A nonlinear evolution equation for pulsating Chapman-Jouguet detonations wi
th chain-branching kinetics is derived. We consider a model reaction system
having two components: a thermally neutral chain-branching induction zone
governed by an Arrhenius reaction, terminating at a location where conversi
on of fuel into chain radical occurs; and a longer exothermic main reaction
layer or chain-recombination zone having a temperature-independent reactio
n rate. The evolution equation is derived under the assumptions of a large
activation energy in the induction zone and a slow evolution time based on
the particle transit time through the induction zone, and is autonomous and
second-order in time in the shock velocity perturbation. It describes both
stable and unstable solutions, the latter leading to stable periodic limit
cycles, as the ratio of the length of the chain-recombination zone to chai
n-induction zone, the exothermicity of reaction, and the specific heats rat
io are varied. These dynamics correspond remarkably well with numerical sol
utions conducted earlier for a model three-step chain-branching reaction.