Covariant Multi-Scale Negative Coupling on Dynamic Riemannian Manifolds: A Geometric Framework for Topological Persistence in Infinite-Dimensional Systems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911495971930112 |
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| author | Hou, Pengyue |
| author_facet | Hou, Pengyue |
| contents | Dimensional reduction is a generic consequence of dissipation in nonlinear evolution equations, often leading to attractor collapse and the loss of dynamical richness. To counteract this, we introduce a geometric framework for Covariant Multi-Scale Negative Coupling Systems (C-MNCS), formulated intrinsically on smooth Riemannian manifolds for a broad class of semilinear dissipative PDEs. The proposed coupling redistributes energy across dynamically separated spectral bands, inducing a scale-balanced feedback that prevents finite-dimensional degeneration. We establish the short-time well-posedness of the coupled state-metric evolution system in Sobolev spaces and derive a priori estimates for phase-space contraction rates. Furthermore, under a global boundedness hypothesis, we prove that the global attractor possesses a strictly finite Hausdorff and Kaplan-Yorke dimension. To bridge abstract topological bounds with physical realizability, we isolate the core Adaptive Spectral Negative Coupling (ASNC) mechanism for numerical validation. High-resolution experiments, utilizing a fully coupled ETDRK4 scheme and continuous QR-based Lyapunov exponent computation on a conformally flat 2D dynamic scalar manifold, corroborate the theoretical predictions. These computations explicitly demonstrate the stabilization of high-dimensional attractors against severe macroscopic dissipation. This geometrically consistent mechanism offers a new paradigm for maintaining structural complexity and multiscale control in infinite-dimensional dynamical systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07325 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Covariant Multi-Scale Negative Coupling on Dynamic Riemannian Manifolds: A Geometric Framework for Topological Persistence in Infinite-Dimensional Systems Hou, Pengyue Chaotic Dynamics Dynamical Systems Computational Physics 37L30, 35B41, 58J35, 65P20 Dimensional reduction is a generic consequence of dissipation in nonlinear evolution equations, often leading to attractor collapse and the loss of dynamical richness. To counteract this, we introduce a geometric framework for Covariant Multi-Scale Negative Coupling Systems (C-MNCS), formulated intrinsically on smooth Riemannian manifolds for a broad class of semilinear dissipative PDEs. The proposed coupling redistributes energy across dynamically separated spectral bands, inducing a scale-balanced feedback that prevents finite-dimensional degeneration. We establish the short-time well-posedness of the coupled state-metric evolution system in Sobolev spaces and derive a priori estimates for phase-space contraction rates. Furthermore, under a global boundedness hypothesis, we prove that the global attractor possesses a strictly finite Hausdorff and Kaplan-Yorke dimension. To bridge abstract topological bounds with physical realizability, we isolate the core Adaptive Spectral Negative Coupling (ASNC) mechanism for numerical validation. High-resolution experiments, utilizing a fully coupled ETDRK4 scheme and continuous QR-based Lyapunov exponent computation on a conformally flat 2D dynamic scalar manifold, corroborate the theoretical predictions. These computations explicitly demonstrate the stabilization of high-dimensional attractors against severe macroscopic dissipation. This geometrically consistent mechanism offers a new paradigm for maintaining structural complexity and multiscale control in infinite-dimensional dynamical systems. |
| title | Covariant Multi-Scale Negative Coupling on Dynamic Riemannian Manifolds: A Geometric Framework for Topological Persistence in Infinite-Dimensional Systems |
| topic | Chaotic Dynamics Dynamical Systems Computational Physics 37L30, 35B41, 58J35, 65P20 |
| url | https://arxiv.org/abs/2603.07325 |