Chemical and biochemical reactions can exhibit surprisingly different behaviours from multiple steady-state solutions to oscillatory solutions and chaotic behaviours. Such behaviour has been of great interest to researchers for many decades. The Briggs-Rauscher, Belousov-Zhabotinskii and Bray-Liebhafsky reactions, for which periodic variations in concentrations can be visualized by changes in colour, are experimental examples of oscillating behaviour in chemical systems. These type of systems are modelled by a system of partial differential equations coupled by a nonlinearity. However, analysing the pattern, one may suspect that the dynamic is only generated by a finite number of spatial Fourier modes. In fluid dynamics, it is shown that for large times, the solution is determined by a finite number of spatial Fourier modes, called determining modes. In the article, we first introduce the concept of determining modes and show that, indeed, it is sufficient to characterise the dynamic by only a finite number of spatial Fourier modes. In particular, we analyse the exact number of the determining modes of $u$ and $v$, where the couple $(u,v)$ solves the following stochastic system \begin{equation*} \partial_t{u}(t) = r_1\Delta u(t) -\alpha_1u(t)- \gamma_1u(t)v^2(t) + f(1 - u(t)) + g(t),\quad \partial_t{v}(t) = r_2\Delta v(t) -\alpha_2v(t) + \gamma_2 u(t)v^2(t) + h(t),\quad u(0) = u_0,\;v(0) = v_0, \end{equation*} where $r_1,r_2,\gamma_1,\gamma_2>0$, $\alpha_1,\alpha_2 \ge 0$ and $g,h$ are time depending mappings specified later.

翻译：化学与生化反应可展现出从多稳态解到振荡解乃至混沌行为等令人惊异的多样性行为。数十年来，这类行为一直备受研究者关注。Briggs-Rauscher反应、Belousov-Zhabotinskii反应和Bray-Liebhafsky反应可通过颜色变化直观呈现浓度的周期性变动，是化学系统中振荡行为的实验范例。此类系统由非线性耦合的偏微分方程组建模。然而，在分析斑图时，可推测其动力学仅由有限个空间Fourier模态生成。在流体动力学中已证明，长时间尺度下，解由有限个空间Fourier模态（称为决定性模态）决定。本文首先引入决定性模态的概念，并证明确实仅需有限个空间Fourier模态即可充分刻画动力学特征。特别地，我们分析了$u$与$v$决定性模态的确切数量，其中耦合函数$(u,v)$求解以下随机系统：\begin{equation*} \partial_t{u}(t) = r_1\Delta u(t) -\alpha_1u(t)- \gamma_1u(t)v^2(t) + f(1 - u(t)) + g(t),\quad \partial_t{v}(t) = r_2\Delta v(t) -\alpha_2v(t) + \gamma_2 u(t)v^2(t) + h(t),\quad u(0) = u_0,\;v(0) = v_0, \end{equation*} 其中$r_1,r_2,\gamma_1,\gamma_2>0$，$\alpha_1,\alpha_2 \ge 0$，而$g,h$为后续定义的时间依赖映射。