It is well-known 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, with this transversality property, there is no non-trivial analytic or meromorphic first integral, i.e., a function constant along each trajectory of the system under consideration. Additional robust conditions are obtained and discussed that guarantee the absence of such an integral in the many-dimensional case, regardless of the finite dimension in question (the strongest analytic non-integrability). These conditions guarantee also the absence of any non-trivial analytic one-parameter symmetry group, and, more generally, analytic or meromorphic vector fields generating a local symmetry, i.e., a local phase flow commuting with the system under consideration. Furthermore, the analytic centralizer of the system is discrete in the compact-open topology. A differential-topological structure of the invariant set of quasi-random motions is studied for this purpose. The approach used is essentially geometrical. Some related topics are also discussed.
@article{urn:eudml:doc:42627,
title = {Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analitic integral in many-dimensional system. I. Basic results: Separatrices of hyperbolic periodic points.},
journal = {Collectanea Mathematica},
volume = {50},
year = {1999},
pages = {119-197},
zbl = {0945.37014},
mrnumber = {MR1706235},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:42627}
}
Dovbysh, Sergei A. Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analitic integral in many-dimensional system. I. Basic results: Separatrices of hyperbolic periodic points.. Collectanea Mathematica, Tome 50 (1999) pp. 119-197. http://gdmltest.u-ga.fr/item/urn:eudml:doc:42627/