The Attractors in the Complex Lorenz Model

摘要

We address the question of finding the attractors of the complex Lorenz model (CLM), which is obtained by extending the space from R~3 to C~3, and defining the model by the same equations as the classical Lorenz model (LM). We have numerical evidence of 2 strong attractors un-related to the Lorenz attractor. We calculate its Lyapunov exponents and show that 2 of them are 0, and the other 4 are double and negative. Hence the attractors are non-chaotic. 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 C~3 by multiplying with i = {the square root of} (-1) the Lorenz equations. Both models locally commute, and with its help we account for the double Lyapunov exponent 0 and show that the basin of attraction of each attractor is a big open set of C~3. The chaotic limit set L_C (is contained in) C~3 obtained from the classical Lorenz attractor L_0 of (LM) by moving it with the (ILM) has 2 positive Lyapunov exponents, but only captures a set of 6D-volume 0 in its basin of attraction.