Bifurcations of limit cycles in a Z4-equivariant planar polynomial vector field of degree 7

One of the main problems in the qualitative theory of real planar differential systems is the determination of number and relative positions of limit cycles. The problem concerns “the most elusive” second part of Hilbert’s 16th problem (see [Smale, 1998; Lloyd, 1988]). In 1983, Jibin Li (see [Li, 2003; Li & Li, 1985; Li & Liu, 1991, 1992]) posed a method of detection functions to investigate possible number and configurations of limit cycles for a Zq-equivariant perturbed polynomial Hamiltonian vector field. By using this method, several results have been obtained. Let H(n) be the maximal number of limit cycles of a polynomial vector field (En) of degree n. We now know that H(2) ≥ 4 (see [Ye, 1986]); H(3) ≥ 12 (see [Yu & Han, 2004]); H(5) ≥ 24 (see [Li et al., 2002]); H(7) ≥ 49 (see [Li & Zhang, 2004]); H(9) ≥ 80 (see Wang et al., 2005]); The numberH(n) of limit cycles of the system (En) grows at least as rapidly as μ(n + 1)2 ln(n + 1), μ = (4 ln 2)(−1) (see [Li et al., 2003; Li, 2003]). And for the weakened Hilbert’s 16th problem (see [Arnold, 1977]), we conjectured that H(2k + 1) ≥ (2k + 1)2 − 1 for k ∈ Z+ in [Li et al., 2002]. We notice that the paper given by [Li & Zhang, 2004] only considered a Z8-equivariant perturbed polynomial Hamiltonian vector field of degree 7. In this paper, we shall use the method of detection functions to study the following Z2-equivariant perturbed polynomial Hamiltonian vector field of degree 7: