V. Koshkin

A problem of chaotic oscillations of a double mathematical pendulum at simultaneous application of parameter extension method and of branching theory methods has been considered; a succession of alternating duplications and quadruplications of the period, which does not posses Feigenbaum universality properties in the examined range, has been obtained.

There has been studied problem of self-similarity of periodical trajectories in Hamilton's systems in the infinite sequence of bifurcations of duplication of the period using as an example the problem of oscillations of satellite on the elliptic orbit relative to the proper center of mass. The universal scale regularities of transformation of periodical movements of the system within the limits of the chaotic movement have been discovered using methods of continuation of solution according to the parameter, the theory of stability by Ljapunov and Floquet, the theory of branching and the methods of scaling, as well. There has been suggested a numeric algorithm of building of the scaling functions of the trajectories (STF) in Hamilton's systems and determination on their basis of universal scale factors of transition to chaos. It has been shown that the STF of Hamilton's systems have a range of qualitative and quantitative dissimilarities from the known dissipative analog.