2019/07/22 by Mardesic, Pavao, Novikov, Dmitry, Ortiz-Bobadilla, Laura +1 · 1 citation
#34C05 #34C07 #34C08 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1907.09627
In this paper we study polynomial Hamiltonian systems dF=0 in the plane and their small perturbations: dF+εω=0. The first nonzero Melnikov function Mμ=Mμ(F,γ,ω) of the Poincaré map along a loop γ of dF=0 is given by an iterated integral. In a previous work (see arXiv 1703.03837), we bounded the length of the iterated integral Mμ by a geometric number k=k(F,γ) which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system F and its orbit γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations dF+εω with arbitrary high length first nonzero Melnikov function Mμ along γ. We construct deformations dF+εω=0 whose first nonzero Melnikov function Mμ is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions Mμ.