A 2D piecewise-linear discontinuous map arising in stock market modeling: Two overlapping period-adding bifurcation structures Article Swipe

We consider a 2D piecewise-linear discontinuous map defined on three partitions that drives the dynamics
\nof a stock market model. This model is a modification of our previous model associated with a map defined
\non two partitions. In the present paper, we add more realistic assumptions with respect to the behavior of
\nsentiment traders. Sentiment traders optimistically buy (pessimistically sell) a certain amount of stocks when
\nthe stock market is sufficiently rising (falling); otherwise they are inactive. As a result, the action of the price
\nadjustment is represented by a map defined by three different functions, on three different partitions. This
\nleads, in particular, to families of attracting cycles which are new with respect to those associated with a
\nmap defined on two partitions. We illustrate how to detect analytically the periodicity regions of these cycles
\nconsidering the simplest cases of rotation number 1∕𝑛, 𝑛 ≥ 3, and obtaining in explicit form the bifurcation
\nboundaries of the corresponding regions. We show that in the parameter space, these regions form two different
\noverlapping period-adding structures that issue from the center bifurcation line. In particular, each point of
\nthis line, associated with a rational rotation number, is an issue point for two different periodicity regions
\nrelated to attracting cycles with the same rotation number but with different symbolic sequences. Since these
\nregions overlap with each other and with the domain of a locally stable fixed point, a characteristic feature of
\nthe map is multistability, which we describe by considering the corresponding basins of attraction. Our results
\ncontribute to the development of the bifurcation theory for discontinuous maps, as well as to the understanding
\nof the excessively volatile boom-bust nature of stock markets.