Symmetry-forced pitchfork bifurcation

Reflection symmetry $x\mapsto -x$ forces the reduced vector field to be odd in $x$, naturally producing paired nonzero equilibrium branches.

Adding a small symmetry-breaking term to a pitchfork bifurcation disconnects its two symmetric daughter branches and generically leaves one or more saddle-node bifurcations. This unfolding is called an imperfect pitchfork.