Stationary pattern emergence in reaction–diffusion systems with unstable equilibria and Lienard kinetics

Pattern formation in reaction–diffusion systems is traditionally understood to arise from Turing instabilities when a stable equilibrium becomes unstable due to diffusion. Here we investigate reaction–diffusion systems with chemical kinetics transformable into Lienard-type equations, enabling stationary spatial patterns to emerge when the equilibrium point of the diffusion-less system is unstable and loses stability through a Hopf bifurcation. Three examples were fully explored: the BVAM model, the van der Pol oscillator, and the Briggs–Rauscher chemical kinetics. Linear and nonlinear analyses were performed for the three cases, and the predictions were tested using one- and two-dimensional numerical simulations. Our results demonstrate that oscillatory chemical kinetics can generate stable spatial patterns, expanding the classical framework of pattern formation. These findings suggest new avenues for exploring pattern formation in oscillatory systems and have implications for understanding oscillation quenching by diffusion in chemical and biological contexts.