2024/03/06 by Erika Hausenblas, Hausenblas, Erika, Tsiry Avisoa Randrianasolo +1
Computer Science · Mathematics · #Nonlinear Dynamics and Pattern Formation #cs.NA #math.AP #math.NA
paper · pdf · doi:10.48550/arxiv.2403.03632
openalex publication_date 2024/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
The goal of this work is to analyze the long-term behavior of reaction-diffusion systems arising in two-species chemical models and to identify finite sets of modes that determine their asymptotic dynamics. The models considered include, as particular cases the Gray--Scott and the Glycolysis models. These systems are described by coupled reaction-diffusion equations and admit a finite-dimensional representation based on a limited number of spatial Fourier modes that capture their essential reduced dynamics. The concept of determining modes, introduced in this context, is closely related to other approaches that seek finite-dimensional representations of infinite-dimensional dynamics, such as the Proper Orthogonal Decomposition and the construction of Approximate Inertial Manifolds. We prove that the dynamics of the system can be completely characterized by a finite number of low modes, since all higher modes are asymptotically determined by them, thus providing an analytical foundation for reduced dynamics in models of pattern formation.