# Breakdown of hyperbolicity for quasiperiodic attracting invariant circles in a family of three-dimensional Henon-like maps

## TL;DR

This paper investigates the breakdown of hyperbolicity in quasiperiodic attracting invariant circles within a family of three-dimensional Henon-like maps, revealing collision and loss of smoothness at parameter edges.

## Contribution

It introduces a numerical approach using the Weighted Birkhoff method to analyze hyperbolicity breakdown in 3D Henon-like maps, extending understanding beyond skew product systems.

## Key findings

- Hyperbolicity breaks down at parameter edges.
- Collision and loss of smoothness of Lyapunov bundles observed.
- Lyapunov exponents remain distinct during breakdown.

## Abstract

We numerically study quasiperiodic normally hyperbolic attracting invariant circles that appear for certain parameter values in a family of three-dimensional Henon-like maps. These parameter values make up contour segments in the parameter space where the invariant circles have constant irrational rotation number. At the edges of these segments we find a breakdown of the hyperbolicity of the invariant circle. We observe the collision and loss of smoothness of two of the invariant Lyapunov bundles while the Lyapunov exponents all remain distinct. This is very similar to the breakdown of quasiperiodic normally hyperbolic invariant circles studied in previous works that have mostly focused on skew product type systems along with a few other special types of systems. The numerical tools we use for finding the invariant circles and calculating rotation numbers, Lyapunov exponents and bundles are based on the recently developed Weighted Birkhoff method. To apply all of these tools we need for the invariant circles to be attracting (or repelling) and for the system to be invertible. This is a severe restriction compared to alternative methods, but it is very numerically efficient and allows us to study even highly irregular circles.