Topological analysis of dynamical systems

75 Pages·1998·1.2 MB·English

by  Gilmore R.

Topological analysis of chaotic dynamical systems
Robert Gilmore
DepartmentofPhysics&AtmosphericScience,DrexelUniversity,Philadelphia,
Pennsylvania19104
Topologicalmethodshaverecentlybeendevelopedfortheanalysisofdissipativedynamicalsystems
thatoperateinthechaoticregime.Theywereoriginallydevelopedforthree-dimensionaldissipative
dynamical systems, but they are applicable to all ‘‘low-dimensional’’ dynamical systems. These are
systems for which the flow rapidly relaxes to a three-dimensional subspace of phase space.
Equivalently, the associated attractor has Lyapunov dimension d ,3. Topological methods
L
supplement methods previously developed to determine the values of metric and dynamical
invariants. However, topological methods possess three additional features: they describe how to
modelthedynamics;theyallowvalidationofthemodelssodeveloped;andthetopologicalinvariants
arerobustunderchangesincontrol-parametervalues.Thetopological-analysisproceduredependson
identifyingthestretchingandsqueezingmechanismsthatacttocreateastrangeattractorandorganize
all the unstable periodic orbits in this attractor in a unique way. The stretching and squeezing
mechanismsarerepresentedbyacaricature,abranchedmanifold,whichisalsocalledatemplateor
aknotholder.Thisturnsouttobeaversionofthedynamicalsysteminthelimitofinfinitedissipation.
This topological structure is identified by a set of integer invariants. One of the truly remarkable
resultsofthetopological-analysisprocedureisthattheseintegerinvariantscanbeextractedfroma
chaotictimeseries.Furthermore,self-consistencycheckscanbeusedtoconfirmtheintegervalues.
These integers can be used to determine whether or not two dynamical systems are equivalent; in
particular, they can determine whether a model developed from time-series data is an accurate
representationofaphysicalsystem.Conversely,theseintegerscanbeusedtoprovideamodelforthe
dynamical mechanisms that generate chaotic data. In fact, the author has constructed a doubly
discreteclassificationofstrangeattractors.Theunderlyingbranchedmanifoldprovidesonediscrete
classification.Eachbranchedmanifoldhasan‘‘unfolding’’orperturbationinwhichsomesubsetof
orbitsisremoved.Theremainingorbitsaredeterminedbyabasissetoforbitsthatforcesthepresence
of all remaining orbits. Branched manifolds and basis sets of orbits provide this doubly discrete
classification of strange attractors. In this review the author describes the steps that have been
developedtoimplementthetopological-analysisprocedure.Inaddition,theauthorillustrateshowto
apply this procedure by carrying out the analysis of several experimental data sets. The results
obtainedforseveralotherexperimentaltimeseriesthatexhibitchaoticbehaviorarealsodescribed.
[S0034-6861(98)00304-3]
CONTENTS 3.Duffingdynamics 1475
4.vanderPol–Shawdynamics 1476
5.Cuspcatastrophedynamics 1476
I. Introduction 1456
V. InvariantsfromTemplates 1477
A. Laserwithmodulatedlosses 1456
A. Locatingperiodicorbits 1477
B. Objectivesofanewanalysisprocedure 1459
B. Topologicalinvariants 1478
C. Previewofresults 1460
1.Linkingnumbers 1478
II. Preliminaries 1460
2.Relativerotationrates 1478
A. Somebasicresults 1461
B. Changeofvariables 1462 C. Dynamicalinvariants 1479
1.Differentialcoordinates 1462 D. Inflatingatemplate 1480
2.Delaycoordinates 1462 VI. UnfoldingaTemplate 1480
C. Qualitativeproperties 1463 A. Topologicalrestrictions 1481
1.Poincare´ program 1463 B. Forcingdiagram 1482
2.Stretchingandsqueezing 1463 1.Zero-entropyorbits 1483
D. Theproblem 1463 2.Positive-entropyorbits 1484
III. TopologicalInvariants 1464 C. Basissetsoforbits 1484
A. Linkingnumbers 1464 D. Routestochaos 1485
B. Relativerotationrates 1465 E. Coexistingbasins 1485
C. Knotholdersortemplates 1467 F. Othertemplateunfoldings 1485
IV. TemplatesasFlowModels 1468 VII. Topological-AnalysisAlgorithm 1486
A. TheBirman-WilliamstheoreminR3 1468 A. Constructanembedding 1486
B. TheBirman-WilliamstheoreminRn 1469 B. Identifyperiodicorbits 1486
C. Templates 1470 C. Computetopologicalinvariants 1487
D. Algebraicdescriptionoftemplates 1471 D. Identifyatemplate 1487
E. Control-parametervariation 1472 E. Validatethetemplate 1487
F. Examplesoftemplates 1474 F. Modelthedynamics 1488
1.Ro¨sslerdynamics 1474 G. Validatethemodel 1488
2.Lorenzdynamics 1474 VIII. Data 1488
ReviewsofModernPhysics,Vol.70,No.4,October1998 0034-6861/98/70(4)/1455(75)/$30.00 ©1998TheAmericanPhysicalSociety 1455
1456 RobertGilmore: Topologicalanalysisofchaoticdynamicalsystems
A. Datarequirements 1489 XIV. Conclusions 1524
1.;100cycles 1489 Acknowledgments 1526
2.;100samples/cycle 1489 References 1526
B. Fastlookatdata 1489
C. Processinginthefrequencydomain 1489
1.High-frequencyfilter 1489 I. INTRODUCTION
2.Low-frequencyfilter 1489
3.Derivativesandintegrals 1490 The subject of this review is the analysis of data gen-
4.Hilberttransform 1490 erated by a dynamical system operating in a chaotic re-
5.Fourierinterpolation 1491
gime.Morespecifically,thisreviewdescribeshowtoex-
6.Hilberttransformandinterpolation 1491
tract, from chaotic data, topological invariants that
D. Processinginthetimedomain 1492
determine the stretching and squeezing mechanisms re-
1.Singular-valuedecompositionfordatafields 1492
sponsible for generating these chaotic data.
2.Singular-valuedecompositionforscalar
In this introductory section we briefly describe, for
timeseries 1492
IX. UnstablePeriodicOrbits 1493 purposes of motivation, a laser that has been operated
A. Closereturnsinflows 1493 under conditions in which it behaved chaotically (see
1.Close-returnsplot 1493 Sec. I.A). The topological tools that we describe in this
2.Close-returnshistogram 1493 review were developed in response to the challenge of
3.Testsforchaos 1494 analyzingthechaoticdatasetsgeneratedbythislaser.In
B. Closereturnsinmaps 1494 Sec.I.Bwelistanumberofquestionsthatwewanttobe
1.First-returnplot 1494
able to answer when analyzing a chaotic signal. None of
2.pth-returnplot 1494
these questions can be addressed by the older tools for
C. Metricmethods 1494
analyzingchaoticdata,whichincludedimensioncalcula-
X. Embedding 1496
tions and estimates of Lyapunov exponents. In Sec. I.C
A. Time-delayembedding 1496
B. Differentialphase-spaceembedding 1497 we preview the results that will be presented during the
1.x,x˙,x¨ 1497 course of this review. It is astonishing that the
2. *x,x,x˙ 1497 topological-analysis tools that we shall describe have
C. Embeddingswithsymmetry 1498 provided answers to more questions than we had origi-
D. Coupled-oscillatorembeddings 1498 nally asked. This analysis procedure has also raised
E. Singular-valuedecompositionembeddings 1499 more questions than we have answered in this review.
F. Singular-valuedecompositionprojections 1499
XI. HorseshoeMechanism(A2) 1499 A. Laserwithmodulatedlosses
A. Belousov-Zhabotinskiireaction 1500
1.Embedding 1500 The possibility of observing deterministic chaos in la-
2.Periodicorbits 1500
serswasoriginallydemonstratedbyArecchietal.(1982)
3.Templateidentification 1501
andGioggiaandAbraham(1983).Theuseoflasersasa
4.Templateverification 1502
testbed for generating deterministic chaotic signals has
5.Basissetoforbits 1502
two major advantages over fluid systems, which had un-
6.Modelingthedynamics 1503
til that time been the principle source for chaotic data:
7.Modelvalidation 1505
B. Laserwithsaturableabsorber 1506 (i) The time scales intrinsic to a laser (1027 to
C. Laserwithmodulatedlosses 1506 1023 sec) are much shorter than the time scales
1.Poincare´ sectionmappings 1506
for fluid experiments.
2.ProjectiontoaPoincare´ section 1507
(ii) Reliable laser models exist in terms of a small
3.Result 1508
D. OthersystemsexhibitingA dynamics 1508 number of ordinary differential equations whose
2
E. ‘‘Invariant’’versus‘‘robust’’ 1508 solutions show close qualitative similarity to the
F. WhyA ? 1510 behavior of the lasers that are modeled (Puccioni
2
XII. LorenzMechanism(A ) 1511 etal., 1985; Tredicce etal., 1986).
3
A. Opticallypumpedmolecularlaser 1511
We originally studied in detail the laser with modu-
1.Models 1511
2.Amplitudes 1512 lated losses. A schematic of this laser is shown in Fig. 1.
3.Intensities 1515 AKerrcellisplacedwithinthecavityofaCO gaslaser.
2
B. Fluids 1515 The electric field within the cavity is polarized by Brew-
C. Inducedattractorsandtemplates 1516 ster angle windows. The Kerr cell allows linearly polar-
D. WhyA3? 1517 ized light to pass through it. An electric field across the
XIII. DuffingOscillator 1517 Kerrcellrotatestheplaneofpolarization.Asthepolar-
A. Background 1517
ization plane of the Kerr cell is rotated away from the
B. Flowapproach 1517
polarization plane established by the Brewster angle
C. Template 1518
windows,controllablelossesareintroducedintothecav-
D. Orbitorganization 1520
ity.IftheKerrcellisperiodicallymodulated,theoutput
1.Nonlinearoscillator 1520
2.Duffingtemplate 1522 intensityisalsomodulated.Whenthemodulationampli-
E. Levelsofstructure 1524 tude is small, the output modulation is locked to the
Rev.Mod.Phys.,Vol.70,No.4,October1998
RobertGilmore: Topologicalanalysisofchaoticdynamicalsystems 1457
FIG. 1. Schematic representation of a laser with modulated
losses.CO :lasertubecontainingCO withBrewsterwindows;
2 2
M:mirrorsformingcavity;P.S.:powersource;K:Kerrcell;S:
signal generator; D: detector; C: oscilloscope and recorder. A
variableelectricfieldacrosstheKerrcellvariesitspolarization
directionandmodulatestheelectric-fieldamplitudewithinthe
cavity. FIG. 2. Bifurcation diagram for model (1.2) of the laser with
modulated losses, with e50.03, e50.009, V51.5. Stable pe-
1 2
riodic orbits (solid lines), regular saddles (dashed lines), and
modulation of the Kerr cell. When the modulation am-
strange attractors are shown. Period-n branches (Pn>2) are
plitude is sufficiently large and the modulation fre-
created in saddle-node bifurcations and evolve through the
quencyiscomparabletothecavity-relaxationfrequency,
Feigenbaumperiod-doublingcascadeasthemodulationampli-
or one of its subharmonics, the laser-output intensity no
tudeTincreases.Twoadditionalperiod-5branchesareshown
longerremainslockedtothesignaldrivingtheKerrcell,
aswellasa‘‘snake’’basedontheperiod-threeregularsaddle.
and can even become chaotic. The period-two saddle orbit created in a period-doubling bi-
The laser with modulated losses has been studied ex- furcation from the period-one orbit (T;0.8) is related by a
tensively both experimentally (Arecchi etal., 1982; snaketotheperiod-twosaddleorbitcreatedatP2.
Gioggia and Abraham, 1983; Puccioni etal., 1985;
Tredicce, Abraham etal., 1985; Tredicce, Arecchi etal.,
1985; Midavaine, Dangoisse, and Glorieux, 1986; stableasTisincreaseduntilT;0.8.Itbecomesunstable
Tredicce etal., 1986) and theoretically (Matorin, Pik- at T;0.8, with a stable period-two orbit emerging from
ovskii, and Khanin, 1984; Solari etal., 1987; Solari and it in a period-doubling bifurcation. Contrary to what
Gilmore, 1988). The rate equations governing the laser mightbeexpected,thisisnottheearlystageofaperiod-
intensity S and the population inversion N are doublingcascade,fortheperiod-twoorbitisannihilated
dS/dt52k S@~12N!1m cos~vt!#, at T;0.85 in an inverse saddle-node bifurcation with a
0
period-two regular saddle. This saddle-node bifurcation
dN/dt52g@~N2N0!1~N021!SN#. (1.1) destroys the basin of attraction of the period-two orbit.
Here m and vare the modulation amplitude and angu- Any point in that basin is dumped into the basin of a
lar frequency, respectively, of the Kerr cell; N is the period 452321 orbit, even though there are two other
0
pump parameter, normalized to N 51 at the laser coexistingbasinsofattractionforstableorbitsofperiods
0
threshold; and k and gare loss rates. In scaled form, 653321 and 4.
0
this equation is Subharmonics of period n (Pn,n>2) are created in
saddle-node bifurcations at increasing values of T and S
du/dt5@z2T cos~Vt!#u,
(P2 at T;0.1, P3 at T;0.3, P4 at T;0.7, P5 and
dz/dt5~12ez!2~11ez!u, (1.2) highershownininset).Allsubharmonicsinthisseriesto
1 2
periodn511havebeenseenbothexperimentallyandin
where the scaled variables are u5S, z5k k(N21), t
0 simulations of (1.2). The evolution (‘‘perestroika,’’
5kt, T5k m, V5vk, e5kg, e51/kk , and k2
0 1 2 0 Arnol’d, 1986) of each subharmonic follows a standard
51/gk (N 21). The bifurcation behavior exhibited by
0 0 scenario as T increases (Eschenazi, Solari, and Gilmore,
the simple models (1.1) and (1.2) is qualitatively, if not
1989):
quantitatively,inagreementwiththeexperimentallyob-
served behavior of this laser. (i) A saddle-node bifurcation creates an unstable
A bifurcation diagram for the laser, and the model saddle and a node which is initially stable.
(1.2), is shown in Fig. 2. The bifurcation diagram is con- (ii) Each node becomes unstable and initiates a
structed by varying the modulation amplitude T and period-doubling cascade as T increases. The cas-
keeping all other parameters fixed. This bifurcation dia- cade follows the standard Feigenbaum (1978,
gram is similar to experimentally observed bifurcation 1980) scenario. The ratios of T intervals between
diagrams. successive bifurcations, and of geometric sizes of
This diagram shows that a period-one solution exists the stable nodes of periods n32k, have been es-
abovethelaserthreshold(N .1) forT50 andremains timateduptok<6 forsomeofthesesubharmon-
0
Rev.Mod.Phys.,Vol.70,No.4,October1998
1458 RobertGilmore: Topologicalanalysisofchaoticdynamicalsystems
FIG. 3. Multiple basins of attraction coexisting over a broad
rangeofcontrol-parametervalues.Thestableorbitsorstrange
attractors within these basins have a characteristic organiza-
tion. The coexisting orbits shown above are, from inside to
FIG.4. Timeseriesfromlaserwithmodulatedlossesshowing
outside:periodtwobifurcatedfromperiod-onebranch,period
alternation between noisy period-two and noisy period-three
two, period three, period four. The two inner orbits are sepa- behavior(T;1.3inFig.2).
rated by an unstable period-two orbit (not shown); all three
arepartofa‘‘snake.’’ 32k11orbitcollideswiththeattractortoproduceanoisy
period-halving bifurcation.
External crisis: A regular saddle of period n8 in the
ics, both from experimental data and from the
boundary of a period-n (n8(cid:222)n) strange attractor col-
simulations. These ratios are compatible with the
lideswiththeattractor,therebyannihilatingorenlarging
universal scaling ratios.
the basin of attraction.
(iii) Beyond accumulation, there is a series of noisy
Figure5(a)providesaschematicrepresentationofthe
orbits of period n32k that undergo inverse
bifurcation diagram shown in Fig. 2. The different kinds
period-halving bifurcations. This scenario has
of bifurcations encountered in both experiments and
been predicted by Lorenz (1980).
simulations are indicated here. These include direct and
We have observed additional systematic behavior inverse saddle-node bifurcations, period-doubling bifur-
sharedbythesubharmonicsshowninFig.2.Highersub- cations, and boundary and external crises. As the laser-
harmonics are generally created at larger values of T. operating parameters (k ,g,V) change, the bifurcation
0
They are created with smaller basins of attraction. The diagram changes. In Figs. 5(b) and 5(c) we show sche-
rangeofTvaluesoverwhichtheFeigenbaumscenariois matics of bifurcation diagrams obtained for slightly dif-
played out becomes smaller as the period (n) increases. ferentvaluesoftheseoperating(orcontrol)parameters.
In addition, the subharmonics show an ordered pattern In addition to the subharmonic orbits of periodn cre-
in space. In Fig. 3 we show four stable periodic orbits ated at increasing T values (Fig. 2), there are orbits of
thatcoexistundercertainoperatingconditions.Roughly period n that do not appear to belong to that series of
speaking, the larger-period orbits exist ‘‘outside’’ the subharmonics. The clearest example is the period-two
smaller-period orbits. These orbits share many other orbit, which bifurcates from period one at T;0.8. An-
systematics, which have been described by Eschenazi, other is the period-three orbit pair created in a saddle-
Solari, and Gilmore (1989). node bifurcation, which occurs at T;2.45. These bifur-
In Fig. 4 we show an example of a chaotic time series cations were seen in both experiments and simulations.
takenforT;1.3afterthechaoticattractorbasedonthe We were able to trace the unstable orbits of period two
period-twoorbithascollidedwiththeperiod-threeregu- (0.1,T,0.85) and period three (0.4,T,2.5) in simu-
lar saddle. lationsandfoundthattheseorbitsarecomponentsofan
The period-doubling, accumulation, inverse noisy orbit ‘‘snake’’ (Alligood, 1985; Alligood, Sauer, and
period-halving scenario described above is often inter- Yorke, 1997). This is a single orbit that folds back and
rupted by a crisis (Grebogi, Ott, and Yorke, 1983) of forth on itself in direct and inverse saddle-node bifurca-
one type or another: tions as T increases. The unstable period-two orbit (0.1
Boundary crisis: A regular saddle on a period-n ,T,0.85) is part of a snake. By changing operating
branch in the boundary of a basin of attraction sur- conditions,bothsnakescanbeeliminated[seeFig.5(c)].
rounding either the period-n node or one of its periodic As a result, the ‘‘subharmonic P2’’ is really nothing
or noisy periodic granddaughter orbits collides with the other than the period-two orbit, which bifurcates from
attractor. The basin is annihilated or enlarged. the period-one branch P1. Furthermore, instead of hav-
Internal crisis: A flip saddle of period n32k in the ing saddle-node bifurcations creating four inequivalent
boundary of a basin surrounding a noisy period n period-three orbits (at T;0.4 and T;2.45) there is re-
Rev.Mod.Phys.,Vol.70,No.4,October1998
RobertGilmore: Topologicalanalysisofchaoticdynamicalsystems 1459
which are not. These tools suggested that the Smale
(1967) horseshoe mechanism was responsible for gener-
atingthenonlinearphenomenaobtainedinboththeex-
periments and the simulations. This mechanism predicts
that additional inequivalent subharmonics of period n
can exist for n>5. Since we observed that the size of a
basinofattractiondecreasesrapidlywithn,wesearched
for the two additional saddle-node bifurcations involv-
ing period-five orbits that are allowed by the horseshoe
mechanism. Both were located in simulations. Their lo-
cations are indicated in Fig. 2 at T;0.6 and T;2.45.
One was also located experimentally. The other may
also have been seen, but the basin was too small to be
certain of its existence.
Bifurcationdiagramshavebeenobtainedforavariety
of physical systems: other lasers (Wedding, Gasch, and
Jaeger, 1984; Waldner etal., 1986; Rolda´n etal., 1997);
electric circuits (Bocko, Douglas, and Frutchy, 1984;
Klinker, Meyer-Ilse, and Lauterborn, 1984; Satija,
Bishop, and Fesser, 1985; van Buskirk and Jeffries,
1985); a biological model (Schwartz and Smith, 1983); a
bouncing ball (Tufillaro, Abbott, and Reilly, 1992); and
a stringed instrument (Tufillaro etal., 1995). These bi-
furcation diagrams are similar, but not identical, to the
ones shown above. This raised the question of whether
similar processes were governing the description of this
large variety of systems.
During these analyses, it became clear that standard
tools (dimension calculations and Lyapunov exponent
estimates) were not sufficient for a satisfying under-
standing of the stretching and squeezing processes that
occur in phase space and which are responsible for gen-
erating chaotic behavior. In the laser we found many
coexisting basins of attraction, some containing a peri-
odic attractor, others a strange attractor. The rapid al-
ternationbetweenperiodicandchaoticbehaviorascon-
trolparameters(e.g.,TandV)werechangedmeantthat
dimension and Lyapunov exponents varied at least as
rapidly.
For this reason, we sought to develop additional tools
thatwereinvariantundercontrolparameterchangesfor
theanalysisofdatageneratedbydynamicalsystemsthat
exhibit chaotic behavior.
B. Objectivesofanewanalysisprocedure
FIG. 5. (a) Schematic of bifurcation diagram shown in Fig. 2.
Various bifurcations are indicated: ↓, saddle node; m, inverse
In view of the experiments just described, and the
saddle node; d, boundary crisis; !, external crisis. Period-
data that they generated, we hoped to develop a proce-
doubling bifurcations are identified by a small vertical line
dure for analyzing data that achieved a number of ob-
separatingstableorbitsofperiodsdifferingbyafactoroftwo.
jectives. These included an ability to answer the follow-
Accumulation points are identified by A. Strange attractors
ing questions:
basedonperiodnareindicatedbyCn.Ascontrolparameters
(i) Is it possible to develop a procedure for under-
change, the bifurcation diagram is modified, as in (b) and (c).
Thesequence(a)to(c)showstheunfoldingofthe‘‘snake’’in standing dynamical systems and their evolution (‘‘per-
theperiod-twoorbit. estroika’’) as the operating parameters (e.g., m, k0, and
g) change?
ally only one pair of period-three orbits, the other pair (ii) Is it possible to identify a dynamical system by
being components of a snake. means of topological invariants, following suggestions
Topological tools (relative rotation rates, Solari and proposed by Poincare´ (1892)?
Gilmore, 1988) were developed to determine which or- (iii) Can selection rules be constructed under which it
bits might be equivalent, or components of a snake, and is possible to determine the order in which periodic or-
Rev.Mod.Phys.,Vol.70,No.4,October1998
1460 RobertGilmore: Topologicalanalysisofchaoticdynamicalsystems
bitscanbecreatedand/orannihilatedbystandardbifur- these topological invariants for surrogate orbits to the
cations? Or when different orbits might belong to the topological invariants for corresponding periodic orbits
same snake? on branched manifolds of various types.
(iv) Is it possible to determine when two strange at- (vii) The identification of a branched manifold is con-
tractors are (a) equivalent (one can be transformed into firmed or rejected by using the branched manifold to
theother,bychangingparameters,forexample,without predict topological invariants of additional periodic or-
creating or annihilating any periodic orbits); (b) adia- bitsextractedfromthedataandcomparingthesepredic-
batically equivalent (one can be deformed into the tions with those computed from the surrogate orbits.
other,bychangingparameters,andonlyasmallnumber (viii) Topological constraints derived from the linking
of orbit pairs below any period are created or de- numbersandtherelativerotationratesprovideselection
stroyed); or (c) inequivalent (there is no way to trans- rules for the order in which orbits can be created and
form one into the other)? must be annihilated as control parameters are varied.
(ix) A basis set of orbits can be identified that defines
the spectrum of all unstable periodic orbits in a strange
C. Previewofresults
attractor, up to any period.
(x) The basis set determines the maximum number of
A procedure for analyzing chaotic data has been de-
coexisting basins of attraction that a small perturbation
veloped that addresses many of the questions presented
of the dynamical system can produce.
above. This procedure is based on computing the topo-
(xi)Ascontrolparameterschange,theperiodicorbits
logical invariants of the unstable periodic orbits that oc-
in the dynamical system are determined by a sequence
cur in a strange attractor. These topological invariants
of different basis sets. Each such sequence represents a
are the orbits’ linking numbers and their relative rota-
tion rates. Since these are defined in R3, we originally ‘‘route to chaos.’’
The information described above can be extracted
thought this topological analysis procedure was re-
from time-series data. Experience shows that the data
stricted to the analysis of three-dimensional dissipative
need not be exceptionally clean and the data set need
dynamical systems. However, it is applicable to higher-
not be exceptionally long.
dimensional dynamical systems, provided points in
There is now a doubly discrete classification for
phase space relax sufficiently rapidly to a three-
strange attractors generated by low-dimensional dy-
dimensional manifold contained in the phase space.
namical systems. The gross structure is defined by an
Such systems can have any dimension, but they are
underlying branched manifold. This can be identified by
‘‘strongly contracting’’ and have Lyapunov dimension
(Kaplan and Yorke, 1979) d ,3. a set of integers that is robust under control-parameter
L variation. The fine structure is defined by a basis set of
The results are as follows:
orbits. This basis set changes as control parameters
(i) The stretching and squeezing mechanisms respon-
change.Asequenceofbasissetscanrepresentarouteto
sible for creating a strange attractor and organizing all
chaos. Different sequences represent distinct routes to
unstable periodic orbits in it can be identified by a par-
chaos.
ticular kind of two-dimensional manifold (‘‘branched
manifold’’). This is an attractor that is obtained in the
‘‘infinitedissipation’’limitoftheoriginaldynamicalsys-
tem.
(ii) All such manifolds can be identified and classified II. PRELIMINARIES
by topological indices. These indices are integers.
(iii) Dynamical systems classified by inequivalent A dynamical system is a set of ordinary differential
branched manifolds are inequivalent. They cannot be equations,
deformed into each other. dx
(iv)Inparticular,thefourmostwidelycitedexamples 5x˙5F~x,c!, (2.1)
dt
oflow-dimensionaldynamicalsystemsexhibitingchaotic
behavior [Lorenz equations (Lorenz, 1963), Ro¨ssler wherexPRn andcPRk (Arnol’d,1973;Gilmore,1981).
equations (Ro¨ssler, 1976a), Duffing oscillator (Thomp- Thevariablesxarecalledstatevariables.Theyevolvein
son and Stewart, 1986; Gilmore, 1981), and van der Pol- time in the space Rn, called a state space or a phase
Shaw oscillator (Thompson and Stewart, 1986; Gilmore, space. The variables cPRk are called control param-
1984)]areassociatedwithdifferentbranchedmanifolds, eters. They typically appear in ordinary differential
and are therefore intrinsically inequivalent. equations as parameters with fixed values. In Eq. (1.1)
(v)Thecharacterizationofabranchedmanifoldisun- thevariablesS,N,andtarestatevariablesandthe‘‘con-
changed as the control parameters are varied. stants’’ k , g, v, m, and N are control parameters.
0 0
(vi) The branched manifold is identified by (a) identi- Ordinary differential equations arise quite naturally
fying segments of the time series that can act as surro- to describe a wide variety of physical systems. The sur-
gatesforunstableperiodicorbitsbythemethodofclose veys by Cvitanovic (1984) and Hao (1984) present a
returns; (b) computing the topological invariants (link- broad spectrum of physical systems that are described
ing numbers and relative rotation rates) of these surro- by nonlinear ordinary differential equations of the form
gates for unstable periodic orbits; and (c) comparing (2.1).
Rev.Mod.Phys.,Vol.70,No.4,October1998
RobertGilmore: Topologicalanalysisofchaoticdynamicalsystems 1461
FIG. 6. Smooth transformation that reduces the flow to the
verysimplenormalform(2.4)locallyintheneighborhoodofa
nonsingularpoint.
FIG.7. Orthogonaltransformationthatreducestheflowtothe
A. Somebasicresults
local normal form (2.5) in the neighborhood of a nonsingular
point.
We review a few fundamental results that lie at the
heart of dynamical systems. U U
The existence and uniqueness theorem (Arnol’d, n 1/2
(
1973) states that through any point in phase space there y˙15uF~x0,c!u5 Fk~x0,c!2 ,
k51
is a solution to the differential equations, and that the
solution is unique: y˙ 5l y j52,3,...,n. (2.5)
j j j
x~t!5f t;x~t50!,c . (2.2) Thelocaleigenvaluesl dependonx anddescribehow
(cid:132) (cid:133) j 0
theflowdeformsthephasespaceintheneighborhoodof
This solution depends on time t, the initial conditions
x(t50), and the control-parameter values c. x0. This is illustrated in Fig. 7. The constant associated
with the y direction shows how a small volume is dis-
It is useful to make a distinction between singular 1
placed by the flow in a short time Dt. If l .0 and l
points x* and nonsingular points in the phase space. A 2 3
,0, the flow stretches the initial volume in the y direc-
singularpointx* isapointatwhichtheforcingfunction 2
F(x*,c)50 inEq.(2.1).Sincedx/dt5F(x,c)50 atasin- tionandshrinksitinthey3 direction.Theeigenvalueslj
are called local (they depend on x ) Lyapunov expo-
gular point, a singular point is also a fixed point, 0
dx*/dt50: nents.Weremarkherethatoneeigenvalueofaflowata
nonsingular point always vanishes, and the associated
x~t!5x~0!5x*. (2.3) eigenvector is in the flow direction.
The divergence theorem relates the time rate of
The distribution of the singular points of a dynamical
change of a small volume of the phase space to the di-
system provides more information about a dynamical
vergence of the function F(x;c). We assume a small vol-
system than we have learned to exploit (Gilmore, 1981,
ume V is surrounded by a surface S5]V at time t and
1996), even when these singularities are ‘‘off the real
ask how the volume changes during a short period of
axis’’ (Eschenazi, Solari, and Gilmore, 1989). That is,
time. The volume will change because the flow will dis-
even before these singularities come into existence,
place the surface. The change in the volume is equiva-
there are canonical precursors that indicate their immi-
lent to the flow through the surface, which can be ex-
nent creation.
pressed as (Gilmore, 1981)
Alocalnormal-formtheorem(Arnol’d,1973)guaran-
R
tees that at a nonsingular point x there is a smooth
0
transformation to a new coordinate system y5y(x) in V~t1dt!2V~t!5 dxi(cid:217) dSi. (2.6)
]V
which the flow (2.1) assumes the canonical form
HeredS isanelementofsurfaceareaorthogonaltothe
y˙ 51, i
1 displacement dx and (cid:217) is the standard mathematical
i
y˙ 50, j52,3,...,n. (2.4) generalization in Rn of the cross product in R3. The
j
time rate of change of volume is
This transformation is illustrated in Fig. 6. The local R R
form (2.4) tells us nothing about how phase space is dV dx
5 i(cid:217) dS 5 F (cid:217) dS . (2.7)
stretched and squeezed by the flow. To this end, we dt ]V dt i ]V i i
present a version of this normal-form theorem that is
much more useful for our purposes. If x is not a singu- The surface integral is related to the divergence of the
0
lar point, there is an orthogonal (volume-preserving) flow F by (Gilmore, 1981)
transformation centered at x to a new coordinate sys- R
0 1 dV 1 def
tem y5y(x) in which the dynamical system equations lim 5lim F (cid:217) dS 5 div F5„ F.
assumethefollowinglocalcanonicalforminaneighbor- V→0 V dt V→0 V ]V i i (cid:149)
hood of x : (2.8)
0
Rev.Mod.Phys.,Vol.70,No.4,October1998
1462 RobertGilmore: Topologicalanalysisofchaoticdynamicalsystems
In a locally cartesian coordinate system, divF5„(cid:149)F dx
5(n ]F /]x . Thedivergencecanalsobeexpressedin 52sx1sy,
i51 i i dt
terms of the local Lyapunov exponents,
n dy
„ F5( l , (2.9) dt 5rx2y2xz,
(cid:149) j
j51
where l 50 (flow direction) and l (j.1) are the local dz
1 j 52bz1xy. (2.13)
Lyapunov exponents in the direction transverse to the dt
flow(seeFig.7).Thisisadirectconsequenceofthelocal
normal form result (2.5). Thenthedifferentialcoordinates(X,Y,Z)canberelated
to the original coordinates by
X5x,
B. Changeofvariables
dX
Wepresentheretwoexamplesofchangesofvariables 5Y,
dt
thatareimportantfortheanalysisofdynamicalsystems,
but which are not discussed in generic differential equa-
dY
tionstexts.Theauthorsofsuchtextstypicallystudyonly 5Z,
pointtransformationsx→y(x). Thecoordinatetransfor- dt
mations we discuss are particular cases of contact trans-
dZ
formationsandnonlocaltransformations.Wetreatthese 5~YZ1sY21Y22sXZ2XZ2X3Y2sX4
transformations because they are extensively used to dt
construct embeddings of scalar experimental data into 2bXZ2sbXY1sbrX22bXY2sbX2!/X.
multidimensional phase spaces. This is done explicitly
for three-dimensional dynamical systems. The extension (2.14)
to higher-dimensional dynamical systems is straightfor-
ward. 2. Delaycoordinates
In this case we define the new coordinate system as
1. Differentialcoordinates follows:
If the dynamical system is y ~t!5x ~t!,
1 1
dx5F~x! x5~x ,x ,x !, (2.10) y2~t!5x1~t2t!,
dt 1 2 3
y ~t!5x ~t22t!, (2.15)
then we define y as follows: 3 1
where tis some time that can be specified by various
y 5x ,
1 1 criteria.Inthedelaycoordinatesystem,theequationsof
motiondonothavethesimpleform(2.12).Rather,they
y 5x˙ 5dy /dt5F ,
2 1 1 1 are
]F dx
y35dy2/dt5x¨15F˙15]x1i dti5~F(cid:149)„!F1. (2.11) ddyti5Hi~y!, (2.16)
The equations of motion assume the form
where it is probably impossible to construct the func-
dy tions H (y) explicitly in terms of the original functions
dt15y2, Fi(x). i
When attempting to develop three-dimensional mod-
dy elsfordynamicalsystemsthatgeneratechaoticdata,itis
dt25y3, necessary to develop models for the driving functions
[theF(x)ontheright-handsideofEq.(2.10)].Whenthe
variables used are differential coordinates [see Eq.
dy
35G~y ,y ,y !5~F „!2F . (2.12) (2.11)],twoofthethreefunctionsthatmustbemodeled
dt 1 2 3 (cid:149) 1
in Eq. (2.12) are trivial and only one is nontrivial. On
In this coordinate system, modeling the dynamics re- the other hand, when delay coordinates [see Eq. (2.15)]
ducestoconstructingthesinglefunctionGofthreevari- areused,allthreefunctions[theH (y) ontheright-hand
i
ables,ratherthanthreeseparatefunctions,eachofthree side of Eq. (2.16)] are nontrivial. This is one of the rea-
variables. sons that we prefer to use differential coordinates—
To illustrate this idea, we consider the Lorenz (1963) rather than delay coordinates—when analyzing chaotic
equations: data, if it is possible.
Rev.Mod.Phys.,Vol.70,No.4,October1998
RobertGilmore: Topologicalanalysisofchaoticdynamicalsystems 1463
C. Qualitativeproperties
1. Poincare´ program
The original approach to the study of differential
equations involved searches for exact analytic solutions.
If they were not available, one attempted to use pertur-
bation theory to approximate the solutions. While this
approachisusefulfordeterminingexplicitsolutions,itis
not useful for determining the general behavior pre-
dicted by even simple nonlinear dynamical systems.
Poincare´ realized the poverty of this approach over a
century ago (Poincare´, 1892). His approach involved
studying how an ensemble of nearby initial conditions
FIG. 8. Stretching and squeezing under a flow. A cube of ini-
(an entire neighborhood in phase space) evolved.
tial conditions (a) evolves under the flow. The cube moves in
Poincare´’s approach to the study of differential equa-
thedirectionoftheflow[seeEq.(2.5)].Thesidesstretchinthe
tions evolved into the mathematical field we now call
directions of the positive Lyapunov exponents and shrink
topology.
along the directions of the negative Lyapunov exponents (b).
Topologicaltoolsareusefulforthestudyofbothcon-
Eventually,twoinitialconditionsreachamaximumseparation
servative and dissipative dynamical systems. In fact,
(c) and begin to get squeezed back together (d). A boundary
Poincare´ was principally interested in conservative
layer (d) separates two distant parts of phase space that are
(Hamiltonian) systems. However, the most important beingsqueezedtogether.
tool—the Birman-Williams theorem—on which our to-
pological analysis method is based is applicable to dissi-
creases, the cube stretches in directions with positive lo-
pative dynamical systems. It is for this reason that the
cal Lyapunov exponents and shrinks in directions with
tools presented in this review are applicable to three-
negative local Lyapunov exponents. Two typical nearby
dimensional dissipative dynamical systems. At present,
points (a) separate at a rate determined by the largest
theycanbeextendedto‘‘low’’(d ,3) dimensionaldis-
L positive local Lyapunov exponent (b). Eventually these
sipative dynamical systems, where d is the Lyapunov
L two points reach a maximum separation (c), and there-
dimension of the strange attractor.
after are squeezed to closer proximity (d). We make a
distinction between ‘‘shrinking,’’ which must occur in a
2. Stretchingandsqueezing dissipativesystemsincesomeeigenvaluesmustbenega-
tive ((n l ,0), and ‘‘squeezing,’’ which forces distant
Inthisreviewweareprincipallyinterestedindynami- j51 j
parts of phase space together. When squeezing occurs,
cal systems that behave chaotically. Chaotic behavior is
the two parts of phase space being squeezed together
defined by two properties:
must be separated by a boundary layer, which is indi-
(a) sensitivity to initial conditions and catedinFig.8(d).Boundarylayersindynamicalsystems
(b) recurrent nonperiodic behavior. are important but have not been extensively studied.
If a dynamical system is dissipative („ F,0 every-
Sensitivity to initial conditions means that nearby (cid:149)
where)allvolumesinphasespaceshrinktozeroasymp-
points in phase space typically ‘‘repel’’ each other. That
toticallyintime.Ifthemotioninphasespaceisbounded
is, the distance between the points increases exponen-
and exhibits sensitivity to initial conditions, then almost
tially, at least for a sufficiently small time:
all initial conditions will asymptotically gravitate to a
d~t!5d~0!elt ~l.0,0,t,t!. (2.17) strange attractor.
Repeatedapplicationsofthestretchingandsqueezing
Hered(t) isthedistanceseparatingtwopointsattimet,
d(0) is the initial distance separating them at t50, t is mechanisms build up an attractor with a self-similar
sufficiently small, and the ‘‘Lyapunov exponent’’ l is (fractal) structure. Knowing the fractal structure of the
attractor tells us nothing about the mechanism that
positive. To put it graphically, the two initial conditions
buildsitup.Ontheotherhand,knowingthemechanism
are ‘‘stretched apart.’’
allowsustodeterminethefractalstructureoftheattrac-
If two nearby initial conditions diverged from each
tor and to estimate its invariant properties.
other exponentially in time for all times, they would
Our efforts in this review are concentrated on deter-
eventually wind up at opposite ends of the universe. If
mining the stretching and squeezing mechanisms that
motion in phase space is bounded, the two points will
generate strange attractors, rather than determining the
eventually reach a maximum separation and then begin
fractal structures of these attractors.
to approach each other again. To put it graphically
again, the two initial conditions are then ‘‘squeezed to-
gether.’’ D. Theproblem
WeillustratetheseconceptsinFig.8foraprocessthat
develops a strange attractor in R3. We take a set of Beginning with equations for a low-dimensional dy-
initial conditions in the form of a cube. As time in- namical system [see Eq. (2.1)], it is possible, sometimes
Rev.Mod.Phys.,Vol.70,No.4,October1998
1464 RobertGilmore: Topologicalanalysisofchaoticdynamicalsystems
with difficulty, to determine the stretching and squeez- It is clear that there is some K-dimensional cube (i
1
ing mechanisms that build up strange attractors and to 51,i 52,...,i 5K) whosevolumegrowsintime,fora
2 K
determine the properties of these strange attractors. short time, but that every K11 dimensional cube de-
In experimental situations, we usually have available creases in volume under the flow.
measurements on only a subset of coordinates in the Wecanprovideabettercharacterizationifwereplace
phase space. More often than not, we have available the cube with a fractal structure. In this case, a conjec-
only a single (scalar) coordinate: x (t). Furthermore,
1 ture by Kaplan and Yorke (1979) (see also Alligood,
the available data are discretely sampled at times t , i
i Sauer, and Yorke, 1997), states that every fractal whose
51,2,...,N.
dimensionisgreaterthand isvolumedecreasingunder
L
The problem we discuss is how to determine, using a
the flow, and that this dimension is
finite length of discretely sampled scalar time-series
data, (a) the stretching and squeezing mechanisms that d 5K1(iK51li. (3.4)
builduptheattractorand(b)adynamicalsystemmodel L ul u
K11
thatreproducestheexperimentaldatasettoan‘‘accept-
able’’ level. If l150, then K51 and dL51; if K5n, then dL5n.
This dimension obeys the inequalities K<d ,K11.
L
Topological invariants generally depend on the peri-
III. TOPOLOGICALINVARIANTS
odicorbitsthatexistinastrangeattractor.Unstablepe-
riodic orbits exist in abundance in a strange attractor;
Every attempt to classify or characterize strange at-
tractors should begin with a list of the invariants that theyaredenseinhyperbolicstrangeattractors(Devaney
attractors possess. These invariants fall into three and Nitecki, 1979). In nonhyperbolic strange attractors
classes: (a) metric invariants, (b) dynamical invariants, their numbers grow exponentially with their period ac-
and (c) topological invariants. cordingtotheattractor’stopologicalentropy.Fromtime
Metric invariants include dimensions of various kinds to time, as control parameters are varied, new periodic
(Grassberger and Procaccia, 1983) and multifractal scal- orbits are created. Upon creation, some orbits may be
ing functions (Halsey etal., 1986). Dynamical invariants stable,buttheyaresurroundedbyopenbasinsofattrac-
include Lyapunov exponents (Oseledec, 1968; Wolf tion that insulate them from the attractor (Eschenazi,
etal., 1985). The properties of these invariants have Solari, and Gilmore, 1989). Eventually, the stable orbits
been discussed in recent reviews (Eckmann and Ruelle, usually lose their stability through a period-doubling
1985; Abarbanel etal., 1993), so they will not be dis- cascade.
cussedhere.Theserealnumbersareinvariantunderco- The stretching and squeezing mechanisms that act to
ordinate transformations but not under changes in
create a strange attractor also act to uniquely organize
control-parametervalues.Theyarethereforenotrobust
all the unstable periodic orbits embedded in the strange
under perturbation of experimental conditions. Finally,
attractor.Thereforetheorganizationoftheunstablepe-
these invariants provide no information on ‘‘how to
riodic orbits within the strange attractor serves to iden-
model the dynamics’’ (Gunaratne, Linsay, and Vinson,
tify the stretching and squeezing mechanisms that build
1989).
up the attractor. It might reasonably be said that the
Although metric invariants play no role in the
organization of period orbits provides the skeleton on
topological-analysisprocedurethatwepresentinthisre-
which the strange attractor is built (Auerbach etal.,
view,theLyapunovexponentsdoplayarole.Inparticu-
1987;Cvitanovic,Gunaratne,andProcaccia,1988;Solari
lar, it is possible to define an important dimension, the
and Gilmore, 1988; Gunaratne, Linsay, and Vinson,
Lyapunov dimension d , in terms of the Lyapunov ex-
L 1989; Lathrop and Kostelich, 1989).
ponents.Weassumeann-dimensionaldynamicalsystem
Inthreedimensionstheorganizationofunstableperi-
has n Lyapunov exponents ordered according to
odic orbits can be described by integers or rational frac-
l >l > >l . (3.1) tions. In higher dimensions we do not yet know how to
1 2 fl n
make a topological classification of orbit organization.
We determine the integer K for which
As a result, we confine ourselves to the description of
K K dissipative dynamical systems that are three dimen-
( (
li>0 li1lK11,0. (3.2) sional, or ‘‘effectively’’ three dimensional. For such sys-
i51 i51
tems, we describe three kinds of topological invariants:
We now ask: Is it possible to characterize subsets of (a) linking numbers, (b) relative rotation rates, and (c)
the phase space whose volume decreases under the knot holders or templates.
flow? To provide a rough answer to this question, we
construct a p-dimensional ‘‘cube’’ in the n-dimensional A. Linkingnumbers
phase space, with edge lengths along p eigendirections
i ,i ,...,i and with eigenvalues l ,l ,...,l . Then the Linking numbers were introduced by Gauss to de-
1 2 p i i i
1 2 p scribe the organization of vortex tubes in the ‘‘ether.’’
volume of this cube will change over a short time t ac-
Given two closed curves x and x in R3 that have no
cording to [see Eqs. (2.8) and (2.9)] A B
pointsincommon,Gaussprovedthattheintegral(Rolf-
V~t!;V~0!e~li11li21fl1lip!t. (3.3) son, 1976)
Rev.Mod.Phys.,Vol.70,No.4,October1998