The Complex Lorenz Equations

where x and y ace co~aplex and z is real. The complex parameters r and a are defined by r = rl + it,,: a = 1 ie and o" and b are real. Behaviour ~markab ly different from the real Lo~,~nz model occurs. Only the origin is a fixed point except for the ~pecial case e + ~ = 0. We have been able to determine analytically two critical values of rt, namely r~ and rb. The or ion is a stable fixed point for 0 < r~ < r~, but for r) > r~, a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if tr < b + I. If o > b + I then this limit is only stable in the region r~¢ < r~ < rf~. When r~ > tie, a transiEon to a finite amplitude oscillation about the limit cycle occurs, The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the Stuart-Landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency. This latter bifurcation is either a subor super-critical Hopf-like bifurcation to a doubly periodic motion, the direction of bifurcation depending on the parameter v:dues. The nature of the bifurcation is complicated by the existence of a zero eigenvalue.