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### Open Access Journals

DCDS

This paper aims at providing an example of a cubic Hamiltonian 2-saddle cycle
that after bifurcation can give rise to an alien limit cycle; this is a limit
cycle that is not controlled by a zero of the related Abelian integral. To
guarantee the existence of an alien limit cycle one can verify generic
conditions on the Abelian integral and on the transition map associated to the
connections of the 2-saddle cycle. In this paper, a general method is
developed to compute the first and second derivative of the transition map
along a connection between two saddles. Next, a concrete generic Hamiltonian
2-saddle cycle is analyzed using these formula's to verify the generic
relation between the second order derivative of both transition maps, and a
calculation of the Abelian integral.

DCDS

The paper deals with the cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. We generalize the results in [1] (case $m=1$), providing a substantially simpler and more transparant proof than the one used in [1].

CPAA

It is known that perturbations from a Hamiltonian 2-saddle cycle
$\Gamma $can produce limit cycles that are not covered by the
Abelian integral, even when the Abelian integral is generic. These
limit cycles are called alien limit cycles. In this paper,
extending the results of [6] and [2], we investigate
the number of alien limit cycles in generic multi-parameter rigid
unfoldings of the Hamiltonian 2-saddle cycle, keeping one
connection unbroken at the bifurcation.

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