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Speeding up chaos and limit cycles in evolutionary language and learning processes
Author(s) -
Doungmo Goufo Emile Franc
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4220
Subject(s) - replicator equation , mathematics , chaotic , kernel (algebra) , dynamical systems theory , limit cycle , population , limit (mathematics) , stability (learning theory) , convergence (economics) , computer science , mathematical economics , artificial intelligence , pure mathematics , mathematical analysis , machine learning , physics , demography , quantum mechanics , sociology , economics , economic growth
Evolution of human language and learning processes have their foundation built on grammar that sets rules for construction of sentences and words. These forms of replicator–mutator (game dynamical with learning) dynamics remain however complex and sometime unpredictable because they involve children with some predispositions. In this paper, a system modeling evolutionary language and learning dynamics is investigated using the Crank–Nicholson numerical method together with the new differentiation with non‐singular kernel. Stability and convergence are comprehensively proven for the system. In order to seize the effects of the non‐singular kernel, an application to game dynamical with learning dynamics for a population with five languages is given together with numerical simulations. It happens that such dynamics, as functions of the learning accuracy μ , can exhibit unusual bifurcations and limit cycles followed by chaotic behaviors. This points out the existence of fickle and unpredictable variations of languages as time goes on, certainly due to the presence of learning errors. More interestingly, this chaos is shown to be dependent on the order of the non‐singular kernel derivative and speeds up as this derivative order decreases. Hence, can people use that order to control chaotic behaviors observed in game dynamical systems with learning! Copyright © 2016 John Wiley & Sons, Ltd.