Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cubic perturbations of elliptic Hamiltonian vector fields of degree
  three   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.

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The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: econ