Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays

Şeyma Bılazeroğlu; Huseyin Merdan; Luca Guerrini

doi:10.3934/dcdss.2021150

Article Contents

2022, Volume 15, Issue 3: 535-554. Doi: 10.3934/dcdss.2021150

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Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays

1.
Çankaya University, Department of Mathematics, Eskişehir Yolu 29.km, Yukarıyurtçu Mahallesi, Mimar Sinan Caddesi, No: 4, 06790, Ankara, Turkey
2.
TOBB University of Economics and Technology, Department of Mathematics, Söğütözü Caddesi, No: 43, 06560, Ankara, Turkey
3.
Polytechnic University of Marche, Department of Management, Piazza Martelli 8, 60121, Ancona, Italy

^* Corresponding author: Huseyin Merdan
^* Corresponding author: Huseyin Merdan

Received: January 2020

Revised: February 2021

Early access: November 30, 2021

Published online: November 30, 2021

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Abstract

Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays are investigated. First, stability analysis of the model is given, and then the conditions on parameters at which the system has a Hopf bifurcation are determined. Second, bifurcation analysis is given by taking one of delay parameters as a bifurcation parameter while fixing the other in its stability interval to show the existence of Hopf bifurcations. The normal form theory and the center manifold reduction for functional differential equations have been utilized to determine some properties of the Hopf bifurcation including the direction and stability of bifurcating periodic solution. Finally, numerical simulations are performed to support theoretical results. Analytical results together with numerics present that time delay has a crucial role on the dynamical behavior of Chlorine Dioxide-Iodine-Malonic Acid (CIMA) reaction governed by a system of nonlinear differential equations. Delay causes oscillations in the reaction system. These results are compatible with those observed experimentally.

Keywords:

Mathematics Subject Classification: Primary: 34K13, 34K18, 34K20; Secondary: 37C75, 37G15.

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Figure 1. When $ \tau_{2} = 0.005 < \tau_{2_0} $ and $ \tau_{1} = 0.0350 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation

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Figure 2. When $ \tau_{2} = \tau_{2_0} \doteq 0.0135 $ and $ \tau_{1} = 0.0350 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation

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Figure 3. When $ \tau_{2} = 0.0140\in \lbrack 0.0135,0.0135+\varepsilon) $ and $ \tau_{1} = 0.0350 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation

Download: Full-size image PowerPoint slide

Figure 4. When $ \tau_{2} = 0.0145\in \lbrack 0.0135,0.0135+\varepsilon) $ and $ \tau_{1} = 0.0350 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation

Download: Full-size image PowerPoint slide

Figure 5. When $ \tau_{2} = 0.02 > \tau_{2_0} $ and $ \tau_{1} = 0.0350 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation

Download: Full-size image PowerPoint slide

Figure 6. When $ \tau_{1} = 0.01 < \tau_{1_0} $ and $ \tau_{2} = 0.02 $, they are phase portrait of (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation

Download: Full-size image PowerPoint slide

Figure 7. When $ \tau_{1} = \tau_{1_0} \doteq 0.0287 $ and $ \tau_{2} = 0.02 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation

Download: Full-size image PowerPoint slide

Figure 8. When $ \tau_{1} = 0.0288\in \lbrack 0.0287,0.0287+\varepsilon) $ and $ \tau_{2} = 0.02 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation

Download: Full-size image PowerPoint slide

Figure 9. When $ \tau_{1} = 0.0290\in \lbrack 0.0287,0.0287+\varepsilon) $ and $ \tau_{2} = 0.02 $, they are portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation

Download: Full-size image PowerPoint slide

Figure 10. When $ \tau_{1} = 0.04 > \tau_{1_0} $ and $ \tau_{2} = 0.02 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation

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