reviews of nonlinear dynamics and complexity || frontmatter

Reviews of Nonlinear Dynamics and Complexity

Edited by Heinz Georg Schuster

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Reviews of Nonlinear Dynamicsand Complexity

Edited by Heinz Georg Schuster

WILEY-VCH Verlag GmbH & Co. KGaA

The Editor

Heinz Georg Schuster University of Kiel [email protected]

Editorial Board

Christoph Adami California Institute of Technology Pasadena

Stefan Bornholdt University of Bremen

Wolfram Just Queen Mary University of London

Kunihiko Kaneko University of Tokyo

Ron Lifshitz Tel Aviv University

Bernhard Mehlig University of Goeteborg

Ernst Niebur Johns Hopkins University Baltimore

Günter Radons Technical University of Chemnitz

Eckehard Schöll Technical University of Berlin

Hong Zhao Xiamen University

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for

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Bibliographic information published by the Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de

2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

Printed in the Federal Republic of Germany Printed on acid-free paper

Typesetting Uwe Krieg, Berlin Printing betz-druck GmbH, Darmstadt Bookbinding Litges & Dopf GmbH, Heppenheim

ISBN: 978-3-527-40729-3

Review of Nonlinear Dynamics and Complexity. Edited by Heinz Georg SchusterCopyright © 2008 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-40729-3

V

Contents

Preface IX

List of Contributors XI

1 Nonlinear Dynamics of Nanomechanical and MicromechanicalResonators 1Ron Lifshitz and M. C. Cross

1.1 Nonlinearities in NEMS and MEMS Resonators 11.1.1 Why Study Nonlinear NEMS and MEMS? 21.1.2 Origin of Nonlinearity in NEMS and MEMS Resonators 21.1.3 Nonlinearities Arising from External Potentials 31.1.4 Nonlinearities Due to Geometry 51.2 The Directly-driven Damped Duffing Resonator 81.2.1 The Scaled Duffing Equation of Motion 81.2.2 A Solution Using Secular Perturbation Theory 91.2.3 Addition of Other Nonlinear Terms 161.3 Parametric Excitation of a Damped Duffing Resonator 181.3.1 Driving Below Threshold: Amplification and Noise Squeezing 211.3.2 Linear Instability 241.3.3 Nonlinear Behavior Near Threshold 251.3.4 Nonlinear Saturation Above Threshold 271.3.5 Parametric Excitation at the Second Instability Tongue 311.4 Parametric Excitation of Arrays of Coupled Duffing Resonators 341.4.1 Modeling an Array of Coupled Duffing Resonators 341.4.2 Calculating the Response of an Array 361.4.3 The Response of Very Small Arrays – Comparison of Analytics

and Numerics 401.4.4 Response of Large Arrays – Numerical Simulation 421.5 Amplitude Equation Description for Large Arrays 431.5.1 Amplitude Equations for Counter Propagating Waves 44

VI Contents

1.5.2 Reduction to a Single Amplitude Equation 461.5.3 Single Mode Oscillations 47

References 49

2 Delay Stabilization of Rotating Waves Without Odd NumberLimitation 53Bernold Fiedler, Valentin Flunkert, Marc Georgi, Philipp Hövel,and Eckehard Schöll

2.1 Introduction 532.2 Mechanism of Stabilization 54

2.3 S1-Symmetry and Stability of Rotating Waves 582.4 Conditions on the Feedback Gain 602.5 Tori 622.6 Conclusion 64

References 67

3 Random Boolean Networks 69Barbara Drossel

3.1 Introduction 693.2 Model 703.2.1 Topology 713.2.2 Update Functions 713.2.3 Dynamics 743.2.4 Applications 763.2.5 Problems 783.3 Annealed Approximation and Phase Diagrams 793.3.1 The Time Evolution of the Proportion of 1s and 0s 793.3.2 The Time Evolution of the Hamming Distance 823.3.3 The Statistics of Small Perturbations in Critical Networks 853.3.4 Problems 873.4 Networks with K = 1 883.4.1 Topology of K = 1 Networks 883.4.2 Dynamics on K = 1 Networks 903.4.2.1 Cycles on Loops 913.4.2.2 K = 1 Networks in the Frozen Phase 913.4.2.3 Critical K = 1 Networks 923.4.3 Dynamics on K = N Networks 933.4.4 Application: Basins of Attraction in Frozen, Critical and

Chaotic Networks 943.4.5 Problems 953.5 Critical Networks with K = 2 953.5.1 Frozen and Relevant Nodes 96

Contents VII

3.5.2 Analytical Calculations 1003.5.3 Problems 1053.6 Networks with Larger K 1053.7 Outlook 1073.7.1 Noise 1073.7.2 Scale-free Networks and Other Realistic Network Structures 1083.7.3 External Inputs 1083.7.4 Evolution of Boolean Networks 1083.7.5 Beyond the Boolean Approximation 109

References 109

4 Return Intervals and Extreme Events in Persistent Time Serieswith Applications to Climate and Seismic Records 111Armin Bunde, Jan F. Eichner, Shlomo Havlin, Jan W. Kantelhardt,and Sabine Lennartz

4.1 Introduction 1114.2 Statistics of Return Intervals 1144.2.1 Data Generation and Mean Return Interval 1144.2.2 Stretched Exponential Behavior and Finite-Size Effects for

Large Return Intervals 1164.2.3 Power-Law Regime and Discretization Effects for Small Return

Intervals 1194.2.4 long-Term Correlations of the Return Intervals 1204.2.5 Conditional Return Intervals 1214.2.6 Conditional Mean Return Intervals 1234.3 Statistics of Maxima 1264.3.1 Extreme Value Statistics for i.i.d. Data 1274.3.2 Effect of Long-Term Persistence on the Distribution of the

Maxima 1284.3.3 Effect of Long-Term Persistence on the Correlations of the

Maxima 1294.3.4 Conditional Mean Maxima 1314.3.5 Conditional Maxima Distributions 1334.4 Long-Term Memory in Earthquakes 1344.5 Conclusions 140

References 143

5 Factorizable Language: From Dynamics to Biology 147Bailin Hao and Huimin Xie

5.1 Coarse-Graining and Symbolic Description 1475.2 A Brief Introduction to Language Theory 1495.2.1 Formal Language 1495.2.2 Factorizable Language 151

VIII Contents

5.3 Symbolic Dynamics 1525.3.1 Dynamical Language 1535.3.2 Grammatical Complexity of Unimodal Maps 1535.4 Sequences Generated from Cellular Automata 1555.4.1 Encounter the Cellular Automata 1555.4.2 Limit Complexity of Cellular Automata 1585.4.3 Evolution Complexity of Cellular Automata 1645.5 Avoidance Signature of Bacterial Complete Genomes 1695.5.1 Visualization of Long DNA Sequences 1695.5.2 Avoided K-Strings in Complete Genomes 1705.5.3 True and Redundant Avoided Strings 1715.5.4 Factorizable Language Defined by a Genome 1725.6 Decomposition and Reconstruction of Protein Sequences 1745.6.1 A Few Notions of Graph Theory 1755.6.2 The K-Peptide Representation of Proteins 1765.6.3 Uniquely Reconstructible Sequences and Factorizable

Language 1795.6.4 Automaton that Accepts a Uniquely Reconstructible Language 1805.6.5 Other Applications of the Sequence Reconstruction Problem 183

References 184

6 Controlling Collective Synchrony by Feedback 187Michael Rosenblum and Arkady Pikovsky

6.1 What is Collective Synchrony? 1876.2 Why to Control? 1906.3 Controlling Neural Synchrony 1916.3.1 Electrical Stimulation of Neural Ensembles 1916.3.2 What Does the Control Theory Say? 1926.3.3 Control of Brain Rhythms: Specific Problems and Assumptions 1936.4 Delayed Feedback Control 1946.4.1 An Example 1946.4.2 Theoretical Description 1956.4.3 Beyond Idealized Model 1976.5 Suppression with a Non-delayed Feedback Loop 1986.5.1 Construction of a Feedback Loop 1996.5.2 Example: Desynchronization of a Neuronal Ensemble with

Synaptic Coupling 2016.6 Determination of Stimulation Parameters by a Test Stimulation 2046.7 Discussion and Outlook 205

References 206

Index 209


IX

Preface

Nonlinear behavior is ubiquitous in nature and ranges from fluid dynamics,via neural and cell dynamics to the dynamics of financial markets. The mostprominent feature of nonlinear systems is that small external disturbances caninduce large changes in their behavior. This can and has been used for ef-fective feedback control in many systems, from Lasers to chemical reactionsand the control of nerve cells and heartbeats. A new hot topic are nonlineareffects that appear on the nanoscale. Nonlinear control of the atomic forcemicroscope has improved it accuracy by orders of magnitude. Nonlinear elec-tromechanical oscillations of nano-tubes, turbulence and mixing of fluids innano-arrays and nonlinear effects in quantum dots are further examples.

Complex systems consist of large networks of coupled nonlinear devices.The observation that scalefree networks describe the behavior of the internet,cell metabolisms, financial markets and economic and ecological systems haslead to new findings concerning their behavior, such as damage control, op-timal spread of information or the detection of new functional modules, thatare pivotal for their description and control.

This shows that the field of Nonlinear Dynamics and Complexity consists ofa large body of theoretical and experimental work with many applications,which is nevertheless governed and held together by some very basic prin-ciples, such as control, networks and optimization. The individual topics aredefinitely interdisciplinary which makes it difficult for researchers to see whatnew solutions – which could be most relevant for them – have been found bytheir scientific neighbors. Therefore its seems quite urgent to provide Reviewsof Nonlinear Dynamics and Complexity where researchers or newcomers to thefield can find the most important recent results, described in a fashion whichbreaks the barriers between the disciplines.

In this first volume topics range from nano-mechanical oscillators, via ran-dom Boolean networks and control to extreme climatic and seismic events.I would like to thank all authors for their excellent contributions. If readerstake from these interdisciplinary reviews some inspiration for their furtherresearch this volume would fully serve its purpose.

X Preface

I am grateful to all members of the Editorial Board and the staff from Wiley-VCH for their excellent help and would like to invite my colleagues to con-tribute to the next volumes.

Kiel, February 2008 Heinz Georg Schuster


XI

List of Contributors

Armin BundeInstitut für Theoretische Physik IIIJustus-Liebig-Universität GiessenHeinrich-Buff-Ring 1635392 [email protected]

M. C. CrossDepartment of PhysicsCalifornia Institute of TechnologyPasadena, CA [email protected]

Barbara DrosselInstitut für FestkörperphysikTechnische Universität DarmstadtHochschulstrasse 664289 [email protected]

Jan F. EichnerInstitut für Theoretische Physik IIIJustus-Liebig-Universität GiessenHeinrich-Buff-Ring 1635392 [email protected]

Bernold FiedlerInstitut für Mathematik IFU BerlinArnimallee 2–614195 [email protected]

Valentin FlunkertInstitut für Theoretische PhysikTechnische Universität BerlinHardenbergstrasse 3610623 [email protected]

Marc GeorgiInstitut für Mathematik IFU BerlinArnimallee 2–614195 [email protected]

Bailin HaoThe Institute of Theoretical Physics55 Zhongguancun East RoadHaidan DistrictBeijing 100080ChinaandThe T-Life Research CenterFudan University220 Handan RoadShanghai [email protected]

Shlomo HavlinMinerva Center and Departmentof PhysicsBar-Ilan [email protected]

XII List of Contributors

Philipp HövelInstitut für Theoretische PhysikTechnische Universität BerlinHardenbergstrasse 3610623 [email protected]

Jan W. KantelhardtInstitut für PhysikMartin-Luther-Universitä[email protected]

Sabine LennartzInstitut für Theoretische Physik IIIJustus-Liebig-Universität GiessenHeinrich-Buff-Ring 1635392 [email protected]

Ron LifshitzRaymond and Beverly Sackler School ofPhysics and AstronomyTel Aviv UniversityTel Aviv [email protected]

Arkady PikovskyDepartment of PhysicsPotsdam UniversityAm Neuen Palais 1014469 [email protected]

Michael RosenblumDepartment of PhysicsPotsdam UniversityAm Neuen Palais 1014469 [email protected]

Eckehard SchöllInstitut für Theoretische PhysikTechnische Universität BerlinHardenbergstrasse 3610623 [email protected]

Huimin XieDepartment of MathematicsSuzhou UniversitySuzhou [email protected]

reviews of nonlinear dynamics and complexity || frontmatter

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