Chaotic Behaviour of Nonlinear Dynamical Systems in Economics

The purpose of this paper is to present a new method for studying qualitative behaviour of unknown nonlinear dynamical systems. Analysing economic time se-ries we will present several interesting conclusions about the Czech capital market represented by the PX 50 Index. This paper is based on the diploma thesis [Pos99] which deals with the chaotic behaviour of nonlinear dynamical systems in general. 1 Nonlinear Dynamics in Economics Economics and investment analysis have only recently begun to pay attention to nonlin-ear dynamics. The implications of chaotic behaviour make the techniques controversial. In essence, a chaotic system can produce random-looking results that are not truly ran-dom. Long-term forecasting is impossible. Markets are not efficient and they are not forecastable. In [Pos99] we can find the basics of nonlinear dynamical systems as they apply to systems with known equations. This is a necessary preliminary to introducing the analysis of real systems. In real life, we end up knowing very little. We can learn the concepts necessary to understand dynamical systems. In this paper we will apply these concepts to time series (in Section 2) and discuss actual analysis of capital market time series (in Section 3). One characteristic of chaotic systems is the sensitive dependence on initial conditions. This property makes the dynamical system to be unpredictable in the long term. The unpredictability occurs for two reasons. Dynamical systems are for us now feedback systems. What comes out goes back in again, is transformed and comes back out, endlessly. Feedback systems are for example compounding interests, except that the transformation is exponential (it has a power higher than 1). Any differences in initial values will grow exponentially as well. We will show some examples later. A second characteristic of complex systems involves the concept of critical levels and bifurcations. There is sometimes included so called a fractal dimension1 to be a crucial element o