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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]

We study a simple financial market model with interacting chartists and fundamentalists that may give rise to multiband chaotic attractors. In particular, asset prices fluctuate erratically around their fundamental values, displaying a significant bull and bear market behavior. An in-depth analytical and numerical study of our model furthermore reveals the emergence of a new bifurcation structure, a phenomenon that we call a bandcount accretion bifurcation structure. The latter consists of regions associated with chaotic dynamics only, the boundaries of which are not defined by homoclinic bifurcations, but mainly by contact bifurcations of particular type where two distinct critical points of certain ranks coincide. [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]