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(2016)
Exact estimates of the number of limit cycles of autonomous systems with three stationary points on a plane, Vestsi Nats
A. Grin (2006)
Reduction to transversality of curves in the construction of a Dulac functionDifferential Equations, 42
F. Dumortier, Joan Ferragud, J. Llibre (2006)
Qualitative Theory of Planar Differential Systems
Y. Il'yashenko (2002)
Centennial History of Hilbert’s 16th ProblemBulletin of the American Mathematical Society, 39
A. Grin, K. Schneider, L. Cherkas (2011)
Dulac-Cherkas functions for generalized Liénard systemsElectronic Journal of Qualitative Theory of Differential Equations
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Dulac function of polynomial autonomous systems on a plane, Differ
A. Grin, S. Rudevich (2019)
Dulac-Cherkas Test for Determining the Exact Number of Limit Cycles of Autonomous Systems on the CylinderDifferential Equations, 55
A. Gasull, H. Giacomini (2006)
Upper bounds for the number of limit cycles through linear differential equationsPacific Journal of Mathematics, 226
A. Grin, A. Kuz’mich (2017)
Dulac–Cherkas criterion for exact estimation of the number of limit cycles of autonomous systems on a planeDifferential Equations, 53
(1966)
Kachestvennaya teoriya dinamicheskikh sistem vtorogo poryadka (Qualitative Theory of Dynamical Systems of the Second Order)
(2011)
A new approach to study limit cycles on a cylinder
For autonomous systems of differential equations with smooth right-hand sides, in asimply connected domain of the real phase plane we consider the problem of finding the exactnumber of limit cycles surrounding one or several stationary points with total Poincaréindex \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$+1$$\end{document}. A two-step method is proposed for solving thisproblem. At the first step, using the Dulac–Cherkas test, we find nested closed curves that haveno common points and split the simply connected domain into doubly connected subdomains,with each curve being transversal to the vector field of the system and surrounding all stationarypoints. This allows finding an upper bound for the number of limit cycles, because the system hasexactly one limit cycle in each of the interior doubly connected subdomains, while the exteriordoubly connected subdomain contains at most one limit cycle. The second step is performed tocheck the existence of a limit cycle in the exterior subdomain. At this step, using once more theDulac–Cherkas test, or the Dulac test, or their generalizations, we construct an auxiliary closedcurve transversal to the vector field and surrounding the previously found closed curves. Theefficiency of the method is demonstrated by examples of polynomial Liénard systems,including a generalized van der Pol system, for which we establish the uniqueness of the limit cyclein the entire phase plane.
Differential Equations – Springer Journals
Published: Apr 1, 2020
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