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.

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 50.2 (1999): 119-197. <http://eudml.org/doc/42627>.

@article{Dovbysh1999,
abstract = {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.},
author = {Dovbysh, Sergei A.},
journal = {Collectanea Mathematica},
keywords = {Dinámica topológica; Funciones analíticas; Análisis multivariante; Problemas hiperbólicos; separatrices of periodic points; homoclinic points; symbolic dynamics; first integrals; non-integrability; heteroclinic points; invariant manifolds},
language = {eng},
number = {2},
pages = {119-197},
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.},
url = {http://eudml.org/doc/42627},
volume = {50},
year = {1999},
}

TY - JOUR
AU - Dovbysh, Sergei A.
TI - 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.
JO - Collectanea Mathematica
PY - 1999
VL - 50
IS - 2
SP - 119
EP - 197
AB - 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.
LA - eng
KW - Dinámica topológica; Funciones analíticas; Análisis multivariante; Problemas hiperbólicos; separatrices of periodic points; homoclinic points; symbolic dynamics; first integrals; non-integrability; heteroclinic points; invariant manifolds
UR - http://eudml.org/doc/42627
ER -