Codimension one and two bifurcations in Cattaneo-Christov heat flux model

Zhouchao Wei; Wei Zhang; Irene Moroz; Nikolay V. Kuznetsov

doi:10.3934/dcdsb.2020344

Article Contents

2021, Volume 26, Issue 10: 5305-5319. Doi: 10.3934/dcdsb.2020344

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Codimension one and two bifurcations in Cattaneo-Christov heat flux model

1.
School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China, Zhejiang Institute, China University of Geosciences, Hangzhou, Zhejiang 311305, China
2.
College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, China
3.
Mathematical Institute, University of Oxford, Oxford OX2 6GG, England
4.
Faculty of Mathematics and Mechanics, St. Petersburg State University, Peterhof, St. Petersburg, Russia, Faculty of Information Technology, University of Jyväskylä, Jyväskylä, Finland

^* Corresponding author: weizhouchao@163.com
^* Corresponding author: weizhouchao@163.com

Received: April 30, 2020

Revised: September 30, 2020

Early access: November 2020

Published: October 2021

The first author is supported by National Natural Science Foundation of China (Grant No. 11772306), Zhejiang Provincial Natural Science Foundation of China under Grant (No.LY20A020001), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (CUGGC05). The second author is supported by National Natural Science Foundation of China (Grant No. 11832002). The last author is supported by the Russian Leading Scientific School (Center of Excellence) program (2624.2020.1)

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Abstract

Layek and Pati (Phys. Lett. A, 2017) studied a nonlinear system of five coupled equations, which describe thermal relaxation in Rayleigh-Benard convection of a Boussinesq fluid layer, heated from below. Here we return to that paper and use techniques from dynamical systems theory to analyse the codimension-one Hopf bifurcation and codimension-two double-zero Bogdanov-Takens bifurcation. We determine the stability of the bifurcating limit cycle, and produce an unfolding of the normal form for codimension-two bifurcation for the Layek and Pati's model.

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Mathematics Subject Classification: Primary: 34D20, 34D45; Secondary: 34K18.

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Figure 1. When $ \sigma = 10, r = 28, \delta = 15, b = 3 $ and initial condition is $ (5.1, 6.2, 7.3, 8.4, 9.5) $, chaotic attractors are shown and corresponding Lyapunov exponents are (0.9263, -0.0000, -11.8503, -14.3371, -14.7389): (a) X-Y-Z space; (b) Z-P-W space

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Figure 2. (a) Let $ (\sigma, b) = (10, 8/3) $. The equilibrium $ O $ of system (1) is asymptotically stable in the green region; the equilibria $ E_{1,2} $ of system (1) is asymptotically stable in the yellow region

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Figure 3. First Lyapunov coefficient $ l_1 $ will be negative for $ \sigma = 10, b = 8/3, 0<\delta <10/11 $

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Figure 4. Stable periodic orbit near $ O $ of system (1) from Hopf bifurcation with parameter values $ (\sigma, b, r, \delta) = (10, 8/3, 0.53, 0.5) $, and initial values $ (0.002, 0.002, 0.001, 0.02, 0.001) $ : (a) stable periodic orbit; (b) time series of state variables

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Figure 5. Stable periodic orbit near $ E_1 $ of system (1) from Hopf bifurcation with parameter values $ (\sigma, b, r, \delta) = (10, 8/3, 4.92, 0.5) $, and initial values $ (1.65, 1.6, 9, -6.5, 2.6) $: (a) stable periodic orbit; (b) time series of state variables

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Figure 6. Stable periodic orbit near $ E_1 $ of system (1) from Hopf bifurcation with parameter values $ (\sigma, b, r, \delta) = (10, 8/3, 6.638712, 1.5268) $, and initial values $ (2.33,2.46,10.41,-10.90,9.59) $ : (a) stable periodic orbit; (b) time series of state variables

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Figure 7. Bogdanov-Takens bifurcation of system (25)

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