A generalization of Françoise's algorithm for calculating higher order Melnikov functions
Let \(F(x,y)\in \mathbb{C}[x,y]\) be a complex polynomial. It is known that a finite set \(A\) of atypical values such that \(F\) defines a locally trivial fibration \(F: \mathbb{C}^2\setminus F^2(B)\to B\), \(B= \mathbb{C}\setminus A\), is called Milnor fibration. Taking the first homology groups \(H_1\) of the fibers, \(F_t= F^{-1}(t)\), \(t\in B\), gives the homology fibration, \(F_*: \bigcup_{t\in B}H_1(B)\). Let \(C\) be a section of the homology fibration defined on \(B\). Consider a polynomial one-form \(\omega\). We study the perturbation of the Hamiltonian form \(dF\), \(dF+\varepsilon\omega\). Choose a transveral parameterized by the values of the Hamiltonian, and denote \(P_\varepsilon(t)\) the holonomy of the foliation, \(dF+\varepsilon\omega\) along \(C\). It is analytic and can be written as \(P_\varepsilon(t)= t+\sum_{l\geq 1} \varepsilon^l M_l(t)\). If everything is real, then \(P_\varepsilon(t)\) is the Poincaré map. The functions \(M_\varepsilon(t)\) are called the Melnikov functions. Françoise gives an algorithm for calculating the first nonvanishing Melnikov function, \(M_l\), of a small polynomial perturbation of a Hamiltonian vector field and shows that \(M_l\) is given by an Abelian integral. This is done under the condition that vanishing of an Abelian integral of any polynomial form on the family of cycles implies that the form is algebraically relatively exact. The authors study a simple example where Françoise's condition is not verified. They generalize Françoise's algorithm to this case and show that \(M_l\) belongs to the \(C[\log t,t,1/t]\) module above the Abelian integrals. Wonderful illustrations are given when lifting the path corresponds to lifting four layers in the fiber. The paper is very well written and contains much of the necessary mathematics to make very understandable.
scientific article; zbMATH DE number 1888297
scientific article