(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:
the isolation of their first approximations by means of the Newton polyhedrons and their analogous;
simplification of the first approximations by means of the power and logarithmic transformations;
finding self-similar solutions to quasihomogeneous systems (to which belong all first approximations);
finding asymptotic forms of their solutions and
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
A. B. Batkhin, A. D. Bruno, “Real normal form of a binary polynomial at a second-order critical point”, Zh. Vychisl. Mat. Mat. Fiz., 63:1 (2023), 3–15; Comput. Math. Math. Phys., 63:1 (2023), 1–13
A. D. Bruno, A. B. Batkhin, “Computation of asymptotic forms of solutions to system of nonlinear partial differential equations”, Keldysh Institute preprints, 2022, 048, 36 pp.
3.
A. D. Bruno, A. A. Azimov, “Computation of unimodular matrices”, Keldysh Institute preprints, 2022, 046, 20 pp.
A. D. Bruno, A. B. Batkhin, Z. Kh. Khaydarov, “Examples of computation of level lines of polynomials in a plane”, Keldysh Institute preprints, 2021, 098, 36 pp.
A. D. Bruno, A. B. Batkhin, “Normal form of a binary polynomial in the critical point of the second order”, Keldysh Institute preprints, 2021, 065, 20 pp.
6.
A. D. Bruno, A. B. Batkhin, “Level lines of a polynomial in the plane”, Keldysh Institute preprints, 2021, 057, 24 pp.
A. D. Bruno, “Families of periodic solutions and invariant tori of Hamiltonian system without parameters”, Keldysh Institute preprints, 2020, 071, 15 pp.
10.
A. D. Bruno, “On types of stability in Hamiltonian systems”, Keldysh Institute preprints, 2020, 021, 24 pp.
A. D. Bruno, “Normal form of a Hamiltonian system with a periodic perturbation”, Zh. Vychisl. Mat. Mat. Fiz., 60:1 (2020), 39–56; Comput. Math. Math. Phys., 60:1 (2020), 36–52
A. D. Bruno, “Calculation of complicated asymptotic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2017, 055, 27 pp.
A. D. Bruno, “Calculation of fundamental units of number rings by means of the generalized continued fraction”, Keldysh Institute preprints, 2017, 046, 28 pp.
27.
A. D. Bruno, “Solving the polynomial equations by algorithms of power geometry”, Keldysh Institute preprints, 2017, 034, 28 pp.
A. D. Bruno, “Power Geometry and elliptic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2013, 088, 28 pp.
35.
A. D. Bruno, “Asymptotic solving nonlinear equations and idempotent mathematics”, Keldysh Institute preprints, 2013, 056, 31 pp.
2012
36.
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.
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.
39.
A. D. Bruno, “Power-elliptic expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 060, 19 pp.
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.
44.
A. D. Bruno, “On complicated expansions of solutions to ODE”, Keldysh Institute preprints, 2011, 015, 26 pp.
A. D. Bruno, “Structure of the best diophantine approximations and multidimensional generalizations of the continued fraction”, Chebyshevskii Sb., 11:1 (2010), 68–73
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.
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
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.
60.
A. D. Bruno, V. I. Parusnikov, “Two-sided generalization of the continued fraction”, Keldysh Institute preprints, 2008, 058, 25 pp.
61.
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
62.
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
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.
64.
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.
65.
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.
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.
67.
A. D. Bruno, I. V. Goryuchkina, “Review of all asymptotic expansions of solutions to the equation P6”, Keldysh Institute preprints, 2007, 060, 16 pp.
68.
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.
69.
A. D. Bruno, V. P. Varin, “Family $c$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 051, 14 pp.
A. D. Bruno, V. P. Varin, “Complicated families of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 035, 18 pp.
71.
A. D. Bruno, V. P. Varin, “Periodic solutions of the restricted three-body problem for small $\mu$”, Keldysh Institute preprints, 2007, 034, 22 pp.
72.
A. D. Bruno, “Power Geometry as a new mathematics”, Keldysh Institute preprints, 2007, 028, 24 pp.
73.
A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the sixth Painlevé equation”, Keldysh Institute preprints, 2007, 019, 19 pp.
74.
A. D. Bruno, V. F. Edneral, “Computation of normal forms of the Euler–Poisson equations”, Keldysh Institute preprints, 2007, 001, 17 pp.
2006
75.
A. D. Bruno, “Complicated expansions of solutions to an ODE system”, Keldysh Institute preprints, 2006, 081, 13 pp.
76.
A. D. Bruno, “Exotic expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2006, 066, 31 pp.
77.
A. D. Bruno, V. Yu. Petrovich, “Desingularizations of the restricted three-body problem”, Keldysh Institute preprints, 2006, 053, 12 pp.
78.
A. D. Bruno, V. P. Varin, “The generating family $i$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2006, 036
A. D. Bruno, “On movable singular points of solutions to the ordinary differential equations”, Keldysh Institute preprints, 2006, 026, 13 pp.
80.
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.
81.
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
82.
A. D. Bruno, “Theory of normal forms of the Euler-Poisson equations”, Keldysh Institute preprints, 2005, 100
83.
A. D. Bruno, “Properties of the modulus polyhedron”, Keldysh Institute preprints, 2005, 072
A. D. Bruno, N. A. Kudryashov, “Power expansions of solutions to an analogy to the first Painlevé equation”, Keldysh Institute preprints, 2005, 017
95.
A. D. Bruno, V. P. Varin, “On families of periodic solutions to the restricted three-body problem”, Keldysh Institute preprints, 2005, 010
96.
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
97.
A. D. Bruno, V. Yu. Petrovich, “Singularities of solutions to the first Painlevé equation”, Keldysh Institute preprints, 2004, 075, 13 pp.
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
A. D. Bruno, T. V. Shadrina, “Axisymmetric boundary layer on a needle”, Keldysh Institute preprints, 2003, 064, 28 pp.
107.
A. D. Bruno, “Expansions of solutions to an ODE system”, Keldysh Institute preprints, 2003, 059, 24 pp.
108.
A. D. Bruno, “Asymptotically сlose slutions to an ODE system”, Keldysh Institute preprints, 2003, 058, 28 pp.
109.
A. D. Bruno, A. V. Gridnev, “Power and exponential expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2003, 051, 19 pp.
A. D. Bruno, E. S. Karulina, “Power expansions of solutions to the fifth Painlevé equation”, Keldysh Institute preprints, 2003, 050, 26 pp.
111.
A. D. Bruno, I. V. Chukhareva, “Power expansions of solutions to the sixth Painlevé equation”, Keldysh Institute preprints, 2003, 049, 24 pp.
112.
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.
A. D. Bruno, “A new generalization of the continued fraction”, Keldysh Institute preprints, 1999, 082
129.
A. D. Bruno, “On Complexity of Problems of Power Geometry”, Keldysh Institute preprints, 1999, 059
130.
A. D. Bruno, “Finding Self-Similar Solutions by Means of Power Geometry”, Keldysh Institute preprints, 1999, 057
1997
131.
A. D. Bruno, V. J. Petrovich, “Computation of periodic oscillations of a satellite”, Matem. Mod., 9:6 (1997), 82–94
132.
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
A. D. Bruno, “A general approach to asymptotic nonlinear analysis”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1996, no. 6, 24–27
1995
136.
A. D. Bruno, A. Soleev, “Bifurcations of solutions in an invertible system of ordinary
differential equations”, Dokl. Akad. Nauk, 345:5 (1995), 590–592
A. D. Bruno, “Methods for computing the normal form”, Dokl. Akad. Nauk, 344:3 (1995), 298–300
138.
A. D. Bruno, V. P. Varin, “The Second Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 128
139.
A. D. Bruno, V. P. Varin, “The First Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 124
140.
A. D. Bruno, A. Soleev, “The Hamiltonian Truncations of a Hamiltonian System”, Keldysh Institute preprints, 1995, 055
141.
A. D. Bruno, A. Soleev, “Homoclinic Solutions of an Invertible ODE System”, Keldysh Institute preprints, 1995, 054
142.
A. D. Bruno, “The Newton Polyhedron in the Nonlinear Analysis”, Keldysh Institute preprints, 1995, 048
143.
A. D. Bruno, A. Soleev, “Local Analysis of a Singularity of an Invertible ODE System. Complicated Cases”, Keldysh Institute preprints, 1995, 047
144.
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
145.
A. D. Bruno, A. Soleev, “Local Analysis of a Singularity of an Invertible ODE System. Simple Cases”, Keldysh Institute preprints, 1995, 040
146.
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
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
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
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
157.
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
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
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
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
A. I. Aptekarev, A. B. Batkhin, A. D. Bruno, “Vladimir Igorevich Parusnikov”, Chebyshevskii Sb., 17:1 (2016), 286–298
1975
184.
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
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
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