Abstract
The problem of exact nonlocal estimation of the number of limit cycles surrounding one point of rest in a simply connected domain of the real phase space is considered for autonomous systems of differential equations with continuously differentiable right-hand sides. Three approaches to solving this problem are proposed that are based on sequential two-step usage of the Dulac–Cherkas criterion, which makes it possible to find closed transversal curves dividing the connected domain in doubly connected subdomains that surround the point of rest, with the system having precisely one limit cycle in each of them. The effectiveness of these approaches is exemplified with polynomial Liènard systems, a generalized van der Pol system, and a perturbed Hamiltonian system. For some systems, the derived estimate holds true in the entire phase space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ilyashenko, Y., Centennial history of Hilbert’s 16th problem, Bull. Amer. Math. Soc., 2002, vol. 39, pp. 301–354.
Andronov, A.A., Leontovich, E.A., Gordon, I.I., and Maier, A.G., Kachestvennaya teoriya dinamicheskikh sistem vtorogo poryadka (Qualitative Theory of Dynamical Second-Order Systems), Moscow, 1966.
Ye, Y., Theory of limit cycles, Trans. Amer. Math. Soc. Monogr., Providence, RI., 1986, vol. 66.
Zhang, Z., Ding, T., Huang, W., and Dong, Z., Qualitative theory of differential equations, Transl. Math. Monogr., Providence, RI: American Mathematical Society, 1992, vol. 101.
Cherkas, L.A., The Dulac function for polynomial autonomous systems on a plane, Differ. Equations, 1997, vol. 33, no. 5, pp. 692–701.
Cherkas, L.A., Grin’, A.A., and Bulgakov, V.I., Konstruktivnye metody issledovaniya predel’nykh tsiklov avtonomnykh sistem vtorogo poryadka (chislenno-algebraicheskii podkhod) (Constructive Methods for Studying the Limit Cycles of Autonomous Second-Order Systems (Numerical-Algebraic Approach)), Grodno, 2013.
Cherkas, L.A. and Grin’, A.A., On a Dulac function for the Kukles system, Differ. Equations, 2010, vol. 46, no. 6, pp. 818–826.
Cherkas, L.A., Grin, A.A., and Schneider, K.R., Dulac–Cherkas functions for generalized Liènard systems, Electron. J. Qual. Theory Differ. Equ., 2011, no. 35, pp. 1–23; http://www.math.u-szeged.hu/ ejqtde/.
Cherkas, L.A., Grin, A.A., and Schneider, K.R., A new approach to study limit cycles on a cylinder, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 2011, vol. 18, no. 6, pp. 839–851.
Grin’, A.A., Reduction to transversality of curves in the construction of a Dulac function, Differ. Equations, 2006, vol. 42, no. 6, pp. 896–900.
Cherkas, L.A. and Grin’, A.A., Bendixson–Dulac criterion and reduction to global uniqueness in the problem of estimating the number of limit cycles, Differ. Equations, 2010, vol. 46, no. 1, pp. 61–69.
Kuz’mich, A.V., Extraction of the class of Kukles generalized systems with unique limit cycle, Vestn. Grodn. Gos. Univ. Ser. 2 Mat., Fiz., Inform., 2015, no. 3 (199), pp. 18–26.
Grin, A.A. and Schneider, K., On the construction of a class of generalized Kukles systems having at most one limit cycle, J. Math. Anal. Appl., 2013, vol. 408, no. 2, pp. 484–497.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.A. Grin’, A.V. Kuz’mich, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 2, pp. 174–182.
Rights and permissions
About this article
Cite this article
Grin’, A.A., Kuz’mich, A.V. Dulac–Cherkas criterion for exact estimation of the number of limit cycles of autonomous systems on a plane. Diff Equat 53, 171–179 (2017). https://doi.org/10.1134/S0012266117020033
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266117020033