Figure 5. Variation of the largest scaled eigenvalues of the Jacobian matrix (a) including HI and (b) excluding HI.

In (a) we see the transition from a stable node to a stable focus followed by a supercritical Hopf bifurcation from quiescence to a limit cycle. In (b) we see only a stable quiescent state. In the absence of HI the eigenvalue scaling is completely determined by . Comparing (a) and (b) it is clear that hydrodynamic instabilities due to HI are the main mechanism for spontaneous oscillations. (a) also shows that the rate of relaxation, given by the magnitude of the real part of the largest eigenvalue, decreases as we approach the bifurcation from below.