. Bruno, “Calculation of exotic expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2017, 096, 22 pp.  mathnet
15. A. D. Bruno, “Calculation of complicated asymptotic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2017, 055, 27 pp.  mathnet
16. A. D. Bruno, “Calculation of fundamental units of number rings by means of the generalized continued fraction”, Keldysh Institute preprints, 2017, 046, 28 pp.  mathnet
17. A. D. Bruno, “Solving the polynomial equations by algorithms of power geometry”, Keldysh Institute preprints, 2017, 034, 28 pp.  mathnet
2016
18. A. D. Bruno, “From Diophantine approximations to Diophantine equations”, Chebyshevskii Sb., 17:3 (2016),  38–52  mathnet  elib
19. A. D. Bruno, “On solution of an algebraic equation”, Keldysh Institute preprints, 2016, 070, 20 pp.  mathnet
20. A. D. Bruno, “From Diophantine approximations to Diophantine equations”, Keldysh Institute preprints, 2016, 001, 20 pp.  mathnet
2015
21. A. D. Bruno, “Universal generalization of the continued fraction algorithm”, Chebyshevskii Sb., 16:2 (2015),  35–65  mathnet  elib
2014
22. A. B. Batkhin, A. D. Bruno, “On investigation of the certain real algebraic surface”, Keldysh Institute preprints, 2014, 083, 28 pp.  mathnet
2013
23. A. D. Bruno, I. V. Goryuchkina, “Convergence of power expansions of solutions to an ODE”, Keldysh Institute preprints, 2013, 094, 16 pp.  mathnet
24. A. D. Bruno, “Power Geometry and elliptic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2013, 088, 28 pp.  mathnet
25. A. D. Bruno, “Asymptotic solving nonlinear equations and idempotent mathematics”, Keldysh Institute preprints, 2013, 056, 31 pp.  mathnet
2012
26. A. D. Bruno, A. V. Parusnikova, “Expansions and asymptotic forms of solutions to the fifth Painlevé equation near infinity”, Keldysh Institute preprints, 2012, 061, 32 pp.  mathnet
27. A. D. Bruno, “Power-elliptic expansions of solutions to an ordinary differential equation”, Zh. Vychisl. Mat. Mat. Fiz., 52:12 (2012),  2206–2218  mathnet  mathscinet  zmath  elib; Comput. Math. Math. Phys., 52:12 (2012), 1650–1661  isi  elib  scopus
2011
28. A. D. Bruno, A. V. Parusnikova, “Periodic and Elliptic Asymptotic Forms of Solutions to the Fifth Painlev'e Equation”, Keldysh Institute preprints, 2011, 061, 18 pp.  mathnet
29. A. D. Bruno, “Power-elliptic expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 060, 19 pp.  mathnet
30. A. D. Bruno, “Power-exponential expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 054, 11 pp.  mathnet
31. A. B. Batkhin, A. D. Bruno, V. P. Varin, “Sets of stability of multiparameter Hamiltonian systems”, Keldysh Institute preprints, 2011, 042, 32 pp.  mathnet
32. A. D. Bruno, “Exponential expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 036, 16 pp.  mathnet
33. A. D. Bruno, A. V. Parusnikova, “Expansions of solutions to the fifth Painlevé equation near its nonsingular point”, Keldysh Institute preprints, 2011, 018, 16 pp.  mathnet
34. A. D. Bruno, “On complicated expansions of solutions to ODE”, Keldysh Institute preprints, 2011, 015, 26 pp.  mathnet
35. A. D. Bruno, A. B. Batkhin, “Resolution of an algebraic singularity by Power Geometry algorithms”, Keldysh Institute preprints, 2011, 010, 30 pp.  mathnet
2010
36. A. D. Bruno, “Structure of the best diophantine approximations and multidimensional generalizations of the continued fraction”, Chebyshevskii Sb., 11:1 (2010),  68–73  mathnet  mathscinet
37. A. D. Bruno, A. V. Parusnikova, “Local Expansions of Solutions of the Fifth Painlevé Equation”, Keldysh Institute preprints, 2010, 072, 27 pp.  mathnet
38. A. B. Aranson, A. D. Bruno, “Power expansions of the shifted solutions to the N. Kowalewski system”, Keldysh Institute preprints, 2010, 048, 32 pp.  mathnet
39. A. D. Bruno, A. V. Parusnikova, “Asymptotic Expansions of Solutions to the Fifth Painlevé equation”, Keldysh Institute preprints, 2010, 039, 23 pp.  mathnet
40. A. D. Bruno, A. B. Batkhin, V. P. Varin, “Computation of the Sets of Stability in Multiparameter Problems”, Keldysh Institute preprints, 2010, 023, 22 pp.  mathnet
41. A. D. Bruno, “The structure of the multidimensional Diophantine approximations”, Keldysh Institute preprints, 2010, 011, 8 pp.  mathnet
42. A. D. Bruno, A. V. Gridnev, “Nonpower expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2010, 010, 21 pp.  mathnet
43. A. D. Bruno, A. B. Batkhin, V. P. Varin, “The stability set of a gyroscopic problems”, Keldysh Institute preprints, 2010, 004, 30 pp.  mathnet
44. A. D. Bruno, “Sets of stability of multiparameter problems”, Keldysh Institute preprints, 2010, 003, 14 pp.  mathnet
2009
45. A. D. Bruno, V. F. Edneral, “Algorithmic analysis of local integrability”, Dokl. Akad. Nauk, 424:3 (2009),  299–303  mathnet  mathscinet; Dokl. Math., 79:1 (2009), 48–52  isi  scopus
46. A. D. Bruno, I. V. Goryuchkina, “Non-formal solutions to ODE”, Keldysh Institute preprints, 2009, 061, 14 pp.  mathnet
47. A. D. Bruno, I. V. Goryuchkina, “Elliptic asymptotic forms of solutions to the Painlev'e equations”, Keldysh Institute preprints, 2009, 006, 26 pp.  mathnet
48. A. D. Bruno, V. F. Edneral, “On integrability of a planar system of ODEs near a degenerate stationary point”, Zap. Nauchn. Sem. POMI, 373 (2009),  34–47  mathnet; J. Math. Sci. (N. Y.), 168:3 (2010), 326–333  scopus
2008
49. A. D. Bruno, I. V. Goryuchkina, “All expansions of solutions to the sixth Painlevé equation near its nonsingular point”, Keldysh Institute preprints, 2008, 075, 30 pp.  mathnet
50. A. D. Bruno, V. I. Parusnikov, “Two-sided generalization of the continued fraction”, Keldysh Institute preprints, 2008, 058, 25 pp.  mathnet
51. A. D. Bruno, V. P. Varin, “The families c and i of periodic solutions of the restricted problem for $\mu=5\cdot10^{-5}$”, Keldysh Institute preprints, 2008, 022, 26 pp.  mathnet
2007
52. A. D. Bruno, V. F. Edneral, “On integrability of the Euler–Poisson equations”, Fundam. Prikl. Mat., 13:1 (2007),  45–59  mathnet  mathscinet  zmath; J. Math. Sci., 152:4 (2008), 479–489  scopus
53. A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the equation P6 are obtained from base ones”, Keldysh Institute preprints, 2007, 077, 28 pp.  mathnet
54. A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the equation P6 in the case $a\cdot b=0$”, Keldysh Institute preprints, 2007, 070, 30 pp.  mathnet
55. A. D. Bruno, I. V. Goryuchkina, “All base asymptotic expansions of solutions to the equation P6 in the case $a\cdot b\ne0$”, Keldysh Institute preprints, 2007, 062, 33 pp.  mathnet
56. A. D. Bruno, I. V. Goryuchkina, “Methods are used for researching of asymptotic expansions of solutions to the equation P6”, Keldysh Institute preprints, 2007, 061, 30 pp.  mathnet
57. A. D. Bruno, I. V. Goryuchkina, “Review of all asymptotic expansions of solutions to the equation P6”, Keldysh Institute preprints, 2007, 060, 16 pp.  mathnet
58. A. D. Bruno, V. F. Edneral, “Analysis of the local integrability by methods of normal form and power geometry”, Keldysh Institute preprints, 2007, 053, 16 pp.  mathnet
59. A. D. Bruno, V. P. Varin, “Family $c$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 051, 14 pp.  mathnet
60. A. D. Bruno, V. P. Varin, “Complicated families of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 035, 18 pp.  mathnet
61. A. D. Bruno, V. P. Varin, “Periodic solutions of the restricted three-body problem for small $\mu$”, Keldysh Institute preprints, 2007, 034, 22 pp.  mathnet
62. A. D. Bruno, “Power Geometry as a new mathematics”, Keldysh Institute preprints, 2007, 028, 24 pp.  mathnet
63. A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the sixth Painlevé equation”, Keldysh Institute preprints, 2007, 019, 19 pp.  mathnet
64. A. D. Bruno, V. F. Edneral, “Computation of normal forms of the Euler–Poisson equations”, Keldysh Institute preprints, 2007, 001, 17 pp.  mathnet
2006
65. A. D. Bruno, “Complicated expansions of solutions to an ODE system”, Keldysh Institute preprints, 2006, 081, 13 pp.  mathnet
66. A. D. Bruno, “Exotic expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2006, 066, 31 pp.  mathnet
67. A. D. Bruno, V. Yu. Petrovich, “Desingularizations of the restricted three-body problem”, Keldysh Institute preprints, 2006, 053, 12 pp.  mathnet
68. A. D. Bruno, V. P. Varin, “The generating family $i$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2006, 036  mathnet
69. A. D. Bruno, “On movable singular points of solutions to the ordinary differential equations”, Keldysh Institute preprints, 2006, 026, 13 pp.  mathnet
70. A. D. Bruno, I. V. Goryuchkina, “Expansions of solutions to the sixth Painlevé equation near singular points $x=0$ и $x=\infty$”, Keldysh Institute preprints, 2006, 013, 32 pp.  mathnet
71. A. D. Bruno, I. V. Goryuchkina, “Expansions of solutions to the sixth painleve equation in cases $a=0$ and $b=0$”, Keldysh Institute preprints, 2006, 002, 30 pp.  mathnet
2005
72. A. D. Bruno, “Theory of normal forms of the Euler-Poisson equations”, Keldysh Institute preprints, 2005, 100  mathnet
73. A. D. Bruno, “Properties of the modulus polyhedron”, Keldysh Institute preprints, 2005, 072  mathnet
74. A. D. Bruno, I. N. Gashenenko, “Simple finite solutions to the N. Kowalewski equations”, Keldysh Institute preprints, 2005, 068  mathnet
75. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for small $\mu$”, Keldysh Institute preprints, 2005, 067  mathnet
76. A. D. Bruno, “Normal Forms and Integrability of the Euler–Poisson Equations”, Keldysh Institute preprints, 2005, 066  mathnet
77. A. D. Bruno, I. N. Gashenenko, “Last expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2005, 065  mathnet
78. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for big $\mu$”, Keldysh Institute preprints, 2005, 064  mathnet
79. A. D. Bruno, T. V. Shadrina, “On the viscous incompressible fluid flow around a plate”, Keldysh Institute preprints, 2005, 054  mathnet
80. A. D. Bruno, V. I. Parusnikov, “New generalizations of the continued fraction”, Keldysh Institute preprints, 2005, 052, 17 pp.  mathnet
81. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for small $\mu$”, Keldysh Institute preprints, 2005, 048  mathnet
82. A. D. Bruno, V. I. Parusnikov, “Further generalization of the continued fraction”, Keldysh Institute preprints, 2005, 040  mathnet
83. A. D. Bruno, “Complicated expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2005, 036  mathnet
84. A. D. Bruno, N. A. Kudryashov, “Power expansions of solutions to an analogy to the first Painlevé equation”, Keldysh Institute preprints, 2005, 017  mathnet
85. A. D. Bruno, V. P. Varin, “On families of periodic solutions to the restricted three-body problem”, Keldysh Institute preprints, 2005, 010  mathnet
86. A. D. Bruno, I. V. Goryuchkina, “Power expansions of solutions to the sixth Painlevé equation near a regular point”, Keldysh Institute preprints, 2005, 004  mathnet
2004
87. A. D. Bruno, V. Yu. Petrovich, “Singularities of solutions to the first Painlevé equation”, Keldysh Institute preprints, 2004, 075, 13 pp.  mathnet
88. A. D. Bruno, “Algorithm of the generalizationued continued fraction”, Keldysh Institute preprints, 2004, 045, 27 pp.  mathnet
89. A. D. Bruno, T. V. Shadrina, “The compressible heat conductive boundary layer on a needle”, Keldysh Institute preprints, 2004, 037, 38 pp.  mathnet
90. A. D. Bruno, T. V. Shadrina, “About incompressible boundary layer on a needle”, Keldysh Institute preprints, 2004, 036, 26 pp.  mathnet
91. A. D. Bruno, T. V. Shadrina, “Methods of a study of the boundary layer on a needle”, Keldysh Institute preprints, 2004, 035, 23 pp.  mathnet
92. A. D. Bruno, “On generalisations of the continued fraction”, Keldysh Institute preprints, 2004, 010, 19 pp.  mathnet
93. A. D. Bruno, “Asymptotic behaviour and expansions of solutions of an ordinary differential equation”, Uspekhi Mat. Nauk, 59:3(357) (2004),  31–80  mathnet  mathscinet  zmath  elib; Russian Math. Surveys, 59:3 (2004), 429–480  isi  scopus
2003
94. A. D. Bruno, V. I. Parusnikov, “Polyhedra of absolute values for triples of linear forms”, Keldysh Institute preprints, 2003, 093, 21 pp.  mathnet
95. A. D. Bruno, “The сorrect generalization of the continued fraction”, Keldysh Institute preprints, 2003, 086, 19 pp.  mathnet
96. A. D. Bruno, T. V. Shadrina, “Axisymmetric boundary layer on a needle”, Keldysh Institute preprints, 2003, 064, 28 pp.  mathnet
97. A. D. Bruno, “Expansions of solutions to an ODE system”, Keldysh Institute preprints, 2003, 059, 24 pp.  mathnet
98. A. D. Bruno, “Asymptotically сlose slutions to an ODE system”, Keldysh Institute preprints, 2003, 058, 28 pp.  mathnet
99. A. D. Bruno, A. V. Gridnev, “Power and exponential expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2003, 051, 19 pp.  mathnet
100. A. D. Bruno, E. S. Karulina, “Power expansions of solutions to the fifth Painlevé equation”, Keldysh Institute preprints, 2003, 050, 26 pp.  mathnet
101. A. D. Bruno, I. V. Chukhareva, “Power expansions of solutions to the sixth Painlevé equation”, Keldysh Institute preprints, 2003, 049, 24 pp.  mathnet
102. A. D. Bruno, Yu. V. Zavgorodnyaya, “Power series and nonpower asymptotics of solutions to the second Painlevé equation”, Keldysh Institute preprints, 2003, 048, 36 pp.  mathnet
103. A. D. Bruno, “Asymptotically close solutions to an ordinary differential equation”, Keldysh Institute preprints, 2003, 031, 17 pp.  mathnet
104. A. D. Bruno, “The asymptotical solution of nonlinear equations by means of Power Geometry”, Keldysh Institute preprints, 2003, 028, 31 pp.  mathnet
105. A. D. Bruno, “Asymptotics and expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2003, 009, 25 pp.  mathnet
2002
106. A. D. Bruno, V. P. Varin, “Classes of families of generalized periodic solutions to the Beletsky equation”, Keldysh Institute preprints, 2002, 064  mathnet
107. A. D. Bruno, “Analysis of the Euler-Poisson equations by methods of Power Geometry”, Keldysh Institute preprints, 2002, 041  mathnet
108. A. D. Bruno, “Asymptotics of solutions to the ordinary differential equations”, Keldysh Institute preprints, 2002, 040  mathnet
109. A. D. Bruno, V. V. Lunev, “Properties of expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2002, 023  mathnet
2001
110. A. D. Bruno, V. V. Lunev, “Asymptotical expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2001, 090  mathnet
111. A. D. Bruno, V. V. Lunev, “Local expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2001, 073  mathnet
112. A. D. Bruno, V. V. Lunev, “The modified system of equations described motions of a rigid body”, Keldysh Institute preprints, 2001, 049  mathnet
2000
113. A. D. Bruno, “Power Expansions of Solutions of a System of Algebraic and Differential Equations”, Keldysh Institute preprints, 2000, 068  mathnet
114. A. D. Bruno, “Power Expansions of Solutions of One Algebraic or Differential Equation”, Keldysh Institute preprints, 2000, 063  mathnet
115. A. D. Bruno, “Families of Periodic Solutions of the Beletskii Equation”, Keldysh Institute preprints, 2000, 051  mathnet
116. A. D. Bruno, V. Yu. Petrovich, “Normal Forms of the ODE System”, Keldysh Institute preprints, 2000, 018  mathnet
117. A. D. Bruno, “Self-similar solutions and power geometry”, Uspekhi Mat. Nauk, 55:1(331) (2000),  3–44  mathnet  mathscinet  zmath  elib; Russian Math. Surveys, 55:1 (2000), 1–42  isi  elib  scopus
1999
118. A. D. Bruno, “A new generalization of the continued fraction”, Keldysh Institute preprints, 1999, 082  mathnet
119. A. D. Bruno, “On Complexity of Problems of Power Geometry”, Keldysh Institute preprints, 1999, 059  mathnet
120. A. D. Bruno, “Finding Self-Similar Solutions by Means of Power Geometry”, Keldysh Institute preprints, 1999, 057  mathnet
1997
121. A. D. Bruno, V. J. Petrovich, “Computation of periodic oscillations of a satellite”, Matem. Mod., 9:6 (1997),  82–94  mathnet  mathscinet  zmath
122. A. D. Bruno, V. I. Parusnikov, “Comparison of various generalizations of continued fractions”, Mat. Zametki, 61:3 (1997),  339–348  mathnet  mathscinet  zmath; Math. Notes, 61:3 (1997), 278–286  isi
1996
123. A. D. Bruno, “Zero-Multiple and Retrograde Periodic Solutions of the Restricted Three-Body Problem”, Keldysh Institute preprints, 1996, 093  mathnet
124. A. D. Bruno, “A general approach to asymptotic nonlinear analysis”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1996, 6,  24–27  mathnet  mathscinet  zmath
1995
125. A. D. Bruno, V. P. Varin, “The Second Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 128  mathnet
126. A. D. Bruno, V. P. Varin, “The First Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 124  mathnet
127. A. D. Bruno, A. Soleev, “The Hamiltonian Truncations of a Hamiltonian System”, Keldysh Institute preprints, 1995, 055  mathnet
128. A. D. Bruno, A. Soleev, “Homoclinic Solutions of an Invertible ODE System”, Keldysh Institute preprints, 1995, 054  mathnet
129. A. D. Bruno, “The Newton Polyhedron in the Nonlinear Analysis”, Keldysh Institute preprints, 1995, 048  mathnet
130. A. D. Bruno, A. Soleev, “Local Analysis of a Singularity of an Invertible ODE System. Complicated Cases”, Keldysh Institute preprints, 1995, 047  mathnet
131. A. D. Bruno, M. M. Vasiliev, “Newton Polyhedra and the Asymptotic Analysis of the Viscous Fluid Flow Around Flat Plate”, Keldysh Institute preprints, 1995, 044  mathnet
132. A. D. Bruno, A. Soleev, “Local Analysis of a Singularity of an Invertible ODE System. Simple Cases”, Keldysh Institute preprints, 1995, 040  mathnet
133. A. D. Bruno, S. Yu. Sadov, “Formal integral of a divergence-free system”, Mat. Zametki, 57:6 (1995),  803–813  mathnet  mathscinet  zmath; Math. Notes, 57:6 (1995), 565–572  isi
134. A. D. Bruno, A. Soleev, “Local analysis of singularities of an invertible system of ordinary differential equations”, Uspekhi Mat. Nauk, 50:6(306) (1995),  169–170  mathnet  mathscinet  zmath; Russian Math. Surveys, 50:6 (1995), 1258–1259  isi
135. A. Soleev, A. D. Bruno, “Newton polyhedra and Hamiltonian systems”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1995, 6,  84–86  mathnet  mathscinet  zmath
136. A. D. Bruno, “The Newton polyhedron in nonlinear analysis”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1995, 6,  45–51  mathnet  mathscinet  zmath
1994
137. A. D. Bruno, V. I. Parusnikov, “Klein polyhedrals for two cubic Davenport forms”, Mat. Zametki, 56:4 (1994),  9–27  mathnet  mathscinet  zmath; Math. Notes, 56:4 (1994), 994–1007  isi
1992
138. A. D. Bruno, “Nondegeneracy conditions in the Kolmogorov theorem”, Dokl. Akad. Nauk, 322:6 (1992),  1028–1032  mathnet  mathscinet  zmath; Dokl. Math., 45:1 (1992), 221–225
139. A. D. Bruno, “Smooth linearization of differential equations”, Dokl. Akad. Nauk, 322:3 (1992),  446–450  mathnet  mathscinet  zmath; Dokl. Math., 45:1 (1992), 105–109
1991
140. A. D. Bruno, A. Soleev, “Local uniformization of the branches of a space curve, and Newton polyhedra”, Algebra i Analiz, 3:1 (1991),  67–101  mathnet  mathscinet  zmath; St. Petersburg Math. J., 3:1 (1992), 53–82
141. A. D. Bruno, “On a finitely smooth linearization of a system of differential equations near a hyperbolic singular point”, Dokl. Akad. Nauk SSSR, 318:3 (1991),  524–527  mathnet  mathscinet  zmath; Dokl. Math., 43:3 (1991), 711–715
1990
142. A. D. Bruno, “The normal form of a system, close to a Hamiltonian system”, Mat. Zametki, 48:5 (1990),  35–46  mathnet  mathscinet  zmath; Math. Notes, 48:5 (1990), 1100–1108  isi
143. A. D. Bruno, “System, similar to a normal form”, Mat. Zametki, 48:3 (1990),  20–31  mathnet  mathscinet  zmath; Math. Notes, 48:3 (1990), 885–893  isi
1989
144. A. D. Bruno, “Normalization of a Hamiltonian system near an invariant cycle or torus”, Uspekhi Mat. Nauk, 44:2(266) (1989),  49–78  mathnet  mathscinet  zmath; Russian Math. Surveys, 44:2 (1989), 53–89  isi
1988
145. A. D. Bruno, “The normal form of a Hamiltonian system”, Uspekhi Mat. Nauk, 43:1(259) (1988),  23–56  mathnet  mathscinet  zmath; Russian Math. Surveys, 43:1 (1988), 25–66  isi
1986
146. A. D. Bruno, “Stability in a Hamiltonian system”, Mat. Zametki, 40:3 (1986),  385–392  mathnet  mathscinet  zmath; Math. Notes, 40:3 (1986), 726–730  isi
1983
147. A. D. Bruno, “Analytic invariants of a differential equation”, Dokl. Akad. Nauk SSSR, 273:4 (1983),  781–785  mathnet  mathscinet  zmath
148. A. D. Bryuno, “Noncanonical invariants of Hamiltonian systems”, Mat. Zametki, 33:3 (1983),  333–344  mathnet  mathscinet  zmath; Math. Notes, 33:3 (1983), 167–174
1982
149. A. D. Bruno, “Divergence of a real normalizing transformation”, Mat. Zametki, 31:3 (1982),  403–410  mathnet  mathscinet  zmath; Math. Notes, 31:3 (1982), 207–211  isi
1977
150. A. D. Bruno, “Properties of certain functions of celestial mechanics”, Mat. Zametki, 22:1 (1977),  109–116  mathnet  mathscinet  zmath; Math. Notes, 22:1 (1977), 550–554
1976
151. A. D. Bruno, “Normal form and averaging methods”, Dokl. Akad. Nauk SSSR, 230:2 (1976),  257–260  mathnet  mathscinet  zmath
1975
152. A. D. Bruno, “Integral analytic sets”, Dokl. Akad. Nauk SSSR, 220:6 (1975),  1255–1258  mathnet  mathscinet  zmath
153. A. D. Bruno, “Normal form of real differential equations”, Mat. Zametki, 18:2 (1975),  227–241  mathnet  mathscinet  zmath; Math. Notes, 18:2 (1975), 722–731
1974
154. A. D. Bruno, “Analytic integral manifolds”, Dokl. Akad. Nauk SSSR, 216:2 (1974),  253–256  mathnet  mathscinet  zmath
155. A. D. Bruno, “Normal form of differential equations with a small parameter”, Mat. Zametki, 16:3 (1974),  407–414  mathnet  mathscinet  zmath; Math. Notes, 16:3 (1974), 832–836
1973
156. A. D. Bruno, “Local invariants of differential equations”, Mat. Zametki, 14:4 (1973),  499–507  mathnet  mathscinet  zmath; Math. Notes, 14:4 (1973), 844–848
1972
157. A. D. Bruno, “Analytic form of differential equations”, Tr. Mosk. Mat. Obs., 26 (1972),  199–239  mathnet  zmath
1971
158. A. D. Bruno, “Analytic form of differential equations. I, II”, Tr. Mosk. Mat. Obs., 25 (1971),  119–262  mathnet  mathscinet  zmath
1970
159. A. D. Bruno, “Instability in a Hamiltonian system and the distribution of asteroids”, Mat. Sb. (N.S.), 83(125):2(10) (1970),  273–312  mathnet  mathscinet  zmath; Math. USSR-Sb., 12:2 (1970), 271–312
1969
160. A. D. Bruno, “An analytic form of differential equations”, Mat. Zametki, 6:6 (1969),  771–778  mathnet  mathscinet; Math. Notes, 6:6 (1969), 927–931
1967
161. A. D. Bruno, “The divergence of transformations to normal form of differential equations”, Dokl. Akad. Nauk SSSR, 174:5 (1967),  1003–1006  mathnet  mathscinet  zmath
162. A. D. Bryuno, “Formal stability of Hamiltonian systems”, Mat. Zametki, 1:3 (1967),  325–330  mathnet  mathscinet  zmath; Math. Notes, 1:3 (1967), 216–219
1965
163. A. D. Bruno, “On convergence of transforms of differential equations to the normal form”, Dokl. Akad. Nauk SSSR, 165:5 (1965),  987–989  mathnet  mathscinet  zmath
164. A. D. Bruno, “Power asymptotics of solutions of non-linear systems”, Izv. Akad. Nauk SSSR Ser. Mat., 29:2 (1965),  329–364  mathnet  mathscinet  zmath
1964
165. A. D. Bruno, “The normal form of differential equations”, Dokl. Akad. Nauk SSSR, 157:6 (1964),  1276–1279  mathnet  mathscinet  zmath
166. A. D. Bruno, “The expansion of algebraic numbers into continued fractions”, Zh. Vychisl. Mat. Mat. Fiz., 4:2 (1964),  211–221  mathnet  mathscinet  zmath; U.S.S.R. Comput. Math. Math. Phys., 4:2 (1964), 1–15
1962
167. A. D. Bruno, “Asymptotic behavior of solutions of systems of differential equations”, Dokl. Akad. Nauk SSSR, 143:4 (1962),  763–766  mathnet  mathscinet  zmath

2016
168. A. I. Aptekarev, A. B. Batkhin, A. D. Bruno, “Vladimir Igorevich Parusnikov”, Chebyshevskii Sb., 17:1 (2016),  286–298  mathnet  elib
1975
169. É. Dzhusti, M. I. Vishik, A. V. Fursikov, A. S. Schwarz, O. I. Bogoyavlenskii, B. M. Levitan, V. V. Kucherenko, A. G. Kushnirenko, M. V. Fedoryuk, M. A. Shubin, A. D. Bruno, “Sessions of the Petrovskii Seminar on differential equations and mathematical problems of physics”, Uspekhi Mat. Nauk, 30:2(182) (1975),  261–269  mathnet  mathscinet

Presentations in Math-Net.Ru
1. Solving a polynomial equation
A. D. Bruno
XV International Conference «Algebra, Number Theory and Discrete Geometry: modern problems and applications», dedicated to the centenary of the birth of the Doctor of Physical and Mathematical Sciences, Professor of the Moscow State University Korobov Nikolai Mikhailovich
May 29, 2018 10:40
2. Разложение решений ОДУ в трансряды
A. D. Bruno
Seminar on analytic theory of differential equations
April 25, 2018 14:30
3. Calculation of complex asympotics of solutions of Painleve equations
A. D. Bruno
Seminar on analytic theory of differential equations
April 12, 2017 14:30   
4. Solving the Polynomial Equations by Algorithms of Power Geometry
A. D. Bruno
Seminar on Complex Analysis (Gonchar Seminar)
February 20, 2017 17:00
5. From Diophantine approximations to fundamental units of algebraic fields
A. D. Bruno
Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and Applications
January 30, 2016 14:30
6. Asymptotic solutions to algebraic equation
A. D. Bruno, A. B. Batkhin
Seminar of the Department of Algebra
May 10, 2011 15:00
7. Power geometry as new mathematics
A. D. Bruno
Meetings of the St. Petersburg Mathematical Society
April 19, 2005
8. Степенная геометрия как новая математика
A. D. Bruno
Meetings of the Moscow Mathematical Society
April 5, 2005
9. A new generalization of the continued fraction
A. D. Bruno
Meetings of the St. Petersburg Mathematical Society
April 20, 2004

Organisations
 
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