Group-Theoretic Stability Analysis of Hexagonal Lattice

This chapter presents a group-theoretic bifurcation analysis of the uniform state of a hexagonal lattice with periodic boundary conditions. We conduct this analysis on a finite group \({\mathrm {D}}_6 \ \ltimes \ (\mathbb {Z}_n\times {\mathbb {Z}}_n)\) and find bifurcating hexagonal, racetrack, stripe, and upside-down patterns. We theoretically investigate the stability of these bifurcating patterns and reveal an astonishing fact: All of them are asymptotically unstable. We confirm this fact through the numerical bifurcation analysis in an economic geography model. Chapter 4 presents the basics of bifurcation analysis. This chapter provides a mathematical foundation for the numerical analysis of the hexagonal lattice in Chap. 2 , the spectrum analysis of population data in Chap. 6 , and the study of invariant patterns in Chap. 10 .