JournalsaihpcVol. 36, No. 7pp. 1941–1957

Infinite orbit depth and length of Melnikov functions

• Pavao Mardešić

Université de Bourgogne, Institute de Mathématiques de Bourgogne - UMR 5584 CNRS, Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, BP 47870, 21078 Dijon, France
• Dmitry Novikov

Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, 7610001, Israel

Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de la Investigación Científica, Circuito exterior, Ciudad Universitaria, 04510, Ciudad de México, Mexico
• Jessie Pontigo-Herrera

Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, 7610001, Israel

In this paper we study polynomial Hamiltonian systems $dF = 0$ in the plane and their small perturbations: $dF + \epsilon \omega = 0$. The first nonzero Melnikov function $M_{\mu } = M_{\mu }(F,\gamma ,\omega )$ of the Poincaré map along a loop γ of $dF = 0$ is given by an iterated integral [3]. In [7], we bounded the length of the iterated integral $M_{\mu }$ by a geometric number $k = k(F,\gamma )$ 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 $dF + \epsilon \omega$ with arbitrary high length first nonzero Melnikov function $M_{\mu }$ along γ. We construct deformations $dF + \epsilon \omega = 0$ whose first nonzero Melnikov function $M_{\mu }$ is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions $M_{\mu }$.