Cubic perturbations of elliptic Hamiltonian vector fields of degree three

TL;DR

This paper investigates the maximum number of small-amplitude limit cycles that can bifurcate from period annuli of a specific class of perturbed elliptic Hamiltonian vector fields of degree three, providing bounds and examples.

Contribution

It establishes explicit bounds on the number of limit cycles bifurcating from period annuli under cubic perturbations of degree four Hamiltonian systems.

Findings

At most 5, 7, or 8 limit cycles can bifurcate in different geometric cases.

The bounds are exact in the interior eight-loop case.

Examples of systems with 6 limit cycles are provided in the saddle-loop case.

Abstract

The purpose of the present paper is to study the limit cycles of one-parameter perturbed plane Hamiltonian vector field $X_\varepsilon$ $$ X_\varepsilon : \left\{ \begin{array}{llr} \dot{x}=\;\; H_y+\varepsilon f(x,y)\\ \dot{y}=-H_x+\varepsilon g(x,y), \end{array} \;\;\;\;\; H~=\frac{1}{2} y^2~+U(x) \right. $$ which bifurcate from the period annuli of $X_0$ for sufficiently small $\varepsilon$. Here $U$ is a univariate polynomial of degree four without symmetry, and $f, g$ are arbitrary cubic polynomials in two variables. We take a period annulus and parameterize the related displacement map $d(h,\varepsilon)$ by the Hamiltonian value $h$ and by the small parameter $\varepsilon$. Let $M_k(h)$ be the $k$-th coefficient in its expansion with respect to $\varepsilon$. We establish the general form of $M_k$ and study its zeroes. We deduce that the period annuli of $X_0$ can produce for sufficiently small $\varepsilon$, at most 5, 7 or 8 zeroes in the interior eight-loop case, the saddle-loop case, and the exterior eight-loop case respectively. In the interior eight-loop case the bound is exact, while in the saddle-loop case we provide examples of Hamiltonian fields which produce 6 small-amplitude limit cycles. Polynomial perturbations of $X_0$ of higher degrees are also studied.