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

  • * Corresponding author: Weihua Jiang

    * Corresponding author: Weihua Jiang

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

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  • 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.

    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 $\rho_4 = 0.2, ~\phi_1(0) = ( 0.1-0.1\mathrm{{i}}, 0.1+0.5\mathrm{{i}})^{\mathrm{{T}}}$, $h_4 = (0.1, 0.3)^{\mathrm{{T}}}$, $w_4 = 1$, $l = 1$, $n_2 = 3$

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

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

    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 $(\varphi(t, x), \psi(t, x)) = (u_0-0.005\sin x, v_0-0.001\sin x)$

    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 $(\varphi(t, x), \psi(t, x)) = (u_0-0.005\sin x, v_0-0.001\sin x)$

    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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