Isomorphic transformation of the Lorenz equations into a single-control-parameter structure

After decades of intensive attention since their initial derivation in fluid turbulence [1], and their analogy with Laser Physics [2], which resulted in hundreds, most likely thousands of contributions, including entire books and book chapters [3, 4], the Lorenz equations still remain one of the challenging problems, from both the mathematical and the physical points of view. On the one hand, physicists have been mainly concerned with experimental demonstrations that would eventually display deterministic low-dimensional chaotic behavior, in fluid turbulence [5] as well as in coherent-light-matter interactions [6]. On the other hand, mathematicians have dedicated extensive efforts towards computer assisted proofs pertaining to the existence of the so called Lorenz attractor [7-10]. These equations, whose unstable dynamics is governed by three control parameters, exhibit a rich variety of solutions. Yet, it appears that, for historical reasons, the barycenter of attraction spins around narrow zones of the large control-parameter domains, be it in fluid turbulence or in the nonlinear laser-matterinteraction issue. For instance, numerous investigations devoted to fluid dynamics mostly paid attention to the initial set of parameters that was put forward by Lorenz in his primary work, while laser physicists concentrated their efforts on effective constraints that typically yield periodic solutions.  

It is also worth recalling that despite the huge amount of published reports that dealt, and continue to deal with Lorenz dynamics [11-16]; in terms of their mathematical properties, no particular connection between the innumerable results has ever been put forward, leaving one lone justification to each exotic report, which always associates to the unpredictable nonlinear nature that connects the three interacting variables. As a consequence, the reported data seems to go along some quite irregular sequences similar to the chaotic nature of these nonlinear equations. The lone and overall evidence that could somehow be extracted when digging into the countless publications is the fact that, for some control parameters, the solutions exhibit periodic time traces, that deviate into some hierarchical cascading with the increase or decrease of some external excitation level, while for others, the solutions become chaotic, with erratic trajectories, that depict some sort of “strange attractors” when represented in the associated phase space. Probably, the search for any generic law in the solution-structures has been left aside because of the fact that the primary concern has, in most cases, been devoted to the demonstration of deterministic chaos in experimental systems, and the proof of existence of the Lorenz attractor, from a mathematical point of view [7-10]. To the best of our knowledge, no generic study has ever been undertaken to give a complete synopsis of the Lorenz equations, out of which one might precisely forecast its unstable solutions, given the values of its control parameters.