# Global continuation of homoclinic solutions

## TL;DR

This paper investigates the global structure of homoclinic solutions in nonautonomous difference equations without assuming small perturbations, using global implicit function theorems and degree theory for Fredholm operators.

## Contribution

It extends bifurcation theory to analyze the entire set of bounded homoclinic solutions in nonautonomous difference equations beyond small perturbations.

## Key findings

- Established a framework for global continuation of homoclinic solutions.
- Applied degree theory for Fredholm operators to nonautonomous difference equations.
- Provided insights into the structure of solutions without smallness restrictions.

## Abstract

When extending bifurcation theory of dynamical systems to nonautonomous problems, it is a central observation that hyperbolic equilibria persist as bounded entire solutions under small temporally varying perturbations. In this paper, we abandon the smallness assumption and aim to investigate the global structure of the entity of all such bounded entire solutions in the situation of nonautonomous difference equations. Our tools are global implicit function theorems based on an ambient degree theory for Fredholm operators due to Fitzpatrick, Pejsachowicz and Rabier. For this we yet have to restrict to so-called homoclinic solutions, whose limit is 0 in both time directions.