Let H(x, y) be a real cubic polynomial with four distinct critical values (in a complex domain) and let X H = H y ∂ ∂x − H x ∂ ∂y be the corresponding Hamiltonian vector field. We show that there is a neighborhood U of X H in the space of all quadratic plane vector fields, such that any X ∈ U has at most two limit cycles.

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We prove that the higher order Poincaré-Pontryagin functions associated to the perturbed polynomial foliation defined by df − ε (P dx + Qdy) = 0 satisfy a differential equation of Fuchs type. P dx + Qdy( 3) and hence it satisfies a Fuchs equation of order at most r (this bound is exact). The identity (3) goes back at least to Pontryagin [21] and has been probably known to Poincaré. In the case k > 1 the (higher order) Poincaré-Pontryagin function M k is not necessarily of the form (3) with P, Q rational functions. This fact is discussed in Appendix B. We show in section 2 that M k (t) is a linear combination of iterated path integrals of length k along γ(t) whose entries are essentially rational one-forms. This observation is crucial for the proof of Theorem 1. It implies that the monodromy representation of

We study the displacement map associated to small one-parameter polynomial unfoldings of polynomial Hamiltonian vector fields on the plane. Its leading term, the generating function M (t), has an analytic continuation in the complex plane and the real zeroes of M (t) correspond to the limit cycles bifurcating from the periodic orbits of the Hamiltonian flow. We give a geometric description of the monodromy group of M (t) and use it to formulate sufficient conditions for M (t) to satisfy a differential equation of Fuchs or Picard-Fuchs type. As examples, we consider in more detail the Hamiltonian vector fieldṡ z = iz − i(z +z) 3 andż = iz +z 2 , possessing a rotational symmetry of order two and three, respectively. In both cases M (t) satisfies a Fuchs-type equation but in the first example M (t) is always an Abelian integral (that is to say, the corresponding equation is of Picard-Fuchs type) while in the second one this is not necessarily true. We derive an explicit formula of M (t) and estimate the number of its real zeroes.

Scattering theory is a convenient way to describe systems that are subject to time-dependent perturbations which are localized in time. Using scattering theory, one can compute time-dependent invariant objects for the perturbed system knowing the invariant objects of the unperturbed system. In this paper, we use scattering theory to give numerical computations of invariant manifolds appearing in laser-driven reactions. In this setting, invariant manifolds separate regions of phase space that lead to different outcomes of the reaction and can be used to compute reaction rates. V C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4767656] In this paper, we implement numerically the theoretical ideas of Blazevski and de la Llave (2011) using the laser-driven H enon-Heiles system introduced in Kawai et al. (2007). The ideas of this paper are robust and can be used for any system subject to a time-dependent perturbation localized in time. Using the intertwining relations, scattering theory yields a time-dependent conjugacy between the perturbed and unperturbed systems. Assuming one has the relevant invariant manifolds for the unperturbed system, we exploit the conjugacy to compute the corresponding objects for the perturbed system. The algorithm used in this paper to compute invariant manifolds is an alternative to the use of time-dependent normal form theory used in Kawai et al. (2007) and is readily parallelizable.