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Nonlinear Systems and Complexity Series Editor: Albert C. J. Luo Albert C. J. Luo Bifurcation and Stability in Nonlinear Dynamical Systems Nonlinear Systems and Complexity Volume 28 Serieseditor AlbertC.J.Luo SouthernIllinoisUniversity Edwardsville,IL,USA Nonlinear Systems and Complexity provides a place to systematically summarize recentdevelopments,applications,andoveralladvanceinallaspectsofnonlinearity, chaos,andcomplexityaspartoftheestablishedresearchliterature,beyondthenovel and recent findings published in primary journals. The aims of the book series are to publish theories and techniques in nonlinear systems and complexity; stimulate moreresearchinterestonnonlinearity,synchronization,andcomplexityinnonlinear science;andfast-scatterthenewknowledgetoscientists,engineers,andstudentsin thecorrespondingfields.Booksinthisserieswillfocusontherecentdevelopments, findings and progress on theories, principles, methodology, computational tech- niques in nonlinear systems and mathematics with engineering applications. The Series establishes highly relevant monographs on wide ranging topics covering fundamental advances and new applications inthe field. Topical areas include, but arenotlimitedto:Nonlineardynamics;Complexity,nonlinearity,andchaos;Com- putationalmethodsfornonlinearsystems;Stability,bifurcation,chaosandfractalsin engineering; Nonlinear chemical and biological phenomena; Fractional dynamics andapplications;Discontinuity,synchronizationandcontrol. Moreinformationaboutthisseriesathttp://www.springer.com/series/11433 Albert C. J. Luo Bifurcation and Stability in Nonlinear Dynamical Systems AlbertC.J.Luo SouthernIllinoisUniversity Edwardsville,IL,USA ISSN2195-9994 ISSN2196-0003 (electronic) NonlinearSystemsandComplexity ISBN978-3-030-22909-2 ISBN978-3-030-22910-8 (eBook) https://doi.org/10.1007/978-3-030-22910-8 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors, and the editorsare safeto assume that the adviceand informationin this bookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This book systematically presents a fundamental theory for the local analysis of bifurcationandstabilityofequilibriumsinnonlineardynamicalsystems.Untilnow, one does not have any efficient way to investigate stability and bifurcation of dynamical systems with higher-order singularity equilibriums. For instance, infinite-equilibriumdynamicalsystemshavehigherordersingularity,whichdramat- icallychangesdynamicalbehaviorsandpossessessimilarcharacteristicsofdiscon- tinuous dynamical systems. The stability and bifurcation of equilibriums on the specific eigenvector are presented, and the spiral stability and Hopf bifurcation of equilibriumsinnonlinearsystemsarepresentedthroughtheFourierseriestransfor- mation. The bifurcation and stability of higher-order singularity equilibriums are presentedthroughthe(2m)thand(2m+1)th-degreepolynomialsystems.Fromlocal analysis, dynamics of infinite-equilibrium systems is discussed. The research on infinite-equilibrium systemswillbringustotheneweraofdynamicalsystemsand control. Thisbookconsistsofeightchapters.Thefirstchapterdiscussesthelocaltheoryof stability of equilibriums in nonlinear dynamical systems. The spiral stability of equilibriumsontheeigenvectorspaceisdiscussed.Inaddition,basedontheFourier series transformation, the spiral stability of equilibrium is presented. The extended Lyapunovstabilitytheoryisalsopresented.InChap.2,thelocaltheoryofbifurca- tions ofequilibriums in nonlinear systems ispresentedon thespecificeigenvector, andtheHopfbifurcationofequilibriumisdiscussed.InChap.3,thelocalanalysisof stabilityandbifurcationfor1-dimensionaland2-dimensionaldynamicalsystemsis presented.InChap.4,equilibriumhigher-ordersingularityin1-dimensionalsystems is discussed globally. In Chap. 5, the global bifurcation theory for low-degree polynomial systems is presented. For a global view of bifurcation and stability of equilibriumsinnonlineardynamicalsystems,thebifurcationandstabilityofhigher ordersingularityequilibriumsarepresentedthroughthe(2m)thand(2m+1)th-degree polynomialsystemsinChaps.6and7,respectively.InChap.8,infinite-equilibrium systemsarediscussedthroughthelocalanalysisofhigher-order,singulardynamical vii viii Preface systems. The equilibrium computations and normal forms of equilibriums for nonlineardynamicalsystemsarepresented. Finally,Iwouldliketothankmywifeforhersupportforthisresearchwork.The author hopes that the materials presented herein can last long for science and engineering. Such contributions will benefit human beings on their progress and development. Edwardsville,IL,USA AlbertC.J.Luo Contents 1 StabilityofEquilibriums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 ContinuousDynamicalSystems. . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 EquilibriumsandStability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 StabilityandSingularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.1 HyperbolicStabilityonEigenvectors. . . . . . . . . . . . . . . . 18 1.3.2 SpiralStabilityonanInvariantEigenplane. . . . . . . . . . . 30 1.3.3 SpiralStabilityBasedontheFourierSeriesBase. . . . . . . 40 1.4 SpiralStabilityinSecond-OrderNonlinearSystems. . . . . . . . . . 44 1.5 LyapunovFunctionsandStability. . . . . . . . . . . . . . . . . . . . . . . 48 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2 BifurcationsofEquilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.1 Bifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.2 HyperbolicBifurcationsonEigenvectors. . . . . . . . . . . . . . . . . . 60 2.3 HopfBifurcationonanEigenvectorPlane. . . . . . . . . . . . . . . . . 69 2.4 HopfBifurcationBasedontheFourierSeriesBase. . . . . . . . . . . 75 2.5 HopfBifurcationsinSecond-OrderNonlinearSystems. . . . . . . . 80 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3 Low-DimensionalDynamicalSystems. . . . . . . . . . . . . . . . . . . . . . . . 87 3.1 1-DimensionalNonlinearSystems. . . . . . . . . . . . . . . . . . . . . . . 87 3.1.1 StabilityandSingularity. . . . . . . . . . . . . . . . . . . . . . . . . 87 3.1.2 Bifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.1.3 SampledSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.2 2-DimensionalNonlinearSystems. . . . . . . . . . . . . . . . . . . . . . . 109 3.2.1 StabilityandSingularity. . . . . . . . . . . . . . . . . . . . . . . . . 110 3.2.2 HopfBifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 ix x Contents 4 EquilibriumStabilityin1-DimensionalSystems. . . . . . . . . . . . . . . . 123 4.1 SystemClassifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.2 EquilibriumStability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.3 One-EquilibriumSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.4 Two-EquilibriumSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.5 Three-EquilibriumSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5 Low-DegreePolynomialSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.1 LinearSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.2 QuadraticNonlinearSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.3 CubicNonlinearSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.4 QuarticNonlinearSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 6 (2m)th-DegreePolynomialSystems. . . . . . . . . . . . . . . . . . . . . . . . . . 231 6.1 GlobalStabilityandBifurcations. . . . . . . . . . . . . . . . . . . . . . . . 231 6.2 SimpleEquilibriumBifurcations. . . . . . . . . . . . . . . . . . . . . . . . 248 6.2.1 AppearingBifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . 248 6.2.2 SwitchingBifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.2.3 SwitchingandAppearingBifurcations. . . . . . . . . . .. . . . 260 6.3 HigherOrderEquilibriumBifurcations. . . . . . . . . . . . . . . . . . . . 265 6.3.1 AppearingBifurcations. . . . . . . . . . . . . . . . . . . . . . . . . 265 6.3.2 SwitchingBifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . 276 6.3.3 AppearingandSwitchingBifurcations. . . . . . . . . . .. . . . 281 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 7 (2m+1)th-DegreePolynomialSystems. . . . . . . . . . . . . . . . . . . . . . . . 289 7.1 GlobalStabilityandBifurcations. . . . . . . . . . . . . . . . . . . . . . . . 289 7.2 SimpleEquilibriumBifurcations. . . . . . . . . . . . . . . . . . . . . . . . 306 7.2.1 AppearingBifurcations. . . . . . . . . . . . . . . . . . . . . . . . . 306 7.2.2 SwitchingBifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . 319 7.2.3 SwitchingandAppearingBifurcations. . . . . . . . . . .. . . . 324 7.3 HigherOrderEquilibriumBifurcations. . . . . . . . . . . . . . . . . . . . 331 7.3.1 HigherOrderEquilibriumBifurcations. . . . . . . . . . . . . . 331 7.3.2 SwitchingBifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . 351 7.3.3 SwitchingandAppearingBifurcations. . . . . . . . . . .. . . . 356 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 8 Infinite-EquilibriumSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 8.1 EquilibriumComputations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 8.2 NormalForms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 8.3 Infinite-EquilibriumSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . 385 8.3.1 One-Infinite-EquilibriumSystems. . . . . . . . . . . . . . . . . . 386 8.3.2 Two-Infinite-EquilibriumSystems. . . . . . . . . . . . . . . . . . 388 8.3.3 HigherOrderInfinite-EquilibriumSystems. . . . . . . . . . . 393

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