Continuous And Discontinuous Piecewise Smooth One Dimensional Maps Invariant Sets And Bifurcation Structures
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Continuous And Discontinuous Piecewise-smooth One-dimensional Maps: Invariant Sets And Bifurcation Structures
The investigation of dynamics of piecewise-smooth maps is both intriguing from the mathematical point of view and important for applications in various fields, ranging from mechanical and electrical engineering up to financial markets. In this book, we review the attracting and repelling invariant sets of continuous and discontinuous one-dimensional piecewise-smooth maps. We describe the bifurcations occurring in these maps (border collision and degenerate bifurcations, as well as homoclinic bifurcations and the related transformations of chaotic attractors) and survey the basic scenarios and structures involving these bifurcations. In particular, the bifurcation structures in the skew tent map and its application as a border collision normal form are discussed. We describe the period adding and incrementing bifurcation structures in the domain of regular dynamics of a discontinuous piecewise-linear map, and the related bandcount adding and incrementing structures in the domain of robust chaos. Also, we explain how these structures originate from particular codimension-two bifurcation points which act as organizing centers. In addition, we present the map replacement technique which provides a powerful tool for the description of bifurcation structures in piecewise-linear and other form of invariant maps to a much further extent than the other approaches.
Continuous and Discontinuous Piecewise-smooth One-dimensional Maps
"Although the dynamic behavior of piecewise-smooth systems is still far from being understood completely, some significant results in this field have been achieved in the last twenty years. The investigation of these systems is important not only because they represent adequate models for many applications ranging from mechanical and electrical engineering up to financial markets, but also due to the importance of the phenomena observed in other types of dynamical systems as well. It is natural, therefore, to begin the analysis with the most simple subclass of piecewise-smooth systems (namely one-dimensional maps) for which many phenomena can be investigated much more easily than for higher-dimensional systems. In this book, we consider both continuous and discontinuous one-dimensional piecewise-linear maps and summarize the results related to bifurcation structures in regular and robust chaotic domains. The map replacement technique based on symbolic dynamics allows us to offer significantly more analytical proofs than what is usually possible"--
Advances in Discrete Dynamical Systems, Difference Equations and Applications
This book comprises selected papers of the 26th International Conference on Difference Equations and Applications, ICDEA 2021, held virtually at the University of Sarajevo, Bosnia and Herzegovina, in July 2021. The book includes the latest and significant research and achievements in difference equations, discrete dynamical systems, and their applications in various scientific disciplines. The book is interesting for Ph.D. students and researchers who want to keep up to date with the latest research, developments, and achievements in difference equations, discrete dynamical systems, and their applications, the real-world problems.