| Citation: |
Mingzhu Qu, Chunrui Zhang. TURING INSTABILITY AND PATTERNS OF THE FITZHUGH-NAGUMO MODEL IN SQUARE DOMAIN[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1371-1390. doi: 10.11948/20200182
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TURING INSTABILITY AND PATTERNS OF THE FITZHUGH-NAGUMO MODEL IN SQUARE DOMAIN
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Abstract
In this paper, critical conditions of Turing instability for Fitzhugh-Nagumo (FHN) model with diffusion under Neumann boundary conditions are derived. Moreover, different from previous works about the FHN model, we obtain simple bifurcation, double bifurcation, and four-fold bifurcation with stripe pattern, rectangular pattern, spot pattern, square pattern, and highly developed square pattern, respectively. Meanwhile, the theoretical results are applied to two coupled FHN model with diffusion, and the process of the coupling strengths affecting the stability of the model is presented by numerical simulations.
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Keywords:
- Turing instability /
- coupled Fitzhugh-Nagumo /
- diffusion /
- pattern
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References
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Mingzhu Qu, Chunrui Zhang. TURING INSTABILITY AND PATTERNS OF THE FITZHUGH-NAGUMO MODEL IN SQUARE DOMAIN[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1371-1390. doi: 10.11948/20200182
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Figure 1. The figure of functions
$ \sigma = {\sigma _1} $ ,$ \sigma = {\sigma _2}\left( d \right) $ and$ \sigma = {\sigma _{\rm{*}}}\left( {k, d} \right) $ ,$ d > 0 $ ,$ k = 5, 10, 13 \cdots $ , in$ \left( {d, \sigma } \right) $ plane. -
Figure 2. The Turing bifurcation line
$ \mathrm{T} : \sigma = {\sigma _*}\left( d \right) $ ,$ d > 0 $ . -
Figure 3. Stripe patterns when
$ {k_1} = 10 $ ,$ {m_1} = 10 $ ,$ {n_1} = 0 $ . The initial values are$ u0 = 0.00000002\cos \left( {5\pi x} \right) $ ,$ v0 = 0.007\cos \left( {5\pi y} \right) $ . -
Figure 4. Rectangular patterns when
$ {k_1} = 10 $ ,$ {m_3} = 8 $ ,$ {n_3} = 6 $ . The initial values are$ u0 = 0.00000002\cos \left( {4\pi x} \right)\cos \left( {3\pi y} \right) $ ,$ v0 = 0.007\cos \left( {3\pi x} \right)\cos \left( {4\pi y} \right) $ . -
Figure 5. Spot patterns when
$ {k_1} = 10 $ ,$ {m_1} = 10 $ ,$ {n_1} = 0 $ ,$ {m_2} = 0 $ ,$ {n_2} = 10 $ by interacting$ {m_{1, 2}} $ ,$ {n_{1, 2}} $ two modes. The initial values are$ u0 = 0.002\cos \left( {5\pi x} \right) + 0.003\cos \left( {5\pi y} \right) $ ,$ v0 = 0.005\cos \left( {5\pi y} \right) - 0.004\cos \left( {5\pi x} \right) $ . -
Figure 6. Square patterns when
$ {k_1} = 10 $ ,$ {m_3} = 8 $ ,$ {n_3} = 6 $ ,$ {m_4} = 6 $ ,$ {n_4} = 8 $ by interacting$ {m_{3, 4}} $ ,$ {n_{3, 4}} $ two modes. The initial values are$ u0 = 0.007\cos \left( {4\pi x} \right)\cos \left( {3\pi y} \right) + 0.005\cos \left( {3\pi x} \right)\cos \left( {4\pi y} \right) $ ,$ v0 = 0.001\cos \left( {3\pi x} \right)\cos \left( {4\pi y} \right) - 0.002\cos \left( {4\pi x} \right)\cos \left( {3\pi y} \right) $ . -
Figure 7. Highly developed square patterns when
$ {k_1} = 10 $ ,$ {m_1} = 10 $ ,$ {n_1} = 0 $ ,$ {m_2} = 0 $ ,$ {n_2} = 10 $ ,$ {m_3} = 8 $ ,$ {n_3} = 6 $ ,$ {m_4} = 6 $ ,$ {n_4} = 8 $ by interacting$ {m_{1, 2, 3, 4}} $ ,$ {n_{1, 2, 3, 4}} $ four modes. The initial values are$ u0 = 0.007\cos \left( {5\pi x} \right) + 0.005\cos \left( {5\pi y} \right) + 0.006\cos \left( {4\pi x} \right)\cos \left( {3\pi y} \right) + 0.009\cos \left( {3\pi x} \right)\cos \left( {4\pi y} \right) $ ,$ v0 = 0.007\cos \left( {5\pi y} \right) - 0.005\cos \left( {5\pi x} \right) + 0.006\cos \left( {3\pi x} \right)\cos \left( {4\pi y} \right) - 0.009\cos \left( {4\pi x} \right)\cos \left( {3\pi y} \right) $ . -
Figure 8. The figure of functions
$ \sigma = {\sigma _1^{'}} $ ,$ \sigma = {\sigma _2^{'}}\left( d \right) $ and$ {\sigma _{\rm{*}}^{'}}\left( {k, d} \right) $ ,$ d > 0 $ ,$ k = 5, 10, 13 \cdots $ , in$ \left( {d, \sigma } \right) $ plane. -
Figure 9. The Turing bifurcation line
$ \mathrm{T} : \sigma = {\sigma _*^{'}}\left( d \right) $ ,$ d > 0 $ . -
Figure 10. The region of coupling strengths
$ \alpha $ and$ \beta $ when$ d = 0.01 $ ,$ \sigma = 0.14 $ ,$ a = - 2.25 $ ,$ \delta = - 0.6 $ ,$ \varepsilon = 0.153 $ ,$ l = 2 $ . -
Figure 11. Stripe patterns produced by
$ {u_1}\left( {x, y} \right) $ when$ {m_1} = 10 $ ,$ {n_1} = 0 $ . -
Figure 12. Spot patterns produced by
$ {u_2}\left( {x, y} \right) $ when$ {m_1} = 10 $ ,$ {n_1} = 0 $ . -
Figure 13. Spot patterns produced by
$ {v_2}\left( {x, y} \right) $ when$ {m_1} = 10 $ ,$ {n_1} = 0 $ . -
Figure 14. Rectangular patterns produced by
$ {u_1}\left( {x, y} \right) $ when$ {m_3} = 8 $ ,$ {n_3} = 6 $ . -
Figure 15. Square patterns produced by
$ {u_2}\left( {x, y} \right) $ when$ {m_3} = 8 $ ,$ {n_3} = 6 $ . -
Figure 16. Square patterns produced by
$ {v_2}\left( {x, y} \right) $ when$ {m_3} = 8 $ ,$ {n_3} = 6 $ .