A reduced-order modeling of pattern formations (2403.03632v2)

Abstract: Chemical and biochemical reactions can exhibit a wide range of complex behaviors, including multiple steady states, oscillatory patterns, and chaotic dynamics. These phenomena have captivated researchers for many decades. Notable examples of oscillating chemical systems include the Briggs--Rauscher, Belousov--Zhabotinskii, and Bray--Liebhafsky reactions, where periodic variations in concentration are often visualized through observable color changes. These systems are typically modeled by a set of partial differential equations coupled through nonlinear interactions. Upon closer analysis, it appears that the dynamics of these chemical/biochemical reactions may be governed by only a finite number of spatial Fourier modes. We can also draw the same conclusion in fluid dynamics, where it has been shown that, over long periods, the fluid velocity is determined by a finite set of Fourier modes, referred to as determining modes. In this article, we introduce the concept of determining modes for a two-species chemical models, which covers models such as the Brusselator, the Gray-Scott model, and the Glycolysis model \cite{ashkenazi1978spatial,segel1980mathematical}. We demonstrate that it is indeed sufficient to characterize the dynamic of the model using only a finite number of spatial Fourier modes.