Global bifurcation of homoclinic solutions

TL;DR

This paper develops a comprehensive framework combining functional analysis and dynamical systems tools to classify the global bifurcation structures of bounded solutions in parametrized nonautonomous differential equations.

Contribution

It introduces new criteria and methods for analyzing the shape of bifurcating solution branches using Fredholm operator theory and the Evans function.

Findings

Classifies global bifurcation branches of bounded solutions.

Establishes criteria using the parity of Fredholm operator paths.

Utilizes the Evans function for bifurcation analysis.

Abstract

In the analysis of parametrized nonautonomous evolutionary equations, bounded entire solutions are natural candidates for bifurcating objects. Appropriate explicit and sufficient conditions for such branchings, however, require to combine contemporary functional analytical methods from the abstract bifurcation theory for Fredholm operators with tools originating in dynamical systems. This paper establishes alternatives classifying the shape of global bifurcating branches of bounded entire solutions to Carath\'eodory differential equations. Our approach is based on the parity associated to a path of index 0 Fredholm operators, the global Evans function as a recent tool in nonautonomous bifurcation theory and suitable topologies on spaces of Carath\'eodory functions.