(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.
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
