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 Izv. Vyssh. Uchebn. Zaved. Mat., 2015, Number 8, Pages 14–24 (Mi ivm9024)

Application of normalized key functions in a problem of branching of periodic extremals

E. V. Derunova, Yu. I. Sapronov

Chair of Mathematical Modeling, Voronezh State University, 1 University sq., Voronezh, 394006 Russia

Abstract: In this paper we construct a procedure of approximate calculation and analysis of branches of bifurcating solutions to a periodic variational problem. The goal of the work is a study of bifurcation of cycles in dynamic systems in cases of double resonances $1:2:3$, $1:2:4$, $p:q:p+q$ and others. An ordinary differential equation (ODE) of the sixth order is considered as a general model equation. Application of the Lyapunov–Schmidt method and transition to boundary and angular singularities allow to simplify a description of branches of extremals and caustics. Also we list systems of generators of algebraic invariants under an orthogonal semi-free action of the circle on $\mathbb R^6$ and normal forms of the main part of the key functions.

Keywords: Fredholm functionals, extremals, circular symmetry, resonance, bifurcation, Lyapunov–Schmidt method.

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2015, 59:8, 9–18

Document Type: Article
UDC: 517.958

Citation: E. V. Derunova, Yu. I. Sapronov, “Application of normalized key functions in a problem of branching of periodic extremals”, Izv. Vyssh. Uchebn. Zaved. Mat., 2015, no. 8, 14–24; Russian Math. (Iz. VUZ), 59:8 (2015), 9–18

Citation in format AMSBIB
\Bibitem{DerSap15} \by E.~V.~Derunova, Yu.~I.~Sapronov \paper Application of normalized key functions in a~problem of branching of periodic extremals \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2015 \issue 8 \pages 14--24 \mathnet{http://mi.mathnet.ru/ivm9024} \transl \jour Russian Math. (Iz. VUZ) \yr 2015 \vol 59 \issue 8 \pages 9--18 \crossref{https://doi.org/10.3103/S1066369X15080022} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84937931032} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. D. V. Kostin, “Bifurcations of resonance oscillations and optimization of the trigonometric impulse by the nonsymmetry coefficient”, Sb. Math., 207:12 (2016), 1709–1728
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