Theorem: Haken-slaving reduction yields dx/dt = lambda*x + (u*g/mu)*x^3 (pitchfork normal form)

Theorem (Haken slaving canonical reduction): given two-mode system dx/dt = lambda*x + u*x*y (slow), dy/dt = -mu*y + g*x^2 (fast) with mu >> |lambda|, the center-manifold reduction y* = g*x^2/mu (enforcing dy/dt = 0) substituted back into…