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SM ISO690:2012 GRIN, Alexander, KUZMICH, Andrei. The precise estimation of limit cycles number for planar autonomous system. In: Mathematics and IT: Research and Education, 1-3 iulie 2021, Chişinău. Chișinău, Republica Moldova: 2021, pp. 38-39. |
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Mathematics and IT: Research and Education 2021 | ||||||
Conferința "Mathematics and IT: Research and Education " Chişinău, Moldova, 1-3 iulie 2021 | ||||||
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Pag. 38-39 | ||||||
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Rezumat | ||||||
For smooth autonomous systemsformulathe problem of precise non-local estimation of the limit cycles number in a simply-connected domain of a real phase plane containing one or three equilibrium points with a total Poincar´e index +1 is considered. To solve this problem, we present new approaches that are based on a sequential two-step construction of Dulac or Dulac-Cherkas or their modifications [1,2], which provide the closed transversal curves decomposing the simply-connected domain in simply-connected subdomains and doubly-connected subdomains. Theorem 1. Suppose that in a simply-connected domain system (1) has the unique anti-saddle equilibrium point O, and ª is the Dulac-Cherkas function of system (1) with k < 0 in the domain , where the set W = f(x; y) 2 : ª(x; y) = 0g consists of s embedded ovals !i surrounding the point O. Then, system (1) has exactly one limit cycle in each of the s¡1 ring-shaped subdomains I that are bounded by neighboring ovals !i and !i+1 and can have at most 1 limit cycle in the domain s. To obtain a precise estimate of the number of limit cycles, it is necessary to detect existence or absence of a limit cycle in the region s. Our idea is based on the following result. Theorem 2. Suppose that the assumptions of Theorem 1 are valid and system (1) has a closed transversal curve V that lies in a doubly connected subdomain s that surrounds the external oval of the curve W, two of them forming the boundary of a ring-shaped domain ~ s ½ s: If the trajectories of system (1) enter, as t increases, the interior of the domain @~ s from outside (or vice versa) through the boundary ~ s, then there exists the unique stable (or unstable) limit cycle of system (1), in the subdomain ~ s and system (1) has exactly s limit cycles in the domain in total. Our approaches for the construction of the transversal curve V , satisfying to the requirements of Theorem 2, are presented in [1, 2]. The developed approaches are efficiently applied to several polynomial systems of Linard type, for which it is proved that there exist a limit cycle in each of the doubly-connected subdomains. |
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