Centre manifold variety for eight-parameter families of polynomial vector fields of arbitrary degree

Abstract

We characterize the centre variety of the eight-parameter families of real planar polynomial vector fields given, in complex notation, by , where A,B,C,D ∈ ℂ \{0}, (n₁,j₁) ≠ (n₂,j₂) ≠ (n₃,j₃) ≠ (n₄,j₄), n_k + j_k > 1 for k = 1,2,3,4, n₁ + j₁ = n₂ + j₂ = n₃ + j₃ = n₄ + j₄, |1 − n₃ + j₃| = |1 − n₂ + j₂| ≠ |1 − n₁ + j₁| and j₄ = n₄ − 1.