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**Bifurcations and chaos in piecewise-smooth dynamical systems.**
*(English)*
Zbl 1047.34048

World Scientific Series on Nonlinear Science. Series A 44. River Edge, NJ: World Scientific (ISBN 981-238-420-0/hbk). xii, 363 p. (2003).

This book presents a number of important new phenomena related to the description of nonlinear dynamical processes whose behaviour is controlled by piecewise-smooth dynamical systems. The book contains interesting examples of piecewise-smooth dynamical systems and its bifurcation phenomena with direct applications. The practical significance of these phenomena is illustrated by a series of well-documented and realistic applications to switching power converters, relay systems and different types of pulse-width modulated control systems. Other examples are derived from mechanical engineering, digital electronics and economic business cycles theory. At several places of the book, the mode of expression from mathematical point of view is not quite correct.

The book consists of 7 chapters. In chapters 1 and 2, the authors summarize the basic concepts of modern nonlinear dynamical systems theory. Chapters 3 and 4 are dedicated to the relay control systems including bifurcations and chaotic oscillations in these systems. The chaotic oscillations in pulse-width modulated systems are investigated in Chapter 5. Chapter 6 is entitled border-collision on a two-dimensional torus and one of the main ideas of this chapter is to show that the transition to chaos via two-frequency quasiperiodicity in piecewise-smooth systems can differ fundamentally from the mechanisms described in the existing literature. In the last chapter, the methods of piecewise smooth discrete dynamical systems are applied to microeconomies and management science, involving decision theory. The book will be certainly a useful tool for many scientists and engineers.

The book consists of 7 chapters. In chapters 1 and 2, the authors summarize the basic concepts of modern nonlinear dynamical systems theory. Chapters 3 and 4 are dedicated to the relay control systems including bifurcations and chaotic oscillations in these systems. The chaotic oscillations in pulse-width modulated systems are investigated in Chapter 5. Chapter 6 is entitled border-collision on a two-dimensional torus and one of the main ideas of this chapter is to show that the transition to chaos via two-frequency quasiperiodicity in piecewise-smooth systems can differ fundamentally from the mechanisms described in the existing literature. In the last chapter, the methods of piecewise smooth discrete dynamical systems are applied to microeconomies and management science, involving decision theory. The book will be certainly a useful tool for many scientists and engineers.

Reviewer: Alois Klíč (Praha)

### MSC:

34C23 | Bifurcation theory for ordinary differential equations |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

34C28 | Complex behavior and chaotic systems of ordinary differential equations |

37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |

82C05 | Classical dynamic and nonequilibrium statistical mechanics (general) |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |