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Bifurcation structure in the skew tent map and its application as a border collision normal form
No full textThe goal of the present paper is to collect the results related to dynamics of a one-dimensional piecewise linear map widely known as the skew tent map. These results may be useful for the researchers working on theoretical and applied problems in the field of nonsmooth dynamical systems. In particular, we propose the complete description of the bifurcation structure of the parameter space of the skew tent map, providing the related proofs. It is also shown how these results can be used to classify border collision bifurcations in one-dimensional piecewise smooth maps
Nonsmooth one-dimensional maps: some basic concepts and definitions
No full textThe main purpose of the present survey is to contribute to the theory of dynamical systems defined by one-dimensional piecewise monotone maps. We recall some definitions known from the theory of smooth maps, which are applicable to piecewise smooth ones, and discuss the notions specific for the considered class of maps. To keep the presentation clear for the researchers working in other fields, especially in applications, many examples are provided. We focus mainly on the notions and concepts which are used for the investigation of various kinds of attractors of a map and related bifurcation structures observed in its parameter space
The role of constraints in a segregation model: The symmetric case
Get PDFAbstractIn this paper we study the effects of constraints on the dynamics of an adaptive segregation model introduced by Bischi and Merlone (2011) [3]. The model is described by a two dimensional piecewise smooth dynamical system in discrete time. It models the dynamics of entry and exit of two populations into a system, whose members have a limited tolerance about the presence of individuals of the other group. The constraints are given by the upper limits for the number of individuals of a population that are allowed to enter the system. They represent possible exogenous controls imposed by an authority in order to regulate the system. Using analytical, geometric and numerical methods, we investigate the border collision bifurcations generated by these constraints assuming that the two groups have similar characteristics and have the same level of tolerance toward the members of the other group. We also discuss the policy implications of the constraints to avoid segregation
The Role of Constraints in a Segregation Model: The Asymmetric Case
Get PDFWe extend the analysis on the effects of the entry constraints on the dynamics of an adaptive segregation model of Shelling’s type when the two populations involved differ in numerosity, level of tolerance toward members of the other population, and speed of reaction. The model is described by a two-dimensional piecewise smooth dynamical system in discrete time, where
the entry constraints represent possible exogenous controls imposed by an authority in order to regulate the maximum number of members of the two populations allowed to enter the system, usually the district in which they live in. In this paper, we investigate the nature of some particular border
collision bifurcations and discuss the policy implications of the entry constraints in terms of segregation. The investigation reveals that asymmetries in the level of tolerance of the two populations involved may lead to phenomena of overreaction or overshooting in the adjustment process. In
order to avoid the risk of segregation, suitable entry limitations must be
imposed at least on the more tolerant population
Dangerous Bifurcations Revisited
No full textA dangerous border collision bifurcation has been defined as the dynamical instability that occurs when the basins of attraction of stable fixed points shrink to a set of zero measure as the parameter approaches the bifurcation value from either side. This results in almost all trajectories diverging off to infinity at the bifurcation point, despite the eigenvalues of the fixed points before and after the bifurcation being within the unit circle. In this paper, we show that similar bifurcation phenomena also occur when the stable orbit in question is of a higher periodicity or is chaotic. Accordingly, we propose a generalized definition of dangerous bifurcation suitable for any kind of attracting sets. We report two types of dangerous border collision bifurcations and show that, in addition to the originally reported mechanism typically involving singleton saddle cycles, there exists one more situation where the basin boundary is formed by a repelling closed invariant curve
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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