Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system

Qi An; Weihua Jiang

doi:10.3934/dcdsb.2018183

Article Contents

2019, Volume 24, Issue 2: 487-510. Doi: 10.3934/dcdsb.2018183

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Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system

Qi An and
Weihua Jiang^,

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Weihua Jiang
* Corresponding author: Weihua Jiang

Received: November 2017

Early access: May 2018

Published: February 2019

The authors are supported the National Natural Science Foundation of China (No.11371112)

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Abstract

We study the Turing-Hopf bifurcation and give a simple and explicit calculation formula of the normal forms for a general two-components system of reaction-diffusion equation with time delays. We declare that our formula can be automated by Matlab. At first, we extend the normal forms method given by Faria in 2000 to Hopf-zero singularity. Then, an explicit formula of the normal forms for Turing-Hopf bifurcation are given. Finally, we obtain the possible attractors of the original system near the Turing-Hopf singularity by the further analysis of the normal forms, which mainly include, the spatially non-homogeneous steady-state solutions, periodic solutions and quasi-periodic solutions.

Keywords:

Time-delayed reaction-diffusion systems,
Hopf-zero bifurcation,
Turing-Hopf bifurcation,
normal forms,
spatial inhomogeneous patterns.

Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 35B36.

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Figure 1. The spatially non-homogeneous periodic attractors $H = (H_1, H_2)$, under the Case 4. Here $\rho_4 = 0.2, ~\phi_1(0) = ( 0.1-0.1\mathrm{{i}}, 0.1+0.5\mathrm{{i}})^{\mathrm{{T}}}$, $h_4 = (0.1, 0.3)^{\mathrm{{T}}}$, $w_4 = 1$, $l = 1$, $n_2 = 3$

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Figure 2. The spatially non-homogeneous quasi-periodic attractors $H = (H_1, H_2)$, under the case 5. Here $\rho_4 = 0.2, \rho_5 = 0.1, \phi_1(0) = ( 0.1-0.1\mathrm{{i}}, 0.1+0.5\mathrm{{i}})^{\mathrm{{T}}}$, $h_4 = (0.1, 0.3)^{\mathrm{{T}}}$, $h_5 = (0.2, 0.5)^{\mathrm{{T}}}$, $w_5 = 1$, $\varpi = 0.5$, $l = 1$, $n_2 = 3$

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Figure 3. (a) Bifurcation sets in $(\alpha_1, \alpha_2)$ plane. (b) Phase portraits in $D_1-D_6$

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Figure 4. Two spatially inhomogeneous periodic solutions coexist in $D_3$, with $(\alpha_1, \alpha_2) = (0.05, -0.33)$. (a), (b) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0+0.005\sin x, v_0+0.001\sin x)$. (c), (d) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0-0.005\sin x, v_0-0.001\sin x)$

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Figure 5. Two spatially inhomogeneous steady state solutions coexist in $D_5$, with $(\alpha_1, \alpha_2) = (-0.1, -0.4)$. (a), (b) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0+0.005\sin x, v_0+0.001\sin x)$. (c), (d) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0-0.005\sin x, v_0-0.001\sin x)$

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Table 1. The correspondence between the planner and original system

Planar system (29)	The original system (1)
$E_1$	Constant steady state $(0, 0)$
$E_2$	Spatially homogeneous periodic solution
$E_3$	Non-constant steady state
$E_4$	Spatially non-homogeneous periodic solution
Periodic solution	Spatially non-homogeneous quasi-periodic solution

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