The case of stable node E1.

There are thousands of languages in the world, many of which are in danger of extinction due to language competition and evolution. Language is an aspect of culture, the rise, and fall of a language directly affects its corresponding culture. To preserve languages and prevent their mass extinction, it is crucial to develop a mathematical model of language coexistence. In this paper, we use a qualitative theory of ordinary differential equations to analyze the bilingual competition model, and obtain the trivial and non-trivial solutions of the bilingual competition model without sliding mode control, then analyze the stability of solutions and prove that solutions of the model have positive invariance. In addition, to maintain linguistic diversity and prevent mass extinction of languages, we propose a novel bilingual competition model with sliding control. The bilingual competition model is analyzed by proposing a sliding control policy to obtain a pseudo-equilibrium point. Meanwhile, numerical simulations clearly illustrate the effectiveness of the sliding mode control strategy. The results show that the likelihood of successful language coexistence can be increased by changing the status of languages and the value of monolingual-bilingual interaction, provides theoretical analysis for the development of policies to prevent language extinction.

全球现存数千种语言，其中诸多语种因语言竞争与演化正濒临消亡。语言作为文化的组成部分，语种的兴衰直接影响其对应的文化体系。为保护语种、避免大规模消亡，构建语言共存的数学模型至关重要。本文运用常微分方程（Ordinary Differential Equations, ODE）定性理论对双语竞争模型展开分析，得到无滑模控制（Sliding Mode Control, SMC）场景下该模型的平凡解与非平凡解，随后对解的稳定性进行分析，并证明模型解具备正不变性。此外，为维持语言多样性、避免语种大规模消亡，本文提出一种全新的带滑模控制的双语竞争模型。通过引入滑模控制策略对该双语竞争模型进行分析，得到伪平衡点。与此同时，数值仿真结果清晰验证了该滑模控制策略的有效性。研究结果表明，通过调整语种地位与单语-双语交互参数，可提升语言成功共存的概率，为防范语种消亡的政策制定提供了理论依据。