Abstract
We consider the problem of estimating the number of limit cycles and their localization for an autonomous polynomial system on the plane with fixed real coefficients and with a small parameter. At the origin, the system has a structurally unstable focus whose first Lyapunov focal quantity is nonzero for the zero value of the parameter. We develop an algebraic method for constructing a Dulac-Cherkas function in a neighborhood of this focus in the form of a polynomial of degree 4. The method is based on the construction of an auxiliary positive polynomial containing terms of order ≥ 4 in the phase variables. The coefficients of these terms are found from a linear algebraic system obtained by equating the coefficients of the corresponding auxiliary function with zero. We present examples in which the suggested method permits one to find parameter intervals and the corresponding neighborhoods of the focus in each of which the number of limit cycles remains constant for all parameter values in the respective interval.
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Original Russian Text © A.A. Grin’, 2014, published in Differentsial’nye Uravneniya, 2014, Vol. 50, No. 1, pp. 3–9.
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Grin’, A.A. Dulac-Cherkas function in a neighborhood of a structurally unstable focus of an autonomous polynomial system on the plane. Diff Equat 50, 1–7 (2014). https://doi.org/10.1134/S0012266114010017
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DOI: https://doi.org/10.1134/S0012266114010017