• Previous Article
    Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem
  • DCDS-B Home
  • This Issue
  • Next Article
    A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence
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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020344
References:
[1]

F. A. CarrilloF. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd Edition, Springer, New York, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. SotomayorL. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

F. A. CarrilloF. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd Edition, Springer, New York, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. SotomayorL. 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.  Google Scholar

[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.  Google Scholar

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
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
Figure 3.  First Lyapunov coefficient $ l_1 $ will be negative for $ \sigma = 10, b = 8/3, 0<\delta <10/11 $
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
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
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
Figure 7.  Bogdanov-Takens bifurcation of system (25)
[1]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[2]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[3]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[4]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[5]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[6]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[7]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[8]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[9]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[10]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[11]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[12]

Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459

[13]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[14]

Yolanda Guerrero–Sánchez, Muhammad Umar, Zulqurnain Sabir, Juan L. G. Guirao, Muhammad Asif Zahoor Raja. Solving a class of biological HIV infection model of latently infected cells using heuristic approach. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020431

[15]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275

[16]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[17]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[18]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[19]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[20]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]