A method for research of controllability

Controllability of dynamical systems is studied. A dynamical control system is defined by a family of vector fields. Dynamical control systems are considered on arbitrary n-dimensional manifolds. Following notions are used: a skeleton of a state space for dynamical control systems, connected components of a skeleton, a linking property for connected components, and a cycle of connected components. A notion of a skeleton allows one to reduce the research of controllability to the research of finite control system given on the graph. It is shown that the property of return (recurrence) for trajectories of a dynamical control system provides the controllability