# Chaotic Dynamics in Nonautonomous Maps: Application to the Nonautomous Henon Map

## TL;DR

This paper extends the theory of chaotic dynamics to nonautonomous two-dimensional maps, providing new conditions for chaos and hyperbolicity, exemplified through the nonautonomous Hénon map.

## Contribution

It introduces a nonautonomous version of Conley-Moser conditions and derives a new sufficient condition for hyperbolic chaotic invariant sets in nonautonomous maps.

## Key findings

- Established a precise definition of chaotic invariant sets for nonautonomous maps.
- Derived new sufficient conditions for hyperbolic chaos in nonautonomous systems.
- Applied the theory to the nonautonomous Hénon map to identify parameter regimes with chaotic behavior.

## Abstract

In this paper we analyze chaotic dynamics for two dimensional nonautonomous maps through the use of a nonautonomous version of the Conley-Moser conditions given previously. With this approach we are able to give a precise definition of what is meant by a chaotic invariant set for nonautonomous maps. We extend the nonautonomous Conley-Moser conditions by deriving a new sufficient condition for the nonautonomous chaotic invariant set to be hyperbolic. We consider the specific example of a nonautonomous H\'enon map and give sufficient conditions, in terms of the parameters defining the map, for the nonautonomous H\'enon map to have a hyperbolic chaotic invariant set.