ABC-like flows on the 3-torus

10 Pages·2022·19.223 MB·English

by  Mihai Marciu, Radu Slobodeanu

ABC-like flows on the 3-torus
Cite as: Chaos 32, 083145 (2022); https://doi.org/10.1063/5.0090849
Submitted: 09 March 2022 • Accepted: 01 August 2022 • Published Online: 25 August 2022
Mihai Marciu and Radu Slobodeanu
ARTICLES YOU MAY BE INTERESTED IN
Amplitude response, Melnikov’s criteria, and chaos occurrence in a Duffing’s system subjected
to an external periodic excitation with a variable shape
Chaos: An Interdisciplinary Journal of Nonlinear Science 32, 083144 (2022); https://
doi.org/10.1063/5.0082235
Conservative dynamics in a novel class of 3D generalized thermostatted systems
Chaos: An Interdisciplinary Journal of Nonlinear Science 32, 083143 (2022); https://
doi.org/10.1063/5.0101570
Impulsive systems with growing numbers of chaotic attractors
Chaos: An Interdisciplinary Journal of Nonlinear Science 32, 071102 (2022); https://
doi.org/10.1063/5.0102521
Chaos 32, 083145 (2022); https://doi.org/10.1063/5.0090849 32, 083145
© 2022 Author(s).
Chaos ARTICLE scitation.org/journal/cha
ABC-like flows on the 3-torus
Citeas:Chaos32,083145(2022);doi:10.1063/5.0090849
Submitted:9March2022 Accepted:1August2022
· ·
PublishedOnline:25August2022 View Online Export Citation CrossMark
MihaiMarciua) andRaduSlobodeanub)
AFFILIATIONS
FacultyofPhysics,UniversityofBucharest,405Atomi¸stilor,POBMG-11,RO-077125Bucharest-M˘agurele,Romania
a)E-mail:[email protected]
b)Authortowhomcorrespondenceshouldbeaddressed:[email protected]
ABSTRACT
Weintroducea3-parameterfamilyofvectorfieldsonthe3-torusasalinearcombinationofuniteigenfieldsofthecurloperatorforthe
eigenvalue√2.ForthisfamilyreminiscentoftheclassicalABCflow,westudytheexistenceofstationarypoints,wegivenumericalevidence
fortheexistenceofchaoticregions,andwepresentanintegrablecase.Ourmainresultisthatthenon-vanishingmembersofthefamilyare
associatedwithtightcontactstructures.
PublishedunderanexclusivelicensebyAIPPublishing.https://doi.org/10.1063/5.0090849
TheArnold–Beltrami–Childress(ABC)flowsareamongtheold- At the same time, nowhere vanishing Beltrami fields play an
estandmostwell-knownsystemsthatexhibitchaoticdynamics. importantroleinaseeminglyunrelatedareaofmathematics:con-
They correspond to eigenfields of the operator curl on the flat tact topology and geometry. Let us resume this important insight
3-torus,associatedwiththefirstpositiveeigenvalue,and,inpar- suggestedbySullivanandprovedbyEtnyreandGhristinRef.5.On
ticular, they are steady solutions of Euler’s inviscid fluid equa- a Riemannian 3-manifold (M,g), the dual 1-form α of a Beltrami
tions.IntherangeofparametersA,B,Cwheretheyarenowhere fieldVsatisfies,
vanishing,thesevectorfieldscanbeseenasReebfieldsofacon-
dα fα, δα 0, (1)
tact structure that is tight (contrary to overtwisted, according ∗ = =
to the celebrated contact topological dichotomy). Considering for some smooth function f:M R, where is the Hodge
highereigenvaluesofcurlisverytemptingsincetheeigenspaces → ∗
star operator and δ d , the co-differential. However, the first
arelarger,andnewdynamicalandcontacttopologicalbehavior =∗ ∗
equation implies that α dα fα2vol , and, in particular, α
mayarise.Inthiswork,wereportourresultsaboutananalogous ∧ = | | g ∧
dα 0 whenever α and f are nowhere vanishing. The latter con-
flowassociatedwitheigenfieldsforthesecondpositiveeigenvalue 6=
dition is the definition of a contact form on M. One can say that
ofthecurloperator.Weestablishthetightnessoftheassociated
thenonvanishingBeltramifieldV“engenders”acontactstructure,
structures, and we provide numerical evidence for the chaotic
i.e.,amaximallynon-integrable2-planefielddefinedbykerα 0.
behaviorinthenearlyintegrableregime. =
Conversely,foranycontactform,itsassociatedReebfield[uniquely
determinedbyα(R) 1,andi dα 0]isaBeltramifield(witha
= R =
constant proportionality factor) with respect to some Riemannian
metriconM.Formoreinformationoncontacttopology,werefer
I.INTRODUCTION
thereadertoRef.6.
The divergence-free vector fields V aligned with their own A major concept of contact topology is the tight-overtwisted
curl(i.e.,curlV fVforsomefunctionf)arespecialsolutionsof dichotomy: a contact structure is overtwisted if there exist an
=
steady Euler equations on any Riemannian 3-manifold. They are embeddeddisk,which,alongitsboundary,istangenttothecontact
a cornerstone in many important constructions in fluid dynam- planesandtightifitisnotovertwisted.Giventheaforementioned
ics, such as Ref. 1, and represent the door by which complexity correspondence with the Beltrami fields, it became important to
entersintoplay,eitherthroughchaoticbehavior2orthroughTur- know which type of contact structures (tight or overtwisted) are
ingcompleteness(undecidability)3offluid’sevolution.SeeRef.4for engenderedbynonvanishingBeltramifieldsonagivenRiemannian
moreinformationandanuptodateaccountofthetopologicalfluid 3-manifold (as first prompted in Ref. 5). From a complementary
dynamics. perspective,wewanttoknowwhethervarioustopologicalclassesof
Chaos32,083145(2022);doi:10.1063/5.0090849 32,083145-1
PublishedunderanexclusivelicensebyAIPPublishing
Chaos ARTICLE scitation.org/journal/cha
contactstructuresadmitaBeltramirepresentativewithrespecttoa (firstpositive)eigenvalueλ 1,
1=
naturalmetric.Manyresultsareknownbynowinthesedirections,
withfurtherspecializationonBeltramifieldswithaconstantpro- w sin(z)∂ cos(z)∂ , w cos(z)∂ sin(z)∂ ,
portionalityfactor(curleigenfields).TheHopffieldontheround 1= x+ y 4 = x− y
3-sphereS3 (oranyReebfieldonaSasakian3-manifoldforwhich w2=sin(x)∂y+cos(x)∂z, w5 =cos(x)∂y−sin(x)∂z, (3)
weautomaticallyhavecurl R 2R)andtheABCfieldontheflat w sin(y)∂ cos(y)∂ , w cos(y)∂ sin(y)∂ .
3-torusT3inducetightcontac=tstructures;5energyminimizingBel- 3= z+ x 6 = z− x
t1ramgi f)i,ewldhserined6ucingS2oivsearntwoirsiteendtacbolenstaucrtfascteruanctdurgesaownel(lS-c1h×os6en, Themetricdualofw1ispreciselythestandardcontactformη1onT3.
⊕ 6 6= 6 TheArnold–Beltrami–Childress(ABC)flowsare,bydefinition,
metric, are constructed in Ref. 7; Beltrami fields for every even
givenby
eigenvalue that engender overtwisted contact structures on S3 are
constructed in Ref. 8, where it was also proved that for infinitely
w Aw Bw Cw Asin(z) Ccos(y),
manyeigenvalues,thereexistcurleigenfieldsonT3 thatengender = 1+ 2+ 3= +
overtwistedcontactstructures.Formoreresults,seeRefs.9and10. Acos(z) Bsin(x),Bc(cid:0)os(x) Csin(y) (4)
In this paper, we focus on the 3-torus T3 R3/(2πZ)3 + +
= (cid:1)
endowedwithaflatmetric.Weintroduceafamilyofeigenfieldsof andwerefirststudiedinRefs.14and15.
thecurloperator,forthesecondpositiveeigenvalue√2,thatmim- Amoregeneralcurleigenfieldfortheeigenvalueλ 1would
1=
icsthecelebratedArnold–Beltrami–Childress(ABC)fields(corre- be w Aw Bw Cw Dw , but let us notice that it can be
= 1+ 2+ 3+ 4
spondingtothefirstpositiveeigenvalue1).Ourmainresult(Propo- writtenas
sition 4) establishes that any nonvanishing member of this family
engenders a tight contact structure. Along the way, we revisit the π A
w A2 D2sin z arctan Ccos(y),
analogousstatementfortheABCfields(proveninRef.5)andnotice = + 2 + − D +
thatourABC-likefamilyiscoveringonlyhalfofthe√2-eigenspace (cid:18)p (cid:18) (cid:19)
π A
(seeSecs.IIandIII).InSecs.IVandV,wegivenumericalevidence A2 D2cos z arctan Bsin(x),
forchaoticstreamlinesinourABC-likeflows,asexpectedaccording + (cid:18)2 + − D(cid:19)+
p
to Ref. 2, and we discuss the case C 0 when our field is (com-
pletely) integrable and could allow th=e possibility of proving the Bcos(x) Csin(y) , (5)
+
existenceofchaoticstreamlinesusingtheMelnikovmethod.11We (cid:19)
giveourconclusionsandseveralopenquestionsinSec.VI.
soithastheform(4)ofanABCflow.Asimilarargumentcanbe
producedwhenaddingEw ,orFw ;hence,theABCfamilyactually
5 6
coverstheentireeigenspace(asclaimedinp.455ofRef.5):
II.CURLSPECTRUMONTHE3-TORUSANDTHEABC
Proposition1. AnyvectorfieldinE(curl,λ )isanABCfield,
FLOW 1
possibly after a translation in the coordinates and a relabel of the
Inthissection,wefixoursetupandrevisitshortlytheclassi- coefficients.
cal case of ABC flows that are shown to cover actually the whole As shown in Ref. 14, by symmetry, one may assume 1 A
=
eigenspaceofthecurloperatorforthefirstpositiveeigenvalue. B C 0,andtheABCflowhasnoequilibriumpoints[i.e.,the
≥ ≥ ≥
Recall12(Theorem5.1)thatthespectrumofthecurloperator field(4)isnowherevanishing]ifandonlyif0 B2 C2<1.From
curl dontheflat3-torusT3isgivenby thepath-connectednessofthisparameterspac≤eand+fromGraysta-
=∗
bility, one easily obtains the following result that slightly extends5
(Proposition4.8):
λ k :k Z3 k : k2 k2 k2 ,
{ =±| | ∈ } | | = 1+ 2+ 3 Theorem 1. On the flat 3-torus, every nonvanishing curl
(cid:0) q (cid:1) eigenfieldforthefirstpositiveeigenvalueλ1=1istransversetoatight
where an eigenvalue λ has the multiplicity m(curl,λ) ] µ contactstructure.
Z3, µ λ . = {
∈ O|ne|=can| c|}heckthatforanynon-zerovectorb R3,b k,the
vectorfield ∈ ⊥ III.HIGHERORDERABCFLOWS
Inthissection,weintroduceananalogoftheABCflowinthe
1 eigenspaceforthesecondpositiveeigenvalueofthecurloperator.
V cos(k x)b sin(k x)b k (2)
k= · + k · × We study its stagnation points and its symmetries, and we estab-
| |
lishthattheinducedcontactstructure(inthenonvanishingcase)is
isaneigenfieldofthecurloperatorwitheigenvalue k;cf.13.Notice tight.Inthisway,weaddressaquestionraisedinRef.5:“Itwould
| |
that V b (constantnorm);therefore,V isnonvanishing.As beinterestingtodeterminewhetherthehigher-ordereigenfieldsare
such,|eakc|h=V| |inducesacontactstructureonTk3.Moreover,allthese alsotight.”
k
contactstructuresaretightaccordingtoRef.8. An orthogonal L2 basis in the 12-dimensional eigenspace
Consider the following orthogonal L2-basis in the E(curl,λ )ofacurloperatorfortheeigenvalueλ √2isprovided
6-dimensional eigenspace E(curl,λ ) of the curl operator for the bythefol2lowingchoicesofkandbin(2): 2=
1
Chaos32,083145(2022);doi:10.1063/5.0090849 32,083145-2
PublishedunderanexclusivelicensebyAIPPublishing
Chaos ARTICLE scitation.org/journal/cha
[(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(0,1,1),(1,0,0)],
(1,1,0), 1 , 1 ,0 , (1,0,1), 1 ,0, 1 , (0,1,1),(0, 1 , 1 ) ,
√2 −√2 √2 −√2 √2 −√2
(6)
h (cid:16) (cid:17)i h (cid:16) (cid:17)i h i
[(1, 1,0),(0,0,1)], [( 1,0,1),(0,1,0)], [(0,1, 1),(1,0,0)],
− − −
(1, 1,0), 1 , 1 ,0 , ( 1,0,1), 1 ,0, 1 , (0,1, 1), 0, 1 , 1 .
− √2 √2 − √2 √2 − √2 √2
h (cid:16) (cid:17)i h (cid:16) (cid:17)i h (cid:16) (cid:17)i
The unit vector fields on this basis will be denoted by v, withtheassociateddynamicalsystem,
i
i 1,2,...,12 .Forinstance,thefirstthreeelementsare
={ } A B
x sin(x y) sin(x z) Ccos(y z),
1 1 ˙ =−√2 + + √2 + + +
v : sin(x y)∂ sin(x y)∂ cos(x y)∂ ,
1 =−√2 + x+ √2 + y+ + z A C
y sin(x y) Bcos(x z) sin(y z), (9)
1 1 ˙= √2 + + + − √2 +
v : sin(x z)∂ cos(x z)∂ sin(x z)∂ , (7)
2 = √2 + x+ + y− √2 + z B C
z Acos(x y) sin(x z) sin(y z).
1 1 ˙= + − √2 + + √2 +
v : cos(y z)∂ sin(y z)∂ sin(y z)∂ .
3 = + x− √2 + y+ √2 + z
Thefirstdifferencewithrespecttotheclassicalcaseisthefact
WenowintroducethesecondorderABCfieldasthefollowing thatABC-likefields(8)donotcoveranymorethefulleigenspace,
curleigenfieldfortheeigenvalueλ √2: butonly“half”ofit.Indeed,amoregeneralcurleigenfieldforthe
2= eigenvalue λ would be v Av Bv Cv Dv , but it can be
2 = 1+ 2+ 3+ 4
v Av Bv Cv , (8) writtenas
= 1+ 2+ 3
√A2 D2 A B
v + sin x y π arctan sin(x z) Ccos(y z),
= − 2 (cid:18) + − 2 + D(cid:19)+ √2 + + +
√A2 D2 π A C
+ sin x y arctan Bcos(x z) sin(y z),
2 + − 2 + D + + − √2 +
(cid:18) (cid:19)
π A B C
A2 D2cos x y arctan sin(x z) sin(y z) ,
+ (cid:18) + − 2 + D(cid:19)− √2 + + √2 + !
p
soithastheform(8)ofanABC-likeflow.Asimilarargumentcan AsaconsequenceofProposition2(ii,iii),wecanassumethat
bereproducedwhenaddingEv ,orFv ;hence,theABC-likefamily
5 6
actually covers the subspace span v ,v ,...,v . Analogously, the A B C,
{ 1 2 6} ≥ ≥
familyofvectorfieldsoftheform
andduetoProposition2(i),onecanfurtherassumethatAandBare
v Av Bv Cv positive.Finally,sinceaBeltramifieldcanbescaledbyaconstant,
= 7+ 8+ 9 one can take A 1. Let us denote with (cid:127) (B,C) R2 1 B
coversthesubspacespan v ,v ,...,v ofE(curl,λ ). = ={ ∈ | ≥
Remark 1. In Ree{f.7e168, theea1u2t}heor studied2the streamline ≥|C|}therestrictedparameterspacethatwehaveobtained.Inside
the parameter space, we define the subset ϒ (cid:127) of all parame-
structure and symmetry properties of some curl eigenvectors for ⊂
tersforwhichthecorrespondingABC-likefield(8)hasatleastone
tthhee feiieglednvWal2uedeλf−in2e=d −in√(25..3T)hoef cRoerfr.es1p6onidsenint ionurEn(coutarlt,iλon2)s:ovf czeorrore.sIptsoncdoimngplteomneonntvϒancis=hi(cid:127)n\gϒvewctiollrgfiievledsth[eofptahreafmoremter(8s)u]b.space
=−√12(v1+v2+v3−v7−v8−v9). LetusfirstnoticethatifanABC-likevectorfieldvhasazero,
Bydirectcheck,wecanestablishthefollowing: then,bysimpleapplicationofthetriangleinequality[giventhatv’s
i
Proposition2. Equation(9)isinvariant(v0 v)under in(7)areunitvectors],weobtaintheconditionontheparameters,
=
(i) A0 A,B0 B,x0 x πandtheother2analogousones. 1 A B C.
(ii) A =B−,B =C,−C A=,x + y,y z,z x. = ≤ +| |
0 0 0 0 0 0
(iii) B =C,C =B,x =3π y=,y =3π x,=z 3π z. Therefore,ϒ (B,C) R2 1>B> C, B C >1 .
0= 0= 0= 2 − 0= 2 − 0= 2 − ⊂{ ∈ | | | +| | }
Chaos32,083145(2022);doi:10.1063/5.0090849 32,083145-3
PublishedunderanexclusivelicensebyAIPPublishing
Chaos ARTICLE scitation.org/journal/cha
Proposition3. Thesetϒc ofparametersforwhichtheABC- whichinturnleadsusto(here,weskipmanycumbersomecompu-
likevectorfieldsvhavenostagnationpointsispath-connected. tationsandacasebycasediscussionthathavebeenaccomplished
Proof. Letϕ : arctan 1 .Byaddingitsequationstwobytwo usingWolframMathematica.Thedetailedversionisavailablefrom
0 = √2
andusingtheidentityacos(x) bsin(x) √a2 b2cos(x arctan theauthorsuponrequest.)
ab (a>0), we can equivale+ntly recas=t the sy+stem of eq−uations 3(A2 B2)2t4 8√2(A2 B2)C2t3 2 3(A2 B2)2
characterizingstagnationpointsas − − + − − −
(cid:0) (cid:1)(cid:1) (A2 B2)C2 6C4 t2 8√2(A2 B(cid:0)2)C2t 3(A2 B2)2
Acos z0−ϕ0 +Bcos y0+ϕ0 =0, − + + + − − −
2(A2 B2)C2 3C(cid:1)4 0,
Acos(cid:0)z0+ϕ0(cid:1)+Ccos(cid:0)x0−ϕ0(cid:1)=0, (10) wher+et: ta+nx forx− π=(thisisnotarestrictionsincex π
Bcos(cid:0)y0−ϕ0(cid:1)+Ccos(cid:0)x0+ϕ0(cid:1)=0. is the co=ordinat0e of0a6=st±ag2nation point iff B=1). The p0a=ram±e2-
ter space ϒc gathers all coefficients for which the above algebraic
Here, x0=y+z,y0=(cid:0) z+x,(cid:1)andz0=(cid:0)x+y. E(cid:1)liminating y0 and z0 equationhasnorealsolutiont.However,afourthdegreepolynomial
using the second and third equation (always possible due to
hasonlycomplexrootsifandonlyif[1>0and(P>0orD>0)]
A B C)andinjectingtheminthefirstone,weget
≥ ≥ or[1 D R 0andP>0];seetheAppendixfornotations.For
= = =
C A 1,theseconditionsdefineapathconnectedregioninthe(B,C)
Acos arccos −Acos(x0−ϕ0) −2ϕ0 pla=ne,showninFig.1.
(cid:18) (cid:18) (cid:19) (cid:19) (cid:3)
C
+Bcos arccos −Bcos(x0+ϕ0) +2ϕ0 =0, thelinReeamriazraktio2n. ofA(t9a)isstagnationpoint(x,y,z),thematrixgivingus
(cid:18) (cid:18) (cid:19) (cid:19)
Bcos(x+z)√−2Acos(x+y) Aco√s(2x+y) −Bsin(x+z) −Asin(x+y)− Bco√s(2x+z)
J=−Aco√s(2x+y) −Csin(y+z) Acos(x+y)√−2Ccos(y+z) Cco√s(2y+z) −Asin(x+y) .
 Bco√s(2x+z) −Csin(y+z) −Bsin(x+z)− Cco√s(2y+z) Ccos(y+z)√−2Bcos(x+z) 
 
WeremarkthatJissymmetric[duetov(x,y,z) 0]andtrace-
=
free.Therefore,theeigenvaluesarerealandsumuptozero.Moreover,
det(J) 1ABC(3(sin(2x) sin(2y)
=−4 +
sin(2z)) sin(2(x y z))).
+ + + +
IfABC 0,wecanconcludeasinRef.14thatthestagnationpoints
6=
are,generically,unstablefixedpointsofthedynamicalsystem(9).
Wearereadytogivethemainresultofthissection:
Proposition 4. On the flat 3-torus, every nonvanishing curl
eigenfieldoftheform(8)forthesecondpositiveeigenvalueλ √2is
2=
transversetoatightcontactstructureinthesamecontactomorphism
classasthestandardcontactformη .
1
Proof. AccordingtoProposition3,thenonvanishingABC-like
flowsareindexedbyapath-connectedparameterspace.Inparticu-
lar,theinducedcontactstructuresarecontacthomotopic.As“con-
tacthomotopic contactisotopic”bythecelebratedGraystability
⇒
theorem,alltheseinducedcontactstructuresareinthesamecontac-
tomorphicclassandsharethepropertyofbeingtight/overtwisted.
However, in Remark 8 of Ref. 8, it was proved that the curl
√2-eigenfield v in (7) is inducing a tight contact structure being
1
in the same contactomorphism class as the standard contact
formη . (cid:3)
1
IV.FLOWVISUALIZATIONBYPOINCARÉSECTIONS
FIG. 1. The parameter space ϒc evaluated and plotted using the functions
ItwasnumericallyshownbyHénoninRef.15thattheABC ImplicitRegionandRegionPlotinRef.17.
fieldonthe3-torusmayexhibitbothchaoticregionsandanonzero
Chaos32,083145(2022);doi:10.1063/5.0090849 32,083145-4
PublishedunderanexclusivelicensebyAIPPublishing
Chaos ARTICLE scitation.org/journal/cha
FIG.2. TheclassicalABCflowseenthroughtheintersectionswiththeplanesz 0,y 0andx 0,respectively.Here(A,B,C) (1, 2/3, 1/3).
= = = =
p p
measuresetofinvarianttori.Arnold18believedthatthisisageneric InthecaseoftheclassicalABC-flow(4)withA 1,B 2,
behavior,andhisconjecturewasrecentlysettled2inaprobabilistic = = 3
and C 1 , the method of Poincaré sections yields Fig. 2 whqere
sense.Inthissection,wereproduceHénon’splot,andweconstruct = √3
weclearlyseethe“chaotic”andintegrableregions(cf.alsoRefs.14
analogous ones for our ABC-like field. The conclusion is simi-
and15).
lar,butthe“chaoticarea”seemstobelargerforourhigherorder
ForthesamesetofparametersourABC-likeflowhavestagna-
eigenfield.
tionpointsandlookslikeinFig.3.
Werecallthatforthevisualizationofa3-dimensionalflowin
The visualization of the ABC-like flow is considerably
a 2-dimensional picture, the so called method of Poincaré sections
improvedifweconsiderdiagonalplanes,asshowninFig.4.
consists of plotting successive transversal intersections of several
Fortheparameters(A,B,C) (1,0.4,0.01),ourABC-likeflow
streamlines (obtained from various initial conditions) with a sur- =
hasnostagnationpointsandlookslikeinFig.5.
face of section (here a plane). A streamline is identified by its
unique color. In practice, we computed Poincaré sections by first
numerically integrating the dynamical system for various initial
conditions using NDSolve, and then, we collected the transver-
V.ANINTEGRABLECASE
sal intersections of trajectories with the specified planes using
WhenEvent. Our code was inspired by the one provided here: AsinthecaseofthestandardABC-flow,ifoneoftheparameter
https://reference.wolfram.com/language/example/PoincareSections. vanishes, then the flow is integrable. Indeed, assuming C 0, the
=
html. followingfunctionisafirstintegralofourvectorfieldv Av Bv
= 1+ 2
Chaos32,083145(2022);doi:10.1063/5.0090849 32,083145-5
PublishedunderanexclusivelicensebyAIPPublishing
Chaos ARTICLE scitation.org/journal/cha
FIG.3. TheABC-likeflowseenthroughtheintersectionswiththeplanesy 0andx 0,respectively.Here(A,B,C) (1, 2/3, 1/3).
= = =
p p
definedin(8): A.Chaosinthenear-integrablecase
For a dense set of parameters A,B, and C ε>0 small
=
enough,thechaoticbehavioroftheclassicalABCfield(4)wasestab-
H(x,y,z) A cos(x y) √2sin(x y)
lished in Refs. 19 and 20. The associated dynamical system was
= + + +
(cid:16) (cid:17) seen as a periodic perturbation of the integrable C 0 case and
B cos(x z) √2sin(x z) , (11) was casted as a planar Hamiltonian system so that =the Melnikov
+ + − +
(cid:16) (cid:17) methodcouldbeappliedforprovingtheexistenceofatransversal
homoclinicpoint.
or, equivalently, H(x,y,z) √3(Asin(x y ϕ ) Bsin(x z In our case, by introducing new coordinates x x,y x
ϕ )) with ϕ arctan 1 a=s in Sec. III. T+he+con0ser−ved quan+tity y,z x z(whichamountstoapush-forwardofvth˜ro=ugha˜v=ol-
−His0exactlyeq0ua=lto v, √∂2 ∂ ∂ sothatthemovementinthe u+me˜pr=eser+vingdiffeomorphismofT3)andrescalingthecoefficients
direction( 1,1,1)ishun−ifoxrm+.Tyh+epzriojectionofthestreamlineson A,B,Cwith√3/2,thedynamicalsystem(9)takestheform
−
theplanex y z 0looksasinFig.6.
− − =
x 1 (Asin(y) Bsin(z)) C 2cos(2x y z),
˙˜ =−√3 ˜ − ˜ + 3 ˜−˜−˜
q
y Bcos(z ϕ ) Ccos(2x y z ϕ ), (12)
˙˜= ˜− 0 + ˜−˜−˜− 0
z Acos(y ϕ ) Ccos(2x y z ϕ ),
˙˜= ˜+ 0 + ˜−˜−˜+ 0
whilethe(scaled)firstintegral(11)forthecaseC 0isindependent
=
ofx,
˜
H(y,z) Asin(y ϕ ) Bsin(z ϕ ). (13)
˜ ˜ = ˜+ 0 − ˜− 0
e
The function H has four critical points (y ,z ) [0,2π]2, two of
which being of saddle type: π ϕ ,π ˜0ϕ˜0 ∈and 3π ϕ ,3π
2 − 0 2 + 0 2 − 0 2
ϕ . The vaelues taken by H at these two points are (A B),
+ 0 (cid:0) (cid:1) (cid:0) ± −
respectively.ThelevelcurvesofHarepicturedinFig.7foraspecific
(cid:1)
choiceofparameters. e
FIG.4. TheABC-likeflowseenthroughtheintersectionswiththeplanex y Moreover,fortheunperturebedsystem(i.e.,forC 0),wecan
0.Here(A,B,C) (1, 2/3, 1/3). + explicitlycomputethehomoclinicorbit(yh(t),zh(t))co=nnectingto
= = itselfthehyperbolicpoint 3π ϕ ,3π ϕ˜ an˜dthecorresponding
p p 2 − 0 2 + 0
(cid:0) (cid:1)
Chaos32,083145(2022);doi:10.1063/5.0090849 32,083145-6
PublishedunderanexclusivelicensebyAIPPublishing
Chaos ARTICLE scitation.org/journal/cha
FIG.5. TheABC-likeflowseenthroughtheintersectionswiththeplanesz 0andx 0,respectively.Here(A,B,C) (1,0.4,0.01).
= = =
FIG. 6. Streamlines in the plane x y z 0 in the integrable case
− − =
A 1,B 2/3,C 0. FIG.7. LevelcurvesofHforA 1,B 1/2intheplane(y,z).
= = = = = ˜ ˜
p
e
Chaos32,083145(2022);doi:10.1063/5.0090849 32,083145-7
PublishedunderanexclusivelicensebyAIPPublishing
Chaos ARTICLE scitation.org/journal/cha
structure and their chaotic behavior. We obtained that the vec-
torfieldsconsideredare(whennonvanishing)associatedwithtight
contact structures contactomorphic to the standard one, and we
have provided numerical evidence (via Poincaré sections and the
evaluationofMelnikov’sdistancefunction)thatthechaoticregions
coexistwithinvarianttoriandseemtooccupyalargeramountof
space than in the ABC case. Despite the neat similitude with the
standardcaseofABCfieldscorrespondingtothefirstpositiveeigen-
value,astrikingincreaseindifficultyhasbeenencounteredinthe
proof of their tightness, as well as in the proof of the existence
of chaotic streamlines in the nearly integrable regime. This dark-
ens the perspective of employing this type of analytic methods in
FjoIiGne.d8b.yGlirnaepsh.Foofrtt0h7→ezeMrop(ot0i)ntbtu,ildtheonfun3c2tiopnoinFtisncdoRmopuotetdyineuldmsearnicaalplypraonxdi- the study of higher eigenspaces. Nevertheless, there remain some
0 openquestionsthatseemtousbothapproachableandinteresting
matedvalueof2.82736,andM0(t0)isevaluatedby−0.43656. (onT3):
doesthereexist“overtwistedfields”(i.e.,novanishingBeltrami
evolutioninthex-direction, •
˜ fields that engender an overtwisted contact structure) in this
1 B second eigenspace E(curl,λ √2)?; if not, identify the next
xh(t) √2(A B)t 2arctan 1 sinh √ABt 2=
˜ = 3" − − r − A ! highereigenspacewhereovertwistedfieldsexistand,ifpossible,
(cid:16) (cid:17) constructexplicitexamples;
A identify the tight contactomorphism class of Beltrami fields
2arctan 1cosh √ABt , • defined in (2) (recall that these classes are indexed by an
+ rB − (cid:16) (cid:17)!# integer22).
4B
yh(t) π ϕ arcsin 1 ,
˜ = − 0+ − A B (A B)cosh(2√ABt)
(cid:18) + + − (cid:19) ACKNOWLEDGMENTS
4A
ϕ0−arcsin 1− A B (A B)cosh(2√ABt) WearegratefultoD.Peralta-SalasandF.Schlenkforinsight-
 ift 0, (cid:18) + + − (cid:19) fulfeedbacks.Thecomputationalpartofthisworkwasperformed
z˜h(t)=π+ϕ≥0+arcsin 1− A B (A 4BA)cosh(2√ABt) uCsNinFgISth-FeDcoI-m20p2u0t-e0r3s5ta5t.ionsprovidedbyCNFISthroughtheproject
 ift<0. (cid:18) + + − (1(cid:19)4) AUTHORDECLARATIONS
Byinspection,weseethatoursystem(12)isoftheform[(54)
ConflictofInterest
in Ref. 21]. According to (51) in Ref. 21, the Melnikov distance
functionisgivenby Theauthorshavenoconflictstodisclose.
M(t ) ∞ Bcos(zh(τ) ϕ )cos 2xh(τ t ) yh(τ)
0 = − ˜ − 0 ˜ + 0 −˜
Z−∞(cid:8) (cid:0) AuthorContributions
−z˜h(τ)+ϕ0 +Acos(y˜h(τ)+ϕ0)cos 2x˜h(τ +t0) MihaiMarciu:Writing–originaldraft(equal).RaduSlobodeanu:
Writing–originaldraft(equal).
yh(τ) zh((cid:1)τ) ϕ dτ. (cid:0)
−˜ −˜ − 0
Forthehomoclinic(14),theabove(cid:1)M(cid:9)elnikovintegralseemsvirtually
impossibletocomputeexplicitly.Nevertheless,usingthecapabilities DATAAVAILABILITY
ofRef.17,wecanproducenumericalevidencefortheexistenceofa
simplezeroofM.SeeFig.8fortheparticularcasewhenA 1and Datasharingisnotapplicabletothisarticleasnonewdatawere
= createdoranalyzedinthisstudy.
B 1/2. According to the discussion in Ref. 21 (Sec. 6.4, case 1),
=
thepresenceofsimplezerosimpliesthatourABC-likeflowexhibits
“smalehorseshoe”chaoswhenCisclosetozero.
APPENDIX:FOURTHDEGREEEQUATIONS
VI.EPILOGUE Considerageneralquarticequationwithrealcoefficientsand
a 0,
Wehaveinvestigatedtheeigenfieldsofacurloperatoronthe 6=
3-torus, associated with the second positive eigenvalue √2 under
twoaspects:thetype(tight/overtwisted)oftheirtransversalcontact ax4 bx3 cx2 dx e 0.
+ + + + =
Chaos32,083145(2022);doi:10.1063/5.0090849 32,083145-8
PublishedunderanexclusivelicensebyAIPPublishing
Chaos ARTICLE scitation.org/journal/cha
Thediscriminant(whosesignmainlydeterminesthenatureofthe 4V.I.ArnoldandB.Khesin,TopologicalMethodsinHydrodynamics,2nded.
roots)is (Springer,2021).
5J.B.EtnyreandR.Ghrist,“ContacttopologyandhydrodynamicsI:Beltrami
1 256a3e3 192a2bde2 128a2c2e2 144a2cd2e 27a2d4 fieldsandtheSeifertconjecture,”Nonlinearity13,441–458(2000).
= − − + − 6H.Geiges,AnIntroductiontoContactTopology(CambridgeUniversityPress,
144ab2ce2 6ab2d2e 80abc2de 18abcd3 16ac4e 2008).
+ − − + +
7R.GhristandR.Komendarczyk,“Overtwistedenergy-minimizingcurleigen-
4ac3d2 27b4e2 18b3cde 4b3d3 4b2c3e b2c2d2.
− − + − − + fields,”Nonlinearity19,41–51(2006).
8D.Peralta-SalasandR.Slobodeanu,“ContactstructuresandBeltramifieldson
Thediscussionofthenatureoftherootsisrefinedbyconsidering
thetorusandthesphere,”arXiv:2004.10185(2020)[toappearinIndianaUniv.
thefollowingquantities:
Math.J.(2022)].
P 8ac 3b2, R b3 8da2 4abc, 9J.B.Etnyre,R.Komendarczyk,andP.Massot,“Tightnessincontactmetric3-
= − = + − manifolds,”Invent.Math.188,621–657(2012).
D 64a3e 16a2c2 16ab2c 16a2bd 3b4. 10R.Komendarczyk,“TightBeltramifieldswithsymmetry,”Geom.Dedicata134,
= − + − − 217–238(2008).
For example, if the quartic equation has two double roots, then 11J.GuckenheimerandP.Holmes,NonlinearOscillations,DynamicalSystems,
D 0.Formoredetails,seehttps://en.wikipedia.org/wiki/Quartic_ andBifurcationsofVectorFields(Springer-Verlag,NewYork,1990).
= 12Ch.Bär,“Thecurloperatoronodd-dimensionalmanifolds,”J.Math.Phys.60,
functionandRef.23.
031501(2019).
ForourquarticequationintheproofofProposition1,wehave 13A.Enciso,R.Lucà,andD.Peralta-Salas,“Vortexreconnectioninthethree
(forA 1) dimensionalNavier–Stokesequations,”Adv.Math.309,452–486(2017).
1= 2985984 B2 1 2C8 81B12 216B10 C2 1 18B8 8C4 14T. Dombre, U. Frisch, J. M. Greene, M. Hénon, A. Mehr, and A. M.
23C2= 8 6B6 C2−1 3C4 31C−2 3 B4+144+C8 204C6 Soward,“ChaoticstreamlinesintheABCflows,”J.FluidMech.167,353–391
+ + − (cid:0) + (cid:1) (cid:0) + + +(cid:0) (cid:1) − (cid:0) (1986).
+241C4−2(cid:1)04C2+(cid:0)144 −(cid:1)6(cid:0)B2 36C10−69C8(cid:1)+34C(cid:0)6+34C4−69C2 15M.Hénon,“Surlatopologiedeslignesdecourantdansuncasparticulier,”C.
36 9 C2 1 2 9C8 6C6 5C4 6C2 9 , R.Acad.Sci.Paris262,312–314(1966).
+ + − (cid:1) − (cid:0) − − + 16T.Nishiyama,“Beltramiflowsequippedwiththesamecrystallographicsymme-
(cid:1) P=(cid:0) 432 B(cid:1)2(cid:0)−1 2 3B4−B2 C2+6 −2(cid:1)C(cid:1)4−C2+3 , triesasbluephasesofcholestericliquidcrystals,”Eur.J.Mech.B/Fluids75,15–28
D 62208 B2 1 4C4 3B4 B2 7C2 9 5C4 7C2 3 , (2019).
= R(cid:0)(cid:0)=−172(cid:1)8√(cid:1) 2(cid:0)(cid:0)1−−B2 3(cid:0)C(cid:0)4 3+B2(cid:1)−(cid:1)+2C2+−3 . +(cid:1)(cid:1) 1187VW.oIl.frAamrnoRleds,e“aSruchr,lIantco.,pMoloAgTiHeEdMesAéTcIoCuAlevmeresniotsns1t2a.t3io,nCnhaaimrepsadigens,fIluLi,d2e0s2p1a.r-
faits,”C.R.Acad.Sci.Paris261,17–20(1965).
(cid:0) (cid:1) (cid:0) (cid:1) 19J.L.Gautero,“Chaoslagrangienpouruneclassed’écoulementsdeBeltrami,”C.
REFERENCES R.Acad.Sci.ParisSér.II301,1095–1098(1985).
20X.-H. Zhao, K.-H. Kwek, J.-B. Li, and K.-L. Huang, “Chaotic and resonant
1A.EncisoandD.Peralta-Salas,“KnotsandlinksinsteadysolutionsoftheEuler streamlinesintheABCflow,”SIAMJ.Appl.Math.53,71–77(1993).
equation,”Ann.Math.175,345–367(2012). 21I. Mezic´ and S. Wiggins, “On the integrability and perturbation of three-
2A.Enciso,D.Peralta-Salas,andA.Romaniega,“Beltramifieldsexhibitknotsand dimensionalfluidflowswithsymmetry,”J.NonlinearSci.4,157–194(1994).
chaosalmostsurely,”arXiv:2006.15033v1(2020). 22Y. Kanda, “The classification of tight contact structures on the 3-torus,”
3R. Cardona, E. Miranda, D. Peralta-Salas, and F. Presas, “Constructing Tur- Commun.Anal.Geom.5,413–438(1997).
ing complete Euler flows in dimension 3,” Proc. Natl. Acad. Sci. U.S.A. 118, 23E.L.Rees,“Graphicaldiscussionoftherootsofaquarticequation,”Am.Math.
e2026818118(2021). Mon.29,51–55(1922).
Chaos32,083145(2022);doi:10.1063/5.0090849 32,083145-9
PublishedunderanexclusivelicensebyAIPPublishing