A Local Spectral Reduction of a Reaction-Diffusion Equation to a Jonah-Type Normal Form

This paper constructs a rigorous local reduction from a scalar reaction-diffusion PDE (ut=uxx+μu−u3u_t = u_{xx} + \mu u - u^3ut=uxx+μu−u3) to a finite-dimensional normal form whose saddle structure matches the expansion and contraction rates of the Jonah triangle. By tuning the domain length LLL and parameter μ\muμ, we ensure the leading eigenvalues of the linearization are α=ln⁡(451/343)\alpha = \ln(451/343)α=ln(451/343) and −β=−ln⁡(343/143)-\beta = -\ln(343/143)−β=−ln(343/143). We prove that on the two-dimensional spectral subspace, the cubic reduced dynamics takes the form y˙=αy(1−y2)−2αyw2\dot{y} = \alpha y(1-y^2) - 2\alpha y w^2y˙=αy(1−y2)−2αyw2, w˙=−βw−2αy2w−αw3\dot{w} = -\beta w - 2\alpha y^2 w - \alpha w^3w˙=−βw−2αy2w−αw3. We establish the complete global phase portrait for this truncated normal form (a hyperbolic saddle with separatrix y=0y=0y=0 separating two stable equilibria). Using the invariant manifold theorem for semilinear parabolic equations, we transfer the local hyperbolic saddle structure to the PDE near the zero equilibrium. We explicitly state that the global portrait of the normal form does not necessarily transfer to the full PDE due to amplitude limitations, and that the sign difference between discrete multipliers and continuous flow exponents is acknowledged but not resolved. No reduction of Navier-Stokes is claimed.