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

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.

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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

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
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
3. G. Haller, “Distinguished material surfaces and coherent structures in three-dimensional fluid flows”, Physica D: Nonlinear Phenomena, 149:4 (2001), 248
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
5. V. V. Kozlov, “Sila Lorentsa i ee obobscheniya”, Nelineinaya dinam., 7:3 (2011), 627–634
6. Jaume Llibre, Clàudia Valls, “A note on the first integrals of the ABC system”, J. Math. Phys, 53:2 (2012), 023505
7. A. Luque, D. Peralta-Salas, “Motion of Charged Particles in ABC Magnetic Fields”, SIAM J. Appl. Dyn. Syst, 12:4 (2013), 1889
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
9. G. V. Gorr, D. N. Tkachenko, E. K. Shchetinina, “Research on the Motion of a Body in a Potential Force Field in the Case of Three Invariant Relations”, Rus. J. Nonlin. Dyn., 15:3 (2019), 327–342
10. Kazuyuki Yagasaki, Shogo Yamanaka, “Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers”, SIGMA, 15 (2019), 049, 17 pp.
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