For either of these solutions, the characteristic equation of the matrix in (29) is 

This equation possesses one real negative root and two complex conjugate roots when r>1; the complex conjugate roots are pure imaginary if the product of the coefficients of λ² and λ equals the constant term, or

This is the critical value of r for the instability of steady convection. Thus if σ<b+1, no positive value of r satisfies (34), and steady convection is always stable, but if σ>b+1, steady convection is unstable for sufficiently high Rayleigh numbers. This result of course applies only to idealized convection governed by (25)-(27), and not to the solutions of the partial differential equations (17) and (18).
The presence of complex roots of (34) shows that if unstable steady convection is disturbed, the motion will oscillate in intensity. What happens when the disturbances become large is not revealed by linear theory. To investigate finite-amplitude convection, and to study the subspace to which trajectories are ultimately confined, we turn to numerical integration. 

TABLE 1. Numerical solution of the convection equations. Values of X, Y, Z are given at every fifth iteration N, for the first 160 iterations.

7. Numerical integration of the convection equations 

To obtain numerical solutions of the convection equations, we must choose numerical values for the constants. Following Saltzman (1962), we shall let σ=10 and a²=1/2, so that b=8/3. The critical Rayleigh number for instability of steady convection then occurs when r=470/19=24,74.
We shall choose the slightly supercritical value r= 28. The states of steady convection are then represented by the points (,27) and (,,27) in phase space, while the state of no convection corresponds to the origin (0,0,0).
We have used the double-approximation procedure for numerical integration, defined by (9), (10), and (14). The value Δτ=0.01 bas been chosen for the dimensionless time increment. The computations have been performed on a Royal McBee LGP-30 electronic computing machine. 

TABLE 2. Numerical solution of the convection equations Values of X, Y, Z are given at every iteration N for which Z possesses a relative maximum, for the first 6000 iterations.