Let $H\in \mathbb{R}[x,y]$, and assume that the Hamiltonian foliation $dH=0$ in $\mathbb{R}^2$ has a continuous family of cycles $\gamma(t)\subset {H=t}$. We consider a deformation \(dH+\varepsilon\eta=0,\) where $\eta$ is a polynomial 1-form and $\varepsilon$ is a small parameter. The question is then what happens to the family of cycles $\gamma(t)$ under this deformation.

To study this problem, we complexify the foliations and consider the displacement map \(\Delta(t,\varepsilon) =\varepsilon M_1(t)+\varepsilon^2 M_2(t)+\cdots,\) where the functions $M_j(t)$ are analytic in a neighborhood of a regular value $t_0$ of $H$ and are called Melnikov functions (or Poincaré–Pontryagin functions).

Depending on whether $\Delta\equiv0$ or $\Delta\not\equiv0$, the family either persists as periodic orbits or gives rise to limit cycles. In this context, the Melnikov functions provide essential information. It follows from Françoise’s algorithm that if $\Delta\not\equiv0$, then the first nonzero Melnikov function $M_\mu$ can be expressed in terms of iterated integrals of length at most $\mu$. However, this bound depends explicitly on the deformation $\eta$.

On the other hand, in 2018 we showed that there exists a constant $\kappa$, depending only on $H$ and on the orbit under monodromy of $\gamma(t_0)$, that bounds the length of the iterated integrals appearing in $M_\mu$. This constant was called the orbit depth. Later, however, we exhibited an example showing that the orbit depth can be infinite. This motivated us to develop new approaches for obtaining bounds on the length of the iterated integrals appearing in Melnikov functions. In this talk, I will explain the problem of bounding the length of Melnikov functions. The talk by P. Mardesic will continue this discussion and present recent joint work in this direction.