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In this work we consider the border collision bifurcations occurring in a one-dimensional piecewise linear map with two discontinuity points. The map, motivated by an economic application, is written in a generic form and considered in the stable regime, with all slopes between zero and one. We prove that the period adding structures occur in maps with more than one discontinuity points and that the Leonov's method to calculate the bifurcation curves forming these structures is applicable also in this case. We demonstrate the existence of particular codimension-2 bifurcation (big-bang bifurcation) points in the parameter space, from which infinitely many bifurcation curves are issuing associated with cycles involving several partitions. We describe how the bifurcation structure of a map with one discontinuity is modified by the introduction of a second discontinuity point, which causes orbits to appear located on three partitions and organized again in a period-adding structure. We also describe particular codimension-2 bifurcation points which represent limit sets of doubly infinite sequences of bifurcation curves and appear due to the existence of two discontinuities. [ABSTRACT FROM AUTHOR]

The aim of this work is to study discontinuous one-dimensional maps in the case of slopes and offsets having opposite signs. Such models represent the dynamics of applied systems in several disciplines. We analyze in particular attracting cycles, their border collision bifurcations and the properties of the periodicity regions in the parameter space. The peculiarity of this family is that we can make use of the technical instrument of the first return map. With this, we can rigorously prove properties which were known numerically, as well as prove new ones, giving a complete characterization of the overlapping periodicity regions. [ABSTRACT FROM AUTHOR]

50 years ago (1959) in a series of publications by Leonov, a detailed analytical study of the nested period adding bifurcation structure occurring in piecewise-linear discontinuous 1D maps was presented. The results obtained by Leonov are barely known, although they allow the analytical calculation of border-collision bifurcation subspaces in an elegant and much more efficient way than it is usually done. In this work we recall Leonov's approach and explain why it works. Furthermore, we slightly improve the approach by avoiding an unnecessary coordinate transformation, and also demonstrate that the approach can be used not only for the calculation of border-collision bifurcation curves. [ABSTRACT FROM AUTHOR]

In this paper we present a contribution toward a dynamic theory of the consumer. Canonical economic theory assumes a perfectly rational agent that acts according to optimization principles of a utility function subject to budget constraints. Here we propose a two-dimensional discrete dynamical system that describes the choice problem of a consumer under the assumption of bounded rationality. The map representing this adaptive process is characterized by the presence of a denominator that can vanish. We use recent results on the global bifurcations of this kind of maps in order to explain the coexistence of different attractors and the structure of the corresponding basins of attraction. The stationary equilibria of the map represent the rational choices of the consumer in the static setting, i.e. solutions of the utility maximization problem under budget constraints. We use geometric and numerical methods to study the problem of coexistence of different attractors and the related problem of the basins of attraction and their global bifurcations. [ABSTRACT FROM AUTHOR]