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.

翻译：化学与生化反应可展现出多种复杂行为，包括多重稳态、振荡模式及混沌动力学。这些现象数十年来一直吸引着研究者的关注。振荡化学系统的典型实例包括Briggs--Rauscher反应、Belousov--Zhabotinskii反应及Bray--Liebhafsky反应，其浓度周期性变化常通过可见的颜色变化呈现。这类系统通常由一组通过非线性相互作用耦合的偏微分方程建模。进一步分析表明，这些化学/生化反应的动力学可能仅由有限个空间傅里叶模态主导。在流体动力学中亦可得出相同结论——已有研究表明，在长时间尺度下流体速度由有限个傅里叶模态决定，这类模态被称为确定模态。本文针对双组分化学模型（涵盖Brusselator、Gray-Scott模型及Glycolysis模型等）引入确定模态概念，并证明仅需有限个空间傅里叶模态即可充分表征该模型的动力学特性。