Blenders in center unstable H\'enon-like families: with an application to heterodimensional bifurcations

TL;DR

This paper introduces a family of polynomial maps called center unstable Hénon-like maps, demonstrating they can exhibit blenders and occur near heterodimensional cycles, with implications for complex dynamical behaviors.

Contribution

It provides an explicit construction of blenders in polynomial maps and links their occurrence to heterodimensional cycles using a renormalization approach.

Findings

Existence of blenders in center unstable Hénon-like maps

Blenders occur near certain heterodimensional cycles

Connection established between blenders and heterodimensional cycles

Abstract

We give an explicit family of polynomial maps called center unstable H\'enon-like maps and prove that they exhibits blenders for some parametervalues. Using this family, we also prove the occurrence of blenders near certain non-transverse heterodimensional cycles under high regularity assumptions. The proof involves a renormalization scheme along heteroclinic orbits. We also investigate the connection between the blender and the original heterodimensional cycle.