Abstract
We propose new sufficient conditions for saddle-node bifurcations in one- and twoparametric dynamical systems obtained with the parameter functionalization method, derive asymptotic formulas for the resulting solutions, and analyze their stability. As an application, we consider the synchronization problem for periodic oscillations of an autonomous Van der Pol generator under an external harmonic influence.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Guckenheimer, J. and Holmes, P.J., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, New York: Springer-Verlag, 1983. Translated under the title Nelineinye kolebaniya, dinamicheskie sistemy i bifurkatsii vektornykh polei, Moscow–Izhevsk: Inst. Komp. Issled., 2002.
Arnol’d, V.I., Geometricheskie metody v teorii obyknovennykh differentsial’nykh uravnenii (Geometric Methods in the Theory of Ordinary Differential Equations), Moscow–Izhevsk: Regulyarnaya i Khaoticheskaya Dinamika, 2000.
Kuznetsov, Yu.A., Elements of Applied Bifurcation Theory, New York: Springer, 1998.
Chow, S.-N., Li, C., and Wang, D., Normal Forms and Bifurcation of Planar Vector Fields, Cambridge: Cambridge Univ. Press, 1994. Translated under the title Normal’nye formy i bifurkatsii vektornykh polei na ploskosti, Moscow: MTsNMO, 2005.
Shil’nikov, L.P., Shil’nikov, A.L., Turaev D.V., et al., Metody kachestvennoi teorii v nelineinoi dinamike (Methods of Qualitative Theory in Nonlinear Dynamics), Part 2, Moscow–Izhevsk: Inst. Komp. Issled., 2009.
Krasnosel’skii, M.A. and Zabreiko, P.P., Geometricheskie metody nelineinogo analiza (Geometric Methods of Nonlinear Analysis), Moscow: Nauka, 1975.
Kozyakin, V.S. and Krasnosel’skii, M.A., Method of Parameter Functionalization in the Problem of Bifurcation Points, Dokl. Akad. Nauk SSSR, 1980, vol. 254, no. 5, pp. 1061–1064.
Ibragimova, L.S. and Yumagulov, M.G., Parameter Functionalization and Its Application to the Problem of Local Bifurcations in Dynamic Systems, Autom. Remote Control, 2007, vol. 68, no. 4, pp. 573–582.
Vyshinskii, A.A., Ibragimova, L.S., Murtazina, S.A., et al., The Operator Method for Approximate Study of Proper Bifurcation in Multiparametric Dynamical Systems, Ufim. Mat. Zh., 2010, vol. 2, no. 4, pp. 3–26.
Krasnosel’skii, M.A., Vainikko, G.M., Zabreiko, P.P., et al., Priblizhennoe reshenie operatornykh uravnenii (Approximate Solution of Operator Equations), Moscow: Nauka, 1969.
Anishchenko, V.S. and Vadivasova, T.E., Lektsii po nelineinoi dinamike (Lectures on Nonlinear Dynamics), Moscow–Izhevsk: NITs “Regulyarnaya i Khaoticheskaya Dinamika,” 2011.
Kato, T., Perturbation Theory for Linear Operators, Berlin: Springer, 1966. Translated under the title Teoriya vozmushchenii lineinykh operatorov, Moscow: Mir, 1972.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © M.G. Yumagulov, E.S. Imangulova, 2017, published in Avtomatika i Telemekhanika, 2017, No. 4, pp. 63–77.
This paper was recommended for publication by A.M. Krasnosel’skii, a member of the Editorial Board
Rights and permissions
About this article
Cite this article
Yumagulov, M.G., Imangulova, E.S. The parameter functionalization method for the problem of saddle-node bifurcations in dynamical systems. Autom Remote Control 78, 630–642 (2017). https://doi.org/10.1134/S0005117917040051
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117917040051