Speciality:
01.02.01 (Theoretical mechanics)
Birth date:
13.09.1962
Keywords: dynamical systems; non-integrability questions of dynamical systems; non-integrability criteria; symbolic dynamics; chaotization of dynamical systems; rigid body dynamics.
Subject:
A structure of the so-called "stochastic layer" was explained and investigated which is observed in numerical experiments. A series of new dynamical effects related to the splitting of separatrices in the perturbed Euler–Poinsot problem was discovered. Qualitative estimates for a set of invariant KAM-tori were obtained in a series of Hamiltonian-type systems with one-and-a-half degrees of freedom which possess no explicit small parameter. Essentially new conditions of geometric character were found which guarantee the non-integrability of multi-dimensional dynamical systems in the strongest analytical sense, i.e. the absence of a non-constant analytic (and even meromorphic) first integral. The conditions obtained possess a few advantages over the non-integrability criteria known before. In particular, they are applicable and constructively verified for wide classes of systems, including problems in mechanics and physics. These non-integrability conditions are related to intersection of separatrices or branching of solutions in the complex domain. For this purpose, the branching of solutions was considered from the standpoint of the symbolic dynamics methods; this allowed the author to obtain efficient non-integrability criteria even in the classical case of Hamiltonian systems with two or one-and-a-half degrees of freedom.
Main publications:
Dovbysh S. A. Branching of solutions as obstructions to the existence of a meromorphic integral in many-dimensional systems // Hamiltonian Systems with Three or More Degrees of Freedom (ed. C. Simo). NATO ASI Series. Series C: Math. and Phys. Sci., v. 533. Dordrecht–Boston–London: Kluwer, 1999, 324–329.
Dovbysh S. A. Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analytic integral in many-dimensional systems. I. Basic result: Separatrices of hyperbolic periodic points // Collectanea Mathematica, 1999, 50(2), 119–197.
