A Variant of 3D Lorenz Model under Gay-Lussac Approximation and its Dynamical Properties

In this work, a variant of the well-known 3D Lorenz system is considered. The Lorenz system can be recovered from the Oberbeck-Boussinesq assumption applied to the 2D

Rayleigh-Bénard fluid flow problem using methods of Fourier-Galerkin truncation. An incompressible generalization of this approach is the Gay-Lussac approximation in

which density variation is extended to terms other than gravity of the governing equations (e.g., advection) and expressed as a linear function of temperature. The Gay-

Lussac approach additionally introduces a quantity known as the Gay-Lussac parameter, which describes the effect of thermal convection for buoyancy and advection and is

physically valid for . Following similar Fourier-Galerkin methods, a variant of the Lorenz model with two additional third-order nonlinear terms involving nondimensionalized is

derived from this approximation. When the accepted range for is extended to , the model has been shown to exhibit various dynamical and structural patterns such as

chaotic attraction and changes in equilibrium stability. Dynamical, topological and physical characteristics such as dissipativity are addressed through insight from linear

stability analysis of equilibrium points, computation of the model's Lyapunov spectrum, bifurcation diagrams, and other considerations.