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

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.

Further data

Item Type: Article in a book
Refereed: Yes
Keywords: non-dissipative Lorenz model; nonlinear feedback loop; quasi-periodicity; incommensurate frequency
Institutions of the University: Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Lehrstuhl Wissenschaftliches Rechnen
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics >
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > >
Research Institutions
Research Institutions > Research Centres
Research Institutions > Research Centres > Forschungszentrum für Modellbildung und Simulation (MODUS)
Research Institutions > Research Centres > Forschungszentrum für Wissenschaftliches Rechnen an der Universität Bayreuth - HPC-Forschungszentrum
Result of work at the UBT: No
DDC Subjects: 000 Computer Science, information, general works
000 Computer Science, information, general works > 004 Computer science
500 Science
500 Science > 500 Natural sciences
500 Science > 510 Mathematics
500 Science > 550 Earth sciences, geology
600 Technology, medicine, applied sciences
600 Technology, medicine, applied sciences > 600 Technology
Date Deposited: 14 Nov 2019 08:17
Last Modified: 14 Nov 2019 08:17
URI: https://eref.uni-bayreuth.de/id/eprint/53218