Modeling learning and teaching interaction by a map with vanishing denominators:Fixed points stability and bifurcations

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. (C) 2019 Elsevier Ltd. All rights reserved

Synchronization of coupled neural oscillators with heterogeneous delays

We investigate the effects of heterogeneous delays in the coupling of two excitable neural systems. Depending upon the coupling strengths and the time delays in the mutual and self-coupling, the compound system exhibits different types of synchronized oscillations of variable period. We analyze this synchronization based on the interplay of the different time delays and support the numerical results by analytical findings. In addition, we elaborate on bursting-like dynamics with two competing timescales on the basis of the autocorrelation function.Comment: 18 pages, 14 figure

The conceptual bases of introduction of foresight marketing into business management

The article outlines a critical analysis of theoretical approaches to the development of the concept of foresight marketing on the basis of which the own definition of “foresight marketing” has been developed. Modern enterprises are interested in receiving reasonable foresight forecasts in marketing, which was not sufficiently distributed in the practical activity of the enterprise.Theoretical and practical aspects of the foresight framework are still explored insufficiently. Therefore, the purpose of the article is to improve the theoretical framework of foresight marketing and to create the conceptual model of foresight marketing.The following scientific methods have been used in this research: system analysis, content analysis, comparative method, method of logical generalization, morphological method, dialectics of the relationship between fundamental and applied knowledge.The analysis revealed imperfection of the theoretical apparatus of foresight marketing, thus, the modern approach to the interconnection and balance of foresight marketing and strategic marketing on the enterprises was suggested. To develop a conceptual model of foresight marketing, the authors have considered and described its major components. A conceptual model of foresight marketing is a systematic combination of certain elements, namely: conditions, barriers, and prerequisites; subject and object; methodology of foresight marketing; principles, tools, and categories.The article presents a generalized model of foresight marketing process, which shows the basic subjects and bases on four consecutive stages. The authors provide a new perspective as for the concept of foresight marketing and the basic prerequisites for the implementation of foresight in marketing management

Bifurcation Structures in a Bimodal Piecewise Linear Map

In this paper we present an overview of the results concerning dynamics of a piecewise linear bimodal map. The organizing principles of the bifurcation structures in both regular and chaotic domains of the parameter space of the map are discussed. In addition to the previously reported structures, a family of regions closely related to the so-called U-sequence is described. The boundaries of distinct regions belonging to these structures are obtained analytically using the skew tent map and the map replacement technique

Qualitative Theory of Dynamical Systems, Tools and Applications for Economic Modelling: Lectures Given at the COST Training School on New Economic Complex Geography at Urbino, Italy, 17-19 September 2015

The book presents the lectures delivered during a short course held at Urbino University in summer 2015 on qualitative theory of dynamical systems, included in the activities of the COST Action IS1104 “The EU in the new economic complex geography: models, tools and policy evaluation”. It provides a basic introduction to dynamical systems and optimal control both in continuous and discrete time, as well as some numerical methods and applications in economic modelling. Economic and social systems are intrinsically dynamic, characterized by interdependence, nonlinearity and complexity, and these features can only be approached using a qualitative analysis based on the study of invariant sets (equilibrium points, limit cycles and more complex attractors, together with the boundaries of their basins of attraction), which requires a trade-off between analytical, geometrical and numerical methods. Even though the early steps of the qualitative theory of dynamical systems have been in continuous time models, in economic and social modelling discrete time is often used to describe event-driven (often decision-driven) evolving systems. The book is written for Ph.D. and master’s students, post-doctoral fellows, and researchers in economics or sociology, and it only assumes a basic knowledge of calculus. However it also suggests some more advanced topics

A dynamical model of proximal development: Multiple implementations

Dynamical systems are quite important in psychological research as virtually all psychological processes occur in time. In this chapter we show how to implement a dynamical model of proximal development using a spreadsheet, R and C++. We discuss strengths and weakness of each approach. Using a spreadsheet or a statistical software such as R make these approaches palatable both for people with background in economics and psychology; on the other hand, using C++ provides better efficiency at the cost of requiring some more competencies. Last but not least, all the approaches we propose use free and open source software

Global dynamic scenarios in a discrete-time model of renewable resource exploitation: a mathematical study

We consider the two-dimensional map introduced in Bischi et al. (J Differ Equ Appl 21(10):954–973, 2015) formulated as a model for a renewable resource exploitation process in an evolutionary setting. The global dynamic scenarios displayed by the model are not so often encountered in smooth two-dimensional dynamical systems. We explain the occurrence of such scenarios at the light of the theory of noninvertible maps. Moreover, complex structures of basins of attraction of coexisting invariant sets are observed. We analyze such structures by examining stability properties of chaotic sets, in the case in which a non-topological Milnor attractor is present. Stability changes of a chaotic set occur through global bifurcations (such as riddling and blowout) and are detected by means of the study of the spectrum of Lyapunov exponents associated with the set