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Consider the Lorenz system $$\dot{x}(t) = \sigma(y-x) \, ,$$ $$\dot{y}(t) = x(\rho-z) - y \, ,$$ $$\dot{z}(t) = xy-\beta z \, .$$ Usually one considers the parameters $\sigma, \rho,$ and $\beta$ to be positive. Is there an inherent problem in the system if these parameters are negative?

(cross posted after 9 days on MSE)

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  • $\begingroup$ Excellent question, I don't understand why people downvoted the question. I accidently put a negative in front of one of the parameters and got some "funky" figures. It would be interesting to know what the physical meaning of such a system is though... $\endgroup$
    – Thomas Wagenaar
    Commented Sep 20, 2020 at 8:43
  • $\begingroup$ @ThomasW agreed, and I search for an answer or a reference quite thoroughly... $\endgroup$
    – Amir Sagiv
    Commented Sep 20, 2020 at 15:13

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There are many well-known ODE systems from various fields of science that can be transformed into the Lorenz system or generalized Lorenz system after a change of variables. After such a transformation the parameters of the original system may become negative in terms of the Lorenz system. The reason for looking for such a transformation is simple: to provide general methods for studying all these systems.

See, for example,

Leonov G. A., Kuznetsov N. V. On differences and similarities in the analysis of Lorenz, Chen, and Lu systems. Applied Mathematics and Computation, 256, 334-343 (2015).

and references therein and around the paper.

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  • $\begingroup$ Thank you! In such cases, what is known about the negative-parameter regime? From a short glimpse at the Leonov-Kuznetsov paper, I don't see a direct discussion on that. $\endgroup$
    – Amir Sagiv
    Commented Mar 9, 2020 at 13:04
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    $\begingroup$ For example, the attractor in the Chen system corresponds to a repeller in the Lorenz system (for some parameters, which include negative). Lorenz-like systems are extensively studied in many papers and it is hard to compare explicitly which regions of parameters are affected because most of theorems include several long inequalities in terms of the parameters. I remembered a paper, which explicitly deals (numerically) with the region of negative paremeters in the Lorenz system, but now I cannot find it. As I remembered it, there is nothing essentially new. $\endgroup$
    – demolishka
    Commented Mar 9, 2020 at 13:59

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