| Citation: |
Hailong Yuan, Yani Ma, Yanfei Dai. BIFURCATION AND SPATIOTEMPORAL PATTERNS IN A HOMOGENEOUS DIFFUSIVE GIERER-MEINHARDT SYSTEM[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 860-894. doi: 10.11948/20230330
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BIFURCATION AND SPATIOTEMPORAL PATTERNS IN A HOMOGENEOUS DIFFUSIVE GIERER-MEINHARDT SYSTEM
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Abstract
This paper investigates an activator-inhibitor system with diffusion under homogeneous Neumann boundary conditions. Firstly, we consider the Hopf bifurcation at the positive equilibrium, and obtain the conditions for determining the bifurcation direction and stability of the bifurcating periodic solutions. Secondly, we demonstrate that the system undergoes a Turing-Hopf bifurcation with codimension-two. By calculating the normal form on the center manifold, we show that the system has the complex spatiotemporal dynamics near the Turing-Hopf bifurcation point. Moreover, the Turing instability of the positive equilibrium is discussed. By the bifurcation theory, we establish the local structure of the steady state bifurcations, and describe some conditions for determining the direction of bifurcations. Finally, some numerical simulations are carried out to explain and supplement the results of various bifurcation analyses.
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References
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Hailong Yuan, Yani Ma, Yanfei Dai. BIFURCATION AND SPATIOTEMPORAL PATTERNS IN A HOMOGENEOUS DIFFUSIVE GIERER-MEINHARDT SYSTEM[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 860-894. doi: 10.11948/20230330
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Figure 1.
Here
$ d=1.5 $ $ b=0.5<b_0^H=1 $ $ u_0(x)=2+0.5\cos(5x) $ $ v_0(x)=4+0.5\cos(5x) $ $ (2, 4) $ -
Figure 2.
Here
$ d=1.5 $ $ b=b_0^H=1 $ $ u_0(x)=1+5\sin(4x/5) $ $ v_0(x)=1+5\sin(4x/5) $ -
Figure 3.
Here
$ d=1.5 $ $ b=0.5<b_0^H=1 $ $ u_0(x)=2+5\sin(6x/5) $ $ v_0(x)=4+5\sin(6x/5) $ -
Figure 4.
Left: Turing-Hopf bifurcation point
$ (b^*, d_{k_*}^{*}) $ $ b $ $ d $ -
Figure 5.
$ (\varepsilon_1, \varepsilon_2)=(0.1, -0.1) $ $ u(x, 0)=u_0+2\cos5x, v(x,0)=v_0-2\cos5x $ $ E_0 $ -
Figure 6.
$ (\varepsilon_1, \varepsilon_2)=(0.1, 0.07), E_0 $ $ E_1^{\pm} $ -
Figure 7.
When
$ (\varepsilon_1, \varepsilon_2)=(-0.1, 0.6) $ $ E_0 $ $ E_2 $ $ E_1^{\pm} $ -
Figure 8.
When
$ (\varepsilon_1, \varepsilon_2)=(-0.1, 0.02) $ $ E_0, E_3^{\pm} $ $ E_2, E_1^{\pm} $ -
Figure 9.
When
$ (\varepsilon_1, \varepsilon_2)=(-0.1, -0.03) $ $ u(x, 0)=u_0+0.01\cos2x, v(x,0)=v_0-0.01\cos2x $ $ E_0, E_1^{\pm} $ $ E_2 $ -
Figure 10.
When
$ (\varepsilon_1, \varepsilon_2)=(-0.1, -0.08) $ $ u(x, 0)=u_0+0.1\cos2x, v(x,0)=v_0+0.1\cos2x $ $ E_0 $ $ E_2 $ -
Figure 11.
The neutral curves
$ d $ $ i\in R $ $ b=6 $ $ b=5.7 $ -
Figure 12.
Steady state bifurcation solutions at the simple eigenvalue for
$ b=6 $ $ d=1.4 $ $ d=1.25. $ $ u $ $ v $ -
Figure 13.
Steady state bifurcation solution at the double eigenvalue for
$ b=5.7 $ $ d=1.425. $ $ u $ $ v $ -
Figure 14.
Positive periodic solution of (1.2) for b=6. Here, d=1.1 and the solution tends to a spatially homogeneous time periodic orbit. Left: u; right: v.