Measuring and Controlling the Chaotic Motion of Profits

The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.

Behaviour and fate of metals in urban wastewater treatment plants: a review

The concerns on metals in urban wastewater treatment plants (WWTPs) are mainly related to its contents in discharges to environment, namely in the final effluent and in the sludge produced. In the near future, more restrictive limits will be imposed to final effluents, due to the recent guidelines of the European Water Framework Directive (EUWFD). Concerning the sludge, at least seven metals (Cd, Cr, Cu, Hg, Ni, Pb and Zn) have been regulated in different countries, four of which were classified by EUWFD as priority substances and two of which were also classified as hazardous substances. Although WWTPs are not designed to remove metals, the study of metals behaviour in these systems is a crucial issue to develop predictive models that can help more effectively the regulation of pre-treatment requirements and contribute to optimize the systems to get more acceptable metal concentrations in its discharges. Relevant data have been published in the literature in recent decades concerning the occurrence/fate/behaviour of metals in WWTPs. However, the information is dispersed and not standardized in terms of parameters for comparing results. This work provides a critical review on this issue through a careful systematization, in tables and graphs, of the results reported in the literature, which allows its comparison and so its analysis, in order to conclude about the state of the art in this field. A summary of the main consensus, divergences and constraints found, as well as some recommendations, is presented as conclusions, aiming to contribute to a more concerted action of future research. © 2015, Islamic Azad University (IAU).