Table Of ContentMathematics for Industry 7
Kazuyuki Aihara
Jun-ichi Imura
Tetsushi Ueta E ditors
Analysis and
Control of Complex
Dynamical
Systems
Robust Bifurcation, Dynamic Attractors,
and Network Complexity
Mathematics for Industry
Volume 7
Editor-in-Chief
MasatoWakayama(KyushuUniversity,Japan)
ScientificBoardMembers
RobertS.Anderssen(CommonwealthScientificandIndustrialResearchOrganisation,Australia)
HeinzH.Bauschke(TheUniversityofBritishColumbia,Canada)
PhilipBroadbridge(LaTrobeUniversity,Australia)
JinCheng(FudanUniversity,China)
MoniqueChyba(UniversityofHawaiiatMānoa,USA)
Georges-HenriCottet(JosephFourierUniversity,France)
JoséAlbertoCuminato(UniversityofSãoPaulo,Brazil)
Shin-ichiroEi(HokkaidoUniversity,Japan)
YasuhideFukumoto(KyushuUniversity,Japan)
JonathanR.M.Hosking(IBMT.J.WatsonResearchCenter,USA)
AlejandroJofré(UniversityofChile,Chile)
KerryLandman(TheUniversityofMelbourne,Australia)
RobertMcKibbin(MasseyUniversity,NewZealand)
GeoffMercer(AustralianNationalUniversity,Australia)(Deceased,2014)
AndreaParmeggiani(UniversityofMontpellier2,France)
JillPipher(BrownUniversity,USA)
KonradPolthier(FreeUniversityofBerlin,Germany)
WilSchilders(EindhovenUniversityofTechnology,TheNetherlands)
ZuoweiShen(NationalUniversityofSingapore,Singapore)
Kim-ChuanToh(NationalUniversityofSingapore,Singapore)
EvgenyVerbitskiy(LeidenUniversity,TheNetherlands)
NakahiroYoshida(TheUniversityofTokyo,Japan)
Aims&Scope
The meaning of “Mathematics for Industry” (sometimes abbreviated as MI or MfI) is different
fromthatof“MathematicsinIndustry”(orof“IndustrialMathematics”).Thelatterisrestrictive:it
tendstobeidentifiedwiththeactualmathematicsthatspecificallyarisesinthedailymanagement
andoperationofmanufacturing.Theformer,however,denotesanewresearchfieldinmathematics
thatmayserveasafoundationforcreatingfuturetechnologies.Thisconceptwasbornfromthe
integrationandreorganizationofpureandappliedmathematicsinthepresentdayintoafluidand
versatileformcapableofstimulatingawarenessoftheimportanceofmathematicsinindustry,as
well as responding to the needs of industrial technologies. The history of this integration and
reorganizationindicatesthatthisbasicideawillsomedayfindincreasingutility.Mathematicscan
beakeytechnologyinmodernsociety.
Theseriesaimstopromotethistrendby(1)providingcomprehensivecontentonapplications
of mathematics, especially to industry technologies via various types of scientific research, (2)
introducingbasic,useful,necessaryandcrucialknowledgeforseveralapplicationsthroughcon-
crete subjects, and (3) introducing new research results and developments for applications of
mathematicsintherealworld.Thesepointsmayprovidethebasisforopeninganewmathematics-
orientedtechnologicalworldandevennewresearchfieldsofmathematics.
More information about this series at http://www.springer.com/series/13254
Kazuyuki Aihara Jun-ichi Imura
(cid:129)
Tetsushi Ueta
Editors
Analysis and Control of
Complex Dynamical Systems
Robust Bifurcation, Dynamic Attractors,
and Network Complexity
123
Editors
Kazuyuki Aihara Tetsushi Ueta
The Universityof Tokyo Tokushima University
Tokyo Tokushima
Japan Japan
Jun-ichiImura
TokyoInstituteof Technology
Tokyo
Japan
ISSN 2198-350X ISSN 2198-3518 (electronic)
Mathematics forIndustry
ISBN 978-4-431-55012-9 ISBN 978-4-431-55013-6 (eBook)
DOI 10.1007/978-4-431-55013-6
LibraryofCongressControlNumber:2015933364
SpringerTokyoHeidelbergNewYorkDordrechtLondon
©SpringerJapan2015
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Preface
Recent advanced communication and information technologies have attracted our
attention to developments of highly dependable, strongly resilient, and energy-
efficientsystemswithapplicationstointelligenttransportationsystems(ITS),smart
grids, high-level medical diagnosis/treatment systems, and so on. Such systems in
general involve complex behavior induced by interactions among subsystems, and
complex network structure as well as a large number of components. We need to
analyzesuchcomplexbehaviorforcapturingtheintrinsicpropertiesofthesystems
and to design control systems for realizing the desirable behavior. Both dynamical
systemstheoryandcontrolsystemstheorywillplayindispensableandcentralroles
in addressing such issues.
Dynamical systems theory originated from Newton’s motion equations in the
seventeenthcentury,andhasbeenfoundedbyPoincaré’sgreatcontributionslatein
the nineteenth century. After that, various mathematical methods such as ergodic
theory, stability theory of periodic solutions including equilibrium points, and
bifurcationtheoryfornonlineardynamicalsystemshavebeendeveloped,andsince
the late 1970s, they have been extended to different research topics on more
complexphenomena/controlsuchasbifurcationstochaos,chaoscontrol,andchaos
synchronization.
Ontheotherhand,JamesWatt’ssteamengineattheindustrialrevolutioninthe
eighteenthcenturyhasopenedthegatetofeedbackcontrol,andMaxwell’sstability
analysis late in the nineteenth century, which theoretically analyzed the instability
phenomena of steam engines, was the occasion of developing control systems
theory.Continuinguponclassicalcontroltheorydealingwithcontrolsystemdesign
mainly in the frequency domain since the 1920s, modern control theory has been
advancingsincethe1960s,whichenablesustoanalyzecontrollability/observability
and to design optimal control by means of state equations in the time domain.
Moreover,adeepunderstandingonrobustnessofthesystembehaviorfordynamic
uncertainty including unmodeled dynamics in addition to parametric uncertainty
has been gained, and then robust control theory has been developed since the
1980s.
v
vi Preface
The above two research fields, however, have been developing almost inde-
pendently so far, although there have been several successes to be related in both
fields such as Pontryagin’s maximum principle and R.E. Kalman’s pioneering
contribution on chaos and control theory. The main focus in dynamical systems
theory is nonlinear autonomous dynamics with a kind of unstable phenomena like
bifurcations and chaos, while the focus in control systems theory is feedback sta-
bilization of linear non-autonomous dynamics at an equilibrium point. This moti-
vatesustodevelopanewparadigmonanalysisandcontrolofcomplex/large-scale
dynamical systems throughout collaborative research between dynamical systems
theory and control systems theory.
Thisbook,whichisthefirsttrialtowarddevelopmentsofsuchanewparadigm,
presents fundamental and theoretical breakthroughs on analysis and control of
complex/large-scale dynamical systems toward their applications to various engi-
neering fields. In particular, this book focuses on the following three topics:
1. Analysis and control of bifurcation under model uncertainty.
2. Analysis and control of complex behavior including quasi-periodic/chaotic
orbits.
3. Modeling of network complexity emerging from dynamical interaction among
subsystems.
Accordingtotheabovethreetopics,thisbookisorganizedasfollows:InPartI,
robust bifurcation analysis, which deals with bifurcation analysis for dynamical
systemssubjecttouncertaintyduetounmodeleddynamics,ispresentedandvarious
kinds ofbifurcationcontrol methodsbased onthedegree ofstability areproposed.
Part II begins with the analysis of chaotic behavior of triangle-folding maps, and
presentsnovel attemptsfor controlling various kinds of complex behavior,namely
feedback stabilization of quasi-periodic orbits and spatial patterns, chaos control,
ultra-discretization based control, and control of unstabilizable switched systems.
Finally, Part III includes research topics on network model reduction and network
structure identification toward control of large-scale network systems.
Thisbookcanbebeneficialtomathematicians,physicists,biophysicistaswellas
researchers on nonlinear science and control engineering for a better fundamental
understanding of analysis and control synthesis of such complex systems.
We would like to thank the contributors: Shun-ichi Azuma, Ken’ichi Fujimoto,
Tomohisa Hayakawa, Yoshito Hirata, Natsuhiro Ichinose, Masaki Inoue, Daisuke
Ito, Masato Ishikawa, Takayuki Ishizaki, Kenji Kashima, Takuto Kita, Hiroyuki
Kitajima, Miki U. Kobayashi, Mio Kobayashi, Motomasa Komuro, Takuji Kou-
saka,JunNishimura,ToshiyukiOgawa,YasuakiOishi,MasayasuSuzuki,Tomomi
Takegami, and Tetsuya Yoshinaga for writing excellent chapters.
This book is the outcome of the Japanese Research Project “the Aihara Inno-
vativeMathematicalModellingProject(2010.3–2014.3)”,oneoftop30projectsof
“FundingProgramforWorld-LeadingInnovativeR&DonScienceandTechnology
(FIRST Program)” initiated by the Council for Science and Technology Policy
(CSTP) in Japan. We would also like to express our sincere gratitude to all
members of the international advisory board of this project for their fruitful
Preface vii
suggestionsandencouragementstowardthedevelopmentsofthisproject:Shun-ichi
Amari, Miwako Doi, Hiroshi Fujii, Celso Grebogi, Seiichi Ishizu, Kunihiko
Kaneko, Hiroshi Kawakami, Hidenori Kimura, Yoshiki Kuramoto, Jürgen Kurths,
Henk Nijmeijer, Hugh Robinson, Ichiro Tsuda, Keiji Yamagami, and James
A. Yorke.
Tokyo, January 2015 Kazuyuki Aihara
Jun-ichi Imura
Tetsushi Ueta
Contents
Part I Robust Bifurcation and Control
1 Dynamic Robust Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . 3
Masaki Inoue, Jun-ichi Imura, Kenji Kashima
and Kazuyuki Aihara
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Problem Formulation: Dynamic Robust
Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Equilibrium, Stability/Instability, and Robustness Analysis. . . . 7
1.3.1 Equilibrium Analysis. . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Robust Hyperbolicity Analysis. . . . . . . . . . . . . . . . . 11
1.3.3 Robust Bifurcation Analysis. . . . . . . . . . . . . . . . . . . 14
1.4 Examples of Robust Bifurcation Analysis . . . . . . . . . . . . . . . 15
1.4.1 Robustness Analysis of Saddle-Node Bifurcation . . . . 15
1.4.2 Robustness Analysis of Hopf Bifurcation. . . . . . . . . . 16
1.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Robust Bifurcation Analysis Based on Degree of Stability. . . . . . . 21
Hiroyuki Kitajima, Tetsuya Yoshinaga, Jun-ichi Imura
and Kazuyuki Aihara
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 System Description and Robust Bifurcation Analysis. . . . . . . . 22
2.2.1 Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Discrete-time Systems. . . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 Robust Bifurcation Analysis. . . . . . . . . . . . . . . . . . . 24
2.3 Method of Robust Bifurcation Analysis. . . . . . . . . . . . . . . . . 25
2.4 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 Equilibrium Point . . . . . . . . . . . . . . . . . . . . . . . . . . 27
ix
x Contents
2.4.2 Periodic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Use of a Matrix Inequality Technique for Avoiding
Undesirable Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Yasuaki Oishi, Mio Kobayashi and Tetsuya Yoshinaga
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Considered Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Avoidance of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6.1 Method for Chaos Avoidance. . . . . . . . . . . . . . . . . . 38
3.6.2 Experimental Result . . . . . . . . . . . . . . . . . . . . . . . . 39
3.7 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 A Method for Constructing a Robust System Against
Unexpected Parameter Variation. . . . . . . . . . . . . . . . . . . . . . . . . 41
Hiroyuki Kitajima and Tetsuya Yoshinaga
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.2 Search for Optimal Parameter Values . . . . . . . . . . . . 43
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.1 Discrete-Time System . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.2 Continuous-Time System. . . . . . . . . . . . . . . . . . . . . 46
4.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 Parametric Control to Avoid Bifurcation Based
on Maximum Local Lyapunov Exponent. . . . . . . . . . . . . . . . . . . 49
Ken’ichi Fujimoto, Tetsuya Yoshinaga, Tetsushi Ueta
and Kazuyuki Aihara
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.4 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
