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Handbook of chaos control


Second completely revised and enlarged edition

849 pages, October 2007  (flyer (PDF, 205,2 KB))  (Wiley - VCH)      


Eckehard Schöll, Technische Universität Berlin
Hans Georg Schuster, Universität Kiel

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During the last decade, control of complex irregular dynamics has evolved as one of the central issues in applied nonlinear science. The number of papers published in this field has been steadily growing since the first pionieering papers appeared in 1990, and has reached an annual number between 600 and 700 during the past few years. The seminal article by Ott, Grebogi, and Yorke (1990), in which they demonstrated that small time dependent changes in the control parameters of a nonlinear system can turn a previously chaotic trajectory into a stable periodic motion, alone was cited much more than 2000 times. Nowadays the notion of chaos control has been extended to a much wider class of problems involving the stabilization of unstable periodic or stationary states in nonlinear dynamic systems. Within the last few years major progress has been made in this field, in particular with respect to

  • extending the methods of chaos control to spatio-temporal patterns
  • extending the methods of control of deterministic dynamic systems to stochastic and noise-mediated systems
  • development of novel control schemes - deepened understanding and analytic insight into different control
  • applications to various areas, e.g. biological, medical, technological systems

Since its first publication in 1999, the Handbook on Chaos Control has become the standard reference in this field. Eight years after the first edition, there is need to present the new material which has been accumulated, and to set new trends and identify new promising directions of research.

The present volume is the second completely revised and enlarged edition, and includes only articles which have been newly written for this volume. It aims at presenting a comprehensive overview of the state of the art in this growing field, containing chapters written by the leading scientists who are active in this area. The focus is put on recent developments like novel control schemes, analytical insights, control of chaotic space-time patterns, control of noisy nonlinear systems and noise-induced dynamics, secure communication with chaos, and applications of chaos control to physics, chemistry, biology, medicine, and engineering. Furthermore, the overlap of chaos control with the traditional field of control theory in the engineering community is identified.

The chapters of the book are state-of-the-art review articles and should be of interest to graduate students and researchers; theoretical and experimental physicists, applied mathematicians, electronic engineers, nonlinear scientists from interdisciplinary fields, e.g. chemistry, biology, medicine, control theory, engineering. The 36 chapters are grouped into nine parts, where the first four parts deal with basic aspects
and extension of methods; controlling space-time chaos; controlling noisy motion; communicating with chaos and chaos synchronisation, and the last five parts contain applications to optics, to electronic systems, to chemical reaction systems, to biology and medicine, and to engineering. Among the topics are, for instance, secure communication with chaotic semiconductor lasers, control of communication networks, noninvasive time-delayed feedback control of laser diodes, electronic circuits, and semiconductor nanostructures, control of chemical turbulence and electrochemical oscillators, suppression of synchronisation as therapeutic tools for neural diseases like
Parkinson and epilepsy, and control of cardiac dynamics.

We hope that this volume will stimulate further developments in this still thrilling area which is centered on the overlap of basic research and far-reaching applications.

We would like to thank all authors who have contributed to this volume as well as the publishers for their excellent cooperation. Special thanks are due to Philipp Hövel for his technical assistence.

Berlin and Kiel, June 2007

Eckehard Schöll and Heinz G. Schuster

Table of Contents

A. Basic aspects and extension of methods


1. C. Grebogi, E. Macau (UK): Controlling Chaos

2. J. Socolar (USA): Time-delay control for discrete maps

3. K. Pyragas (Lithuania): An analytical treatment of the delayed feedback

control algorithm

4. B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, E. Schöll (Germany):

Beyond the odd number limitation of time-delayed feedback control

5. W. Just (UK): On global properties of time-delayed feedback control

6. J.C. Claussen (Germany): Poincare-based chaos control of delayed

measured systems: Limitations and Improved Control

7. A. Fradkov, B. Andrievsky (Russia): Nonlinear and adaptive

control of chaos


B. Controlling space-time chaos


8. R. Grigoriev (USA): Localised Control of spatiotemporal chaos

9. S. Boccaletti (Italy): Controlling spatiotemporal chaos

10. U. Parlitz (Germany): Multiple delay feedback control


C. Controlling noisy motion


11. N. Janson, A. Balanov (UK), E. Schöll(Germany):Control of noise-induced


12. A. Pikovsky, M. Rosenblum, D. Goldobin (Germany): Controlling

coherence of noisy and chaotic oscillations by delayed feedback

13. C. Masoller (Uruguay): Resonances induced by the delay time in nonlinear

autonomous oscillators with feedback



D. Communicating with chaos, chaos synchronisation


14. W. Kinzel (Würzburg): Secure communication with synchronized

chaotic systems

15. T. L. Carroll (USA): Noise robust chaotic system

16. H. Abarbanel (USA): Nonlinear Communications strategies

17. K.A. Shore (UK): Synchronisation and message transmission for networked

chaotic optical communications

18. J. Kurths (Germany): Feedback control principles for phase



E. Applications to optics


19. L. Illing, D. Gauthier (USA): Controlling fast chaos in

opto-electronic delay dynamical systems

20. O. Hess (UK): Control of Broad-Area laser dynamics with Delayed Optical


21. H.J. Wünsche (Germany): Noninvasive control of semiconductor lasers

by delayed optical feedback

22. J. Ohtsubo (Japan): Chaos and control in semiconductor lasers

23. B. Gütlich, C. Denz (Germany): From pattern control to synchronisation:

control techniques in nonlinear optical feedback systems



F. Applications to electronic systems


24. E. Schöll (Germany): Delayed feedback control of chaotic spatiotemporal

patterns in semiconductor nanostructures

25. H. Benner (Germany), W. Just (UK): Observing global properties of

time-delayed feedback control in electronic circuits

26. A. Kittel (Germany): Application of a Black Box Strategy to Chaos Control


G. Applications to chemical reaction systems


27. H. Engel (Germany): Feedback-mediated control of hypermeandering spiral


28. A. Mikhailov, C. Beta (Germany): Control of spatiotemporal chaos in

surface chemical reactions

29. J. L. Hudson, I.Z. Kiss (USA): Forcing and feedback control of arrays of

chaotic electrochemical oscillators


H. Applications to biology


30. P. Tass, O. Popovych, Ch. Hauptmann (Germany): Chaos synchronisation in

oscillatory neural networks

31. D. Christini(USA):control of cardiac electrical nonlinear dynamics,

32. Sitabhra Sinha (India): Controlling Spatiotemporal chaos and spiral

turbulence in excitable media


I. Applications to engineering


33. H. Nijmeijer (Netherlands), H. Huijberts (London, UK): Nonlinear

chaos control and synchronization

34. M. Ogorzalek (Poland): Electronic chaos controllers - from theory to


35. E. Mosekilde, Zh. Zhusubaliyev (Denmark): chaos in pulse width

modulated control systems

36. T. Hikihara (Kyoto, Japan): Transient dynamics of Duffing system under

Time Delayed Feedback Control: Global phase structure and

Application to Engineering





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