11th Chaotic Modeling and Simulation International Conference
Chaotic Modeling and Simulation International Conference
CHAOS 2018: 11th Chaotic Modeling and Simulation International Conference pp 239-255 | Cite as
Coexistence of Chaotic and Non-chaotic Orbits in a New Nine-Dimensional Lorenz Model
B.-W. Shen
T. Reyes
S. Faghih-Naini
First Online: 29 May 2019
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.
Appendix: Solutions of the Non-trivial Critical Points
The time-independent 9DLM can be reduced as follows:
$$\begin{aligned} X^2 - XY_1 - bZ = 0, \end{aligned}$$
$$\begin{aligned} Z = r-1, \end{aligned}$$
$$\begin{aligned} Y_1 = {bXZT_1 \over X^2 + T_2}, \end{aligned}$$
$$\begin{aligned} T_1=\frac{3X^4+2bX^2(d_1+d_2)+b^2d_1d_2}{X^4+(bd_2+2bd_1)X^2+b^2d_1d_2}, \end{aligned}$$
$$\begin{aligned} T_2=bd_0T_1. \end{aligned}$$
Equation (25) along with Eqs. (26)–(29) represents an equation for a single variable X. Therefore, we can solve Eqs. (25)–(29) for the solution of X and for the solutions of the remaining variables since they can be expressed in terms of X, as follows:
$$\begin{aligned} Y = X, \end{aligned}$$
$$\begin{aligned} Z_1 =Y_1\left( \frac{X^5+bX^3(d_2+2d_1)+b^2Xd_2d_1}{6bX^4+4b^2X^2(d_2+d_1)+2b^3d_2d_1}\right) , \end{aligned}$$
$$\begin{aligned} Y_2 =2XZ_1 \left( \frac{2bX^2 +b^2d_2}{X^4 +X^2b(d_2+2d_1)+b^2d_2d_1}\right) , \end{aligned}$$
$$\begin{aligned} Z_2 = 3XY_2\left( \frac{X^2 +bd_2}{18bX^2 + 9b^2d_2}\right) , \end{aligned}$$
$$\begin{aligned} Y_3 = \frac{3bXZ_2}{X^2 +bd_2}, \end{aligned}$$
$$\begin{aligned} Z_3 = \frac{3Z_2}{4} - \frac{d_2Y_3}{4X}. \end{aligned}$$
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1.Department of Mathematics and StatisticsSan Diego State UniversitySan DiegoUSA
Cite this paper as:
Shen BW., Reyes T., Faghih-Naini S. (2019) Coexistence of Chaotic and Non-chaotic Orbits in a New Nine-Dimensional Lorenz Model. In: Skiadas C., Lubashevsky I. (eds) 11th Chaotic Modeling and Simulation International Conference. CHAOS 2018. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-15297-0_22
First Online 29 May 2019
DOI https://doi.org/10.1007/978-3-030-15297-0_22
Publisher Name Springer, Cham
Print ISBN 978-3-030-15296-3
Online ISBN 978-3-030-15297-0
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