In the following article, we construct an interaction model (a variant of the SIR-model) of general language change. In the context of language change it is desirable to deduce the long-term behaviour of the corresponding dynamical system (for example to decide if complete of reversible language change are going to happen). We analyse this dynamical system by first proving non-existence of periodic orbits and then invoking the Poincar\'{e}-Bendixson theorem to show convergence to critical points only. Non-existence of periodic orbits is established by contradiction in showing that the average position of a potential periodic orbit must coincide with a certain critical point $C$ which cannot be encircled by the flow of the dynamical system so that the average position would be pulled to that side. Thus the long-term behaviour of the model for any given initial constellation of speakers depends only on four interaction parameters and can be easily analysed by looking at the four critical points. Subsequent numerical analysis of real data on language change is used to justify the relevance of the constructed model for the practicing quantitative linguist. We show how data-fitting methods can be used to determine the four interaction parameters and predict from them the long-term behaviour of the system, i.e. if complete language change or reversible language change will take place.

翻译：在本文中，我们构建了一个关于一般语言变化的交互模型（SIR模型的一个变体）。在语言变化的背景下，推导相应动力系统的长期行为是至关重要的（例如，用以判断是否会发生完全或可逆的语言变化）。我们通过首先证明周期轨道的不存在性，然后应用Poincaré-Bendixson定理来表明系统仅收敛于临界点，从而分析该动力系统。周期轨道的不存在性通过反证法确立：证明潜在周期轨道的平均位置必须与某个临界点$C$重合，而该临界点不能被动力系统的流所环绕，否则平均位置将被拉向该侧。因此，对于任何给定的说话者初始构型，模型的长期行为仅取决于四个交互参数，并且可以通过考察四个临界点来轻松分析。随后，我们利用语言变化的真实数据进行数值分析，以证明所构建模型对于从事定量语言学研究的实际相关性。我们展示了如何利用数据拟合方法确定四个交互参数，并基于这些参数预测系统的长期行为，即判断是否会发生完全的语言变化或可逆的语言变化。