Sergei DOVBYSH

Institute of Mechanics, Moscow State University,
Michurinskij pr-t, d.1. 
Moscow 117192
RUSSIE
dovbysh@inmech.msu.su

Some non-integrable multi-dimensional systems

Abstract: Recently, the author established the non-integrability conditions for multi-dimensional systems which are related to the presence of a set of quasi-random motions possessing a ``rough structure''.These conditions concerns either 1) homoclinic structure formed by transversally intersecting separatrices of hyperbolic periodic solutions, or 2) the phenomenon of branching of solutions in the complex domain as being considered from the viewpoint of symbolic dynamics methods. Here, the non-integrability is understood in the strongest analytic sense, i.e. as the absence of a non-trivial analytic or meromorphic first integral on a level of a priori first integrals. Moreover, it implies the absence of a non-trivial analytic one-parameter symmetry group, or, more generally, meromorphic vector field generating a local phase flow commuting with the system under consideration. Another new faithful generalization is the discreteness of the analytic centralizer in the compact-open topology. We discuss applications of the non-integrability conditions in various dynamical systems arising in physics and mechanics (such as a three-component homogeneous Yang--Mills field, a spherical pendulum with the horizontally oscillating point of suspension, the planar problem of more than three point vortices in an ideal incompressible liquid, the planar problem of more than two bodies attracting by the Newton law). The following observation can be very useful in the consideration of systems of many weakly interacting particles in an external time-periodic force field. If a given diffeomorphism $S$ possesses a homoclinic structure satisfying the non-integrability conditions then the same is valid for the direct product of any finite set of copies of $S$. Using the robust character of the non-integrability conditions (i.e. the fact that they persist under small perturbations) one can transfer immediately the non-integrability result for a single particle to the system of many particles. This approach was utilized in proving the non-integrability of the many-vortex the and many-body systems.