Compatibility of nonmaximal lips and origin cyclicity constructions

Determine whether a nonmaximal hierarchy of limit-cycle bifurcations near the Ilyashenko–Kotova lips in the perturbed polynomial Liénard family can be combined with additional limit cycles near the origin.

Background

The paper studies a polynomial Liénard perturbation of a Hamiltonian system possessing a two-cuspidal Ilyashenko–Kotova lips configuration. It constructs parameter regimes producing many limit cycles near the lips, near infinity, and near the origin.

Proposition \ref{prop:P3-not-origin-max} shows that the maximal dominance hierarchies used for the lips and origin constructions cannot be realized simultaneously with arbitrarily strong dominance. The subsequent remark clarifies that this obstruction concerns only the simultaneous maximal constructions; it leaves unresolved whether a weaker, nonmaximal lips construction can coexist with additional cycles generated near the origin.