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**Fractional-order nonlinear systems. Modeling, analysis and simulation.**
*(English)*
Zbl 1228.34002

Nonlinear Physical Science. Berlin: Springer; Beijing: Higher Education Press (ISBN 978-3-642-18100-9/hbk; 978-7-04-031534-9/hbk; 978-3-642-18101-6/ebook). xvi, 219 p. (2011).

The principal goal of this monograph is to introduce the reader to a new and rapidly growing branch of dynamical systems called fractional-order chaotic systems. The material is organized into seven chapters. Chapter 1 provides a concise introduction to fractional-order chaotic systems. Fundamentals of fractional calculus are presented in Chapter 2. Fractional linear time-invariant (LTI) and nonlinear systems are introduced in Chapter 3; fractional-order controllers are also discussed there. Stability of fractional-order LTI, nonlinear and interval fractional-order systems forms the subject of Chapter 4. Chapter 5 surveys a variety of fractional-order chaotic systems of lower order (up to three) including, among others, the well-known Chua’s oscillator, Duffing, Rössler and Lorenz systems. Three control strategies for fractional-order chaotic systems – digital state-space proportional feedback control, sliding mode control and synchronization – are discussed in Chapter 6, whereas additional remarks are collected in the final Chapter 7. Each chapter concludes with references. Matlab functions have been written for all systems discussed in the book; these are available for download from the web site of MathWorks and their utilization is described in Appendix A. In addition, Simulink models have been designed for fractional-order Chua and Volta systems to be used as samples for further simulation with this program. Relevant Laplace and inverse Laplace transforms related to fractional-order calculus are collected in Appendix B. The book concludes with a glossary and an index. Although many classes of fractional-order chaotic systems are not discussed, the book is a very nice first step towards a systematic description of this new and interesting direction of research. The monograph can be used by graduate students and researchers working with chaotic dynamical systems, control, synchronization and related fields.

Reviewer: Yuri V. Rogovchenko (Umeå)

### MSC:

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

34H05 | Control problems involving ordinary differential equations |

34-04 | Software, source code, etc. for problems pertaining to ordinary differential equations |

34A08 | Fractional ordinary differential equations |

26A33 | Fractional derivatives and integrals |

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

34D06 | Synchronization of solutions to ordinary differential equations |

34H10 | Chaos control for problems involving ordinary differential equations |