Spontaneous oscillations in two 2-component cells coupled by diffusion.

Mathematical examples are presented of oscillators with two variables which do not oscillate in isolation, but which do oscillate stably when coupled with a twin via diffusion. Two examples are presented, the Lefever-Prigogine Brusselator and a system used to model glycolytic oscillations. The mathematical method is not the usual bifurcation theory, but rather a type of singular perturbation theory combined with bifurcation theory. For both examples, it is shown that all stationary solutions are unstable for appropriate parameter settings. In the case of the Brusselator, it is further shown that there exist limit cycles; i.e. stable oscillations, in this parameter range. A numerical example is presented.