Bifurcations in Continuous Piecewise Linear Differential Systems: Applications to Low-Dimensional Electronic Oscillators

317 Pages·2022·5.805 MB·English

by  Enrique Ponce, Javier Ros, Elísabet Vela

RSME Springer Series 7
Enrique Ponce
Javier Ros
Elísabet Vela
Bifurcations
in Continuous
Piecewise Linear
Differential Systems
Applications to Low-Dimensional
Electronic Oscillators
RSME Springer Series
Volume 7
Editor-in-Chief
MariaA.HernándezCifre,DepartamentodeMatemáticas,UniversidaddeMurcia,
Murcia,Spain
SeriesEditors
Nicolas Andruskiewitsch, FaMAF - CIEM (CONICET), Universidad Nacionalde
Córdoba,Córdoba,Argentina
Francisco Marcellán, Departamento de Matemáticas, Universidad Carlos III de
Madrid,Leganés,Madrid,Spain
Pablo Mira, Departamento de Matematica Aplicada y Estadistica, Universidad
PolitécnicadeCartagena,Cartagena,Spain
TimothyG.Myers,CentredeRecercaMatemàtica,Barcelona,Spain
Joaquín Pérez, Departamento de Geometría y Topología, University of Granada,
Granada,Spain
MartaSanz-Solé,DepartmentofMathematicsandInformatics,BarcelonaGraduate
SchoolofMathematics(BGSMath),UniversitatdeBarcelona,Barcelona,Spain
KarlSchwede,DepartmentofMathematics,UniversityofUtah,SaltLakeCity,UT,
USA
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Enrique Ponce (cid:129) Javier Ros (cid:129) Elísabet Vela
Bifurcations in Continuous
Piecewise Linear Differential
Systems
Applications to Low-Dimensional Electronic
Oscillators
EnriquePonce JavierRos
DepartamentodeMatemáticaAplicadaII DepartamentodeMatemáticaAplicadaII
UniversidaddeSevilla UniversidaddeSevilla
Sevilla,Spain Sevilla,Spain
ElísabetVela
DepartamentodeMatemáticaAplicadaII
UniversidaddeSevilla
Sevilla,Spain
ISSN2509-8888 ISSN2509-8896 (electronic)
RSMESpringerSeries
ISBN978-3-031-21134-8 ISBN978-3-031-21135-5 (eBook)
https://doi.org/10.1007/978-3-031-21135-5
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To ourfamilies,
andspeciallyto ourchildren:
Emma,Álvaro,Javier,andJuan.
Preface
Dynamicalsystemsarerelevantwheneveronewantstodescribeevolutionproblems
with respectto time, usuallygivenbyordinaryor partialdifferentialequations,or
bytheapplicationofdiscretetransformations.Typically,itisveryusefultoconsider
suchevolutionnotonlyinthephasespacebutalsowhilestudyingthedependence
ofthesystemwithrespecttoparameters,whichleadstotheconceptofbifurcations.
Regardingdynamicalsystemsgovernedbyordinarydifferentialequationsandtheir
bifurcations, we reccommend the monographs[58, 65, 78, 116, 122] as excellent
generalreferences.
Within the realm of nonlinear dynamical systems, piecewise-linear differential
systems(PWLsystems,forshort)constituteaninterestingclassfromseveralpoints
of view. First, they naturally appear in realistic nonlinear engineering models, as
certaindevicesareaccuratelymodeledbypiecewiselinearvectorfields,see[32].In
fact,thesekindsofmodelsarefrequentinapplicationstakenfromsimpleelectronic
oscillators (to be the main field chosen in this book for illustrating the theoretical
results) andnonlinearcontrolsystems, wherepiecewise linear systems turnoutto
be veryaccurate models. They are used in mathematicalbiologyas well, see [27,
124–126],wheretheyconstituteroughapproximatemodels.Thus,theyrepresentan
interestingyetelementarysubclassofpiecewise-smoothdynamicalsystems.
Furthermore, since non-smooth piecewise linear characteristics can be consid-
eredastheuniformlimitofsmoothnonlinearities,the globaldynamicsofsmooth
models has been sometimes approximated by piecewise linear models and vice
versa,obtainingagoodqualitativeagreementbetweenbothmodellingapproaches,
see [84, 129]. Note that, in practice, any nonlinear characteristic usually exhibits
a saturated part, which is difficult to be approximated by polynomial models.
Therefore,thispossibilityofwhatwecouldcallgloballinearizationbymeansofa
finitenumberoflinearpiecesemphasizestheusefulnessofPWLsystems,frequently
being the most natural extension of linear systems in order to capture nonlinear
phenomena.
In fact, it is a widely extended feeling among researchers in the field that the
richness of dynamical behaviour to be found in piecewise linear systems covers
vii
viii Preface
almostalltheinstancesofdynamicsexhibitedbygeneralsmoothnonlinearsystems:
limitcycles,homoclinicandheteroclinicorbits,strangeattractors...
The considerationofthis classasan alternativeto smoothnonlinearsystemsis
gainingpopularityduetothefactthatsolutionscanbewritteninclosedformwhen
theyarerestrictedtoaregionofthephasespacewherethesystembecomeslinear.
Nevertheless,theanalysisoftheglobaldynamicsisfarfrombeingtrivialsinceone
mustmatchthedifferentsolutionsineverylinearityregion.Suchmatchingtypically
requiresthe explicit knowingof differentflight times, that is, the times employed
by the orbit in each linearity zone, which is true only by exception. On the other
hand,standardfamiliesofPWLsystemshaveanon-smallnumberofparameters,so
thatthecompleteanalysisofpossibledynamicalbehavioursisusuallyaformidable
task.Inthissense,thedisposalofgoodcanonicalformsisapreliminaryaimofgreat
relevance,asitwillbeevidentthroughoutthisbook.
As far as we know, the pioneering investigation of piecewise-linear systems
in a rigorous way is due to A. Andronov and coworkers. Their book Theory of
Oscillations [4] continues to remain an obligatory reference, still being a source
ofideas. Precisely,fromthe readingofsuch a booktherearose theinspirationfor
the first works of our group about PWL systems, see [43] and [44]. The analysis
ofpiecewise-linearsystemsalsoreceivedgrowingattentionaftertheworkonPWL
chaoticsystems,see[97]andreferencestherein.Forexample,theso-calledclosing
equationsmethodusedalongthebookwasdevisedbyAndronovandwasexploited
by Kriegsmann[75] in the contextof limit cycle bifurcations.This authorstudied
therapidbifurcationintheWienbridgeoscillator,laterrevisitedin[44].
Readers should be aware that the scope of this book is restricted almost
exclusivelytoPWLsystemswhosevectorfieldsarecontinuousandnotdependent
ontime.Forthoseinterestedin thewiderclassof discontinuous,non-autonomous
orevenhybridPWLsystems,whichisalsoaveryactivefieldofresearch,werefer
toreferences[2,32,40,94,95,130].Althoughnotstrictlyneeded,someconcepts
fromthetheoryofdiscontinuoussystemscouldapplytocontinuousones,see[96].
Althoughwepayattentionmainlytothecontinuouscase,thelackofdifferentia-
bility of PWL systems precludes the standard application of the powerful results
from the modern geometric theory of differentiable dynamical systems, see for
instance the celebrated books [58] and [78]. Apparently simple tasks sometimes
becomeintricateproblems,asforinstancethedeterminationofthetopologicaltype
ofanequilibriumpoint,see [15].Thismakesthedevelopmentofa generaltheory
forPWLsystemsanimpressivelylargepuzzle.Onemustproceedviacasestudies,
tryingnottoneglectanyparticularcase.Itisinthiscontextthatthismonographhas
been written: looking for filling in the remaining empty shelves of PWL systems
theory.
ToclassifyPWLsystems,weconsiderthenumberofdifferentlinearityregions,
that is regions where the restriction of the vector field becomes a linear function.
Sometimes, we speak of linear zones instead, with the same meaning. Another
featuretobeconsideredistheexistenceofpossiblesymmetriesforthevectorfield.
Thus,wesaythatthedifferentialsystemx˙ = f(x)issymmetricwithrespecttothe
originiff(−x)=−f(x)forallx∈Rn.
Preface ix
Nowadays, the family of planar, continuous PWL systems (CPWL , for short)
2
seems to be well understood, at least for some frequent subfamilies, as are the
systems with only two zones (2CPWL systems) or with three zones but having
2
symmetry with respect to the origin, to be denoted as S3CPWL systems. Other
2
problemshowever,asisthedeterminationofthemaximumnumberoflimitcycles
in planardiscontinuousPWL systems with onlytwo zones, still are the subjectof
intensive,contemporaryresearch.
Actually, some results for general 2CPWL systems were obtained after the
2
cumbersome consideration of all the possible cases, one-by-one, through the
detailed study of properties of different half-return maps. A paradigmatic case of
this,relatedtotheuniquenessoflimitcycles,isthesocalledLum–Chuaconjecture.
In the article [93] by R. Lum and L. O. Chua, there appeared the following
conjecture:
Conjecture (Lum–Chua).Acontinuouspiecewiselinearvectorfieldwithoneboundary
hasatmostonelimitcycle.Thelimitcycle,ifitexists,iseitherattractingorrepelling.
The Lum–Chua conjecture was shown to be true after the long study made
in [43]. A natural question arose: is there a shorter way to arrive at the same
conclusion? Thanks to some results that take advantage of Massera’s geometric
method, we report how to give a positive answer to such a question, see Chap.3.
Nonetheless, readers interested in such a subject should be aware that during the
revisionprocessofthisbookanewpromisingapproachhasbeenproposedin[17].
As mentioned, for planar continuous systems with three linear zones having
symmetrywithrespecttotheorigin,theirdynamicalcomprehensionisacceptable.
Related references for these S3CPWL vector fields are the work [45] and the
2
thorough analysis made in the PhD thesis of A. Teruel, see [123], following a
differentapproach.Infact,thelaterstudygaverisetothebook[86].
However, for quasi-symmetric 3CPWL vector fields (i.e., symmetric systems
2
shiftedbyaddingjustsomeconstanttoeachcomponent)orgeneralnon-symmetric
systems, there are few results available related to the existence and uniqueness
of limit cycles, a situation we want to amend in this book by resorting again
to the quoted Massera’s geometric method and showing also some outstanding
applications,seeChap.4.
Muchhasbeendoneinthree-dimensionalPWLsystems,bothinthefamiliesof
2CPWL andS3CPWL systemsinthelastdecades,thankstotheaforementioned
3 3
theses by V. Carmona and J. Ros first, see [10, 118], and by E. García-Medina
andS.Fernández-Garcíalater,see[38,54],leadingtoavarietyofpapers.Without
enteringintotheintricateworldofchaoticdynamics,butlookingfornewwaysto
moveaheadtowardsthechaoticfrontier,wewanttostudypartialunfoldingsofthe
analogous to Hopf-pitchfork bifurcations in PWL systems. A particular objective
here is to determine how many limit cycles can bifurcate from such a critical
situation, where three eigenvalues (a real and a complex pair of eigenvalues for
thelinearmatrixassociatedtotheregioncontainingtheinvolvedequilibriumpoint)
aresimultaneouslylocatedattheimaginaryaxisofthecomplexplane.
x Preface
Summarizing,thisbookisorganizedasfollows.InPartI,aftersomeintroductory
chapterto emphasizethe differencesbetweenlinearandpiecewise linearsystems,
we review some terminology and results related to canonical forms in the study
of PWL systems along with certain techniques that are useful for the bifurcation
analysisoftheirperiodicorbits,seeChap.2.First,wedevelopathreezonesLiénard
canonical form able to represent also systems with two zones, to facilitate the
subsequent study on existence and uniqueness of limit cycles. Next, we work in
arbitrarydimensiontoreviewgeneralresultsalthoughwewilllaterdealonlywith
systemsindimensions2and3.
Part II is completely devoted to planar PWL systems. We exploit and extend
recentresultsachievedin[91],whichallowsustopavethewayforashorterproof
of the Lum–Chua conjecture. We also give all the details about the focus-center-
limitcyclebifurcation(FCLCbifurcation,forshort)inplanarsystemswithonlytwo
linearzones,seeChap.3.Suchabifurcationwillbestudiedinothercontextsalong
the book,since it is the analoguefor the PWL setting to the Poincaré–Andronov–
Hopfbifurcationforsmoothsystems.
Other general results for existence and uniqueness of limit cycles in 3CPWL
2
systems,thatisplanarsystemswiththreelinearzones,appearinChap.4,alongwith
thesymmetricversionoftheFCLCbifurcation,justmentioned.Othermechanisms
able to generate limit cycles, as the boundary equilibrium bifurcations (BEB, for
short), are explored in Chap.5. As a consequence, cases with two limit cycles
surrounding the only equilibrium point are detected. At this point, we are in a
position to apply the developed theory in basic realistic circuits coming from
nonlinear electronics, by analyzing the bifurcation set of quasi-symmetric Wien
bridgeoscillators.
In a different direction of research, in Chap.6, a family of algebraically com-
putable piecewise linear nodal oscillators is introduced, and some real electronic
devicesthatbelongtothefamilyareshown.Theoutstandingfeatureofthisfamily
makesitanexceptionalbenchmarkfortestingapproximatemethodsofanalysisof
oscillators.
Another contributionincluded in this part is the study of a specific bifurcation
thatcanonlyappearinPWLsystems,asisthefocus-saddlebifurcation,seeChap.7,
whichcanalsoinvolvetheappearanceofperiodicorbitsingenericconfigurationof
parameters,andhomoclinicconnectionsinsomenon-genericcases.
Part III represents a particular incursion in PWL systems of dimension three,
basically by studying the FCLC bifurcation and its possible degenerationin three
differentcontexts.Inthissense,someresultsinvolving2CPWL vectorfields,that
3
is in PWL systems with only two zones, are offered in Chap.8. These results are
alsoofinterestinsystemswiththreelinearzones,andweshowthisbyanalysingthe
celebratedChua’scircuit,byconsideringtwoadjacentzoneswheretheoscillations
takeplace.ThesymmetriccasefortheresultsofChap.8istackledinChap.9.
We want to emphasize that there is much to be done in 3D systems, since for
instance,nowadays,theirboundaryequilibriumbifurcationsarenotcompletelywell
understood.Pursuingtheaimoffillinginthecatalogofpossiblebifurcationsleading
to limit cycles, we study some unfoldings of the analogous to Hopf-pitchfork