Bifurcation, chaos and multi-stability regions in an asset pricing model with three subsystems

An asset pricing model with two types of chartists and fundamentalists and trend followers is considered, it is driven by a two-dimensional piecewise linear discontinuous map with three subsystems. There are great differences in the dynamic behaviour between expected offset (the expectations of trend followers offset the difference between the expectations of Type 1 traders in bull and bear markets) and expected non-offset. It is proven that there is no chaos in the dynamic of system with expected offset. However, chaos may exist in the dynamic of system with expected non-offset. We present a systematic approach to the problem of analysing the bifurcation phenomena associated with the appearance/disappearance of cycles, which may be related to several bifurcations. The multi-stability regions in parameter plane and related basins of multi-attractors in phase space are investigated. This paper aims to uncover the endogenous law of the unpredictability and excessive volatility in financial markets.