Continuous and discontinuous piecewise-smooth one-dimensional maps. Invariant sets and bifurcation structures (Q2911601)

From the authors preface: ``There are several reasons to be interested in the dynamics of piecewise smooth maps, both continuous and discontinuous. On the one hand, the nonlinear dynamics theory has been mainly developed for smooth systems, therefore many problems associated with nonsmooth functions still remain open. Clearly, the nonsmoothness and, especially, the discontinuity of a system enrich its dynamics, leading to new distinctive properties and bifurcation phenomena which, in addition, may interplay with those characteristic for smooth systems. On the other hand, there is a demand for a theory of piecewise smooth dynamical systems from the applied sciences. It is nowadays widely recognized that many practically important real world processes, characterized by a sharp switching between the states, can hardly be represented adequately without the use of piecewise smooth functions.''NEWLINENEWLINE``To contribute to the theory of piecewise smooth dynamical systems, in the present book, we consider the most simple class of these systems, namely, one-dimnesional piecewise monotone maps defined on two partitions. Clearly, such a study is not only important in itself but also for understanding the dynamics of more generic classes of piecewise smooth maps. We mainly focus on two aspects of the considered class of maps, important both from the theoretical and the applied points of view, namely, on various invariant sets (in particular, attractors) of such maps and on the related bifurcation structures observed in their parameter space.''NEWLINENEWLINEThe book is structured as follows:NEWLINENEWLINEChapter 1 reviews attracting and repelling invariant sets which can be observed in continuous and discontinuous one-dimensional piecewise smooth maps, as well as further basic concepts, such as basins of attraction, homoclinic and heteroclinic orbits, critical points, absorbing intervals, symbolic dynamics, Farey structures, and first return maps. The authors discuss also some properties of chaotic attractors (in particular, their robustness and cyclicity) as well as the properties of one-dimensional piecewise increasing maps, subdivided in three classes (gap maps, circle homeomorphisms and overlapping maps) depending on their invertibility on the absorbing interval.NEWLINENEWLINEChapter 2 analyses bifurcations which can occur in one-dimensional piecewise smooth maps, such as border collision bifurcations, degenerate bifurcations, as well as homoclinic bifurcations and the associated bifurcations of chaotic attractors. These bifurcations appear as bifurcation structures described in the following chapters.NEWLINENEWLINEChapter 3 provides a brief overview of the basic bifurcation scenarios and structures which can be observed in one-dimensional maps, starting with the well-known logistic map and skew tent map scenarios and continuing with less known scenarios occurring in discontinuous maps, such as period adding and period incrementing, as well as bandcount adding and bandcount incrementing scenarios. NEWLINENEWLINEChapter 4 presents the so-called map replacement technique which provides a description of many bifurcation boundaries to a much farther extent than other approaches.NEWLINENEWLINEChapter 5 describes invariant sets and bifurcation structures in the skew tent map as well as the use of this map as a border collision normal form.NEWLINENEWLINEChapter 6 gives an account of the period adding bifurcation structure in the domain of regular dynamics of a discontinuous piecewise linear map, and the related bandcount adding bifurcation structure in the domain of robust chaos.NEWLINENEWLINEChapter 7 presents another pair of bifurcation structures which can be observed in the parameter space of a discontinuous piecewise linear map, namely the period incrementing and bandcount incrementing structures, associated with regular and chaotic attractors, respectively.NEWLINENEWLINEChapter 8 deals with bifurcation structures and their organizing centers in two-dimensional parameter spaces of various piecewise smooth maps. In particular, it is shown how the four bifurcation structures discussed in the previous chapters originate from particular codimension-two border collision bifurcation points.