

Principal scientific achievements
 Proof of the Dulac's conjecture:
Polynomial vector field in the real plane has but a finite number of
limit cycles.
The proof of this short statement requires a book [44]
published by AMS in 1991; it was the subject of the talk in the ICM1990.
This proof was obtained in the competition with the French team:
Ecalle, Martinet, Moussu, Ramis. The proof of the same
conjecture given by Ecalle and based on the ideas of the
four authors appeared in a book published in 1992.
Preliminary studies: [30], [32], [36], [41].
 Investigation of generic properties of polynomial vector
fields in the complex plane (talk in the ICM1978) [12], [17],
[56].
 Solvability of local problems of ODE:
algebraic unsolvability of the center focus problem [7],
analytic unsolvability of the Liapunov stability problem [12],
general investigation of algebraically and analytically
solvable problems of local dynamics [37].
 Geometric theorems on divergence of normalizing series
and related topics in complex analysis [19],[20],[22],[24].
 Upper estimate of the Hausdorff and box counting dimensions
for attractors of dissipative systems, with applications to
NavierStokes and KuramotoSivashinsky equations: [17], [18] (prolonged
by BabinVishik),[29], [47], [48], [46].
 Nonlinear Stokes Phenomena: advisorship of the investigations
of Voronin, Elizarov, Shcherbakov, summarized in [50], including
[51], [52], [53].
 Generation of limit cycles under perturbation of planar
Hamiltonian systems. Related study of zeroes of Abelian integrals
by means of the
theory of Riemann surfaces and other tools of complex analysis
(like RiemannRoch and PicardLefshetz theorems).
Initiated in [1], [2], prolonged [5], [14], [62], [60].
 Normal forms for local families and nonlocal bifurcations.
A complete list of finitely
smooth integrable normal forms for local families
of vector fields and maps [45], [53].
Solution of the HilbertArnold problem for elementary
polycycles [54], [48] (together with Yakovenko).
Systematic exposition of the nonlocal bifurcations theory
in the multidimensional space (together with Li Weigu) [72].
The book [72] contains new proofs of classical
theorems and many new results.
 Relations between random and smooth dynamical systems.
New robust properties of attractors are found (joint work with
A. Gorodetski, [77], [80]).
 Hilbert type numbers for Abel equation: an upper
estimate of the number of limit cycles for a polynomial
nonautonomous eaqution on the line is obtained [79]).


