On the dynamics of a high-order Lorenz–Stenflo system

Abstract

Results presented in a recent paper in this journal concerning a continuous-time dynamical system, namely that involving high-order Lorenz-Stenflo equations, are extended in this paper. More specifically, the present paper reports on nonlinear dynamics of a six-variable, fourparameter high-order Lorenz-Stenflo system. Six cross-sections of a four-dimensional parameter-space are considered. By using Lyapunov exponents spectra to characterize the dynamical behavior at each point of each of these plots, it is shown that different regions are allowed, from equilibrium point to chaos regions. It is also shown that hyperchaos is not an allowed behavior in a high-order Lorenz-Stenflo system. In addition, new results reported here are compared with those obtained for the original Lorenz-Stenflo system.

Periodicity and Chaos of High-Order Lorenz Systems

“…In this sense, the higher its peak number, the closer to chaos a solution gets. Peak numbers, therefore, function as a measure of chaos and thus as an alternative to Lyapunov exponents that were used for this purpose in [Rech, 2016]. Figure 3 shows the periodicity diagrams for our Lorenz systems.…”

“…To this end, some measure of chaos and periodicity corresponding to each pair of two parameters can be plotted on a twodimensional parameter plane. 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].…”

High-dimensional generalizations of the Lorenz system and implications for predictability

“…From a physical point of view, high-dimensional systems can make for a more realistic model either because they include more Fourier modes in the derivation [4][5][6] or because additional elements of the physical reality are considered [7][8][9][10][11][12][13]. There is also a possibility of discovering an entirely new phenomenon exclusive to high-dimensional systems such as hyperchaos [14][15][16][17][18].…”

“…In addition to r, one can also vary σ [24]. Exploring the σ-r parameter space allows one to consider both σ and r simultaneously, which has been done extensively for the original Lorenz system [17,18,[25][26][27][28][29]. Plotted in the σ-r parameter space, elaborate patterns have been reported for the numerical solutions of the three-dimensional system including the so-called 'onionlike structure' consisting of alternating regimes of chaos and periodicity in high-dimensional Lorenz systems [6,27].…”