Relaxation times were measured from the decay to period-one attractors, a period-two attractor, and a period-three attractor in two reaction processes: the voltammetric oxidations of ethanol and 1-propanol. The results provide strong evidence that a periodic state that can undergo a subharmonic bifurcation is most stable near the point where the relaxation changes from a monotone to an alternating approach. The relation between the experimental results and theory is discussed and it is concluded that the reaction processes are in what is called the strongly contracting limit.