Table Of ContentNonlinear 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
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fundamental advances and new applications inthe field. Topical areas include, but
arenotlimitedto:Nonlineardynamics;Complexity,nonlinearity,andchaos;Com-
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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
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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
