## Controlling chaos. Theoretical and practical methods in non-linear dynamics.(English)Zbl 0883.58021

London: Academic Press. viii, 165 p. (1996).
Part I: General outlook (74 pp., 146 items for ‘further reading’, 82 references) is a somewhat long and well readable introduction to chaos control. The author’s bias “Knowledge of the classical control theory is not necessary to understand chaos controlling methods $$[\dots]$$” is not valid one, and it goes through the whole Part I. Typically for the style of Part I is the deterministic chaos in the continuous time defined. The working definition of chaotic behaviour on $$t\in [0,\infty]$$ as the behaviour which is bounded and has at least one positive Lyapunov exponent was possible even at the level of Part I. There are widely used the Lyapunov exponents but even these are nowhere defined. (No reference is given in the Part I on definition of these at reprinted Paper 2 by Romerias et al.)
The item ‘Controlling chaos through feedback’ starts with the Ott-Grebogi-Yorke (OGY) method, followed by Pyragas’s and the classical control method. By chaos control it is understood, as common after the famous OGY method, the change from the chaotic attractor to the limit cycle. In Part I it is illustrated by beautiful color plates. Curiously, “a control by periodic external perturbation” the author considers as an example of “feedback controlling loops”. Further, controlling chaos by chaos is presented.
The Section ‘Controlling chaos without feedback’ starts with control through operating conditions and continues with control by system design. Curiously, the chaotic system with autonomous (i.e., dynamic) controller in the feedback loop is included in this Section. Further the notions of taming chaos (decreasing the largest Lyapunov exponent) and entraintment and migration control (transfer the system dynamic from one corvergent region to another) are stated briefly.
The Section ‘Synchronization of chaos’ is in fact concerned with a very interesting extension of reference following control of classical control theory. It starts with Pecora and Caroll’s approach and ends with secure communication scheme based on the use of two chaotic systems. The real possibilities of these suggested secrete encryption schemes are not settled in the chaos control community.
The Section ‘Engineering applications’ contains an occasional proportional feedback method, sampled input waveform method and control of transient behaviour in mechanical systems. These systems are the professional specialization of the author.
Part II: Selected reprints contains 12 benchmark papers on chaos control by Ott, Grebogi, Yorke, Romeiras, Dayawansa, Dressler, Nitche, Ditto, Rauseo, Spano, Tél, Shinbrod, Ott, Pyragas, Jackson, Pecora, Carroll, Cuomo, Oppenheim, Pérez, Cerdeiral, Kocarev, Parlitz. Almost all these authors are physicists.
The field of chaos analysis has been for a long time very well theoretically covered by mathematicians, especially the Russian ones like Shil’nikov and Mel’nikov. The field of chaos synthesis is a more newer one and was started by the circuit specialist Chua. The papers of Part II on chaos control are papers from physics journals from the years 1989-1994. In fact, the book contains practical and not theoretical methods, contrary to the claim in its subtitle. The chaos control theory remains the great challenge for control theorists.

### MSC:

 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 93C10 Nonlinear systems in control theory 37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory 93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory