Bifurcations of limit cycles in complex networks of Hamiltonian systems and Lloyd's conjecture

Abstract

This paper is motivated by Hilbert's sixteenth problem, concerning the number and distribution of limit cycles in planar polynomial vector fields. We propose an original method, inspired by the complex systems framework, in order to construct a family of Hamiltonian systems admitting non-degenerate centers located on the vertices of a planar network. We introduce two novel polynomial perturbations adapted to those systems, with the aim to produce a high number of limit cycles; the first one is oriented along the gradient of the unperturbed system, and the second one is again constructed with a complex systems approach. Using the Melnikov integral, we reach, in a small number of particular cases, the lower bound of cubical order for the Hilbert number H(n), conjectured by Lloyd in 1988.