Revistes Catalanes amb Accés Obert (RACO)

Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analytic integral in many-dimensional systems I. Basic result: Separatrices of hyperbolic periodic points

Sergei Aleksandrovich Dovbysh


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 manydimensional 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.

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