Bifurcation of 2-periodic orbits from non-hyperbolic fixed points

TL;DR

This paper introduces the concept of 2-cyclicity for one-dimensional maps with non-hyperbolic fixed points, providing a new framework to analyze bifurcations of 2-periodic orbits and applying it to polynomial families.

Contribution

It proposes the novel concept of 2-cyclicity for analyzing bifurcations in one-dimensional maps, extending ideas from planar vector fields.

Findings

Defined 2-cyclicity for families of maps

Analyzed bifurcation of 2-periodic orbits

Applied concept to polynomial map families

Abstract

We introduce the concept of 2-cyclicity for families of one-dimensional maps with a non-hyperbolic fixed point by analogy to the cyclicity for families of planar vector fields with a weak focus. This new concept is useful in order to study the number of 2-periodic orbits that can bifurcate from the fixed point. As an application we study the 2-cyclicity of some natural families of polynomial maps.