We develop a set of mode coupling equations which describe the nonlinear evolution of the Rayleigh-Taylor and E¿B gradient drift instabilities which are relevant to the ionosphere. We show that for a three mode system, the nonlinear equations describing these instabilities correspond exactly to the Lorenz equations which approximately describe the Rayleigh-Benard instability. It is shown that the three mode system can exhibit a strange attractor with chaotic behavior. Ion inertia plays a critical role in this phenomenon in that if it is neglected, in the collisional limit, the three mode system does not exhibit chaos and a stable convection pattern results.