Group-Theoretic Analysis of Square Lattice

After a brief introduction to group-theoretic bifurcation analysis of the racetrack economy in Chap. 3 , this chapter advances a detailed group-theoretic analysis as the foundation for the whole book. We investigate the bifurcation of a scalar field on a square lattice with periodic boundary conditions through the analysis of a finite group \(\mathrm {D}_4 \ \ltimes \ (\mathbb {Z}_n\times \mathbb {Z}_n)\) , where \(\mathbb {Z}_n\) is a cyclic group and \(\ltimes \) is a semidirect product. Bifurcating patterns from a uniform state include square, stripe, and upside-down patterns. We conduct the numerical bifurcation analysis of the square lattice for an economic geography model to observe these patterns. This chapter is a prerequisite for further studies of the square lattice in Chaps. 10 and 13 .