# American Institute of Mathematical Sciences

October  2021, 26(10): 5305-5319. doi: 10.3934/dcdsb.2020344

## 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

Received  May 2020 Revised  October 2020 Published  October 2021 Early access  November 2020

Fund Project: 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)

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.

Citation: Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5305-5319. doi: 10.3934/dcdsb.2020344
##### References:
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##### References:
 [1] F. A. Carrillo, F. Verduzco and J. Delgado, Analysis of the Takens-Bogdanov bifurcation on m-parameter ized vector fields, International Journal of Bifurcation and Chaos, 20 (2010), 995-1005.  doi: 10.1142/S0218127410026277. [2] C. C. Daumann and P. C. Rech, Hyperchaos in convection with the Cattaneo-Christov heat-flux model, European Physical Journal B: Condensed Matter and Complex Systems, 92 (2019), 1-5. [3] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York. 1983. doi: 10.1007/978-1-4612-1140-2. [4] Yu. A. Kuznetsov, Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE's, SIAM Journal on Numerical Analysis, 36 (1999), 1104-1124.  doi: 10.1137/S0036142998335005. [5] Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd Edition, Springer, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [6] G. C. Layek and N. C. Pati, Bifurcations and chaos in convection taking non-Fourier heat-flux, Physics Letters A, 381 (2017), 3568-3575.  doi: 10.1016/j.physleta.2017.09.020. [7] E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. [8] E. N. Lorenz, Irregularity: A fundamental property of the atmosphere, Tellus A: Dynamic Meteorology and Oceanography, 36 (1984), 98-110.  doi: 10.3402/tellusa.v36i2.11473. [9] S. Moon, J. M. Seo, B.-S. Han, J. Park and J.-J. Baik, A physically extended Lorenz system, Chaos, 29 (2019), 063129. doi: 10.1063/1.5095466. [10] L. Stenflo, Generalized Lorenz equations for acoustic-gravity waves in the atmosphere, Physica Scripta, 53 (1996), 83-84.  doi: 10.1088/0031-8949/53/1/015. [11] J. Sotomayor, L. F. Mello and D. D. C. Braga, Bifurcation analysis of the Watt governor system, Computational and Applied Mathematics, 26 (2007), 19-44.  doi: 10.1590/S0101-82052007000100002. [12] Z. Wei and W. Zhang, Hidden hyperchaotic attractors in a modified Lorenz-Stenflo system with only one stable equilibrium, International Journal of Bifurcation and Chaos, 24 (2014), 1450127, 14 pp. doi: 10.1142/S0218127414501272.
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
(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
First Lyapunov coefficient $l_1$ will be negative for $\sigma = 10, b = 8/3, 0<\delta <10/11$
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
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
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
Bogdanov-Takens bifurcation of system (25)
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