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.

1. Fundamentals of piecewise-smooth, continuous systems. 1.1. Applications. 1.2. A framework for local behavior. 1.3. Existence of equilibria and fixed points. 1.4. The observer canonical form. 1.5. Discontinuous bifurcations. 1.6. Border-collision bifurcations. 1.7. Poincaré maps and discontinuity maps. 1.8. Period adding. 1.9. Smooth approximations -- 2. Discontinuous bifurcations in planar systems. 2.1. Periodic orbits. 2.2. The focus-focus case in detail. 2.3. Summary and classification -- 3. Codimension-two, discontinuous bifurcations. 3.1. A nonsmooth, saddle-node bifurcation. 3.2. A nonsmooth, Hopf bifurcation. 3.3. A codimension-two, discontinuous Hopf bifurcation -- 4. The growth of Saccharomyces cerevisiae. 4.1. Mathematical model. 4.2. Basic mathematical observations. 4.3. Bifurcation structure. 4.4. Simple and complicated stable oscillations -- 5. Codimension-two, border-collision bifurcations. 5.1. A nonsmooth, saddle-node bifurcation. 5.2. A nonsmooth, period-doubling bifurcation -- 6. Periodic solutions and resonance tongues. 6.1. Symbolic dynamics. 6.2. Describing and locating periodic solutions. 6.3. Resonance tongue boundaries. 6.4. Rotational symbol sequences. 6.5. Cardinality of symbol sequences. 6.6. Shrinking points. 6.7. Unfolding shrinking points -- 7. Neimark-Sacker-like bifurcations. 7.1. A two-dimensional map. 7.2. Basic dynamics. 7.3. Limiting parameter values. 7.4. Resonance tongues. 7.5. Complex phenomena relating to resonance tongues. 7.6. More complex phenomena

maps whose topological entropy is equal to zero (i.e., maps that have only cyeles of pe 2 riods 1,2,2 , ... ) are studied in detail and elassified. Various topological aspects of the dynamics of unimodal maps are studied in Chap ter 5. We analyze the distinctive features of the limiting behavior of trajectories of smooth maps. In particular, for some elasses of smooth maps, we establish theorems on the number of sinks and study the problem of existence of wandering intervals. In Chapter 6, for a broad elass of maps, we prove that almost all points (with respect to the Lebesgue measure) are attracted by the same sink. Our attention is mainly focused on the problem of existence of an invariant measure absolutely continuous with respect to the Lebesgue measure. We also study the problem of Lyapunov stability of dynamical systems and determine the measures of repelling and attracting invariant sets. The problem of stability of separate trajectories under perturbations of maps and the problem of structural stability of dynamical systems as a whole are discussed in Chap ter 7. In Chapter 8, we study one-parameter families of maps. We analyze bifurcations of periodic trajectories and properties of the set of bifurcation values of the parameter, in eluding universal properties such as Feigenbaum universality.

​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.

This book reflects the state of the art on nonlinear economic dynamics, financial market modelling and quantitative finance. It contains eighteen papers with topics ranging from disequilibrium macroeconomics, monetary dynamics, monopoly, financial market and limit order market models with boundedly rational heterogeneous agents to estimation, time series modelling and empirical analysis and from risk management of interest-rate products, futures price volatility and American option pricing with stochastic volatility to evaluation of risk and derivatives of electricity market. The book illustrates some of the most recent research tools in these areas and will be of interest to economists working in economic dynamics and financial market modelling, to mathematicians who are interested in applying complexity theory to economics and finance and to market practitioners and researchers in quantitative finance interested in limit order, futures and electricity market modelling, derivative pricing and risk management.

Cycles, Growth and the Great Recession is a collection of papers that assess the nature and role of the business cycle in contemporary economies. These assessments are made in the context of the financial market instability that distinguishes the Great Recession from previous post-war slowdowns. Theorists and applied scholars in the fields of economics and mathematical economics discuss various approaches to understanding cycles and growth, and present mathematical and applied macro models to show how uncertainty shapes cycles by affecting the economic agent choice. Also included is an empirical section that investigates how the Great Recession affected households’ housing wealth, labour productivity and migration decisions. This book aims to: Propose a novel understanding of the business cycle by comparing the approaches of various scholars, starting from Hyman Minsky and Piero Ferri. Show that uncertainty is a main feature of the business cycle that affects decision-making and economic behaviour in general. Explain with mathematical models how the behaviour of economic agents can lead to cyclical paths for modern developed economies. Augment theory with empirical analysis of some central issues related to the Great Recession. This book comprises an original view of such widely discussed subjects as business cycles, uncertainty, economic growth and the Great Recession, constructed around theory, models and applications.

This book is devoted to the study of an effective frequency-domain approach, based on systems control theory, to compute and analyze several types of standard bifurcation conditions for general continuous-time nonlinear dynamical systems. A very rich pictorial gallery of local bifurcation diagrams for such nonlinear systems under simultaneous variations of several system parameters is presented. Some higher-order harmonic balance approximation formulas are derived for analyzing the oscillatory dynamics in small neighborhoods of certain types of Hopf and degenerate Hopf bifurcations.The frequency-domain approach is then extended to the large class of delay-differential equations, where the time delays can be either discrete or distributed. For the case of discrete delays, two alternatives are presented, depending on the structure of the underlying dynamical system, where the more general setting is then extended to the case of distributed time-delayed systems. Some representative examples in engineering and biology are discussed.