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Piecewise-smooth dynamical systems. Theory and applications. (English) Zbl 1146.37003
Applied Mathematical Sciences 163. New York, NY: Springer (ISBN 978-1-84628-039-9/hbk). xxi, 481 p. (2008).
This book treats dynamical systems that have a piecewise smooth right-hand-side. The restriction to piecewise smooth (in contrast to general non-smooth) permits the book authors to avoid theoretical overhead and go straight to statements which can be applied in practically relevant problems. In fact, the book starts with a large number of case studies: a mechanical impact oscillator, an oscillator with backlash (a one-sided spring), a dry-friction oscillator, relay control systems, a DC-DC converter, a model for heartbeats etc. The main objects of study in this book are exactly the same as the classical objects of study in the classical theory of autonomous (or periodically forced) smooth dynamical systems: equilibria, periodic orbits and chaotic attractors and their bifurcations in low-dimensional maps and ordinary differential equations (ODEs).
The types of dynamical systems studied in this book all have the following set-up: the phase space (typically $$\mathbb{R}^n$$ with a small $$n$$) is partitioned into several regions. In each of the regions the dynamics are governed by a smooth map (in the discrete-time case) or ODE (in the continuous-time case). The regions are separated by smooth manifolds. In the simplest case one would divide the phase space $$\mathbb{R}^n$$ into two regions defined by a functional $$H(x)$$: $$S_1=\{x: H(x)<0\}$$ and $$S_2=\{x: H(x)>0\}$$. The manifold $$\{x:H(x)=0\}$$ is called the switching manifold. The different kinds of systems are then distinguished by the switching behavior at the switching manifold. Each of the following kinds is given whole-chapter treatment:
1.
piecewise linear maps that are continuous across the switching manifold,
2.
maps that are continuous but have a square-root type singularity of their Jacobian at the border, or maps that are discontinuous at the switching manifold,
3.
flows with a right-hand-side that is discontinuous at the switching manifold (so-called Filippov systems),
4.
flows with a resetting law at the switching manifold (a simple model for a rigid mechanical impact).
For all of these systems the book follows the approach of classical bifurcation theory as taken in [Y. A. Kuznetsov, Elements of applied bifurcation theory, 3rd ed., New York, NY: Springer (2004; Zbl 1082.37002)]: for example, assume that the system has a (typically initially stable) periodic orbit, vary system parameters, and watch out for special parameter values at which the periodic orbit or the dynamics in its neighborhood changes qualitatively. Apart from the classical bifurcations there is a new type of possible event, called discontinuity-induced bifurcation (DIB) in this book. For maps this refers to situations in which the periodic orbit visits the switching manifold. For flows this refers to a non-transversality of the intersection of the periodic orbit with the switching manifold. Most of the book studies what happens to periodic orbits and equilibria at these DIBs for the above mentioned dynamical systems.
Chapters:
Chapter 1: Introduction, introduces case studies motivating the general statements later in the book;
Chapter 2: Background on dynamical systems, general notation and techniques useful for studying piecewise smooth system, specifically introducing the local discontinuity mapping a convenient technique to reduce DIBs of periodic orbits in flows to DIBs of their local return maps;
Chapter 3: Border collisions in piecewise linear continuous maps;
Chapter 4: Bifurcations of general piecewise smooth maps such as discontinuous maps and continuous maps with a one-sided square root type singularity;
Chapter 5: Boundary equilibrium bifurcations in Filippov systems;
Chapters 6 and 7: Limit cycle bifurcations in impacting systems and other non-Filippov flows; derives local return maps for periodic orbits at DIBs (for example, impacts with zero normal velocity), which are the maps studied in Chapter 4;
Chapter 8: Sliding bifurcations in Filippov flows; derives local return maps for periodic orbits of Filippov systems involving so called sliding modes;
Chapter 9: Further applications and extensions; additional more complicated case studies, also exploring a few DIBs of codimension two.
Throughout, the book approaches every scenario via a motivating case study before generalizing the observations to a general theorem. Proofs of the theorems are partially omitted referring to research literature. Graphical sketches are abundant, supporting the presentation of the essential ideas behind arguments and techniques. Overall, the level of presentation makes the book useful as a source of theoretical background knowledge for researchers and postgraduate students in engineering and applied mathematics. It is also suitable as a reference for undergraduate projects or advanced undergraduate reading groups.

##### MSC:
 37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory 34A36 Discontinuous ordinary differential equations 37G10 Bifurcations of singular points in dynamical systems 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 37N05 Dynamical systems in classical and celestial mechanics 70G40 Topological and differential topological methods for problems in mechanics 37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) 37C10 Dynamics induced by flows and semiflows 37C27 Periodic orbits of vector fields and flows 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
##### Software:
MATCONT; SlideCont