Stability and periodicity of high-order Lorenz–Stenflo equations

Abstract

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.

Periodicity and Chaos of High-Order Lorenz Systems

“…Peak numbers are plotted on (σ, r) parameter planes by considering all possible integer pairs within the parameter domains. In each diagram, the numerically computed peak numbers are assigned different colors following the color scheme similar to that of Park et al [2015] and Park et al [2016a].…”

“…Rech [2016] used the Lyapunov exponent for this purpose regarding a related system called Lorenz-Stenflo equations. We instead adopt periodicity diagrams following Dullin and Schmidt [2007], Park et al [2015] and Park et al [2016b].…”

Dynamical properties of acoustic-gravity waves in the atmosphere

“…The latter appear due to various meteorological conditions including different pressure and density gradients, as well as the presence of shear flows [5]. It has been shown that the nonlinear acoustic-gravity waves can appear in the forms of localized solitary vortices [1,5], ordered structures [8], as well as chaos [9] and turbulence [10].…”

Hopf Bifurcations, Periodic Windows and Intermittency in the Generalized Lorenz Model

“…Within the chaotic regimes presented on bifurcation diagrams, one can recognize periodic windows, as regions, in which much more ordered behavior is observed, including strictly periodic variation. In contrast to the chaotic solution, a periodic time series exhibits a repeating pattern of peaks and troughs, thus the number of local maxima of the variable X (X max ) considered here is limited and allows to define the periodicity [Dullin et al, 2007;Park et al, 2016;Moon et al, 2017]. We observe this situation, for example, for r ≈ 81 [ Fig.…”