Transversal intersection of separatrices and non-existence of an analytical integral in multi-dimensional systems

It is well-known, due to G.D. Birkhoff, S. Smale, L.P. Shil'nikov, V.M. Alekseev, and Yu.I. Neimark, that the existence of transversally intersecting separatrices of hyperbolic periodic solutions leads, in a typical situation, to complicated and irregular dynamics. Therefore, in the case of a two-dimensional mapping or a three-dimensional flow there is no non-trivial analytical first integral, i.e. a function constant along each trajectory of the system under consideration. We will obtain and discuss the conditions that guarantee the absence of such an integral in many-dimensional case, regardless of a finite dimension in question. This is the strongest analytical non-integrability and not any Hamiltonian nature (which is of no importance) is supposed here. The conditions obtained guarantee also the absence of non-trivial analytical one-parameter symmetry group, persist under small perturbations, and can be constructively verified for concrete dynamical systems in mechanics and physics. A differential topological structure of the set of ''quasi-random motions'' is studied for this purpose. (author)