Bifurcations of limit cycles forming compound eyes in the cubic system (Q1111744)

Let H(n) be the maximal number of limit cycle of a planar real polynomial differential system of degree n and \(C^ k_ m\) denote the nest of k limit cycles enclosing m singular points. By computing detection functions, the authors study bifurcation and phase diagrams in the class of a planar cubic disturbed Hamiltonian system. In particular, the following conclusion is reached: The planar cubic system \((E_ 3)\) has 11 limit cycles, which form the pattern of compound eyes of \(C^ 1_ 9\supset 2[C'_ 3\supset (2C^ 2_ 1)]\) and have the symmetrical structure; so the Hilbert number H(3)\(\geq 11\).