Modeling learning and teaching interaction by a map with vanishing denominators: Fixed points stability and bifurcations. (September 2019)

• Record Type: Journal Article
• Authors: Merlone, Ugo; Panchuk, Anastasiia; van Geert, Paul
• Is Part Of: Chaos, solitons and fractals. Volume 126(2019)
• Page Start: 253
• Page End: 265
• Publication Date: 2019-09
• Subjects: Maps with vanishing denominator -- Focal points -- Developmental psychology -- Learning and teaching coupling; 37G10 -- 37N99 -- 91E99; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2019.06.008
• Languages: English
• ISSNs: 0960-0779

Abstract

Abstract: Among the different dynamical systems which have been considered in psychology, those modeling the dynamics of learning and teaching interaction are particularly important. In this paper we consider a well known model of proximal development and analyze some of its mathematical properties. The dynamical system we study belongs to a class of 2D noninvertible piecewise smooth maps characterized by vanishing denominators in both components. We determine focal points, among which the origin is particular since its prefocal set contains this point itself. We also find fixed points of the map and investigate their stability properties. Finally, we consider map dynamics for two sample parameter sets, providing plots of basins of attraction for coexisting attractors in the phase plane. We emphasize that in the first example there exists a set of initial conditions of non-zero measure, whose orbits asymptotically approach the focal point at the origin.