## Title data

Faghih-Naini, Sara ; Shen, Bo-Wen:

**On quasi-periodic solutions associated with the extended nonlinear feedback loop in the five-dimensional non-dissipative Lorenz model.**

*In:* Skiadas, Christos H.
(ed.):
CHAOS 2017 Proceedings. -
Barcelona
: International Society for the Advancement of Science and Technology
,
2017
. - pp. 197-213

## Abstract in another language

A recent study suggested that the nonlinear feedback loop of the three- dimensional non-dissipative Lorenz model (3D-NLM) serves as a nonlinear restoring force by producing nonlinear oscillatory solutions as well as linear periodic solutions near a non-trivial critical point. In this study, using a five-dimensional non-dissipative Lorenz model (5D-NLM), we discuss the role of the extension of the nonlinear feedback loop in producing quasi-periodic trajectories. By solving a locally linear 5D-NLM for an analytical solution, we illustrate that the extension of the nonlinear feedback loop in the 5D-NLM can produce two incommensurate frequencies whose rati o is irrational, yielding a quasi-periodic solution. The quasi-periodic solution trajectory moves endlessly on a torus but never intersects itself.

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 additionally introduces two pairs of downscaling and upscaling processes that are enabled by two high wavenumber modes. The second pair of downscaling and upscaling processes provides a two-way interaction between the original (primary) Fourier modes of the 3D-NLM and the newly-added (secondary) Fourier modes of the 5D-NLM. The third pair of downscaling and upscaling processes involves interactions amongst the secondary modes. By comparing the numerical simulations using one- and two-way interactions, we illustrate that the system with a one-way interaction always produces periodic solutions with two commensurate frequencies. We also indicate that the two-way interaction is crucial for producing the quasi-periodic solution. Based on the current study using the 5D-NLM, and a recent study using a 7D non-dissipative LM, the statement "weather never repeats itself" is supported by the appearance of a quasi-periodic motion that results from the extension of nonlinear feedback loop, which is capable of producing two or more incommensurate frequencies, and may appear throughout the spatial mode-mode interactions rooted in the nonlinear temperature advection.