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Di erential Equations, Vol. 36, No. 12, 2000, pp. 1812{1818. Translated from Di erentsial'nye Uravneniya, Vol. 36, No. 12, 2000, pp. 1652{1657. Original Russian Text Copyright c 2000 by Sadovskii. ORDINARY DIFFERENTIAL EQUATIONS A. P. Sadovskii Grodno State University, Grodno, Belarus Received April 1, 1999 Consider the system 2 3 3 dx=dt = x + Ax + Bx +4Cy ; (1) dy=dt = −y + Mxy=3+ Nx y=3; where A, B, C , M,and N are complex constants. For system (1), there exists a formal series H = xy + f (x;y), where f is a homogeneous polynomial of degree k, such that k k k=3 k+1 dH=dt = v (xy) : (2) k=1 The coecients v in (2) are referred to as the focal quantities of system (1). De nition 1. The equilibrium O(0; 0) of system (1) is called a center if v =0, k =1; 2;::: In the case of a center, system (1) has an integral of the from H = xy + analytic in a neighborhood of x = y =0. De nition 2. The equilibrium O(0; 0) of system (1) is called a focus of order n if v =0, k =1;:::;n−
Differential Equations – Springer Journals
Published: Oct 3, 2004
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