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In the present paper, we investigate the complex dynamics arising from a behavioral exchange rate discontinuous model with heterogeneous agents. Unlike previous works explaining the emergence of chaos in the exchange rate models as the resulting of nonlinearity, our model is able to produce endogenous exchange rate dynamics due to the presence of discontinuity induced by a sentiment index, which affects the way investors take their trading decisions. In particular, it affects the level of optimism/pessimism of fundamentalists regarding their perception on the value of fundamental. Moreover, it also affects the strength to which one kind of chartists places her buying/selling orders. We show that our model, represented by a two-dimensional discontinuous map, has the ability to produce interesting endogenous exchange rate dynamics. In addition, when each component of the map is buffeted by a stochastic component, the model closely replicates the stylized facts of the EUR/USD and EUR/JPY exchange rate markets. [ABSTRACT FROM AUTHOR]

Recently, we reported that a chaotic attractor in a discontinuous one-dimensional map may undergo a so-called exterior border collision bifurcation, which causes additional bands of the attractor to appear. In the present paper, we suppose that the chaotic attractor's basin boundary contains a chaotic repeller, and discuss a bifurcation pattern consisting of an exterior border collision bifurcation and an expansion bifurcation (interior crisis). In the generic case, where neither the border point the chaotic attractor collides with, nor any of its images belong to the chaotic repeller, the exterior border collision bifurcation is followed by the expansion bifurcation, and the distance between both bifurcations may be arbitrarily small but positive. In the non-generic (codimension-2) case, these bifurcations occur simultaneously, so that a border collision bifurcation of a chaotic attractor leads directly to its expansion. [ABSTRACT FROM AUTHOR]

In one-dimensional piecewise smooth maps with multiple borders, chaotic attractors may undergo border collision bifurcations, leading to a sudden change in their structure. We describe two types of such border collision bifurcations and explain the mechanisms causing the changes in the geometrical structure of the attractors, in particular, in the number of their bands (connected components). [ABSTRACT FROM AUTHOR]

We continue the investigation of a one-dimensional piecewise linear map with two discontinuity points. Such a map may arise from a simple asset-pricing model with heterogeneous speculators, which can help us to explain the intricate bull and bear behavior of financial markets. Our focus is on bifurcation structures observed in the chaotic domain of the map's parameter space, which is associated with robust multiband chaotic attractors. Such structures, related to the map with two discontinuities, have been not studied before. We show that besides the standard bandcount adding and bandcount incrementing bifurcation structures, associated with two partitions, there exist peculiar bandcount adding and bandcount incrementing structures involving all three partitions. Moreover, the map's three partitions may generate intriguing bistability phenomena. [ABSTRACT FROM AUTHOR]

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique. [ABSTRACT FROM AUTHOR]