Turing instability for nonlocal heterogeneous reaction-diffusion systems: A computer-assisted proof approach
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913780601978880 |
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| author | Breden, Maxime Payan, Maxime Reisch, Cordula Tang, Bao Quoc |
| author_facet | Breden, Maxime Payan, Maxime Reisch, Cordula Tang, Bao Quoc |
| contents | This paper provides a computer-assisted proof for the Turing instability induced by heterogeneous nonlocality in reaction-diffusion systems. Due to the heterogeneity and nonlocality, the linear Fourier analysis gives rise to \textit{strongly coupled} infinite differential systems. By introducing suitable changes of basis as well as the Gershgorin disks theorem for infinite matrices, we first show that all $N$-th Gershgorin disks lie completely on the left half-plane for sufficiently large $N$. For the remaining finitely many disks, a computer-assisted proof shows that if the intensity $δ$ of the nonlocal term is large enough, there is precisely one eigenvalue with positive real part, which proves the Turing instability. Moreover, by detailed study of this eigenvalue as a function of $δ$, we obtain a sharp threshold $δ^*$ which is the bifurcation point for Turing instability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_05066 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Turing instability for nonlocal heterogeneous reaction-diffusion systems: A computer-assisted proof approach Breden, Maxime Payan, Maxime Reisch, Cordula Tang, Bao Quoc Analysis of PDEs Numerical Analysis Dynamical Systems This paper provides a computer-assisted proof for the Turing instability induced by heterogeneous nonlocality in reaction-diffusion systems. Due to the heterogeneity and nonlocality, the linear Fourier analysis gives rise to \textit{strongly coupled} infinite differential systems. By introducing suitable changes of basis as well as the Gershgorin disks theorem for infinite matrices, we first show that all $N$-th Gershgorin disks lie completely on the left half-plane for sufficiently large $N$. For the remaining finitely many disks, a computer-assisted proof shows that if the intensity $δ$ of the nonlocal term is large enough, there is precisely one eigenvalue with positive real part, which proves the Turing instability. Moreover, by detailed study of this eigenvalue as a function of $δ$, we obtain a sharp threshold $δ^*$ which is the bifurcation point for Turing instability. |
| title | Turing instability for nonlocal heterogeneous reaction-diffusion systems: A computer-assisted proof approach |
| topic | Analysis of PDEs Numerical Analysis Dynamical Systems |
| url | https://arxiv.org/abs/2504.05066 |