Parameter-driven stability and equilibria analysis of a coupled Allen–Cahn system

We examine a coupled Allen–Cahn system that generalizes the classical Allen–Cahn equation, consisting of two interacting components with coupling parameters λ , γ > 0 . We provide a complete classification of critical points, identifying nontrivial equilibria for λ γ ≠ 1 and a continuum of equilibria when λ = γ = 1 . Stability is analyzed from temporal, spectral, and spatial steady-state perspectives, demonstrating asymptotic stability for λ γ < 1 and instability for λ γ > 1 . The steady states exhibit center or saddle behavior depending on the coupling regime. Numerical visualizations of equilibrium maps and phase-plane vector fields illustrate how stability and bifurcation patterns vary across the ( λ , γ ) -plane.