New Economic Era Thinking and Stock Market Bubbles: A Two-Dimensional Piecewise Linear Discontinuous Map Approach

Based on a simple chartist-fundamentalist model, we demonstrate that new economic era thinking, i.e., temporarily optimistic views about the state of the economy, may lead to the creation of endogenous stock market bubbles. The dynamics of our stock market model are represented by a two-dimensional piecewise linear discontinuous map. The discontinuity set is quite different from those considered in the recent works on similar maps. Here, we show how to analyze also maps with novel constraints, proving that the main result is still multistability in the parameter regions associated with the stable fundamental fixed point, coexisting with several attracting cycles. However, in the parameter space, the bifurcation structure of the existing periodicity regions is new, and highly dependent on the parameter values. We describe the bifurcations related to a particular family of cycles with rotation number 1/n, n≥3. We also describe in detail several properties of a saddle 2-cycle (which can never be attracting), playing an important role for the dynamics. Before its homoclinic bifurcation, the stable set of the 2-cycle belongs to the boundary of the set, in the phase plane, related to divergent trajectories. Our results evidence the existence of oscillations around the fundamental fixed point for parameter settings in what is usually considered its stability domain, as well as the existence of chaotic attractors when the fundamental fixed point is unstable.