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 En: ż = iz + Rn(z, z̄), where Rn is a homogeneous polynomial of degree n, in this note we construct two concrete examples, E4 and E5, 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: E4: ż= iz+ 2iz4 + izz̄3 + √ 52278 20723 e z̄4, where θ / ∈ {kπ ± π6 , kπ + π2 , k ∈ Z}, and E5: ż= iz+ 3z5 + √ 20(c+ 3) 9c2 − 15 z 4z̄+ zz̄4 + √ 20(c+ 3)c2 9c2 − 15 z̄ 5, where c is the root between (−3,−√5/3) of the equation 4155c6 − 10716c5 − 63285c4 − 18070c3 + 168075c2 + 205450c+ 60375 = 0. © 2007 Elsevier Masson SAS. All rights reserved.