The complex Lorenz equations

• Published in: Physica. D 1982, Vol.4 (2), p.139-163
• Main Authors: Fowler, A.C., Gibbon, J.D., McGuinness, M.J.
• Format: Article
• Language: English
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/0167-2789(82)90057-4

We have undertaken a study of the complex Lorenz equations x ̇ = −σx + σy . y ̇ = (r − z)x − ay . z ̇ = −bz + 1 2 (x ∗y + xy ∗) . where x and y are complex and z is real. The complex parameters r and a are defined by r = r 1 + ir 2; a = 1 − ie and σ and b are real. Behaviour remarkably different from the real Lorenz model occurs. Only the origin is a fixed point except for the special case e + r 2 = 0. We have been able to determine analytically two critical values of r 1, namely r1 c and r1 c . The origin is a stable fixed point for 0 < r 1 < r 1 c , but for r 1 > r 1 c , a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if σ < b + 1. If σ > + 1 then this limit is only stable in the region r 1 c < r 1 < r lc . When r 1 > r lc , a transition 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 sub- or super-critical Hopf-like bifurcation to a doubly periodic motion, the direction of bifurcation depending on the parameter values. The nature of the bifurcation is complicated by the existence of a zero eigenvalue.