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Shen, Bo-Wen ; Faghih-Naini, Sara:
On recurrent solutions within high-dimensional, non-dissipative Lorenz models.
In: Skiadas, Christos H.
(Hrsg.):
CHAOS 2017 Proceedings. -
Barcelona
: International Society for the Advancement of Science and Technology
,
2017
. - S. 773-787
Abstract
A recent study suggested that the nonlinear feedback loop of the three-dimensional, non-dissipative Lorenz model (3D-NLM) plays a role as a nonlinear restoring force in producing nonlinear oscillatory solutions, as well as linear periodic solutions near a non-trivial critical point. A follow-up study using the 5D-NLM examined the role of the extension of the nonlinear feedback loop in producing quasi-periodic solutions with two incommensurate frequencies. In this study, we analyze recurrent and quasi-periodic solutions within higher-dimensional NLMs (e.g., a 7D-NLM) with the goal of understanding how further extension of the nonlinear feedback loop may produce additional incommensurate frequencies.
While the nonlinear feedback loop of the 3D-NLM consists of a pair of downscaling and upscaling processes, the extended feedback loop within the 5D-NLM introduces two additional pairs of downscaling and upscaling processes, as compared to the 3D-NLM. Here, based on the extension of the nonlinear feedback loop within the 5D-NLM, we derive the 7D-NLM that has five pairs of downscaling and upscaling processes with three pairs that are the same as those within the 5D-NLM. In the 7D-NLM, the second and fourth pairs of downscaling and upscaling processes provide two-way interactions amongst the primary (the largest scale), secondary, and tertiary (the smallest scale) modes. By comparing the numerical simulations using one- and two-way interactions, we illustrate that proper representation of two-way interactions is crucial for capturing recurrent solutions with accurate incommensurate frequencies.
We also derive mathematical equations in order to provide an analogy between a linearized high-dimensional NLM and a system with different coupled springs. We show that the locally linear 7D-NLM (5D-NLM) is analogous to a system with three (two) different springs. This analogy can help illustrate how additional incommensurate frequencies may be generated by coupling additional springs with existing springs, indicating a similar impact between the coupling of springs and the extension of the nonlinear feedback loop. At the end, we outline future work designed for comparing various types of solutions for dissipative and non-dissipative LMs in order to understand whether steady-state, chaotic or limit torus solutions may better describe the nature of weather.