From Different Systems to a Single Common Model: A Review of Dynamical Systems Leading to Lorenz Equations - Juan Carlos Chimal-Eguía; Florencio Guzmán-Aguilar; Víctor Manuel Silva-García; Héctor Báez-Medina; Manuel Alejandro Cardona-López (Axioms, 2025)
This paper presents an analytical exploration of how diverse dynamical systems, arising from different scientific domains, can be reformulated (under specific approximations and assumptions) into a common set of equations formally equivalent to the Lorenz system originally derived to model atmospheric convection. Unlike previous studies that focus on analyzing or applying the Lorenz equations, our objective is to show how these equations emerge from distinct models, emphasizing the underlying structural and dynamical similarities. The mathematical steps involved in these reformulations are included. The systems examined include Lorenz’s original atmospheric convection model, the chaotic water wheel, the Maxwell–Bloch equations for lasers, mechanical gyrostat, solar dynamo model, mesoscale reaction dynamics, an interest rate economic model, and a socioeconomic control system. This work includes a discussion of the unifying features that lead to similar qualitative behaviors across seemingly unrelated systems. By highlighting the Lorenz system as a paradigmatic limit of a broad class of nonlinear models, we underscore its relevance as a unifying framework in the study of complex dynamics.