Global Dynamics and Stabilization of Zero-Mode Singularities in Multi-Scale Reaction-Diffusion Systems via Negative Coupling
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912800697221120 |
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| author | Hou, Pengyue |
| author_facet | Hou, Pengyue |
| contents | This paper establishes a rigorous mathematical framework for the Multi-Scale Negative Coupled System (MNCS), a dynamical model describing hierarchical state spaces with directed, sign-structured interactions. We address the stabilization of reaction-diffusion systems on bounded domains $Ω\subset \mathbb{R}^d$ ($d \le 3$) subject to homogeneous Neumann boundary conditions. A critical feature of this setting is the "zero-mode singularity," where the Laplacian operator possesses a trivial zero eigenvalue ($λ_0=0$), providing no linear dissipation for the spatial mean. We rigorously prove the global well-posedness of the system and the existence of a compact global attractor $\mathcal{A}$ in the phase space $\mathbb{H}=(L^2(Ω))^N$. Utilizing the Moser-Alikakos iteration technique, we establish uniform $L^\infty(Ω)$ bounds, overcoming the lack of Sobolev embedding from $H^1$ into $L^\infty$ in three dimensions. These bounds enable the derivation of explicit upper estimates for the fractal dimension of the attractor via the Kaplan-Yorke trace formula. We show that the dimension scales as $d_F(\mathcal{A}) \sim \max\{0, \mathcal{K}_{\mathcal{A}}-γ\}^{d/2}$, confirming that the negative coupling strength $γ$ acts as a global regularizer that compresses the phase space. The theoretical results are validated using a stiff-stable Second-Order Exponential Time Differencing (ETD2) scheme with Discrete Cosine Transform (DCT) to strictly enforce no-flux boundary conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_00638 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global Dynamics and Stabilization of Zero-Mode Singularities in Multi-Scale Reaction-Diffusion Systems via Negative Coupling Hou, Pengyue Analysis of PDEs Dynamical Systems Chaotic Dynamics 35K57, 37L30 This paper establishes a rigorous mathematical framework for the Multi-Scale Negative Coupled System (MNCS), a dynamical model describing hierarchical state spaces with directed, sign-structured interactions. We address the stabilization of reaction-diffusion systems on bounded domains $Ω\subset \mathbb{R}^d$ ($d \le 3$) subject to homogeneous Neumann boundary conditions. A critical feature of this setting is the "zero-mode singularity," where the Laplacian operator possesses a trivial zero eigenvalue ($λ_0=0$), providing no linear dissipation for the spatial mean. We rigorously prove the global well-posedness of the system and the existence of a compact global attractor $\mathcal{A}$ in the phase space $\mathbb{H}=(L^2(Ω))^N$. Utilizing the Moser-Alikakos iteration technique, we establish uniform $L^\infty(Ω)$ bounds, overcoming the lack of Sobolev embedding from $H^1$ into $L^\infty$ in three dimensions. These bounds enable the derivation of explicit upper estimates for the fractal dimension of the attractor via the Kaplan-Yorke trace formula. We show that the dimension scales as $d_F(\mathcal{A}) \sim \max\{0, \mathcal{K}_{\mathcal{A}}-γ\}^{d/2}$, confirming that the negative coupling strength $γ$ acts as a global regularizer that compresses the phase space. The theoretical results are validated using a stiff-stable Second-Order Exponential Time Differencing (ETD2) scheme with Discrete Cosine Transform (DCT) to strictly enforce no-flux boundary conditions. |
| title | Global Dynamics and Stabilization of Zero-Mode Singularities in Multi-Scale Reaction-Diffusion Systems via Negative Coupling |
| topic | Analysis of PDEs Dynamical Systems Chaotic Dynamics 35K57, 37L30 |
| url | https://arxiv.org/abs/2601.00638 |