Speeding up chaos and limit cycles in evolutionary language and learning processes

Abstract

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 , 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 nonsingular 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! 3055-3065 Recent observations by Caputo and Fabrizio [11] stated that the two definitions earlier are more suitable to describe physical phenomena, related to fatigue, damage, and electromagnetic hysteresis but do not correctly portray some behaviors taking place in multi-scale systems and in materials with massive heterogeneities. Hence, the same authors introduced the following new version of fractional order derivative with no-singular kernel:Other non-linear models including those with diffusion or advection-dispersion have been intensively analyzed in several works [23][24][25][26][27] where numerical and explicit schemes were developed and applied. Other authors analyzed the spacial cases where the order of the derivative is variable and depends both on the time and the spacial variables. Hence, Chen et al. [28] proposed the implicit

Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control

“…On the other side, real life phenomena described by models with fractional differentiation or variable order derivative have been analyzed in number of works [2,3,13,17,27,14,37,38,42] with such a derivative proven to have evident memory properties which vary along with time and spatial variable. The fractional differentiation or the variable order derivative has turned out to play the role of one of the model's important parameter able to change de dynamics of the system and ultimately control it [7,1,15,16,17,35,30,20].…”

“…There are several types of definitions for derivatives with fractional order and they include, among others, the variable order derivative (VOD). Such types of models have been discussed in number of research articles [2,3,13,15,16,18,22,35,30,20]. A special attention has been given to the reasonable memory properties that characterize the variable order derivative and that strongly depend on the time or space coordinates [27,37,38,42,4,39,10,8,9].…”

“…Any reader interested in more properties and interpretations of this type fractional VOD may consult the papers [3,13,15,16,27,37,38,42,18,22,35,30,20] and the references therein.…”