A route to chaos in the Boros-Moll map

TL;DR

This paper investigates the dynamics of the Boros-Moll map, revealing a unique unbounded chaotic region and analyzing bifurcations, thus providing insights into its complex chaotic behavior within a family of related maps.

Contribution

It uncovers the specific conditions leading to chaos in the Boros-Moll map and details the bifurcation phenomena in its unfolding family.

Findings

Existence of an unbounded invariant chaotic region in the Boros-Moll map.

Identification of key bifurcations, including contact and homoclinic bifurcations.

Description of additional bifurcation phenomena in the unfolding family.

Abstract

The Boros-Moll map appears as a subsystem of a Landen transformation associated to certain rational integrals and its dynamics is related to the convergence of them. In the paper, we study the dynamics of a one-parameter family of maps which unfolds the Boros-Moll one, showing that the existence of an unbounded invariant chaotic region in the Boros-Moll map is a peculiar feature within the family. We relate this singularity with a specific property of the critical lines that occurs only for this special case. In particular, we explain how the unbounded chaotic region in the Boros-Moll map appears. Special attention is devoted to explain the main contact/homoclinic bifurcations that occur in the family. We also report some other bifurcation phenomena that appear in the considered unfolding.