# Shilnikov-type Dynamics in Three-Dimensional Piecewise Smooth Maps

## TL;DR

This paper demonstrates the presence of Shilnikov-type chaotic dynamics in three-dimensional piecewise smooth maps, providing analytical insights and methods to analyze border return times, highlighting complex bifurcation behaviors.

## Contribution

It introduces analytical results for Shilnikov-type dynamics in 3D piecewise maps and presents two methods for analyzing border return times, including a novel two-sided Shilnikov phenomenon.

## Key findings

- Existence of two-sided Shilnikov dynamics in 3D maps
- Analytical conditions for Shilnikov bifurcations
- Two methods for border return time analysis

## Abstract

We show the existence of Shilnikov-type dynamics and bifurcation behaviour in general discrete three-dimensional piecewise smooth maps and give analytical results for the occurence of such dynamical behaviour. Our main example in fact shows a `two-sided' Shilnikov dynamics, i.e. simultaneous looping and homoclinic intersection of the one-dimensional eigenmanfolds of fixed points on both sides of the border. We also present two complementary methods to analyse the return time of an orbit to the border: one based on recursion and another based on complex interpolation.