## Double-diffusive convection via 2 by 2 matrices

Convection due to competing or cooperating mechanisms, displays a variety of dynamical phenomena. One mechanism is usually a thermal gradient; typical examples of the second are rotation, a magnetic field, or a concentration gradient. The transition from conduction to convection is via a steady or a Hopf bifurcation; the point separating them is the best known codimension-two point. The steady bifurcation may be super or sub-critical, and the amplitude may undergo a qualitative transition from weak to strong.

All these features -- linear and nonlinear -- can be explained as manifestations of the behavior of the eigenvalues of a generic 2 x 2 matrix near the point where the eigenvalue branches intersect. For the stability problem, the eigenvalues have the conventional interpretation as growth rates, while for the nonlinear steady-state problems, they can be interpreted as the energy of steady states. Thus, there is a strong analogy between the stability problem and the bifurcation diagram.