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Izv. Vyssh. Uchebn. Zaved. Mat., 2014, Number 1, Pages 31–40 (Mi ivm8860)  

Two-party graphs and monotonicity properties of the Poincaré mapping

V. G. Il'icheva, A. A. Zeleninb

a Department of Sea and Ecosystem Research, Southeren Scientific Center of Russian Academy of Sciences, 41 Chekhov Ave., Rostov-on-Don, 344006 Russia
b Chair of Aviation Electrosystems and Aviation Instruments, Moscow Technical State University of Civil Aviation (Rostov Branch), 268 Sholokhov Ave., Rostov-on-Don, 344009 Russia

Abstract: The local differential of a system of nonlinear differential equations with a $T$-periodic right-hand side is representable as a directed sign interaction graph. Within the class of balanced graphs, where all paths between two fixed vertices have the same signs, it is possible to estimate the sign structure of the differential of the global Poincaré mapping (a shift in time $T$). In this case all vertices of a strongly connected graph naturally break into two sets (two parties). As appeared, the influence of variables within one party is positive, while that of variables from different parties is negative. Even having simplified the structure of a local two-party graph (by eliminating its edges), one can still exactly describe the sign structure of the differential of the Poincare mapping. The obtained results are applicable in the mathematical competition theory.

Keywords: graph, signs of paths, equation, two-party property, strong connectivity, monotonicity, Poincaré mapping.

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2014, 58:1, 27–34

UDC: 517.711+577.4
Received: 03.09.2012

Citation: V. G. Il'ichev, A. A. Zelenin, “Two-party graphs and monotonicity properties of the Poincaré mapping”, Izv. Vyssh. Uchebn. Zaved. Mat., 2014, no. 1, 31–40; Russian Math. (Iz. VUZ), 58:1 (2014), 27–34

Citation in format AMSBIB
\Bibitem{IliZel14}
\by V.~G.~Il'ichev, A.~A.~Zelenin
\paper Two-party graphs and monotonicity properties of the Poincar\'e mapping
\jour Izv. Vyssh. Uchebn. Zaved. Mat.
\yr 2014
\issue 1
\pages 31--40
\mathnet{http://mi.mathnet.ru/ivm8860}
\transl
\jour Russian Math. (Iz. VUZ)
\yr 2014
\vol 58
\issue 1
\pages 27--34
\crossref{https://doi.org/10.3103/S1066369X14010034}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84892505098}


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  • Известия высших учебных заведений. Математика Russian Mathematics (Izvestiya VUZ. Matematika)
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