# 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 (1999)

- Volume: 50, Issue: 2, page 119-197
- ISSN: 0010-0757

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topDovbysh, 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 -

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