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Quasi-Periodic Orbits in the Five-Dimensional Nondissipative Lorenz Model: The Role of the Extended Nonlinear Feedback Loop

Sara Faghih-Naini

Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182, USA

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and

Bo-Wen Shen

Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182, USA

E-mail Address: bshen@mail.sdsu.edu

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https://doi.org/10.1142/S0218127418500724Cited by:24 (Source: Crossref)

Abstract

A recent study suggested that the nonlinear feedback loop (NFL) of the three-dimensional nondissipative Lorenz model (3D-NLM) serves as a nonlinear restoring force by producing nonlinear oscillatory solutions as well as linear periodic solutions near a nontrivial critical point. This study discusses the role of the extension of the NFL in producing quasi-periodic trajectories using a five-dimensional nondissipative Lorenz model (5D-NLM). An analytical solution to the locally linear 5D-NLM is first obtained to illustrate the association of the extended NFL and two incommensurate frequencies whose ratio is irrational, yielding a quasi-periodic solution. The quasi-periodic solution trajectory moves endlessly on a torus but never intersects itself.

While the NFL of the 3D-NLM consists of a pair of downscaling and upscaling processes, the extended NFL within the 5D-NLM additionally introduces two new pairs of downscaling and upscaling processes that are enabled by two high wavenumber modes. One 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 other 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 two-way interaction is crucial for producing the quasi-periodic solution. A follow-up study using a 7D nondissipative LM shows that a further extension of NFL, which may appear throughout the spatial mode-mode interactions rooted in the nonlinear temperature advection, is capable of producing one more incommensurate frequency.

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Vol. 28, No. 06

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Received 28 January 2018

Revised 11 April 2018

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This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited.

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Quasi-Periodic Orbits in the Five-Dimensional Nondissipative Lorenz Model: The Role of the Extended Nonlinear Feedback Loop

Abstract

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