### 4 Recherche d'orbites périodiques d'un champ hamiltonien associé à une structure symplectique non standard

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The global convergence of a direct method for determining turning (limit) points of a parameter-dependent mapping is analysed. It is assumed that the relevant extended system has a singular root for a special parameter value. The singular root is clasified as a $bifurcationsingularity$ (i.e., as a $degenerate$ turning point). Then, the Theorz for Imperfect Bifurcation offers a particular scenario for the split of the singular root into a finite number of regular roots (turning points) due to a given parameter imperfection. The relationship...

Saddle connections and subharmonics are investigated for a class of forced second order differential equations which have a fixed saddle point. In these equations, which have linear damping and a nonlinear restoring term, the amplitude of the forcing term depends on displacement in the system. Saddle connections are significant in nonlinear systems since their appearance signals a homoclinic bifurcation. The approach uses a singular perturbation method which has a fairly broad application to saddle...

We prove that the lowest upper bound for the number of isolated zeros of the Abelian integrals associated to quadratic Hamiltonian vector fields having a center and an invariant straight line after quadratic perturbations is one.

Let $\mathcal{A}$ be the real vector space of Abelian integrals$$I\left(h\right)=\int {\int}_{\{H\le h\}}R(x,y)dx\wedge dy,\phantom{\rule{0.277778em}{0ex}}h\in [0,\tilde{h}]$$where $H(x,y)=({x}^{2}+{y}^{2})/2+...$ is a fixed real polynomial, $R(x,y)$ is an arbitrary real polynomial and $\{H\le h\}$, $h\in [0,\tilde{h}]$, is the interior of the oval of $H$ which surrounds the origin and tends to it as $h\to 0$. We prove that if $H(x,y)$ is a semiweighted homogeneous polynomial with only Morse critical points, then $\mathcal{A}$ is a free finitely generated module over the ring of real polynomials $\mathbb{R}\left[h\right]$, and compute its rank. We find the generators of $\mathcal{A}$ in the case when $H$ is an arbitrary cubic polynomial. Finally we...