A PDE Reduction to a Jonah-Type Normal Form

We tune the parameters of the scalar reaction–diffusion equation ∂ₜu = ∂ₓₓu + μu − u³ on (0, L) with Dirichlet conditions so that the linearisation at u = 0 has one unstable eigenvalue α = ln(451/343) and one stable eigenvalue −β = −ln(343/143). These are the continuous-time counterparts of the absolute values of the multipliers of the Jonah triangle, a hyperbolic saddle of the pedal map on the Nagel line at t = 7/10. On the two-dimensional invariant manifold tangent to the first two eigenmodes, the reduced dynamics is an odd cubic normal form with a hyperbolic saddle at the origin, two stable equilibria, and a separatrix given by the stable manifold of the saddle. A numerical integration of the PDE agrees with the reduced system, with an error scaling like the fifth power of the amplitude (script included). Scope: the construction works for any pair of positive rates, so the Jonah numbers enter as a parameter choice and the structure is the classical Chafee–Infante picture. A flow cannot reproduce the negative sign of the multiplier −143/343. The reduction is local; no claim is made about the Navier–Stokes equations.