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Izv. Akad. Nauk SSSR Ser. Mat., 1987, Volume 51, Issue 5, Pages 1088–1103 (Mi izv1332)  

This article is cited in 8 scientific papers (total in 8 papers)

Splitting of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom

S. L. Ziglin


Abstract: An investigation is made of the phenomenon of splitting of the real and complex separatrices of hyperbolic cycles of systems of differential equations of Hamiltonian type with two degrees of freedom, and of its connection with the absence of additional meromorphic first integrals for these systems. The results obtained are used to prove the absence of a nonconstant meromorphic first integral in a system describing a stationary flow of an ideal incompressible fluid with periodic boundary conditions and with velocity field collinear with its curl.
Bibliography: 15 titles.

Full text: PDF file (2084 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1988, 31:2, 407–421

Bibliographic databases:

UDC: 517.933
MSC: Primary 58F05, 58F21; Secondary 34C35, 76C05, 58F27
Received: 10.10.1985

Citation: S. L. Ziglin, “Splitting of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom”, Izv. Akad. Nauk SSSR Ser. Mat., 51:5 (1987), 1088–1103; Math. USSR-Izv., 31:2 (1988), 407–421

Citation in format AMSBIB
\Bibitem{Zig87}
\by S.~L.~Ziglin
\paper Splitting of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1987
\vol 51
\issue 5
\pages 1088--1103
\mathnet{http://mi.mathnet.ru/izv1332}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=925095}
\zmath{https://zbmath.org/?q=an:0679.34029}
\transl
\jour Math. USSR-Izv.
\yr 1988
\vol 31
\issue 2
\pages 407--421
\crossref{https://doi.org/10.1070/IM1988v031n02ABEH001082}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. L. Ziglin, “The $ABC$-flow is not integrable for $A=B$”, Funct. Anal. Appl., 30:2 (1996), 137–138  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. S. L. Ziglin, “The Absence of an Additional Real-Analytic First Integral in Some Problems of Dynamic”, Funct. Anal. Appl., 31:1 (1997), 3–9  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. G. Haller, “Distinguished material surfaces and coherent structures in three-dimensional fluid flows”, Physica D: Nonlinear Phenomena, 149:4 (2001), 248  crossref
    4. S. L. Ziglin, “An Analytic Proof of the Nonintegrability of the ABC-flow for $A=B=C$”, Funct. Anal. Appl., 37:3 (2003), 225–227  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. V. V. Kozlov, “Sila Lorentsa i ee obobscheniya”, Nelineinaya dinam., 7:3 (2011), 627–634  mathnet
    6. Jaume Llibre, Clàudia Valls, “A note on the first integrals of the ABC system”, J. Math. Phys, 53:2 (2012), 023505  crossref
    7. A. Luque, D. Peralta-Salas, “Motion of Charged Particles in ABC Magnetic Fields”, SIAM J. Appl. Dyn. Syst, 12:4 (2013), 1889  crossref
    8. L. V. Lokutsievskiy, Yu. L. Sachkov, “Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater”, Sb. Math., 209:5 (2018), 672–713  mathnet  crossref  crossref  adsnasa  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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