Limit Sets of Trajectories
Dynamical polysystems defined on closed manifolds and consisting of finite number of smooth vector fields are considered. ω-limit sets of their trajectories are studied. If a polysystem consists of one vector field we have an old and difficult problem in the Qualitative Theory of differential equations. In the general case it is known that for any trajectory the ω-limit set exists, is compact and connected. In the general topology such sets are called continuums. There are no other restrictions on the structure of the ω-limit set because it may be proved that any continuum on a manifold is the ω-limit of a polysystem, that is, the ω-limit set for one of its trajectories. A polysystem is said to be universal if any continuum on the manifold is its ω-limit set. It is proved that if a polysystem D is universal then the polysystem D is also universal. Thus, an universal polysystem has a trajectory defined on the real line with prescribed behavior at +∞ and −∞. Now consider the following problem: what is the minimal number of vector fields which form an universal polysystem?