PRECISE ESTIMATIONS OF LIMIT CYCLES NUMBER OF AUTONOMOUS SYSTEMS WITH THREE EQUILIBRIUM POINTS IN THE PLANE
Abstract
For autonomous systems with smooth right sides the problem of precise non-local estimation of the limit cycles number is considered in a simply-connected domain of a real phase plane containing three equilibrium points with a total Poincaré index +1. To solve this problem, we are constructing successively two Dulac-Cherkas functions which provide the closed transversal curves decomposing the simply-connected domain in simply-connected subdomains, doubly-connected subdomains, and possibly a three-connected subdomain. The efficiency of the developed approach is demonstrated by the examples of
the polynomial Liènard systems, for which it is proved that there exist a limit cycle in each of the doubly-connected subdomains and two limit cycles in the three-connected subdomain. We determine the configurations of these limit cycles. The obtained results can be applied in the qualitative theory and in the theory of bifurcations of ordinary differential equations, as well as in the theory of nonlinear oscillations.
the polynomial Liènard systems, for which it is proved that there exist a limit cycle in each of the doubly-connected subdomains and two limit cycles in the three-connected subdomain. We determine the configurations of these limit cycles. The obtained results can be applied in the qualitative theory and in the theory of bifurcations of ordinary differential equations, as well as in the theory of nonlinear oscillations.
About the Authors
A. A. Hryn
Yanka Kupala State University of Grodno
Russian Federation
Ph. D. (Physics and Mathematics), Assistant Professor, Head of the Department of Mathematical Analysis, Differential Equations and Algebra, Faculty of Mathematics and Informatics
Russian Federation
Ph. D. (Physics and Mathematics), Assistant Professor, Head of the Department of Mathematical Analysis, Differential Equations and Algebra, Faculty of Mathematics and Informatics
A. V. Kuzmich
Yanka Kupala State University of Grodno
Belarus
Senior lecturer, Department of Fundamental and Applied Mathematics, Faculty of Mathematics and Informatics
Belarus
Senior lecturer, Department of Fundamental and Applied Mathematics, Faculty of Mathematics and Informatics
References
1. Ilyashenko Y. Centennial history of Hilbert’s 16th problem. Bulletin of the American Mathematical Society, 2002, vol. 39, pp. 301–355. doi: 10.1090/s0273-0979-02-00946-1.
2. Andronov A.A., Leontovich E.A., Gordon I.I., Majer A.G. Qualitative theory of second order dynamic systems. Moscow, Nauka Publ., 1966. 568 p. (in Russian)
3. Ye Y., Cai S.L., Chen L.S., Huang K.C., Luo D.J., Ma Z.E., Yang X.A. Theory of limit cycles, Translations of Mathematical Monographs, vol. 66. Providence, American Mathematical society, 1986. 435 p.
4. Zhang Z., Ding T., Huang W., Dong Z. Qualitative theory of differential equations. Translations of Mathematical monographs vol.101. Providence, American Mathematical Society, 1992. 461 p.
5. Cherkas L.A. Dulac function for polynomial autonomous system in plane. Differerntial equations, 1997, vol. 33, no. 5, pp. 692–701.
6. Cherkas L.A., Grin A.A., Bulgakov V.I. Constructive methods of investigation of limit cycles of second order autonomous systems (numerical-algebraic approach). Grodno, 2013. 489 p. (in Russian)
7. Cherkas L.A., Grin A.A. On a Dulac function for the Kukles system. Differentsial’nye uravneniia - Differerntial equations, 2010, vol. 46, no. 6, pp. 818–826. doi:10.1134/S0012266110060066.
8. Cherkas, L.A., Grin, A.A., Schneider, K.R. Dulac-Cherkas functions for generalized Lienard systems. Electronic Journal of Qualitative Theory of Differential Equations, 2011, no. 35. Available at: http://www.math.u-szeged.hu/ejqtde/.
9. Kuzmich A.V. Emphasizing of a class of Kukles generalized systems with the unique limit cycle. Vestnik GrGU im. Ianki Kupaly. Ser. 2. Matematika. Fizika. Informatika, vychislitel’naya technika i upravlenie [Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Сomputer Technology and its Сontrol], 2015, no. 3 (199), pp.18–26. (in Russian)
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