A quartic system and a quintic system with fine focus of order 18

By using an effective complex algorithm to calculate the Lyapunov constants of polynomial systems E,,: z = iz + R-n(z,(z) over bar), where R-n, is a homogeneous polynomial of degree n, in this note we construct two concrete examples, E-4 and E-5, such that in both cases, the corresponding orders of fine focus can be as high as 18. The systems are given, respectively, by the following ordinary differential equations: E-4: z = iz + 2iz(4) + iz (z) over bar (3) + root 52278/20723ei theta(z) over bar (4) where theta is not an element of {k pi +/- pi/6, k pi + pi/2, k epsilon Z}, and E-5: z = iz + 3z(5) + root 20(c + 3)/9c(2) - 15 z(4)(z) over bar + z (z) over bar + root 20(c + 3)c(2/9)c(2) - 15 (z) over bar (5,) where c is the root between (-3, -root 5/3) of the equation 4155c(6) - 10716c(5) - 63285c(4) - 18070c(3) + 168075c(2) + 205450c + 60375 = 0. (c) 2007 Elsevier Masson SAS. All rights reserved.Mathematics, AppliedSCI(E)0ARTICLE3205-21713