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**From chaos to order. Methodologies, perspectives and applications. With a foreword by Alistair Mees.**
*(English)*
Zbl 0908.93005

World Scientific Series on Nonlinear Science. Series A. 24. Singapore: World Scientific Publishing Co. xxii, 753 p. (1998).

Let us quote the authors’ wise and balanced informal definition: “controlling chaos can be understood as a process or mechanism which enhances existing chaos or creates chaos in a dynamical system when it is useful or beneficial, and supresses it when it is harmful”.

As an introduction to nonlinear systems near a hyperbolic equilibrium, the Grobman-Hartman theorem in \(R^2\) is stated and centre manifold theory in \(R^n\) is sketched. As an introduction to chaos in \(R^3\), the Šilnikov theorem is stated. To anticipate identification of nonlinear systems from time series of measurements, the delayed-coordinates method and the embedding dimension are presented. The very readable Chapter 2 – Nonlinear dynamical systems, contains 88 pp.

The next two chapters are concerned with transition from chaos to order not using control theoretic methods: Chapter 3 – Parameter dependent approaches to chaos control (45 pp.) and Chapter 4 – Open-loop strategies for chaos control (38 pp.). Nevertheless the mostly quoted method, the OGY method consists in an application and a reinvention of the poles shift method known from control theory. On the other hand the very new tasks of entrainment and migration control may be inspiring reading to control theorists.

The following chapters 5-8, Engineering feedback control I, II, III, and Adaptive control of chaos (235 pp.), are a recollection and application of methods known from control theory books and will be mainly useful as a control theory textbook for scientists and circuit specialists. The only control method not used, is the exact linearization. The assertion on p. 192, “the fundamental concept of ‘controllability’ is typically defined by using a finite and fixed \(t_f\)”, is rectified on p. 244 showing the controllability matrix. Only on p. 315 one finds such an important concept as the Lur’e form – and this is missing in the Index.

The chapters 9, 10 – Intelligent control of chaos and Chaos control in distributed systems (128 pp.), are of some encyclopedic character – like the previous four chapters. The topics are artificial neural networks, the Wiener-type model (but not the dual Hammerstein-type model), fuzzy systems, spacio-temporal systems, cellular neural networks, isotropic systems, lattices, distributed intelligent systems – always in connection with chaos and its control. Only on p. 469 one finds one very important form for the linear dynamical part of systems – the observable Frobenius form.

Chapter 11, Chaos synchronization, has 81 pp. and is most closely connected with physics and communications. The future oriented section represents a biologically inspired account of chaotic signal encoding. The promises of technological exploration are great but not realized till now. As a whole, Chapter 11 illustrates both the experimental and the encyclopedic character of the book: what is missing are criteria of stability of synchronization and the derivation of these criteria. But irrespective of this, the chapter may be a source of inspiration for mathematicians and system theorists.

Chapter 12, More on chaos control, has 73 pp. It presents mainly some case studies. Last but not least both the chapter and the book end by ‘anticontrol of chaos’ which means synthesis or construction of chaos given some dynamical linear part and a statical nonlinear part, in the most simple case of the Lur’e system. This construction was pioneered by Chua in the mid eighties. Still a more interesting book would be one with the title: ‘From Order to Chaos.’

The reference section contains 728 items the most recent dating from before 1997.

As an introduction to nonlinear systems near a hyperbolic equilibrium, the Grobman-Hartman theorem in \(R^2\) is stated and centre manifold theory in \(R^n\) is sketched. As an introduction to chaos in \(R^3\), the Šilnikov theorem is stated. To anticipate identification of nonlinear systems from time series of measurements, the delayed-coordinates method and the embedding dimension are presented. The very readable Chapter 2 – Nonlinear dynamical systems, contains 88 pp.

The next two chapters are concerned with transition from chaos to order not using control theoretic methods: Chapter 3 – Parameter dependent approaches to chaos control (45 pp.) and Chapter 4 – Open-loop strategies for chaos control (38 pp.). Nevertheless the mostly quoted method, the OGY method consists in an application and a reinvention of the poles shift method known from control theory. On the other hand the very new tasks of entrainment and migration control may be inspiring reading to control theorists.

The following chapters 5-8, Engineering feedback control I, II, III, and Adaptive control of chaos (235 pp.), are a recollection and application of methods known from control theory books and will be mainly useful as a control theory textbook for scientists and circuit specialists. The only control method not used, is the exact linearization. The assertion on p. 192, “the fundamental concept of ‘controllability’ is typically defined by using a finite and fixed \(t_f\)”, is rectified on p. 244 showing the controllability matrix. Only on p. 315 one finds such an important concept as the Lur’e form – and this is missing in the Index.

The chapters 9, 10 – Intelligent control of chaos and Chaos control in distributed systems (128 pp.), are of some encyclopedic character – like the previous four chapters. The topics are artificial neural networks, the Wiener-type model (but not the dual Hammerstein-type model), fuzzy systems, spacio-temporal systems, cellular neural networks, isotropic systems, lattices, distributed intelligent systems – always in connection with chaos and its control. Only on p. 469 one finds one very important form for the linear dynamical part of systems – the observable Frobenius form.

Chapter 11, Chaos synchronization, has 81 pp. and is most closely connected with physics and communications. The future oriented section represents a biologically inspired account of chaotic signal encoding. The promises of technological exploration are great but not realized till now. As a whole, Chapter 11 illustrates both the experimental and the encyclopedic character of the book: what is missing are criteria of stability of synchronization and the derivation of these criteria. But irrespective of this, the chapter may be a source of inspiration for mathematicians and system theorists.

Chapter 12, More on chaos control, has 73 pp. It presents mainly some case studies. Last but not least both the chapter and the book end by ‘anticontrol of chaos’ which means synthesis or construction of chaos given some dynamical linear part and a statical nonlinear part, in the most simple case of the Lur’e system. This construction was pioneered by Chua in the mid eighties. Still a more interesting book would be one with the title: ‘From Order to Chaos.’

The reference section contains 728 items the most recent dating from before 1997.

Reviewer: A.Vaněček (Praha)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

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

93C10 | Nonlinear systems in control theory |