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

We present a new economic geography (NEG) model with a linear demand function where firms may change the perception of their relative dimension with respect to the local market. If they perceive themselves as big, they behave as Cournot oligopolists, otherwise they behave as monopolistic competitors. We first compare the pure cases in which only one market form prevails in the two-region economy. Comparing these pure cases of monopolistic and oligopolistic competition only a quantitative difference emerges. Subsequently, we assume that firms switch behavior and start to interact strategically when the number of local firms is below a threshold; in that case, the market form evolves endogenously. Results change substantially. ‘Break’ and ‘sustain’ points are separated as in standard NEG models with an isoelastic demand function leading to a co-existence of equilibria. Stable partial agglomeration and oscillations with small amplitude are possible. The dynamics are described by a 1D piecewise smooth map, which can be continuous (in the pure monopolistic or oligopolistic cases) or discontinuous (when the market structure evolves endogenously). We analyze the bifurcation structure of the parameter space of the map comparing these cases. We show that the continuous maps have rather standard dynamics, while the discontinuous map is characterised by border collision bifurcations of fixed points and cycles, which lead to rich and complex bifurcation structures.

We consider a discrete dynamical system, a two-dimensional real map which represents a one-dimensional complex map. Depending on the parameters, its bounded dynamics can be restricted to an invariant circle, cyclic invariant circles, invariant annular regions or disks. We show that on such invariant sets the trajectories are always either periodic of the same period, or quasiperiodic and dense. Moreover, the invariant sets may be transversely attracting or repelling, and undergo the typical cascade of period doubling bifurcations. Homoclinic bifurcations can also occur, leading to chaotic rings, annular regions filled with dense repelling cyclical circles and aperiodic trajectories.

We study a four-parameter family of 2D piecewise linear maps with two discontinuity lines. This family is a generalization of the discrete-time version of the fashion cycle model by Matsuyama, which was originally formulated in continuous time. The parameter space of the considered map is characterised by quite a complicated bifurcation structure formed by the periodicity regions of various attracting cycles. Besides the standard period adding and period incrementing structures, there exist incrementing structures with some distinctive properties, as well as novel mixed structures, which we study in detail. The boundaries of many periodicity regions associated with border collision bifurcations of the related cycles are obtained analytically. Several mixed structures are qualitatively described.