The real and complex Lorenz equations in rotating fluids and lasers

The Lorenz equations are derived systematically from amplitude equations of weakly nonlinear dispersively unstable physical systems near criticality when weak dissipation is added. This derivation is only valid if the undamped neutral curve is not destabilised by the addition of weak dissipation. The addition of extra weak dispersive effects make some of the coefficients complex and yields a complex set of Lorenz equations. Both sets of equations are derived in examples in laser optics and baroclinic instability.