A method of bifurcation graph 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 two-dimensional manifolds. First of all, dynamical control systems on a sphere are analyzed. Some new notions are introduced for dynamical control systems: a skeleton of a state space, irreducible components of a skeleton, a linking property for irreducible components, a cycle of irreducible components, a bifurcation graph. A piece of a path on the bifurcation graph is called a scenario of control. Some pieces of paths on the bifurcation graph are selected. They correspond to controllable systems.