Periodicity and Chaos of High-Order Lorenz Systems

Abstract

High-order Lorenz systems with five, six, eight, nine, and eleven equations are derived by choosing different numbers of Fourier modes upon truncation. For the original Lorenz system and for the five high-order Lorenz systems, solutions are numerically computed, and periodicity diagrams are plotted on (σ, r) parameter planes with σ, r ∈ [0, 1000], and b = 8/3. Dramatic shifts of patterns are observed among periodicity diagrams of different systems, including the appearance of expansive areas of period 2 in the fifth-, eighth-, ninth-, and 11th-order systems and the disappearance of the onion-like structure beyond order 5. Bifurcation diagrams along with phase portraits offer a closer look at the two phenomena.

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.…”

A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006

“…As discussed below, the 5DLM requires higher values of r for the onset of chaos (e.g., [13]). Recently, the 5DLM has been re-derived and analyzed in detail by Moon et al [15] and Felicio and Rech [16], showing the model's robustness. Furthermore, the 5DLM has been extended to higherdimensional LMs (e.g., [17][18][19][20]) and a generalized LM (e.g., [21,22]).…”

Systematic comparison between the generalized Lorenz equations and DNS in the two-dimensional Rayleigh-Bénard convection

“…solutions with at least two positive Lyapunov exponents, which was not seen in the original Lorenz equations). For a systematic comparison between the classic Lorenz equations and the higher-order extensions, Moon et al 12 thoroughly investigated the dynamical behaviors and bifurcation structures of the extended systems obtained by considering higher-order harmonics at dimensions 5, 6, 8, 9, and 11 in wide ranges of parameters, which was later generalized 13 into explicit ODE expressions for (3N)and (3N + 2)-dimensional Lorenz systems for any positive integer N.…”