Principal scientific achievements

1. 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 ICM-1990. 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].
2. Investigation of generic properties of polynomial vector fields in the complex plane (talk in the ICM-1978) [12], [17], [56].
3. 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].
4. Geometric theorems on divergence of normalizing series and related topics in complex analysis [19],[20],[22],[24].
5. Upper estimate of the Hausdorff and box counting dimensions for attractors of dissipative systems, with applications to Navier-Stokes and Kuramoto-Sivashinsky equations: [17], [18] (prolonged by Babin-Vishik),[29], [47], [48], [46].
6. Nonlinear Stokes Phenomena: advisorship of the investigations of Voronin, Elizarov, Shcherbakov, summarized in [50], including [51], [52], [53].
7. 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 Riemann-Roch and Picard-Lefshetz theorems). Initiated in [1], [2], prolonged [5], [14], [62], [60].
8. 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 Hilbert-Arnold 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.
9. Relations between random and smooth dynamical systems. New robust properties of attractors are found (joint work with A. Gorodetski, [77], [80]).
10. 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]).