Branches of localized patterned states

TL;DR

This paper introduces a framework for analyzing spatially localized patterned states, providing a topological criterion to distinguish between different types of localized structures and connecting with continuation algorithms for verification.

Contribution

It offers a new theoretical framework and topological criterion for understanding the structure and emergence of localized patterned states in spatial systems.

Findings

Provides a topological criterion for localized states

Connects theoretical analysis with continuation algorithms

Distinguishes between different types of localized structures

Abstract

Motivated by theoretical analyses of spatially localized structures with arbitrarily long periodic plateaus, we provide a framework of assumptions that simplifies their analysis and leads to a topological criterion for when localized patterned structures lie on a discrete stack of loops or on a single unbounded branch. The framework proposed here also connects closely with continuation algorithms that are often used to verify the hypotheses that guarantee the emergence of localized patterned states.