Mathematical application of a non-local operator in language evolutionary theory

Many natural processes where an object/organism can make copies of itself or mutate to another similar object/organism are widely used in applied sciences such as mathematical modeling, population genetics or models of language evolutionary dynamic. The latter process remains complex with an impressive and unpredictable behavior that has never stopped changing over the time. In this paper, we make use of model combination of a non-local operator, its derivative order and the learning accuracy in a population language dynamic to indicate mathematical means of hastening the occurrence of more unpredictable trajectories in the system. We prove that those trajectories do form bifurcations that lead to an eventual chaotic process in the dynamic of language with learning for a chosen population with five languages. The model is investigated principally with the help of approximation method followed by its stability and convergence analysis. Numerical simulations show bifurcation diagrams that reveal a kind of symmetry in the evolution process of the frequency towards chaos. The evolution is done versus the mutation parameter as it increases. However, other simulations in the form of section's projections of the cascade diagram show a dynamic characterized by more chaotic trajectories as both the learning accuracy and the derivative order decrease. This result unfolds another great feature of non-local operators with possible impact in control theory.