Abstract. The complex Lorenz equations are a nonlinear fifth-order set of physically derived differential equations which exhibit an exact analytic limit cycle which subsequently bifurcates to a torus. In this paper we build upon previously derived results to examine a connection between this torus at high and low (bifurcation parameter) and between zero and nonzero (complexity parameter); in so doing, we are able to gain insight on the effect of the rotational invariance of the system, and on how extra weak dispersion (r 0) affects the chaotic behavior of the real Lorenz system (which describes a weakly dissipative, dispersive instability). 1. Introduction. Very recently, Gibbon and McGuinness  have shown that a fifth-order differential system of amplitude equations may be derived from weakly dissipative systems which exhibit a primarily dispersive instability, if extra ("detuning") dispersive effects are present. Moreover, when spatial variations are excluded, this system may be viewed as a complex generalization of the Lorenz equations .