NONLINEAR PHYSICAL SCIENCE

NONLINEAR PHYSICAL SCIENCE

Nonlmear PhySIcal SCIence focuses on recent advances of fundamental theones andprmciples, analytIcal and symbohc approaches, as weB as computatIOnal techmquesm nonhnear phySIcal SCIence and nonhnear mathematICS WIth engmeenng apphca­nons.

TOpICS of mterest m Nonlinear Physical Science mclude but are not hmited to:

- New findmgs and discovenes m nonhnear phYSICS and mathematIcs- Nonhneanty, compleXIty and mathematIcal structures m nonhnear phYSICS- Nonhnear phenomena and observatIOns m nature and engmeenng- ComputatIOnal methods and theones m complex systems- LIe group analySIS,new theones and pnnciples m mathematIcal modehng- StabIhty, bIfurcatIOn, chaos and fractals m phySIcal SCIence and engmeenng- Nonhnear chemIcal and bIOlogIcal phYSICS- Discontinuity, synchromzatIOn and natural compleXIty m the phySIcal SCIences

SERIES EDITORS

Albert C.l. Luo

Department of Mechamcal and IndustrIalEngineering

Southern IllinOiS Umverslty EdwardSVIlleEdwardSVille, IL 62026-1805, USAEmail' alno@sjneedn

Nan H. IbraglIDov

Department of Mathematics and SCIenceBlekmge Institute of TechnologyS-371 79 Karlskfona, SwedenEmaIl: mb@bth.se

INTERNATIONAL ADVISORY BOARD

Ping Ao, University of Washington, USA; Email: aoping@u.washington.edu

Jan AwreJcewlcz, 1he Iechmcal Umverslty of [odz, Poland; Email: awreJcew@p.!odz.pl

Eugene Benilov, University of Limerick, Ireland; Email: Eugene.Benilov@uI.ie

Eshel Ben-Jacob, lei AVIV Umverslty, Israel; Email: eshel@tamar.tau.ac.il

Maurice Courbage, Universite Paris 7, France; Email: maurice.courbage@univ-paris-diderot.fr

Marian Gidea, Northeastern Illinois University, USA; Email: mgidea@neiu.edu

James A. GlaZIer, Indiana UmvefSlty, USA; Email: glazler@mdlana.edu

Shijun Liao, Shanghai Jiaotong University, China; Email: sjIiao@sjtu.edu.cn

Jose AntOnIO'Ienrelro Machado, ISEP-Inslitute of Engmeenng of Porto, Portugal; Email: Jtm@dee.lsep.lpp.pt

Nikolai A. Magnitskii, Russian Academy of Sciences, Russia; Email: nmag@isa.ru

Josep J. Masdemont, UmvefSltat Pohtecmca de Catalunya (UPe), Spam; Email: losep@barqums.upc.edu

Dnntry E. Pelillovsky, McMaster Umverslty, Canada; Email: dmpeh@math.mcmaster.ca

Sergey Prants, Y.I.Il'ichev Pacific Oceanological Institute of the Russian Academy of Sciences, Russia;

Email: prants@pOl.dvo.ru

Victor I. Shrira, Keele University, UK; Email: v.i.shrira@keele.ac.uk

Jl3n Q130 Sun, Umversily of Cahfornla, USA; Email: jqsun@ucmerced.edu

Abdul-Majid Wazwaz, Saint Xavier University, USA; Email: wazwaz@sxu.edu

Pei Yu, The University of Western Ontario, Canada; Email: pyu@uwo.ca

Albert C.J. LuoJian-Qiao Sun

Complex SystemsFractionality, Time-delayand Synchronization

With 154 figures

Ei:IJ1iirSAlbert C.l. Luo

Department of Mechanical and Industrial Engineering

Southern IllmOis Umverslty, Edwardsville

Edwardsville, IL 62026-1805, USA

Email: aluo@sme.edu

Jian-Qiao Sun

University of California, Merced

5200 N. Lake Road

P.O. Box 2039, Merced, CA 95344, USA

Email: J9sun@ucmerced.edu

ISSN 1867-8440

Nonlmear Physical SCience

ISBN 978-7-04-029710-2

Higher Education Press, BelJmg

ISBN 978-3-642-17592-3

e-ISSN 1867-8459

e-ISBN 978-3-642-17593-0

Spnnger Heidelberg Dordrecht London New York

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is

concerned, speCifically the fights of translation, reprmtmg, reuse of IllustratIOns, recitatIOn, broadcastmg,

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Preface

This edited book covers recent developments on fractional dynamics, time-delaysystems, system synchromzatIOn, and neuron dynamIcs.

FractIOnal calculus IS extensIvely used as a powefful tool to mvestigate complexphenomena in engmeenng and SCIence, and has receIved renewed attention recently.Chapter I of the book investigates fractional dynamics of complex systems. Somerecent results and applications in fractional dynamics are presented. In Chapter 2,the synthesIs and applicatIOn of tractIOnal-order controllers are presented. ThIs ISan active area of research. The tractIOnal-order PID controllers are desIgned for thevelOCity control of an expenmental modular servo system. The system consists ofa digital servomechanism and open-architecture software environment for real-timeimplementation. Experimental results of fractional-order controllers are presentedand analyzed. The effectiveness and superior perfonnance of the fractional-ordercontrols are compared WIth c1asslcalmteger-order PID controllers.

TIme delay IS a common phenomenon in engmeenng, economIcal and bIOlogIcalsystems, and has become a popular research topic in recent years. InChapter 3, equi­librium stability, Lindsedt's method and Hopf bifurcation, and transient behaviorsin differential-delay equations are presented. Multiple-scale and the center mani­fold analysIs are addressed. These methods are applied to mvestigate dynamIcalbehavIOrs of a differential-delay system modeling a sectIOn of the DNA molecule.Chapter 4 focuses on the methodologIes for time-domam solutIOns and control de­sign of time-delayed systems. Method of semi-discretization and continuous timeapproximation are discussed. The spectral properties of the methods will be investi­gated. A comparative study of stabIlity of time-delayed linear time mvanant systemsIS carned out by the Lyapunov method, Pad approxImatIOn and semI-dIscretizatIOn.The methods of solutIOn for stochastic dynamIcal systems WIth time delay are alsodiscussed, and a number of control examples and an experimental validation arepresented.

Chapter 5 develops a theory for synchromzatIOn of multiple dynamIcal systemsunder constraints The metric functionals based on the constraints are introducedto descnbe the synchromCity of two or more dynamIcal systems. The chapter pro-

vi

vides a theoretic framework for designing controllers of slave systems which can besynchronized with master systems.

Finally in Chapter 6, complex dynamics of neurons with time-delay, stochas­tICIty and ImpulsIve dIscontmUIty are presented. Complex dynamIcal behavIOrs m­clude perIOdIC spIkmg, chaotIc spIkmg, perIOdIC and chaotIc burstmg, and synchro­nization. In this chapter, a comprehensive review on recent developments and newresults in nonlinear neural dynamics are presented.

It is our hope that the book presents a reasonably broad view of the state-of­the-art of complex systems, and provIdes a useful reference volume to SCIentIsts,engmeers and students. Furthermore, we hope that the book wIll stImulate moreresearches m the rapIdly evolvmg and mterestmg field of complex systems.

EdwardsvIlle, IllInOISMerced, CalIforma

Albert C.J. LuolIan-QIao Sun

June, 2010

Contents

1 New Treatise in Fractional DynamicsDumitru Baleanu 11 1 Introduction 11.2 BaSIC defimtIOns and properties of tractIOnalderIvatIves

and integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31.3 Fractional variational principles and their applications. . . . . . . . . . . .. 10

1.3.1 FractIOnalEuler-Lagrange equatIOnsfor dIscrete systems. .. 111 32 Fractional Hamiltonian formulation 131.3.3 LagrangIan formulatIOnof field systems WIth tractIOnal

derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 201.4 Fractional optimal control formulation. . . . . . . . . . . . . . . . . . . . . . . . . .. 23

1.4.1 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 241.5 FractIOnalcalculus m nuclear magnetIc resonance. . . . . . . . . . . . . . .. 271.6 FractIOnalwavelet method and ItS applIcatIOns m

drug analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35

2 Realization of Fractional-Order Controllers: Analysis, Synthesis andApplication to the Velocity Control of'a Servo System

Ramiro S. Barbosa. Isabel S. Jesus. Manuel F. Silva.lA Tenreiro MachadO 432 1 Introduction 432.2 FractIOnal-order control systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45

2.2.1 BaSIC theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 452.2.2 Fractional-Order controllers and their implementation.. . . .. 47

2.3 Oustaloup's frequency approximation method 492.4 The experImental modular servo system. . . . . . . . . . . . . . . . . . . . . . . .. 502.5 MathematIcal modellIng and IdentIficatIOn of the servo system. . . .. 502.6 FractIOnal-order real-tIme control system. . . . . . . . . . . . . . . . . . . . . . .. 532.7 ZIegler-NIchols tunmg rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54

V111

2.8

Contents

2.7.1 Ziegler-Nichols tuning rules: quarter decay ratio 552.7.2 Ziegler-Nichols tuning rules: oscillatory behavior 592.7.3 Comments on the results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61A simple analytical method for tumng fractiOnal-ordercontrollers 632.8.1 The proposed analytical tuning method. . . . . . . . . . . . . . . . . .. 65

2.9 Application of optimal fractional-order controllers. . . . . . . . . . . . . . .. 692.9.1 Tuning ofthe PID and Pl:l: D controllers.. . . . . . . . . . . . . . . .. 70

3 Differential-Delay EquationsRichard Rand 833 1 Introduction 833.2 Stability of equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 843.3 Lindstedt's method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 853.4 Hopfbifurcation formula 88

3.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 903.4.2 Denvation............................................. 913.4.3 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92344 Discussion 93

3 5 Transient behavior 943.5.1 Example 943.5.2 Exact solutiOn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 953.5.3 Two vanable expanSiOnmethod (also known as multiple

scales) 953.5.4 Approach to hmit cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97

3.6 Center mamfold analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 973.6.1 Appendix: The ad]omt operator A* . . . . . . . . . . . . . . . . . . . . .. 107

3.7 ApphcatiOn to gene expreSSiOn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1083.7.1 Stability of equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1093.7.2 Lindstedt's method.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. III3.7.3 Numerical example.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 113

3 8 Exercises 114References 115

4 Analysis and Control of Deterministic and Stochastic DynamicalSystems with Time Delay

]ian-Qiao Sun, Bo Song 1194 1 Introduction 119

4.1.1 Deterministic systems 1204.1.2 Stochastic systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1224 1 3 Methods of solution 1224.1.4 Outhne of the chapter. 124

Contents IX

4.2 Abstract Cauchy problem for DDE. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1244.2.1 Convergence with Chebyshev nodes. . . . . . . . . . . . . . . . . . . .. 126

4.3 Method of semi-discretization.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1274.3.1 General time-varymg systems. . . . . . . . . . . . . . . . . . . . . . . . .. 1294 3 2 Feedback controls 1304.3.3 Analysis of the method of semi-discretization. . . . . . . . . . .. 1334.3.4 High order control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1384.3.5 Optimal estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1394.3.6 ComparIson of semI-dIscretizatIOn and hIgher order

control 1404.4 Method of contmuous time approximanon . . . . . . . . . . . . . . . . . . . . .. 143

4.4.1 Control problem formulations. . . . . . . . . . . . . . . . . . . . . . . . .. 1444.5 Spectral properties of the CTA method. . . . . . . . . . . . . . . . . . . . . . . .. 146

4.5.1 A low-pass filter based CTA method. . . . . . . . . . . . . . . . . . . .. 1494.5.2 Example of a first order hnear system. . . . . . . . . . . . . . . . . .. 150

4.6 Stablhty studIes of time delay systems. . . . . . . . . . . . . . . . . . . . . . . . .. 1534.6.1 Stability with Lyapunov-Krasovskii functional. . . . . . . . . .. 1534.6.2 Stability with Pade approximation.. . . . . . . . . . . . . . . . . . . . .. 1554.6.3 Stability with semi-discretization. . . . . . . . . . . . . . . . . . . . . . .. 1564.6.4 Stablhty of a second order LTI system. . . . . . . . . . . . . . . . . .. 156

4.7 Control of LTI systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1634.8 Control ofthe Mathieu system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1674.9 An experimental validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1724.10 Supervisory controL 174

4.10.1 SuperVISory Control of the LTI System. . . . . . . . . . . . . . . .. 1754.10.2 SuperVISory control of the perIOdIC system. . . . . . . . . . . . .. 178

4.11 Method of semI-dIscretizatIOn for stochastic systems. . . . . . . . . . . .. 1814.11.1 Mathematical background.. . . . . . . . . . . . . . . . . . . . . . . . . . .. 1814.11.2 Stability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 183

4.12 Method of fillIte-dimensIOnalmarkov process (FDMP). . . . . . . . . .. 1844.12.1 Fokker-Planck-kolmogorov (FPK) equatIOn 1854.12.2 Moment equations 1864.12.3 Reliability 1874.12.4 First-passage time probability......................... 1884.12.5 Pontryagm-VIttequatIOns 189

4.13 AnalYSIS of stochastic systems WIth time delay.. . . . . . . . . . . . . . . .. 1904.13.1 Stablhty of second order stochastic systems. . . . . . . . . . . .. 1904.13.2 One DImenSIOnal Nonhnear System. . . . . . . . . . . . . . . . . . .. 196

References 198

5 Synchronization of Dynamical Systems in Sense of MetricFunctionals of'Specific Constraints

Albert C.l. Luo 2055 1 Introduction 205

x Contents

5.2 System synchronization 2085.2.1 Synchronization of slave and master systems. . . . . . . . . . . .. 2085.2.2 Generalized synchronization. . . . . . . . . . . . . . . . . . . . . . . . . . .. 2145.2.3 Resultant dynamIcal systems. . . . . . . . . . . . . . . . . . . . . . . . . .. 2165 2 4 Metric functionals 220

5.3 Single-constraint synchronization.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2235.3.1 Synchronicity 2235.3.2 Singularity to constraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2275.3.3 Synchromcity WIth smgulanty 2315.3.4 HIgher-order smgulanty 2325.3.5 SynchromzatIOnto constramt 2365.3.6 Desynchronization to constraint. . . . . . . . . . . . . . . . . . . . . . . .. 2525.3.7 Penetration to constraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 257

5.4 MultIple-constramt synchromzatIOn 2615.4.1 Synchromcity to multIple-constramts 2615.4.2 Smgulanty to constramts 2645.4.3 Synchronicity with singularity to multiple constraints. . . .. 2675.4.4 Higher-order singularity to constraints. . . . . . . . . . . . . . . . . .. 2705.4.5 Synchronization to all constraints. . . . . . . . . . . . . . . . . . . . . .. 2745.4.6 DesynchromzatIOnto all constramts 279547 Penetration to all constraints 2845.4.8 Synchronization-desynchronization-penetration.......... 287

5 5 Conclusions 294References 294

6 The Complexity in Activity of Biological NeuronsYang Xie, Jian-Xue Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2996.1 ComplIcated finng patterns m bIOlogIcal neurons. . . . . . . . . . . . . . .. 300

6.1.1 TIme senes of membrane potential. . . . . . . . . . . . . . . . . . . . .. 3006.1.2 Fmng patterns: spIkmg and burstmg . . . . . . . . . . . . . . . . . . . .. 300

6.2 Mathematical models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3066.2.1 HH model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3066.2.2 FItzHugh-Nagumo model. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30762 3 Hindmarsh-Rose model 308

6.3 NonlInear mechamsms of finng patterns. . . . . . . . . . . . . . . . . . . . . . .. 3096.3.1 DynamIcal mechamsms underlymg Type I excitabIlIty and

Type II excitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3096.3.2 DynamIcal mechamsm for the onset of finng m

the HH model 3IO6.3.3 Type I eXCItabIlIty and Type II eXCItabIlIty dIsplayed m the

Morris-I.ecarmodel 3II6.3.4 Change m types of neuronal eXCItabIlIty VIa bIfurcatIOn

control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3146.3.5 Burstmg and Its topologIcal classIficatIOn. . . . . . . . . . . . . . .. 322

Contents Xl

6.3.6 Bifurcation, chaos and Crisis. . . . . . . . . . . . . . . . . . . . . . . . . .. 3246.4 Sensitive responsiveness of aperiodic firing neurons to external

stimuli. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3266.4.1 Expenmental phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 326642 Nonlinear mechanisms 328

6.5 Synchronization between neurons. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3346.5.1 Significance of synchronization in the nervous system. . . .. 3346.5.2 Coupling: electrical coupling and chemical coupling. . . . .. 335

6.6 Role of nOIse m the nervous system. . . . . . . . . . . . . . . . . . . . . . . . . . .. 3376 6 1 Constmctive role' stochastic resonance and coherence

resonance 3376.6.2 Stochastic resonance: When does it not occur in neuronal

models? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3386.6.3 Global dynamiCsand stochastIc resonance of the forced

FItzHugh-Nagumo neuron model. . . . . . . . . . . . . . . . . . . . . .. 3396.6.4 A novel dynamical mechamsm of neural eXCitabilIty for

integer multiple spiking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3426.6.5 A Further Insight into Stochastic Resonance in an

Integrate-and-fire Neuron with Noisy Periodic Input. . . . .. 3456.6.6 Signal-to-nOIseratIo gam of a nOISY neuron that transmits

subthreshold penOdIC spike trams. . . . . . . . . . . . . . . . . . . . . .. 3526.6.7 Mechanism of bifurcation-dependent coherence resonance

of Morns-Lecar Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3526.7 AnalySIS of tIme senes of mterspike mtervals . . . . . . . . . . . . . . . . . .. 353

6.7.1 Return map 3536.7.2 Phase space reconstructIOn 3536.7.3 ExtractIOn of unstable penOdIC orbIts. . . . . . . . . . . . . . . . . . .. 3556.7.4 NonlInear predictIon and surrogate data methods. . . . . . . .. 3566.7.5 NonlInear charactenstIc numbers. . . . . . . . . . . . . . . . . . . . . . .. 358

6.8 ApplIcatIon.................................................. 3626.9 ConclusIons................................................. 363References 363

Contributors

Dumitru Baleanu Department of MathematICS and Computer SCIence, CankayaUniversity, 06530 Ankara, Turkey; Institute of Space Sciences, P.O. BOX, MG-23,R 76900, Magurele-Bucharest, Romama, EmaIl: dumItru@cankaya.edu.tr.baleanu@venus.mpne.ro

Ramiro S. Barbosa DIpartImento dl FlSlca, UmversIta dl FIrenze, and INFN, VIaSansone 1,50019 Sesto Eno (Firenze), Italy, Email: antoniaciani@gmail.com

M. Courbage Institute of Engineering of Porto, Dept. of Electrical Engineering,Rua Dr. Antomo Bernardmo de AlmeIda, 431, 4200-072 Porto, Portugal, EmaIl:rsb@isep.ipp.pt

Albert C.J. Luo Department of Mechanical and Industrial Engineering, SouthernIllInOIS UmversIty EdwardsvIlle, IL 62026-1805, USA, EmaIl: aluo@sme.edu

l.A. Tenreiro Machado InstItute of Engmeenng of Porto, Dept. of Electncal En­gineering, Rua Dr. Antonio Bernardino de Almeida, 431, 4200-072 Porto, Portu­gal, Email: jtm@isep.ipp.pt

Richard Rand: Cornell University, Ithaca, NY 14853, USA, Email: rrand@cornell.edu

Manuel F. Silva InstItute of Engmeenng of Porto, Dept. of Electncal Engmeenng,Rua Dr. Antomo Bernardmo de AlmeIda, 431, 4200-072 Porto, Portugal, EmaIl:mss@Isep.Ipp.pt

Bo Song School of Engmeenng, UmversIty of CalIfornIa, Merced, CA 95344, USA,EmaIl: bosongl979@gmml.com

Jian-Qiao Sun School of Engineering, University of California, Merced, CA 95344,USA, Email: jqsun@ucmerced.edu

Yong Xie MOE Key Laboratory of Strength and Vibration, School of Aerospace,XI'an JIaotong UmversIty, XI'an 710049, Chma, EmaIl: YXIe@mml.xjtv.edv.cn

lian-Xue Xu MOE Key Laboratory of Strength and VIbratIOn, School of Aerospace,XI'an JIaotong UmversIty, XI'an 710049, Chma, EmaIl: jxxu@maIl.xjtu.edv.cn