  We analyze a class of network motifs in which a short, two-node positive
feed- back motif is inserted in a three-node negative feedback loop. We
demonstrate that such networks can undergo a bifurcation to a state where a
stable fixed point and a stable limit cycle coexist. At the bifurcation point
the period of the oscillations diverges. Further, intrinsic noise can make the
system switch between oscillatory state and the stationary state spontaneously.
We find that this switching also occurs in previous models of circadian clocks
that use this combination of positive and negative feedback. Our results
suggest that real- life circadian systems may need specific regulation to
prevent or minimize such switching events.
