Spatial Period-Doubling Cascade on Square Lattice

We investigate the self-organization of patterns on a square lattice, highlighting the spatial period-doubling cascade among invariant patterns. A group-theoretic bifurcation analysis is conducted in light of the racetrack economy analogy. We have found the existence of two types of cascades in the square lattice, whereas the racetrack economy has only one type of cascade. In the numerical analysis of an economic geography model, we observe the transition of stable invariant patterns as the transportation cost decreases. Several prerequisites include Chaps. 3 and 4 for the direct bifurcation from the uniform state, Chap. 9 for the hierarchical bifurcation in the racetrack economy, and Chaps. 9 and 10 for invariant patterns.