Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities

175 Pages·2017·4.682 MB·English

by  Marat Akhmet, Ardak Kashkynbayev (auth.)

Nonlinear Physical Science
Marat Akhmet
Ardak Kashkynbayev
Bifurcation in Autonomous
and Nonautonomous
Differential Equations with
Discontinuities
Nonlinear Physical Science
Nonlinear Physical Science
Nonlinear Physical Science focuses on recent advances offundamental theories and principles,
analytical and symbolic approaches, as well as computational techniques in nonlinear physical
scienceandnonlinearmathematicswithengineeringapplications.
TopicsofinterestinNonlinearPhysicalScienceincludebutarenotlimitedto:
– Newfindingsanddiscoveriesinnonlinearphysicsandmathematics
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Nonlinearity,complexityandmathematicalstructuresinnonlinearphysics
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Nonlinearphenomenaandobservationsinnatureandengineering
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Computationalmethodsandtheoriesincomplexsystems
–
Liegroupanalysis,newtheoriesandprinciplesinmathematicalmodeling
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Stability,bifurcation,chaosandfractalsinphysicalscienceandengineering
–
Nonlinearchemicalandbiologicalphysics
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Discontinuity,synchronizationandnaturalcomplexityinthephysicalsciences
Series editors
AlbertC.J.Luo NailH.Ibragimov
DepartmentofMechanicalandIndustrial DepartmentofMathematicsandScience
Engineering BlekingeInstituteofTechnology
SouthernIllinoisUniversityEdwardsville S-37179Karlskrona,Sweden
Edwardsville,IL62026-1805,USA e-mail:[email protected]
e-mail:[email protected]
International Advisory Board
PingAo,UniversityofWashington,USA;Email:[email protected]
JanAwrejcewicz,TheTechnicalUniversityofLodz,Poland;Email:[email protected]
EugeneBenilov,UniversityofLimerick,Ireland;Email:[email protected]
EshelBen-Jacob,TelAvivUniversity,Israel;Email:[email protected]
MauriceCourbage,UniversitéParis7,France;Email:[email protected]
MarianGidea,NortheasternIllinoisUniversity,USA;Email:[email protected]
JamesA.Glazier,IndianaUniversity,USA;Email:[email protected]
ShijunLiao,ShanghaiJiaotongUniversity,China;Email:[email protected]
JoseAntonioTenreiroMachado,ISEP-InstituteofEngineeringofPorto,Portugal;
Email:[email protected]
NikolaiA.Magnitskii,RussianAcademyofSciences,Russia;Email:[email protected]
JosepJ.Masdemont,UniversitatPolitecnicadeCatalunya(UPC),Spain;
Email:[email protected]
DmitryE.Pelinovsky,McMasterUniversity,Canada;Email:[email protected]
Sergey Prants, V.I.Il’ichev Pacific Oceanological Institute of the Russian Academy of Sciences,
Russia;Email:[email protected]
VictorI.Shrira,KeeleUniversity,UK;Email:[email protected]
JianQiaoSun,UniversityofCalifornia,USA;Email:[email protected]
Abdul-Majid Wazwaz, Saint Xavier University, USA; Email: [email protected]
PeiYu,TheUniversityofWesternOntario,Canada;Email:[email protected]
More information about this series at http://www.springer.com/series/8389
Marat Akhmet Ardak Kashkynbayev
(cid:129)
Bifurcation in Autonomous
and Nonautonomous
Differential Equations
with Discontinuities
123
Higher
Education
Press
Marat Akhmet Ardak Kashkynbayev
Department ofMathematics Department ofMathematics
Middle EastTechnical University Middle EastTechnical University
Ankara Ankara
Turkey Turkey
ISSN 1867-8440 ISSN 1867-8459 (electronic)
Nonlinear Physical Science
ISBN978-981-10-3179-3 ISBN978-981-10-3180-9 (eBook)
DOI 10.1007/978-981-10-3180-9
JointlypublishedwithHigherEducationPress,Beijing
ISBN978-7-04-047450-3
LibraryofCongressControlNumber:2016963160
©SpringerNatureSingaporePteLtd.andHigherEducationPress2017
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To our beloved families
Preface
Thisbookisdevotedtobifurcationtheoryindiscontinuousdynamicalsystems.The
mainnoveltyistheconsiderationofbifurcationindifferentialandhybridequations
bymeansofmethodsdevelopedbyauthorsinrecentyears.Hopfbifurcationresults
are obtained for planar and three-dimensional systems. Results in nonautonomous
bifurcationtheoryarepresentedfordifferentialequationswithdiscontinuities.This
is the first time illustrations for nonautonomous bifurcation are provided.
This theory is among vast developing subjects in the recent years. The subjects in
this book are evolved from:
(cid:129) Bifurcation theory for autonomous and nonautononmous ODEs;
(cid:129) B—equivalence method is developed for impulsive differential equations with
nonfixed moments of impacts and principles of discontinuous dynamical
systems;
(cid:129) Theory of differential equations with piecewise constant argument of general-
ized type; and
(cid:129) Theory of differential equations with discontinuous right-hand side.
Weexpectthattheresultsobtainedinthisbookwillbeappliedtovariousfields
suchasneuralnetworks,braindynamics,mechanicalsystems,weatherphenomena,
andpopulationdynamics.Thus,wethinkthatinnearfuturethistheorywillbeone
ofthemostattractingareasindynamicalsystemsanditsapplications.Withoutany
doubt,bifurcationtheoryshouldbefurtherdevelopedintoothertypesofdifferential
equation. In this sense, we strongly believe that the present book will be a leading
one in this field. Bifurcation of periodic solutions and nonautonomous systems is
yet to develop in multidimensional case. Center manifold theory is one of the
interesting topics to investigate.
We have published several papers and books related to bifurcation theory in
recent years. In this book, we provide results in discontinuous dynamical systems
that are developed parallel to ODEs. The reader will benefit from recent results
obtainedinthetheoryofbifurcationandwilllearnintheveryconcretewayhowto
apply this theory to differential equations with various types of discontinuity:
impulsive differential equations, differential equations with piecewise constant
vii
viii Preface
argument,anddifferentialequationswithdiscontinuousright-handside.Moreover,
the reader will learn how to analyze nonautonomous bifurcation scenarios in these
equations. The present book is devoted to Hopf, transcritical, and pitchfork bifur-
cations, and it is reasonable to discuss a new possibilities in other types of bifur-
cation such as Neimark–Sacker bifurcation, Shilnikov bifurcation, Bautin
bifurcation, Bogdanov–Taken bifurcation, and bifurcation of almost periodic
solutions.
Thisbookwillbe ofabiginterest both for beginners andexpertsinthefieldof
bifurcation theory. For the former group of specialists, that is, undergraduate and
graduate students, this book will be useful since it provides strong impression that
bifurcation theory can be developed not only for discrete and continuous systems
but also for those which combine these systems in very different ways. The latter
group of specialists will find in this book several powerful instruments developed
for the theory of discontinuous dynamical systems with variable moments of
impacts, differential equations with piecewise constant argument of generalized
type,andFilippovsystems.Asignificantbenefitofthispresentbookisexpectedto
beforthosewhoconsiderbifurcationsinsystemswithdiscontinuitiessincetheyare
presumablynonautonomoussystems.Consequently,nonautonomousbifurcationis
compulsory subject to discuss.
The authors would like to offer their sincere thanks to those who contributed to
thepreparationofthisbook,DuyguAruğaslan ÇinçinandMehmetTuranfor their
collaboration,andtheserieseditorProf.AlbertLuoandeditorofHEPLipingWang
fortheirinterestinthemonographandpatienceduringthepublicationofthebook.
Ankara, Turkey Marat Akhmet
Ardak Kashkynbayev
Contents
1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 General Description of Differential Equations
with Discontinuities .. ..... .... .... .... .... .... ..... .... 2
1.1.1 Impulsive Differential Equations.... .... .... ..... .... 3
1.1.2 Differential Equations with Piecewise Constant
Argument .... ..... .... .... .... .... .... ..... .... 4
1.1.3 Differential Equations with Discontinuous
Right-Hand Sides ... .... .... .... .... .... ..... .... 5
1.2 Nonautonomous Bifurcation. .... .... .... .... .... ..... .... 6
1.3 The Bernoulli Equations.... .... .... .... .... .... ..... .... 8
1.4 Organization of the Book... .... .... .... .... .... ..... .... 8
2 Hopf Bifurcation in Impulsive Systems ... .... .... .... ..... .... 11
2.1 Hopf Bifurcation of a Discontinuous Limit Cycle .... ..... .... 11
2.1.1 The Nonperturbed System. .... .... .... .... ..... .... 11
2.1.2 The Perturbed System.... .... .... .... .... ..... .... 14
2.1.3 Foci of the D-System .... .... .... .... .... ..... .... 16
2.1.4 The Center and Focus Problem. .... .... .... ..... .... 19
2.1.5 Bifurcation of a Discontinuous Limit Cycle ... ..... .... 21
2.1.6 Examples .... ..... .... .... .... .... .... ..... .... 25
2.2 3D Discontinuous Cycles... .... .... .... .... .... ..... .... 26
2.2.1 Introduction... ..... .... .... .... .... .... ..... .... 26
2.2.2 The Nonperturbed System. .... .... .... .... ..... .... 26
2.2.3 The Perturbed System.... .... .... .... .... ..... .... 30
2.2.4 Center Manifold .... .... .... .... .... .... ..... .... 34
2.2.5 Bifurcation of Periodic Solutions ... .... .... ..... .... 37
2.2.6 Examples .... ..... .... .... .... .... .... ..... .... 41
2.3 Periodic Solutions of the Van der Pol Equation.. .... ..... .... 43
2.3.1 Introduction and Preliminaries.. .... .... .... ..... .... 43
2.3.2 Theoretical Results .. .... .... .... .... .... ..... .... 46
ix
x Contents
2.3.3 Center Manifold .... .... .... .... .... .... ..... .... 57
2.4 Notes . .... .... .... ..... .... .... .... .... .... ..... .... 62
3 Hopf Bifurcation in Filippov Systems .... .... .... .... ..... .... 65
3.1 Nonsmooth Planar Limit Cycle from a Vertex... .... ..... .... 65
3.1.1 Introduction... ..... .... .... .... .... .... ..... .... 65
3.1.2 The Nonperturbed System. .... .... .... .... ..... .... 67
3.1.3 The Perturbed System.... .... .... .... .... ..... .... 69
3.1.4 The Focus-Center Problem .... .... .... .... ..... .... 73
3.1.5 Bifurcation of Periodic Solutions ... .... .... ..... .... 75
3.1.6 An Example .. ..... .... .... .... .... .... ..... .... 78
3.2 3D Filippov System .. ..... .... .... .... .... .... ..... .... 80
3.2.1 Introduction... ..... .... .... .... .... .... ..... .... 80
3.2.2 The Nonperturbed System. .... .... .... .... ..... .... 81
3.2.3 The Perturbed System.... .... .... .... .... ..... .... 84
3.2.4 Center Manifold .... .... .... .... .... .... ..... .... 89
3.2.5 Bifurcation of Periodic Solutions ... .... .... ..... .... 91
3.2.6 An Example .. ..... .... .... .... .... .... ..... .... 94
3.3 Notes . .... .... .... ..... .... .... .... .... .... ..... .... 96
4 Nonautonomous Bifurcation in Impulsive Bernoulli Equations. .... 97
4.1 The Transcritical and the Pitchfork Bifurcations.. .... ..... .... 97
4.1.1 Introduction... ..... .... .... .... .... .... ..... .... 97
4.1.2 Preliminaries.. ..... .... .... .... .... .... ..... .... 98
4.1.3 The Pitchfork Bifurcation . .... .... .... .... ..... .... 100
4.1.4 The Transcritical Bifurcation... .... .... .... ..... .... 104
4.2 Impulsive Bernoulli Equations: The Transcritical
and the Pitchfork Bifurcations ... .... .... .... .... ..... .... 110
4.2.1 Introduction and Preliminaries.. .... .... .... ..... .... 110
4.2.2 Bounded Solutions .. .... .... .... .... .... ..... .... 112
4.2.3 The Pitchfork Bifurcation . .... .... .... .... ..... .... 115
4.2.4 The Transcritical Bifurcation... .... .... .... ..... .... 117
4.2.5 Illustrative Examples. .... .... .... .... .... ..... .... 120
4.3 Notes . .... .... .... ..... .... .... .... .... .... ..... .... 122
5 Nonautonomous Bifurcations in Nonsolvable Impulsive Systems........ 123
5.1 The Transcritical and the Pitchfork Bifurcations.. .... ..... .... 123
5.1.1 Introduction... ..... .... .... .... .... .... ..... .... 123
5.1.2 Preliminaries.. ..... .... .... .... .... .... ..... .... 124
5.1.3 Attractivity and Repulsivity in a Linear Impulsive
Nonhomogeneous Systems .... .... .... .... ..... .... 126
5.1.4 The Transcritical Bifurcation... .... .... .... ..... .... 129
5.1.5 The Pitchfork Bifurcation . .... .... .... .... ..... .... 131
5.2 Finite-Time Nonautonomous Bifurcations... .... .... ..... .... 133
5.2.1 Introduction and Preliminaries.. .... .... .... ..... .... 133