- Creation and developing new calculus “Nonlinear Analysis”.
- Applications of Nonlinear Analysis in complicated problems of Mathematics, Mechanics, Celestial Mechanics and Hydromechanics.
- Number Theory.

Below more detaled…

**1. Nonlinear Analysis. **(a) Equations with a linear part: for systems of ordinary differential equations (ODE), I completed the theory of normal forms, began by Poincar´e (1879) and Dulac (1912) for general systems and began by Birkhoff (1929) for Hamiltonian systems. (b) Equations without linear part: I proposed to study properties of solutions to equations (algebraic, ordinary differential (ODE) and partial differential) by studying sets of vectorial power exponents of terms of these equations. Namely to select more simple (“truncated”) equations by means of generalization to polyhedra the Newton (1678) and Hadamard (1893) polygons. By means of power transformations, proposed by me, the truncated equations can be strongly simplified and often solved. Solutions of the truncated equations are asymptotically the first approximations of the solutions to the full equations. Continuing that process, we can obtain approximations of any precision to solutions of initial equations. Basing on the developed Nonlinear Analysis, I proposed algorithms for solutions of a wide set of singular problems. In particular, for computation of six different types of asymptotic expansions of solutions to ODE, including expansions into transseries.

**2. Applications.** (a) In Mathematics: together with my students I found all asymptotic expansions of five types of solutions to the Painlev´e equations (1906) and also gave very effective method of determination of integrabilityof ODE system. (b) In Mechanics: I computed with high precision influence of small nutation oscillations on velocity of precession of a gyroscope and also studied values of parameters of a centrifuge, ensuring stability of its rotation. (c) In Celestial Mechanics: together with my students I studied periodic solutions of the Beletsky equation (1956), describing motion of satellite around its masscenter, moving along an elliptic orbit. I found new families of periodic solutions, which are important for passive orientation of the satellite, including cases with big values of the eccentricity of the orbit, inducing a singularity. Besides, simultaneously with H´enon (1997), I found all regular and singular generating families of periodic solutions of the restricted three-body problem and studiedbifurcations of generated families. It allowed to explain some singularities of motions of small bodies of the Solar System. In particular, I found orbits of 1 periodic flies round planets with close approach to Earth. (d) In Hydromechanics: I studied small surface waves on a water and a boundary layer on a needle, where equations of a flow have a singularity.

**3. Number Theory.** (a) In a situation of small divisors, I proposed new condition on eigenvalues of an ODE system, ensuring convergence of the normalizing transformation. It was called as “Bruno condition”, and it is betterthan the Siegel’s condition (1941). In one-dimensional case, the Bruno condition can be expressed via the continued fraction. Numbers, satisfying to that one-dimensional condition, were called as the \»Brjuno Numbers\». There wereconferences and courses of lectures devoted to them. (b) In 1775 Euler began a search of a multidimensional generalization of the continued fraction. Almost all large mathematicians of XIX century tried to find such generalization. But in contrast to the usual continued fraction, all proposed generalizations either did not give the best Diophantine approximations either did not allow to compute units of number rings or given neither the first neither the second (as the Poincar´e algorithm (1884)). I have found a such generalization, which gives boththese things. (c) I applied the Number Theory for solving problems of Mechanics and Celestial Mechanics.

Born 26 June 1940 in Moscow, USSR (Russia).

- Mathematician:
- MS, Moscow State University, 1962;
- PhD, Institute of Applied Mathematics, 1966;
- Professor, Institute of Applied Mathematics, 1970.

- Keldysh Institute of Applied Mathematics of RAS:
- Junior Researcher, 1965;
- Senior Researcher, 1971;
- Leading Researcher, 1987;
- Head of Mathematical Department, 1995;
- Head of Sector of Singular Problems, 2008.

- 3rd Prize at the Moscow Mathematical Olympiade, 1956;
- 1st Prize at the Moscow Mathematical Olympiade, 1957;
- 2nd Prizes for Students Papers in Moscow St University, 1960 and 1961.
- Included in list of the 500 most influential people of the XIX century.

- Moscow Mathematical Society;
- American Mathematical Society;
- Academy of Nonlinear Sciences.

- Universal Nonlinear Analysis and its applications;
- Number Theory.

- Analytical form of differential equations (I, II). Trans. Moscow Math. Soc. 25 (1971) 131–288, 26 (1972) 199–239;
- Local Methods in Nonlinear Differential Equations. Springer-Verlag: Berlin-Heidelberg-New York-London-Paris-Tokyo, 1989. 350pp;
- The Restricted 3-Body Problem: Plane Periodic Orbits. Walter de Gruyter, Berlin-New York, 1994. 362pp;
- Power Geometry in Algebraic and Differential Equations. Elsevier Science (North-Holland), Amsterdam, 2000. 385pp.