On Dynamic Investigations of Cournot Duopoly Game: When Firms Want to Maximize Their Relative Profits

This paper studies a Cournot duopoly game in which firms produce homogeneous goods and adopt a bounded rationality rule for updating productions. The firms are characterized by an isoelastic demand that is derived from a simple quadratic utility function with linear total costs. The two competing firms in this game seek the optimal quantities of their production by maximizing their relative profits. The model describing the game’s evolution is a two-dimensional nonlinear discrete map and has only one equilibrium point, which is a Nash point. The stability of this point is discussed and it is found that it loses its stability by two different ways, through flip and Neimark–Sacker bifurcations. Because of the asymmetric structure of the map due to different parameters, we show by means of global analysis and numerical simulation that the nonlinear, noninvertible map describing the game’s evolution can give rise to many important coexisting stable attractors (multistability). Analytically, some investigations are performed and prove the existence of areas known in literature with lobes.

One-Layer vs. Two-Layer SOM in the Context of Outlier Identification: A Simulation Study

In this work, we applied a stochastic simulation methodology to quantify the power of the detection of outlying mixture components of a stochastic model, when applying a reduced-dimension clustering technique such as Self-Organizing Maps (SOMs). The essential feature of SOMs, besides dimensional reduction into a discrete map, is the conservation of topology. In SOMs, two forms of learning are applied: competitive, by sequential allocation of sample observations to a winning node in the map, and cooperative, by the update of the weights of the winning node and its neighbors. By means of cooperative learning, the conservation of topology from the original data space to the reduced (typically 2D) map is achieved. Here, we compared the performance of one- and two-layer SOMs in the outlier representation task. The same stratified sampling was applied for both the one-layer and two-layer SOMs; although, stratification would only be relevant for the two-layer setting—to estimate the outlying mixture component detection power. Two distance measures between points in the map were defined to quantify the conservation of topology. The results of the experiment showed that the two-layer setting was more efficient in outlier detection while maintaining the basic properties of the SOM, which included adequately representing distances from the outlier component to the remaining ones.

A 2D Hyperchaotic Map: Amplitude Control, Coexisting Symmetrical Attractors and Circuit Implementation

An absolute value function was introduced for chaos construction, where hyperchaotic oscillation was found with amplitude rescaling. The nonlinear absolute term brings the convenience for amplitude control. Two regimes of amplitude control including total and partial amplitude control are discussed, where the attractor can be rescaled separately by two independent coefficients. Symmetrical pairs of coexisting attractors are captured by corresponding initial conditions. Circuit implementation by the platform STM32 is consistent with the numerical exploration and the theoretical observation. This finding is helpful for promoting discrete map application, where amplitude control is realized in an easy way and coexisting symmetrical sequences with opposite polarity are obtained.

Namibian teachers’ perceptions and practices of teaching mapwork

This article addresses the problem of consistently poor learner performance in mapwork in secondary school geography in Namibia from the perspective of teachers. It presents the findings of a qualitative case study focused on understanding geography teachers’ perceptions and pedagogical practices of mapwork. Data were generated through a questionnaire administered to thirty teachers in fifteen secondary schools in the Ohangwena Region of Northern Namibia, and interviews and classroom observations were done with a purposive sample of three teachers. The study draws on Shulman’s ideas of teachers’ pedagogical content knowledge (1986, 1987) to interpret what the three teachers say about the teaching of mapwork and how they teach it. The findings reveal that the teachers are conscientious but ill-equipped to teach mapwork. Their classroom practices focus on teaching discrete map skills and procedural knowledge with little if any, attention given to spatial conceptual understanding and application of knowledge to solve problems. The study provides insights that may be of value to teachers, teacher educators and Senior Education Officers in Namibia and other southern African contexts when addressing the problem of low learning outcomes in mapwork.

Periodic Orbits and Chaos in Nonsmooth Delay Differential Equations

In this paper, we present a method to find periodic solutions for certain types of nonsmooth differential equations or nonsmooth delay differential equations. We apply the method to three examples, the first is a second-order differential equation with a nonsmooth term, in this case the method allows us to find periodic orbits in a nonlinear center. The two remaining examples are first-order nonsmooth delay differential equations. In the first one, there is a stable periodic solution and in the second, the presence of a chaotic attractor was detected. In the latter, the method allows us to obtain unstable periodic orbits within the attractor. For large values of the delay, both examples can be seen as singularly perturbed delay differential equations. For them, an analysis is performed with an associated discrete map which is obtained in the limit of large delays.