Measuring and controlling the chaotic motion of profits

Abstract

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 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 gathering 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 reg- ular cycle). The orbit sabillization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1]. This work provides another illustration of how our under- standing of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modelled by ordinary differential equations.