Global Bifurcations and Continuation in the Presence of Symmetry with an Application to Solid Mechanics

A group-theoretic approach to global bifurcation and continuation for one-parameter problems with symmetry is presented. The basic theme is the construction of a reduced problem, having solutions with specified symmetries, that can be analyzed by global or local techniques. A global analysis of a general class of reduced problems via well-established continuation techniques shows that symmetry is preserved on global continua of solutions. The approach is illustrated in the analysis of large post-buckling solutions of a nonlinearly elastic ring with $O(2)$ symmetry under uniform hydrostatic pressure, and yields several new results. Specific symmetries of global bifurcating solution branches are enumerated, which enables a detailed qualitative analysis.