Quiescent phases and stability

If an autonomous system of differential equations is diffusively coupled to a quiescent phase then at stationary points the stability properties change. If the coupling matrices are multiples of the identity then introducing a quiescent phase stabilizes against the onset of oscillations, in particular high frequency oscillations are damped.For arbitrary (diagonal) coupling matrices the situation gets much more complex. For dimension two it can be shown that stability at a stationary point is preserved for arbitrary rates if and only if the Jacobian is strongly stable in the sense of Turing stability.