2017/03/10 by Mardesic, Pavao, Novikov, Dmitry, Ortiz-Bobadilla, Laura +1 · 1 citation
#34C05 #34C07 #34C08 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1703.03837
We consider small polynomial deformations of integrable systems of the form dF=0, F∈ℂ[x,y] and the first nonzero term Mμ of the displacement function Δ(t,ε)=∑i=μMi(t)εi along a cycle γ(t)∈ F-1(t). It is known that Mμ is an iterated integral of length at most μ. The bound μ depends on the deformation of dF. In this paper we give a universal bound for the length of the iterated integral expressing the first nonzero term Mμ depending only on the topology of the unperturbed system dF=0. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for Mμ to be given by an abelian integral i.e. by an iterated integral of length 1. We conjecture that our bound is optimal.