Table Of ContentSymmetries and Semi-invariants in the Analysis
of Nonlinear Systems
Laura Menini (cid:2) Antonio Tornambè
Symmetries
and Semi-invariants
in the Analysis
of Nonlinear Systems
Prof.LauraMenini Prof.AntonioTornambè
Dipto.InformaticaSistemieProduzione Dipto.InformaticaSistemieProduzione
UniversitàdiRoma-TorVergata UniversitàdiRoma-TorVergata
ViadelPolitecnico1 ViadelPolitecnico1
00133Rome 00133Rome
Italy Italy
[email protected] [email protected]
ISBN978-0-85729-611-5 e-ISBN978-0-85729-612-2
DOI10.1007/978-0-85729-612-2
SpringerLondonDordrechtHeidelbergNewYork
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Preface
The goal of this book is to present several concepts useful for the analysis of dy-
namicalsystems,andtoillustrate,inthelasttwochapters,howtheycanbeactually
appliedtoimprovethestateoftheartfortwoclassicaltopicsinnonlinearsystems
theory:thelinearizationofanonlinearsystembystateimmersionandthestudyof
stabilityofequilibriumpoints.
The main reasoning that led us to writing this book is that some concepts that
are already well developed in the literature become more important if presented
together.Threeofsuchconceptsarehomogeneity,symmetries(andorbitalsymme-
triesforcontinuous-timesystems)andLiealgebras,which,inouropinion,canbe
better understood if symmetries are seen as a generalization of homogeneity, and
Lie algebras (seen as generators of Lie groups) as a generalization of symmetries.
Anotherverywellknownconceptisthatoffirstintegral,thatisparticularlyhelpful
for researchers working on Hamiltonian systems, or on stability of switched sys-
tems. In our opinion, similar attention should be paid to the generalization of first
integralsrepresentedbysemi-invariants,which,inturn,haveaspecialrelation,that
willbeexploredinthebook,withorbitalsymmetries.
Nonlinear systems theory was traditionally developed for continuous-time sys-
tems, i.e., systems of ordinary differential equations. Only most recently, with the
growthofthe“digitalworld”,theattentionofmanyresearchersisconcentratedon
discrete-timesystems, i.e., systems of difference equations.For linear systems the
similaritybetweencontinuous-timeanddiscrete-timesystemsisnowadayswellun-
derstood and, with some important exceptions, the study of both kinds of systems
canbeactuallyperformedinparallel,obtainingverysimilarresults.Sincethisisnot
sotruefornonlinearsystems,inthisbookwehavemadeaspecialefforttoextend
someoftheconceptsthatarestandardandwellknownforcontinuous-timesystems
to discrete-time ones; in some cases, we report some results, already existing for
discrete-timesystems,butnotsowellknowninthecontrolliterature,thatturnout
tobetheanalogousofwellknownresultsincontinuous-time.
Wehavetriedtobeself-containedasmuchaspossible,andsometimeswehave
reportednotonlythestatements,butalsotheproofsofsomeverystandardresults,
for two reasons: first because we would like the book to reach a wider audience,
v
vi Preface
secondly because such derivations are often very similar to those that are needed
todevelopthelessstandardtopics.Mostofthematerialinthefirstsixchaptersof
thebookisnotnew,but,togetherwithsomenewresults,wesometimesproposean
alternativederivation of some known result that we considermore useful to better
understandthetopicoritsrelationshipwithotherresultspresentedearlier.
Finally,wewouldliketoapologizefortheinevitableerrorsandomissions,espe-
ciallyingivingcreditfortheresultspresentedinthebook.
Rome,Italy LauraMenini
AntonioTornambè
Contents
1 NotationandBackground . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 AnalyticandMeromorphicFunctions . . . . . . . . . . . . . . . . 2
1.3 DifferentialandDifferenceEquations . . . . . . . . . . . . . . . . 4
1.4 DifferentialForms . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 TheCauchy–KovalevskayaTheorem . . . . . . . . . . . . . . . . 19
1.6 TheFrobeniusTheorem . . . . . . . . . . . . . . . . . . . . . . . 20
1.7 Semi-simple,NormalandNilpotentSquareMatrices . . . . . . . . 25
2 AnalysisofLinearSystems . . . . . . . . . . . . . . . . . . . . . . . 29
2.1 TheLinearCentralizerandLinearNormalizerofaSquareMatrix . 29
2.2 DarbouxPolynomialsandFirstIntegrals . . . . . . . . . . . . . . 45
3 AnalysisofContinuous-TimeNonlinearSystems . . . . . . . . . . . 55
3.1 Semi-invariantsandDarbouxPolynomialsofContinuous-Time
NonlinearSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Symmetries and Orbital Symmetries of Continuous-Time
NonlinearSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Continuous-TimeHomogeneousNonlinearSystems . . . . . . . . 70
3.4 Characteristic Solutions of Continuous-Time Homogeneous
NonlinearSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5 ReductionofContinuous-TimeNonlinearSystems . . . . . . . . . 82
3.6 Continuous-TimeNonlinearPlanarSystems . . . . . . . . . . . . 84
3.7 ParameterizationofContinuous-TimeNonlinearPlanarSystems
HavingaGivenOrbitalSymmetry . . . . . . . . . . . . . . . . . 90
3.8 TheInverseJacobiLastMultiplier . . . . . . . . . . . . . . . . . 94
3.9 MatrixIntegratingFactors . . . . . . . . . . . . . . . . . . . . . . 98
3.10 LaxPairsforContinuous-TimeNonlinearSystems . . . . . . . . . 100
3.11 A “Computational” Result for the Darboux Polynomials
ofContinuous-TimeNonlinearSystems . . . . . . . . . . . . . . . 107
3.12 ThePoincaré–DulacNormalFormofContinuous-TimeNonlinear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
vii
viii Contents
3.13 HomogeneityandResonanceofContinuous-TimeNonlinear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.14 TheBelitskiiNormalFormofContinuous-TimeNonlinear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.15 NonlinearTransformationsofLinearSystems . . . . . . . . . . . 135
3.16 InvariantDistributionsandDualSemi-Invariants . . . . . . . . . . 137
3.17 DecompositionofContinuous-TimeNonlinearSystems . . . . . . 140
3.18 SymmetriesofAlgebraicEquations . . . . . . . . . . . . . . . . . 143
3.19 SymmetriesandDimensionalAnalysis . . . . . . . . . . . . . . . 144
3.20 SymmetriesofScalarOrdinaryDifferentialEquations . . . . . . . 146
4 AnalysisofDiscrete-TimeNonlinearSystems . . . . . . . . . . . . . 153
4.1 Semi-invariants and Darboux Polynomials of Discrete-Time
NonlinearSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.2 A “Computational” Result for the Darboux Polynomials
ofDiscrete-TimeNonlinearSystems . . . . . . . . . . . . . . . . 154
4.3 SymmetriesofDiscrete-TimeNonlinearSystems . . . . . . . . . . 158
4.4 SymmetriesofScalarDiscrete-TimeNonlinearSystems . . . . . . 161
4.5 ReductionofDiscrete-TimeNonlinearSystems . . . . . . . . . . 165
4.6 APropertyofDiscrete-TimeNonlinearPlanarSystems . . . . . . 166
4.7 LaxPairsforDiscrete-TimeNonlinearSystems . . . . . . . . . . 168
4.8 ThePoincaré–DulacNormalFormforDiscrete-TimeNonlinear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.9 LinearizationofDiscrete-TimeNonlinearSystems . . . . . . . . . 178
4.10 HomogeneityandResonanceofDiscrete-TimeNonlinear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.11 TheBelitskiiNormalFormofDiscrete-TimeNonlinearSystems . 182
4.12 DecompositionofDiscrete-TimeNonlinearSystems . . . . . . . . 184
5 AnalysisofHamiltonianSystems . . . . . . . . . . . . . . . . . . . . 187
5.1 Euler–LagrangeEquations . . . . . . . . . . . . . . . . . . . . . . 187
5.2 HamiltonianSystems . . . . . . . . . . . . . . . . . . . . . . . . 189
5.3 NormalFormsofHamiltonianSystems . . . . . . . . . . . . . . . 202
5.4 HamiltonianPlanarSystems . . . . . . . . . . . . . . . . . . . . . 207
5.5 SystemsHavinganInverseJacobiLastMultiplierEqualto1 . . . 214
6 LieAlgebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.1 AbstractLieAlgebras . . . . . . . . . . . . . . . . . . . . . . . . 221
6.2 LieAlgebrasofMatrices . . . . . . . . . . . . . . . . . . . . . . 224
6.3 LieAlgebrasofVectorFunctions . . . . . . . . . . . . . . . . . . 226
6.4 RepresentationofLieAlgebrasbyVectorFunctions . . . . . . . . 229
6.5 NonlinearSuperposition . . . . . . . . . . . . . . . . . . . . . . . 231
6.6 NonlinearSuperpositionFormulasforSolvableLieAlgebras . . . 242
6.6.1 Two-DimensionalLieAlgebras . . . . . . . . . . . . . . . 244
6.7 DarbouxPolynomialsofaLieAlgebra . . . . . . . . . . . . . . . 247
6.8 TheJointPoincaré–DulacNormalForm . . . . . . . . . . . . . . 252
6.9 TheExponentialNotation . . . . . . . . . . . . . . . . . . . . . . 254
Contents ix
6.10 TheWei–NormanEquations . . . . . . . . . . . . . . . . . . . . . 262
6.11 CommutationRules . . . . . . . . . . . . . . . . . . . . . . . . . 267
7 LinearizationbyStateImmersion . . . . . . . . . . . . . . . . . . . . 275
7.1 SufficientConditionsfortheExistenceofaLinearizingState
Immersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
7.1.1 Linearization of Continuous-Time Systems by State
Immersion . . . . . . . . . . . . . . . . . . . . . . . . . . 276
7.1.2 LinearizationofDiscrete-TimeSystemsbyState
Immersion . . . . . . . . . . . . . . . . . . . . . . . . . . 280
7.2 ComputationoftheFlowbyStateImmersion . . . . . . . . . . . . 281
7.3 Computation of a Linearizing Diffeomorphism by Using
Semi-invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
7.4 LinearizationofHamiltonianPlanarSystems . . . . . . . . . . . . 285
7.5 LinearizationofHigherOrderHamiltonianSystems . . . . . . . . 290
8 StabilityAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
8.2 ScalarNonlinearSystems . . . . . . . . . . . . . . . . . . . . . . 297
8.3 Semi-invariantsandCenterManifoldforPlanarSystems . . . . . . 300
8.4 StabilityofContinuous-TimeCriticalPlanarSystems . . . . . . . 303
8.4.1 LinearPartwithImaginaryEigenvalues. . . . . . . . . . . 304
8.4.2 A Simple Proof of a Bendixson Result for Planar
Continuous-TimeSystems . . . . . . . . . . . . . . . . . . 304
8.4.3 StabilityAnalysisfor Planar Systemsin the Belitskii
NormalForm . . . . . . . . . . . . . . . . . . . . . . . . 309
8.5 Construction of Lyapunov Functions Through Darboux
PolynomialsforLinearSystems . . . . . . . . . . . . . . . . . . . 316
8.6 Construction of Lyapunov Functions Through Darboux
PolynomialsforNonlinearSystems . . . . . . . . . . . . . . . . . 319
8.7 ExamplesofConstructionofLyapunovFunctions . . . . . . . . . 324
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
