RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB

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
List of publications on Google Scholar
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. 2019 2. A. D. Bruno, “Orbital stability of the periodic solution of a Hamiltonian system”, Keldysh Institute preprints, 2019, 120, 16 pp. 3. A. D. Bruno, “The newest methods of celestial mechanics”, Keldysh Institute preprints, 2019, 079, 18 pp. 4. A. D. Bruno, “Normalization of the periodic Hamiltonian system”, Keldysh Institute preprints, 2019, 064, 18 pp. 5. A. D. Bruno, “Normal form of a Hamiltonian system with a periodic perturbation”, Keldysh Institute preprints, 2019, 057, 27 pp. 6. A. D. Bruno, “On the Parametrization of an Algebraic Curve”, Mat. Zametki, 106:6 (2019),  837–847  ; Math. Notes, 106:6 (2019), 885–893 2018 7. A. D. Bruno, “Normal form of the periodic Hamiltonian system with $n$ degrees of freedom”, Keldysh Institute preprints, 2018, 223, 15 pp. 8. A. D. Bruno, “Complicated and exotic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2018, 118, 44 pp. 9. A. D. Bruno, “Expansion of ODE solutions into transseries”, Keldysh Institute preprints, 2018, 117, 19 pp. 10. A. D. Bruno, “Asymptotic solution of some nonlinear problems”, Keldysh Institute preprints, 2018, 035, 24 pp. 11. A. D. Bruno, “Power geometry and expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2018, 021, 15 pp. 12. A. D. Bruno, “On complicated expansions of solutions to ODES”, Zh. Vychisl. Mat. Mat. Fiz., 58:3 (2018),  346–364    ; Comput. Math. Math. Phys., 58:3 (2018), 328–347 2017 13. A. D. Bruno, “Complicated and exotic expansions of solutions to the fifth Painlevé equation”, Keldysh Institute preprints, 2017, 107, 18 pp. 14. A. D. Bruno, “Calculation of exotic expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2017, 096, 22 pp. 15. A. D. Bruno, “Calculation of complicated asymptotic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2017, 055, 27 pp. 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. 17. A. D. Bruno, “Solving the polynomial equations by algorithms of power geometry”, Keldysh Institute preprints, 2017, 034, 28 pp. 2016 18. A. D. Bruno, “From Diophantine approximations to Diophantine equations”, Chebyshevskii Sb., 17:3 (2016),  38–52 19. A. D. Bruno, “On solution of an algebraic equation”, Keldysh Institute preprints, 2016, 070, 20 pp. 20. A. D. Bruno, “From Diophantine approximations to Diophantine equations”, Keldysh Institute preprints, 2016, 001, 20 pp. 2015 21. A. D. Bruno, “Universal generalization of the continued fraction algorithm”, Chebyshevskii Sb., 16:2 (2015),  35–65 2014 22. A. B. Batkhin, A. D. Bruno, “On investigation of the certain real algebraic surface”, Keldysh Institute preprints, 2014, 083, 28 pp. 2013 23. A. D. Bruno, I. V. Goryuchkina, “Convergence of power expansions of solutions to an ODE”, Keldysh Institute preprints, 2013, 094, 16 pp. 24. A. D. Bruno, “Power Geometry and elliptic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2013, 088, 28 pp. 25. A. D. Bruno, “Asymptotic solving nonlinear equations and idempotent mathematics”, Keldysh Institute preprints, 2013, 056, 31 pp. 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. 27. A. D. Bruno, “Power-elliptic expansions of solutions to an ordinary differential equation”, Zh. Vychisl. Mat. Mat. Fiz., 52:12 (2012),  2206–2218        ; Comput. Math. Math. Phys., 52:12 (2012), 1650–1661 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. 29. A. D. Bruno, “Power-elliptic expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 060, 19 pp. 30. A. D. Bruno, “Power-exponential expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 054, 11 pp. 31. A. B. Batkhin, A. D. Bruno, V. P. Varin, “Sets of stability of multiparameter Hamiltonian systems”, Keldysh Institute preprints, 2011, 042, 32 pp. 32. A. D. Bruno, “Exponential expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 036, 16 pp. 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. 34. A. D. Bruno, “On complicated expansions of solutions to ODE”, Keldysh Institute preprints, 2011, 015, 26 pp. 35. A. D. Bruno, A. B. Batkhin, “Resolution of an algebraic singularity by Power Geometry algorithms”, Keldysh Institute preprints, 2011, 010, 30 pp. 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 37. A. D. Bruno, A. V. Parusnikova, “Local Expansions of Solutions of the Fifth Painlevé Equation”, Keldysh Institute preprints, 2010, 072, 27 pp. 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. 39. A. D. Bruno, A. V. Parusnikova, “Asymptotic Expansions of Solutions to the Fifth Painlevé equation”, Keldysh Institute preprints, 2010, 039, 23 pp. 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. 41. A. D. Bruno, “The structure of the multidimensional Diophantine approximations”, Keldysh Institute preprints, 2010, 011, 8 pp. 42. A. D. Bruno, A. V. Gridnev, “Nonpower expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2010, 010, 21 pp. 43. A. D. Bruno, A. B. Batkhin, V. P. Varin, “The stability set of a gyroscopic problems”, Keldysh Institute preprints, 2010, 004, 30 pp. 44. A. D. Bruno, “Sets of stability of multiparameter problems”, Keldysh Institute preprints, 2010, 003, 14 pp. 2009 45. A. D. Bruno, V. F. Edneral, “Algorithmic analysis of local integrability”, Dokl. Akad. Nauk, 424:3 (2009),  299–303    ; Dokl. Math., 79:1 (2009), 48–52 46. A. D. Bruno, I. V. Goryuchkina, “Non-formal solutions to ODE”, Keldysh Institute preprints, 2009, 061, 14 pp. 47. A. D. Bruno, I. V. Goryuchkina, “Elliptic asymptotic forms of solutions to the Painlev'e equations”, Keldysh Institute preprints, 2009, 006, 26 pp. 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  ; J. Math. Sci. (N. Y.), 168:3 (2010), 326–333 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. 50. A. D. Bruno, V. I. Parusnikov, “Two-sided generalization of the continued fraction”, Keldysh Institute preprints, 2008, 058, 25 pp. 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. 2007 52. A. D. Bruno, V. F. Edneral, “On integrability of the Euler–Poisson equations”, Fundam. Prikl. Mat., 13:1 (2007),  45–59      ; J. Math. Sci., 152:4 (2008), 479–489 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. 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. 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. 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. 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. 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. 59. A. D. Bruno, V. P. Varin, “Family $c$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 051, 14 pp. 60. A. D. Bruno, V. P. Varin, “Complicated families of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 035, 18 pp. 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. 62. A. D. Bruno, “Power Geometry as a new mathematics”, Keldysh Institute preprints, 2007, 028, 24 pp. 63. A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the sixth Painlevé equation”, Keldysh Institute preprints, 2007, 019, 19 pp. 64. A. D. Bruno, V. F. Edneral, “Computation of normal forms of the Euler–Poisson equations”, Keldysh Institute preprints, 2007, 001, 17 pp. 2006 65. A. D. Bruno, “Complicated expansions of solutions to an ODE system”, Keldysh Institute preprints, 2006, 081, 13 pp. 66. A. D. Bruno, “Exotic expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2006, 066, 31 pp. 67. A. D. Bruno, V. Yu. Petrovich, “Desingularizations of the restricted three-body problem”, Keldysh Institute preprints, 2006, 053, 12 pp. 68. A. D. Bruno, V. P. Varin, “The generating family $i$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2006, 036 69. A. D. Bruno, “On movable singular points of solutions to the ordinary differential equations”, Keldysh Institute preprints, 2006, 026, 13 pp. 70. A. D. Bruno, I. V. Goryuchkina, “Expansions of solutions to the sixth Painlevé equation near singular points $x=0$ è $x=\infty$”, Keldysh Institute preprints, 2006, 013, 32 pp. 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. 2005 72. A. D. Bruno, “Theory of normal forms of the Euler-Poisson equations”, Keldysh Institute preprints, 2005, 100 73. A. D. Bruno, “Properties of the modulus polyhedron”, Keldysh Institute preprints, 2005, 072 74. A. D. Bruno, I. N. Gashenenko, “Simple finite solutions to the N. Kowalewski equations”, Keldysh Institute preprints, 2005, 068 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 76. A. D. Bruno, “Normal Forms and Integrability of the Euler–Poisson Equations”, Keldysh Institute preprints, 2005, 066 77. A. D. Bruno, I. N. Gashenenko, “Last expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2005, 065 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 79. A. D. Bruno, T. V. Shadrina, “On the viscous incompressible fluid flow around a plate”, Keldysh Institute preprints, 2005, 054 80. A. D. Bruno, V. I. Parusnikov, “New generalizations of the continued fraction”, Keldysh Institute preprints, 2005, 052, 17 pp. 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 82. A. D. Bruno, V. I. Parusnikov, “Further generalization of the continued fraction”, Keldysh Institute preprints, 2005, 040 83. A. D. Bruno, “Complicated expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2005, 036 84. A. D. Bruno, N. A. Kudryashov, “Power expansions of solutions to an analogy to the first Painlevé equation”, Keldysh Institute preprints, 2005, 017 85. A. D. Bruno, V. P. Varin, “On families of periodic solutions to the restricted three-body problem”, Keldysh Institute preprints, 2005, 010 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 2004 87. A. D. Bruno, V. Yu. Petrovich, “Singularities of solutions to the first Painlevé equation”, Keldysh Institute preprints, 2004, 075, 13 pp. 88. A. D. Bruno, “Algorithm of the generalizationued continued fraction”, Keldysh Institute preprints, 2004, 045, 27 pp. 89. A. D. Bruno, T. V. Shadrina, “The compressible heat conductive boundary layer on a needle”, Keldysh Institute preprints, 2004, 037, 38 pp. 90. A. D. Bruno, T. V. Shadrina, “About incompressible boundary layer on a needle”, Keldysh Institute preprints, 2004, 036, 26 pp. 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. 92. A. D. Bruno, “On generalisations of the continued fraction”, Keldysh Institute preprints, 2004, 010, 19 pp. 93. A. D. Bruno, “Asymptotic behaviour and expansions of solutions of an ordinary differential equation”, Uspekhi Mat. Nauk, 59:3(357) (2004),  31–80        ; Russian Math. Surveys, 59:3 (2004), 429–480 2003 94. A. D. Bruno, V. I. Parusnikov, “Polyhedra of absolute values for triples of linear forms”, Keldysh Institute preprints, 2003, 093, 21 pp. 95. A. D. Bruno, “The ñorrect generalization of the continued fraction”, Keldysh Institute preprints, 2003, 086, 19 pp. 96. A. D. Bruno, T. V. Shadrina, “Axisymmetric boundary layer on a needle”, Keldysh Institute preprints, 2003, 064, 28 pp. 97. A. D. Bruno, “Expansions of solutions to an ODE system”, Keldysh Institute preprints, 2003, 059, 24 pp. 98. A. D. Bruno, “Asymptotically ñlose slutions to an ODE system”, Keldysh Institute preprints, 2003, 058, 28 pp. 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. 100. A. D. Bruno, E. S. Karulina, “Power expansions of solutions to the fifth Painlevé equation”, Keldysh Institute preprints, 2003, 050, 26 pp. 101. A. D. Bruno, I. V. Chukhareva, “Power expansions of solutions to the sixth Painlevé equation”, Keldysh Institute preprints, 2003, 049, 24 pp. 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. 103. A. D. Bruno, “Asymptotically close solutions to an ordinary differential equation”, Keldysh Institute preprints, 2003, 031, 17 pp. 104. A. D. Bruno, “The asymptotical solution of nonlinear equations by means of Power Geometry”, Keldysh Institute preprints, 2003, 028, 31 pp. 105. A. D. Bruno, “Asymptotics and expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2003, 009, 25 pp. 2002 106. A. D. Bruno, V. P. Varin, “Classes of families of generalized periodic solutions to the Beletsky equation”, Keldysh Institute preprints, 2002, 064 107. A. D. Bruno, “Analysis of the Euler-Poisson equations by methods of Power Geometry”, Keldysh Institute preprints, 2002, 041 108. A. D. Bruno, “Asymptotics of solutions to the ordinary differential equations”, Keldysh Institute preprints, 2002, 040 109. A. D. Bruno, V. V. Lunev, “Properties of expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2002, 023 2001 110. A. D. Bruno, V. V. Lunev, “Asymptotical expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2001, 090 111. A. D. Bruno, V. V. Lunev, “Local expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2001, 073 112. A. D. Bruno, V. V. Lunev, “The modified system of equations described motions of a rigid body”, Keldysh Institute preprints, 2001, 049 2000 113. A. D. Bruno, “Power Expansions of Solutions of a System of Algebraic and Differential Equations”, Keldysh Institute preprints, 2000, 068 114. A. D. Bruno, “Power Expansions of Solutions of One Algebraic or Differential Equation”, Keldysh Institute preprints, 2000, 063 115. A. D. Bruno, “Families of Periodic Solutions of the Beletskii Equation”, Keldysh Institute preprints, 2000, 051 116. A. D. Bruno, V. Yu. Petrovich, “Normal Forms of the ODE System”, Keldysh Institute preprints, 2000, 018 117. A. D. Bruno, “Self-similar solutions and power geometry”, Uspekhi Mat. Nauk, 55:1(331) (2000),  3–44        ; Russian Math. Surveys, 55:1 (2000), 1–42 1999 118. A. D. Bruno, “A new generalization of the continued fraction”, Keldysh Institute preprints, 1999, 082 119. A. D. Bruno, “On Complexity of Problems of Power Geometry”, Keldysh Institute preprints, 1999, 059 120. A. D. Bruno, “Finding Self-Similar Solutions by Means of Power Geometry”, Keldysh Institute preprints, 1999, 057 1997 121. A. D. Bruno, V. J. Petrovich, “Computation of periodic oscillations of a satellite”, Matem. Mod., 9:6 (1997),  82–94 122. A. D. Bruno, V. I. Parusnikov, “Comparison of various generalizations of continued fractions”, Mat. Zametki, 61:3 (1997),  339–348      ; Math. Notes, 61:3 (1997), 278–286 1996 123. A. D. Bruno, “Zero-Multiple and Retrograde Periodic Solutions of the Restricted Three-Body Problem”, Keldysh Institute preprints, 1996, 093 124. A. D. Bruno, “A general approach to asymptotic nonlinear analysis”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1996, 6,  24–27 1995 125. A. D. Bruno, V. P. Varin, “The Second Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 128 126. A. D. Bruno, V. P. Varin, “The First Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 124 127. A. D. Bruno, A. Soleev, “The Hamiltonian Truncations of a Hamiltonian System”, Keldysh Institute preprints, 1995, 055 128. A. D. Bruno, A. Soleev, “Homoclinic Solutions of an Invertible ODE System”, Keldysh Institute preprints, 1995, 054 129. A. D. Bruno, “The Newton Polyhedron in the Nonlinear Analysis”, Keldysh Institute preprints, 1995, 048 130. A. D. Bruno, A. Soleev, “Local Analysis of a Singularity of an Invertible ODE System. Complicated Cases”, Keldysh Institute preprints, 1995, 047 131. A. D. Bruno, M. M. Vasiliev, “Newton Polyhedra and the Asymptotic Analysis of the Viscous Fluid Flow Around Flat Plate”, Keldysh Institute preprints, 1995, 044 132. A. D. Bruno, A. Soleev, “Local Analysis of a Singularity of an Invertible ODE System. Simple Cases”, Keldysh Institute preprints, 1995, 040 133. A. D. Bruno, S. Yu. Sadov, “Formal integral of a divergence-free system”, Mat. Zametki, 57:6 (1995),  803–813      ; Math. Notes, 57:6 (1995), 565–572 134. 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      ; Russian Math. Surveys, 50:6 (1995), 1258–1259 135. A. Soleev, A. D. Bruno, “Newton polyhedra and Hamiltonian systems”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1995, 6,  84–86 136. A. D. Bruno, “The Newton polyhedron in nonlinear analysis”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1995, 6,  45–51 1994 137. A. D. Bruno, V. I. Parusnikov, “Klein polyhedrals for two cubic Davenport forms”, Mat. Zametki, 56:4 (1994),  9–27      ; Math. Notes, 56:4 (1994), 994–1007 1992 138. A. D. Bruno, “Nondegeneracy conditions in the Kolmogorov theorem”, Dokl. Akad. Nauk, 322:6 (1992),  1028–1032      ; Dokl. Math., 45:1 (1992), 221–225 139. A. D. Bruno, “Smooth linearization of differential equations”, Dokl. Akad. Nauk, 322:3 (1992),  446–450      ; 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      ; 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      ; 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      ; Math. Notes, 48:5 (1990), 1100–1108 143. A. D. Bruno, “System, similar to a normal form”, Mat. Zametki, 48:3 (1990),  20–31      ; Math. Notes, 48:3 (1990), 885–893 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      ; Russian Math. Surveys, 44:2 (1989), 53–89 1988 145. A. D. Bruno, “The normal form of a Hamiltonian system”, Uspekhi Mat. Nauk, 43:1(259) (1988),  23–56      ; Russian Math. Surveys, 43:1 (1988), 25–66 1986 146. A. D. Bruno, “Stability in a Hamiltonian system”, Mat. Zametki, 40:3 (1986),  385–392      ; Math. Notes, 40:3 (1986), 726–730 1983 147. A. D. Bruno, “Analytic invariants of a differential equation”, Dokl. Akad. Nauk SSSR, 273:4 (1983),  781–785 148. A. D. Bryuno, “Noncanonical invariants of Hamiltonian systems”, Mat. Zametki, 33:3 (1983),  333–344      ; 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      ; Math. Notes, 31:3 (1982), 207–211 1977 150. A. D. Bruno, “Properties of certain functions of celestial mechanics”, Mat. Zametki, 22:1 (1977),  109–116      ; 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 1975 152. A. D. Bruno, “Integral analytic sets”, Dokl. Akad. Nauk SSSR, 220:6 (1975),  1255–1258 153. A. D. Bruno, “Normal form of real differential equations”, Mat. Zametki, 18:2 (1975),  227–241      ; Math. Notes, 18:2 (1975), 722–731 1974 154. A. D. Bruno, “Analytic integral manifolds”, Dokl. Akad. Nauk SSSR, 216:2 (1974),  253–256 155. A. D. Bruno, “Normal form of differential equations with a small parameter”, Mat. Zametki, 16:3 (1974),  407–414      ; Math. Notes, 16:3 (1974), 832–836 1973 156. A. D. Bruno, “Local invariants of differential equations”, Mat. Zametki, 14:4 (1973),  499–507      ; 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 1971 158. A. D. Bruno, “Analytic form of differential equations. I, II”, Tr. Mosk. Mat. Obs., 25 (1971),  119–262 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      ; 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    ; 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 162. A. D. Bryuno, “Formal stability of Hamiltonian systems”, Mat. Zametki, 1:3 (1967),  325–330      ; 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),  987–989 164. A. D. Bruno, “Power asymptotics of solutions of non-linear systems”, Izv. Akad. Nauk SSSR Ser. Mat., 29:2 (1965),  329–364 1964 165. A. D. Bruno, “The normal form of differential equations”, Dokl. Akad. Nauk SSSR, 157:6 (1964),  1276–1279 166. A. D. Bruno, “The expansion of algebraic numbers into continued fractions”, Zh. Vychisl. Mat. Mat. Fiz., 4:2 (1964),  211–221      ; 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 2016 168. A. I. Aptekarev, A. B. Batkhin, A. D. Bruno, “Vladimir Igorevich Parusnikov”, Chebyshevskii Sb., 17:1 (2016),  286–298 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

Presentations in Math-Net.Ru
 1 Solving a polynomial equationA. 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 MikhailovichMay 29, 2018 10:40 2 Ðàçëîæåíèå ðåøåíèé ÎÄÓ â òðàíñðÿäûA. D. Bruno Seminar on analytic theory of differential equationsApril 25, 2018 14:30 3 Calculation of complex asympotics of solutions of Painleve equationsA. D. Bruno Seminar on analytic theory of differential equationsApril 12, 2017 14:30 4 Solving the Polynomial Equations by Algorithms of Power GeometryA. D. Bruno Seminar on Complex Analysis (Gonchar Seminar)February 20, 2017 17:00 5 From Diophantine approximations to fundamental units of algebraic fieldsA. D. Bruno Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and ApplicationsJanuary 30, 2016 14:30 6 Asymptotic solutions to algebraic equationA. D. Bruno, A. B. Batkhin Seminar of the Department of AlgebraMay 10, 2011 15:00 7 Power geometry as new mathematicsA. D. Bruno Meetings of the St. Petersburg Mathematical SocietyApril 19, 2005 8 Ñòåïåííàÿ ãåîìåòðèÿ êàê íîâàÿ ìàòåìàòèêàA. D. Bruno Meetings of the Moscow Mathematical SocietyApril 5, 2005 9 A new generalization of the continued fractionA. D. Bruno Meetings of the St. Petersburg Mathematical SocietyApril 20, 2004

Organisations