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Differential Equations, Vol. 39, No. 6, 2003, pp. 797–806. Translated from Differentsial'nye Uravneniya, Vol. 39, No. 6, 2003, pp. 759–768. Original Russian Text Copyright c 2003 by Cherkas. ORDINARY DIFFERENTIAL EQUATIONS A Precise Estimate of the Number of Limit Cycles of Autonomous Systems on the Plane L. A. Cherkas Belarus State University of Computer Studies and Radioelectronics, Minsk, Belarus Received November 6, 2002 1. INTRODUCTION A new method for estimating the number of limit cycles of the system dx dy = P (x;y); = Q(x;y); (x;y) 2 ;P (x;y);Q(x;y) 2 C ( ); (1) dt dt 1=k 1 with the use of a discontinuous Dulac function of the form B = j (x;y)j , 2 C ( ), k< 0, was widely discussed in [1{5]. In the present paper, we consider practical methods for nding the function (x;y), including the case of an unbounded domain , and present new examples of quadratic systems and polynomial Lienard systems, for which a precise estimate of the number of limit cycles is obtained. 2. AN ALGEBRAIC PROCEDURE FOR FINDING THE FUNCTION (x;y) A method for precisely estimating the number of limit cycles of a structurally stable system (1) in a closed bounded
Differential Equations – Springer Journals
Published: Oct 5, 2004
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