Stability and periodicity of high-order Lorenz-Stenflo equations

In this paper, we derive high-order Lorenz-Stenflo equations with 6 variables and investigate periodic behaviors as well as stability of the equations. The stability of the high-order Lorenz-Stenflo equations is investigated by the linear stability analysis for various parameters. A periodicity diagram is also computed and it shows that the high-order Lorenz-Stenflo equations exhibit very different behaviors from the original Lorenz-Stenflo equations for both periodic and chaotic solutions. For example, period 3 regime for large parameters and scattered periodic regime are newly observed, and chaotic regimes exist for smaller values of r but for larger values of s than the original equations. In contrast, similarities such as the enclosure of the chaotic regime by the periodic regime or complex periodic regimes inside the chaotic regime are also observed for both the original and high-order Lorenz-Stenflo equations.