Scaling Solution Branches of One-Parameter Bifurcation Problems

We consider an algorithm for analyzing bifurcation structure and for branch switching in solution branches of one-parameter dependent problems. Based on Liapunov-Schmidt methods and analyses of scales of solutions, we verify the existence of bifurcating solution branches successively via truncated Taylor expansions of the reduced bifurcation equations and an enlarged system. As examples, bifurcations of semilinear elliptic differential equations are studied on square and hexagon domains.