Global and Conditional Stability of the Steady and Periodic Solutions of the Franck-FitzHugh Model of Electrodissolution of Fe in H₂SO₄

Analysis of the mathematical model proposed by Franck and FitzHugh for the oscillations observed during the electrodissolution of iron in sulfuric acid shows that the number of steady solutions is always exactly one. Under some conditions, there may also exist one or two time-periodic solutions. Depending on the values assumed by three dimensionless parameters, the steady and periodic solutions may be globally or conditionally stable, or unstable. The parameter space is divided into four regions by a stability boundary. In one region the steady solution is globally attracting, and in another a periodic solution is globally attracting. In the other two regions, conditionally stable steady and periodic solutions are separated by an unstable periodic solution. On one part of the stability boundary, the steady solution loses its stability via infinitesimal oscillations and a supercritical Hopf-like bifurcation. On another part, the loss of stability is associated with an infinite period homoclinic bifurcation. The results are compared to those of Franck and FitzHugh and to those of a more recent study by Talbot, Oriani, and DiCarlo.