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Bendixson-Dulac criterion and reduction to global uniqueness in the problem of estimating the number of limit cycles

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Abstract

For autonomous systems on the real plane, we develop a regular method for localizing and estimating the number of limit cycles surrounding the unique singular point. The method is to divide the phase plane into annulus-shaped domains with transversal boundaries in each of which a Dulac function is constructed by solving an optimization problem, which permits one to use the Bendixson-Dulac criterion. We state the principle of reduction to global uniqueness and use it in the case of existence of an Andronov-Hopf function of limit cycles to obtain a sharp global estimate of the number of limit cycles for an individual system as well as for a one-parameter family of such systems in an unbounded domain.

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Authors and Affiliations

  1. Belarus State University of Computer Science and Radio Electronics, Minsk, Belarus

    L. A. Cherkas & A. A. Grin’

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  1. L. A. Cherkas
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  2. A. A. Grin’
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Original Russian Text © L.A. Cherkas, A.A. Grin’, 2010, published in Differentsial’nye Uravneniya, 2010, Vol. 46, No. 1, pp. 59–67.

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Cherkas, L.A., Grin’, A.A. Bendixson-Dulac criterion and reduction to global uniqueness in the problem of estimating the number of limit cycles. Diff Equat 46, 61–69 (2010). https://doi.org/10.1134/S0012266110010076

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  • DOI: https://doi.org/10.1134/S0012266110010076

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