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 lambda,gamma>0. We provide a complete classification of critical points, identifying nontrivial equilibria for lambda gamma not equal 1 and a continuum of equilibria when lambda=gamma=1. Stability is analyzed from temporal, spectral, and spatial steady-state perspectives, demonstrating asymptotic stability for lambda gamma<1 and instability for lambda gamma>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 (lambda,gamma)-plane.