Stanislav Furta

The article deals with the problem of stability of rectilinear motions of multi-link chains of rigid bodies moving in a resisting medium. The authors propose a simple mechanical model that allows them to give a reasonable explanation of instability phenomena observed in reality. The main idea of the article is based on the assumption that lateral forces acting upon lengthy bodies are large enough. This assumption results in the consideration of a nonintegrable constraint as a limiting case. It is well-known that in studies of nonintegrable constraints two different approaches are distinguished. The classical one leads to a nonholonomic constraint, while the other one developed recently by V.V. Kozlov leads to a vakonomic constraint. The authors have shown that instability is typical (in some sense) in both cases though the phase portraits are completely different from the topological point of view.

The paper deals with problems of existence of periodic travelling wave solutions with non-small amplitudes of a PDE describing oscillations of an infinite beam, which lies on a non-linearly elastic support. Such solutions are in fact critical points of a functional on a suitable functional space. By means of a minimax variational technique, the authors found a domain in the parameter space for which there exist periodic travelling waves of a certain fixed period $\Sigma$.

The article deals with discrete dynamical systems defined by iterations of a certain smooth map being a local diffeomorphism of a neighbourhood of the coordinate origin, for which the origin is a fixed point. Criteria to the existence of locally invariant curves adherent to the fixed point are obtained and the formulae for the expansion of the above curves into generalised power series are deduced. The author shows also the connection between the existence of those curves and the existence of the so-called asymptotic trajectories, going to the fixed point as the number of iterations infinitely increases or decreases. Of partitucular interest is the problem of the existence of the invariant curves the asymptotic of the motion along which is generalised power one. As an illustration, the author obtainsed criteria to the existence of asymptotic trajectories in several critical cases, when eigen values of the linear approximation matrix lie on a unit circle. The author considers also the problem of the existence of the motions of a material particle in the homogeneous gravity field, which are asymptotic to periodic skips over a critical point of a certain smooth curve.