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In this paper we study polynomial Hamiltonian systems in the plane and their small perturbations: . The first nonzero Melnikov function of the Poincaré map along a loop γ of is given by an iterated integral . In , we bounded the length of the iterated integral by a geometric number 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 with arbitrary high length first nonzero Melnikov function along γ. We construct deformations whose first nonzero Melnikov function is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions .
Cite this article
Pavao Mardešić, Dmitry Novikov, Laura Ortiz-Bobadilla, Jessie Pontigo-Herrera, Infinite orbit depth and length of Melnikov functions. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 7, pp. 1941–1957DOI 10.1016/J.ANIHPC.2019.07.003