А series of the works «Functional-analytical methods in modern theory of nonlinear dynamical systems and their applications»

Work number - M 107 AWARDED

ProposedbyTarasShevchenkoNationalUniversityofKyiv.

Series of scientific worksconsists of 25 scientific articles.

Series of scientific worksdevoted to the study and modeling of dynamic systems which are described by differential and difference equations. One of the problems is devoted to the justification of the kinetic equations from systems particles interacting via the singular interaction potential. Another problem is devoted to investigation of qualitative behaviorof solutions of linear and weakly nonlinear differential and difference equations in Banach and Hilbert spaces, Hamilton’sequations.

In this work,a method of kinetic cluster expansions of the cumulants of groups of operators for systems of elastic spheres is developed. Using this method we solved one of the actual problems of modern mathematical physics – the problem of mathematical study of kinetic equations, namely, the Enskog kinetic equation and the Fokker-Planck kinetic equation for systems particlesvia the singular potentialinteraction.

The conditions for theexistence of bounded on the whole line of solutions of linear, weakly nonlinear differential and difference equations in locally convex and Banach spaces are investigated.

Newsufficientconditionsforexistenceofperiodicsolutions(ultrasubharmonics) inperturbedlow-dimensionalHamiltoniansystemsareobtained; setsoftori, which appear in KAM-theory, in terms ofHausdorff dimension areinvestigated, in particular, estimatesfor the Hausdorff dimension of a set of Kolmogorov tori for Hamiltonian system close areobtained.