The methods of various moving average rules remain popular with financial market practitioners. These rules have recently become the focus of empirical studies. However there seem to have been very few studies on the analysis of the type of financial market dynamics resulting from the fact that some agents engage in such strategies. In this paper we seek to fill this gap in the literature by proposing a market of financial market dynamics in which demand for traded assets has both a fundamentalist and a chartist component. The chartist demand is governed by the difference between a long run and a short run moving average. Both types of traders are bounded rational in the sense that, based on a certain fitness measure, traders switch from strategy with low fitness to the one with high fitness. We characterise first the stability and bifurcation properties of the underlying deterministic model via the reaction coefficient of the fundamentalists, the extrapolation rate of the chartists and the lengths used for the moving averages. By increasing the switching intensity, we then examine various rational routes to randomness for different, but fixed, long run moving average. The price dynamics of moving average is also examined and it is found that an increase of the window length of the long moving average can destabilize an otherwise stable system, leading to more complicated, even chaotic behaviour
