Dynamical system synchronization

244 Pages·2013·4.733 MB·English

by  Albert C J Luo

Nonlinear Systems and Complexity
SeriesEditor
AlbertC.J.Luo
SouthernIllinoisUniversity
Edwardsville,IL,USA
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Albert C. J. Luo
Dynamical System
Synchronization
AlbertC.J.Luo
SchoolofEngineering
SouthernIllinoisUniversityEdwardsville
Edwardsville,IL,USA
ISBN978-1-4614-5096-2 ISBN978-1-4614-5097-9(eBook)
DOI10.1007/978-1-4614-5097-9
SpringerNewYorkHeidelbergDordrechtLondon
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Preface
Thisbookpresentedatheoryofdynamicalsystemssynchronizationfromadifferent
pointofview.Suchsynchronizationtheoryisbasedonthetheoryofdiscontinuous
dynamicalsystems.Thesynchronizationofdiscretedynamicalsystemsisbasedon
the Ying-Yang theoryofdiscrete dynamical systems.The objective of this book
is to throw out the different points of view to look into dynamical systems
synchronizationwithouttheLyapunovstabilitymethod.
This book consists of six chapters. In Chap. 1, a brief history of dynamical
systemssynchronizationsispresented.Thecurrentmethodsfordynamicalsystems
synchronization are mainly based on the Lyapunov stability theory. Thus, the
dynamical systems for synchronization should be similar systems, which is far
away for practical applications. To solve such difficulty, a theory of dynamical
systemssynchronizationwithspecificconstraintsispresentedinthisbookfromthe
theoryofdiscontinuousdynamicalsystems.InChap.2,theswitchabilityofaflow
to the boundary in discontinuous dynamical systems is presented in order to help
oneunderstandthesynchronizationtheoryoftwodynamicalsystemswithspecific
constraints. In Chap. 3, the basic concepts of dynamical systems synchronization
are presented first, and the theory of dynamical systems synchronization with a
specific constraint is presented. For a further development of dynamical systems
synchronization, thetheoryoftwodynamical systems with multiple constraintsis
discussed in Chap. 4. In Chap. 5, the function synchronization of two distinct
dynamical systems is discussed to show how to apply the theory of dynamical
systems synchronization to practical problems. In Chap. 6, the theory for discrete
dynamical systems synchronization is presented from the Ying-Yang theory of
discretedynamicalsystems.
Finally, I would like to appreciate my students (Yu Guo and Fuhong Min) for
completing numerical computations. Herein, I thank my wife (Sherry X. Huang)
and my children (Yanyi Luo, Robin Ruo-Bing Luo, and Robert Zong-Yuan Luo)
for tolerance, patience, understanding, and support. This is what I can bring them
forhappiness.
Edwardsville,IL,USA AlbertC.J.Luo
v

Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 ABriefHistory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 BookLayout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 DiscontinuityandLocalSingularity. . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 DiscontinuousDynamicalSystems. . . . . . . . . . . . . . . . . . . . . . . 11
2.2 G-Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 PassableFlows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Non-passableFlows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 GrazingFlows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.6 FlowSwitchingBifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3 SingleConstraintSynchronization. . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1 IntroductiontoSynchronization. . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1.1 GeneralizedSynchronization. . . . . . . . . . . . . . . . . . . . . . 77
3.1.2 ResultantDynamicalSystems. . . . . . . . . . . . . . . . . . . . . 79
3.2 SynchronizationwithaSingleConstraint. . . . . . . . . . . . . . . . . . . 83
3.2.1 Synchronicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2.2 SingularitytoConstraint. . . . . . . . . . . . . . . . . . . . . . . . . 87
3.3 SynchronicitywithSingularity. . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.4 HigherOrderSingularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.5 SynchronizationtoConstraint. . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.6 DesynchronizationtoConstraint. . . . . . . . . . . . . . . . . . . . . . . . . 111
3.7 PenetrationtoConstraint. . .. . . .. . . . .. . . .. . . . .. . . .. . . . .. 117
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4 MultipleConstraintsSynchronization. . . . . . . . . . . . . . . . . . . . . . . 121
4.1 SynchronicitytoMultipleConstraints. . . . . . . . . . . . . . . . . . . . . 121
4.2 SingularitytoConstraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.3 SynchronicitywithSingularitytoConstraints. . . . . .. . . . . . . . .. 127
vii
viii Contents
4.4 Higher-OrderSingularitytoConstraints. . . . . . . . . . . . . . . . . . . . 130
4.5 SynchronizationtoAllConstraints. . . . . . . . . . . . . . . . . . . . . . . 133
4.6 DesynchronizationtoAllConstraints. .. . . . .. . . . .. . . .. . . . .. 138
4.7 PenetrationtoAllConstraints. . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.8 Synchronization–Desynchronization–Penetration. . . . . . . . . . . . . 145
4.9 ComplexitybySystemSynchronization. . . . . . . . . . . . . . . . . . . . 151
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5 FunctionSynchronizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.1 SynchronizationConstraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.2 SynchronizationMechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.3 SinusoidalSynchronization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.3.1 SynchronizationDynamics. . . . . . . . . . . . . . . . . . . . .. . . 171
5.3.2 SinusoidalSynchronizationofChaoticMotions. . . . . . . . 178
5.3.3 SinusoidalSynchronizationsofPeriodicMotions. . . . . . . 182
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6 DiscreteSystemsSynchronization. . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.1 DiscreteSystemswithaSingleNonlinearMap. . . . . . . . . . . . . . 197
6.2 DiscreteSystemswithMultipleMaps. . . . . . . . . . . . . . . . . . . . . 203
6.3 CompleteDynamicsofaHenonMapSystem. . . . . . . . . . . . . . . 207
6.4 CompanionandSynchronization. . . . . . . . . . . . . . . . . . . . . . . . . 213
6.5 AnApplicationofDiscreteSystemsSynchronization. . . . . . . . . . 229
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Chapter 1
Introduction
Withhuman-beingdevelopmentandprogress,coordinatesystemsareusedinorder
that the characteristics and behaviors of everything in nature can be described
quantitatively. In other words, because the coordinates systems are used exten-
sively, one gradually understands and improves the objective world. Similarly, to
describethecomplexityofachangingprocessofathingwithtime,oneoftenadopts
a given or known process to compare with such a changing process with time.
Further, one obtains the similarity, instantaneous similarity and differences
between the two dynamical processes for a time interval, and one determines the
complexity of a dynamical system to another known dynamical system. Such
similarity in a certain time interval is a kind of synchronization. In addition, the
synchronization of two or more dynamical systems is a basis to understand an
unknown dynamical system from one or more well-known dynamical systems.
Inotherwords,theresponsecomplexityofanunknownsystemtooneormorewell-
knownsystemscanbemeasuredandcomparedthroughsuchsynchronicity.Thus,
the synchronization in dynamical systems should be treated as an important con-
cept,andtheconceptof“synchronization”isalsoauniversalconceptfordynamical
systems.Basedontheaforesaidreasons,inthisbook,atheoryofsynchronizationof
dynamicalsystemswillbepresentedasatheoreticframework.
A theory for synchronization of multiple dynamical systems under specific
constraints was developed from a theory of discontinuous dynamical systems in
Luo [1]. The concepts on synchronization of two or more dynamical systems to
specificconstraintsweregiven.Thesynchronization,desynchronizationandpene-
tration of multiple dynamical systems to multiple specified constraints were
discussed,andthenecessaryandsufficientconditionsforsuchsynchronicitywere
developed. The synchronicity of two dynamical systems to a single specific con-
straint and tomultiple specific constraints was discussed, and the synchronization
andthecorrespondingcomplexityformultipleslavesystemswithmultiplemaster
systems were presented. The meaning of synchronization for dynamical systems
with constraints is extended as a generalized, universal concept. The theory
presented in this book may be as a universal theory for dynamical systems. The
book provides a theoretic frame work in order to control the slave systems which
A.C.J.Luo,DynamicalSystemSynchronization,NonlinearSystemsandComplexity3, 1
DOI10.1007/978-1-4614-5097-9_1,#SpringerScience+BusinessMedia,LLC2013