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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. [ABSTRACT FROM AUTHOR]

In this work, our goal is to show how a properly constructed first return map can help to describe the dynamics of a two-dimensional piecewise smooth map. In particular, to understand some formation principles of the bifurcation structure of its parameter space. The considered map is important from the applied point of view: it models the effects of fraud in a public procurement procedure and allows one to investigate whether honest or dishonest behavior prevails in society. We present the bifurcation structure of the map formed by the periodicity regions associated with attracting cycles of different periods. By means of the first return map, we show that the boundaries of these regions are related to border collision bifurcations typical for nonsmooth maps, as well as standard fold and flip bifurcations. The main economic result is related to the emergence of unexpected changes in qualitative non-compliant behavior over time. • We describe the bifurcation structures of the two-dimensional piecewise smooth map. • We prove there exists a closed invariant set made up of the critical line segments. • We show the first return map simplifies analysis of stable cycle birth mechanisms. [ABSTRACT FROM AUTHOR]

In the present work we study asymptotic dynamics of a multi-dimensional piecewise smooth map which models an oligopoly market where competitors use adaptive scheme for reaction choice. Each competitor also defines the moment for renewing the capital equipment depending on how intensively the latter is used. Namely, the larger output is produced, the quicker the capital exhausts. It is shown then that the asymptotic dynamics of the map allows coexistence of different metric attractors in which case it is sensitive to initial conditions. We also investigate stability of trajectories representing Cournot equilibria which are here not fixed but periodic points. In particular, it is shown that several such Cournot equilibria, belonging to different invariant manifolds, may coexist some of them being locally asymptotically stable and some being unstable. [ABSTRACT FROM AUTHOR]

Tâtonnement processes are usually interpreted as auctions, where a fictitious agent sets the prices until an equilibrium is reached and the trades are made. The main purpose of such processes is to explain how an economy comes to its equilibrium. It is well known that discrete time price adjustment processes may fail to converge and may exhibit periodic or even chaotic behavior. To avoid large price changes, a version of the discrete time tâtonnement process for reaching an equilibrium in a pure exchange economy based on a cautious updating of the prices has been proposed two decades ago. This modification leads to a one dimensional bimodal piecewise smooth map, for which we show analytically that degenerate bifurcations and border collision bifurcations play a fundamental role for the asymptotic behavior of the model. [ABSTRACT FROM AUTHOR]

We consider a Minskyan type model of a closed economy with autonomous public expenditure formulated in a discrete time framework, where an endogenous debt adjustment process is considered and where income variations account for real world physical and social constraints. The model is characterized by a unique nontrivial fixed point matching the economic equilibrium. We study its stability properties in terms of the constant factor that fixes the firms' desired debt level at a certain proportion of the current income. The stability loss of the fixed point is associated with either a subcritical flip or a supercritical Neimark–Sacker bifurcation. The latter implies occurrence of self-sustained oscillations interpreted as business cycles. We present three possible dynamic scenarios right after the Neimark–Sacker bifurcation: in a generic case and in two resonant cases. We also describe modifications of the attractor when propensity of firms to get into debt grows. In addition, we highlight that the increase of instabilities and complexities of dynamic outcomes is paired with the rise of the so called financial fragility indicator, which is a measure of fragility of the financial structure of the economy. [ABSTRACT FROM AUTHOR]