Non-local decomposition of vector fields

We construct a decomposition of a vector field as a sum of a gradient field and a complementary vector field satisfying some orthogonality condition. The aim of this decomposition is to distinguish between the conservative and the circulative components of the vector field. Complex behavior of dynamical system could be uniquely represented as a composition of a spherical oscillation and a gradient flow. This approach is closely related to a statistical investigation of dynamical systems. We introduce a synchronization index that allows one to investigate internal links for chaotic dynamical system. In numerical experiments we illustrate applications of this approach to some particular dynamical systems with strange attractors.