Global Stability of the Lengyel–Epstein Systems

We study the global (asymptotic) stability of the Lengyel–Epstein differential systems, sometimes called Belousov–Zhabotinsky differential systems. Such systems are topologically equivalent to a two-parameter family of cubic systems in the plane. We show that for each pair of admissible parameters, the unique equilibrium point of the corresponding system is not globally (asymptotically) stable. On the other hand, we provide explicit conditions for this unique equilibrium point to be asymptotically stable and we study its basin of attraction. We also study the generic and degenerate Hopf bifurcations and highlight a subset of the set of admissible parameters for which the phase portraits of the systems have two limit cycles.