For an analytic function f:ℝⁿ,0 → ℝ,0 having a critical point at the origin, we describe the topological properties of the partition of the family of trajectories of the gradient equation ẋ = ∇f(x) attracted by the origin, given by characteristic exponents and asymptotic critical values.

We provide an effective uniform upper bound for the number of zeros of the first non-vanishing Melnikov function of a polynomial perturbations of a planar polynomial Hamiltonian vector field. The bound depends on degrees of the field and of the perturbation, and on the order $k$ of the Melnikov function. The generic case $k=1$ was considered by Binyamini, Novikov and Yakovenko [BNY10]. The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals.