Bifurcation theory classifies how the qualitative behavior of dynamical systems changes as parameters vary. The codimension of a bifurcation is the number of parameters that must be tuned simultaneously to observe it — generic (cod-1) bifurcations occur on codimension-1 manifolds in parameter space. Normal forms are the simplest ODEs exhibiting each bifurcation type: Saddle-node: ẋ = μ + x². Transcritical: ẋ = μx − x². Pitchfork: ẋ = μx − x³. Hopf: ṙ = r(μ − r²). The cusp catastrophe (cod-2) has x³ normal form with two control parameters — its bifurcation set is a cusp in (μ,β) space. Left: phase portrait / trajectory flow. Right: bifurcation diagram (fixed points vs μ). Solid = stable, dashed = unstable.