Some attractors in the extended complex Lorenz model

We address the question of finding the attractors of the extended complex Lorenz model (ℂLM), which is obtained by extending the space from ℝ³ to ℂ³, and defining the model by the same equations as the classical Lorenz model (LM). We have numerical evidence of two strong attractors unrelated to the Lorenz attractor. We calculate its Lyapunov exponents and show that two of them are 0, and the other four are double and negative. Hence the attractors are nonchaotic. We show that they have a quasi-periodic nature. To decipher the structure of these attractors, we introduce the imaginary Lorenz model (ILM), which is defined in the same space ℂ³ by multiplying with i = √-1 the Lorenz equations. Both models locally commute, and with its help we account for the double Lyapunov exponent 0 and show evidence that the basin of attraction of each attractor is a big open set of ℂ³. The chaotic limit set L_ℂ ⊂ ℂ³ obtained from the classical Lorenz attractor L ₀ of (LM) by moving it with the (ILM) has two positive Lyapunov exponents, but only captures a set of 6D-volume 0 in its basin of attraction. Hence this attractor may be hyperchaotic in ℝ⁵.