Bifurcation of limit cycles and integrability conditions for 6-parameter families of polynomial vector fields of arbitrary degree

摘要

We characterize the center variety and the maximum order of any focus of the 6-parameter family of real planar polynomial vectors given, in complex notation, by 2(over dot) = iz + Az(n1) (Z) over bar (j1) + Bz(n2) (Z) over bar (j2) + Cz(n3) (Z) over bar (j3), where A, B, C is an element of C {0}, (n(1), j(1)) not equal (n(2), j(2)) not equal (n(3,) j(3)), n(k) + j(k) > 1 for k = 1,2,3n(1) + j(1) = n(2) + j(2) = n(3) + j(3,) vertical bar 1 - n(3) + j(3)vertical bar = vertical bar 1 - n(2) + j(2)vertical bar = vertical bar 1 - n(2) + j(2)vertical bar not equal vertical bar 1 - n(1) + j(1)vertical bar and (1 - n(1) + j(1))(1 - n(2) + j(2)) not equal 0. (C) 2008 Elsevier Ltd. All rights reserved.