Coexistence of Chaotic and Non-chaotic Orbits in a New Nine-Dimensional Lorenz Model

  • B.-W.  ShenEmail author
  • T. Reyes
  • S. Faghih-Naini
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)


In this study, we present a new nine-dimensional Lorenz model (9DLM) that requires a larger critical value for the Rayleigh parameter (a rc of 679.8) for the onset of chaos, as compared to a rc of 24.74 for the 3DLM, a rc of 42.9 for the 5DLM, and a rc 116.9 for the 7DLM. Major features within the 9DLM include: (1) the coexistence of chaotic and non-chaotic orbits with moderate Rayleigh parameters, and (2) the coexistence of limit cycle/torus orbits and spiral sinks with large Rayleigh parameters. Version 2 of the 9DLM, referred to as the 9DLM-V2, is derived to show that: (i) based on a linear stability analysis, two non-trivial critical points are stable for all Rayleigh parameters greater than one; (ii) under non-dissipative and linear conditions, the extended nonlinear feedback loop produces four incommensurate frequencies; and (iii) for a stable orbit, small deviations away from equilibrium (e.g., the stable critical point) do not have a significant impact on orbital stability. Based on our results, we suggest that the entirety of weather is a superset that consists of both chaotic and non-chaotic processes.


Lorenz model Limit cycle Nonlinear feedback loop Coexistence Incommensurate frequencies Aggregated negative feedback 



We are grateful for support from the College of Science at San Diego State University.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSan Diego State UniversitySan DiegoUSA

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