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 $𝒜$ be the real vector space of Abelian integrals$$I\left(h\right)=\int {\int }_{\{H\le h\}}R(x,y)dx\wedge dy,h\in [0,\tilde{h}]$$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 $𝒜$ is a free finitely generated module over the ring of real polynomials $ℝ\left[h\right]$, and compute its rank. We find the generators of $𝒜$ in the case when $H$ is an arbitrary cubic polynomial. Finally we...

We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on $E={ℝ}^{n+1}$ for which $\partial E={ℝ}^n\times 0$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on ${ℝ}^n\times [0,\infty )$.