PHYSICAL REVIEW E 97, 032219 (2018)
Di ec ed mo ion o sphe es induced by unbiased d i ing o ces in
iscous luids beyond he S okes’ law egime
Jesús Casado-Pascual*
Física Teó ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, 41080 Se illa, Spain
(Recei ed 17 Sep embe 2017; e ised manusc ip ecei ed 27 Feb ua y 2018; published 30 Ma ch 2018)
The eme gence o di ec ed mo ion is in es iga ed in a sys em consis ing o a sphe e imme sed in a iscous
luid and subjec ed o ime-pe iodic o ces o ze o a e age. The di ec ed mo ion a ises om he combined ac ion
o a nonlinea d ag o ce and he applied d i ing o ces, in he absence o any pe iodic subs a e po en ial.
Necessa y condi ions o he exis ence o such di ec ed mo ion a e ob ained and an analy ical exp ession o he
a e age e minal eloci y is de i ed wi hin he adiaba ic app oxima ion. Special a en ion is paid o he case o
wo mu ually pe pendicula o ces wi h sinusoidal ime dependence, one wi h wice he pe iod o he o he . I is
shown ha , al hough nei he o hese wo o ces induces di ec ed mo ion when ac ing sepa a ely, when added
oge he , he esul an o ce gene a es di ec ed mo ion along he di ec ion o he o ce wi h he sho es pe iod. The
dependence o he a e age e minal eloci y on he sys em pa ame e s is analyzed nume ically and compa ed wi h
ha ob ained using he adiaba ic app oxima ion. Among o he esul s, i is ound ha , o app op ia e pa ame e
alues, he di ec ion o he a e age e minal eloci y can be e e sed by a ying he o cing s eng h. Fu he mo e,
ce ain aspec s o he obse ed phenomenology a e explained by means o symme y a gumen s.
DOI: 10.1103/PhysRe E.97.032219
I. INTRODUCTION
The gene a ion o di ec ed mo ion o pa icles in nonlinea
sys ems subjec ed o de e minis ic and/o s ochas ic unbiased
d i ing o ces has been an ac i e esea ch opic o e he
pas decades [1–3]. This phenomenon—commonly known
as a che e ec —has been s udied ex ensi ely in ields as
di e se as biophysics [4], nano echnology [5], g anula media
[6], and spa ially ex ended nonlinea sys ems [7,8]. On he
heo e icalside,muchwo khasbeen ocusedonunde s anding
he mechanisms by which nonlinea i y and symme y b eaking
coope a e o a o mo ion in one di ec ion [2,3].
One class o a che models ha has ecei ed conside able
a en ion in he li e a u e is he so-called ocking a che [9]. In
he ocking a che , pa icles mo ing in a pe iodic subs a e
po en ial also expe ience a pe iodic o quasipe iodic ime-
dependen o ce o ze o a e age. Al hough mos o he wo ks
also conside he p esence o andom o ces, noise is no a
c ucial elemen [10,11]. In his class o models, he mechanism
behind he gene a ion o di ec ed mo ion is basically ha monic
mixing [3,8,12]. Fo his mechanism o be e ec i e, ce ain
spa io empo al symme ies [13], supe symme ies [14], and
hidden symme ies [15] mus be b oken. Al e na i e ways o
b eaking hese symme ies ha e also been in es iga ed in a
wo-s a e B ownian mo o , ealized wi h B ownian pa icles
al e na ing be ween wo phase-shi ed, symme ic po en ials
[16]. I should be no ed ha , in he adi ional ocking a che ,
he ic ion o ces, i any, a e assumed o be linea in he
eloci ies o he pa icles. Consequen ly, he appea ance o
ha monic mixing is solely due o he subs a e po en ial, which
is he only sou ce o nonlinea i y in he sys em.
*[email p o ec ed]
Howe e , he e a e si ua ions in which ic ion canno be
assumed o be linea in he eloci ies o he pa icles. This is
he case, o ins ance, o he d ag o ce—a ype o ic ion ha
ac s on bodies mo ing in iscous luids—a Reynolds numbe s
beyond he S okes ange [17]. Nonlinea ic ion o ces also
appea in o he ields such as in he ac i e B ownian mo ion
[18], which desc ibes he mo ion o sel -p opelled o ganisms,
in he ela i is ic B ownian mo ion [19], as well as in he
B ownian mo ion wi h d y ic ion [20]. In hese si ua ions,
ic ion cons i u es an addi ional sou ce o nonlinea i y ha
may gi e ise o ha monic mixing, hus playing a majo ole in
he a che e ec . In pa icula —and con a y o wha in ui ion
migh sugges —nonlinea ic ion may con ibu e posi i ely
o he eme gence o di ec ed mo ion unde app op ia e condi-
ions. In he case o andom o ces, his possibili y has been
analyzedinRe .[21],whe e he ic ionconside edisnonlinea
and aniso opic.
The p ima y aim o he p esen wo k is o in es iga e
how di ec ed mo ion eme ges om he combined ac ion o
nonlinea ic ion and ze o-mean oscilla ing o ces. To his
end, we ocus ou analysis on a simple bu ealis ic model o a
physical sys em in which ic ion is he only nonlinea elemen
p esen . Mo e speci ically, we examine he mo ion o a sphe e
imme sed in a iscous luid and subjec ed o a ime-pe iodic
o ce o ze o a e age, co e ing a wide ange o Reynolds
numbe s. We pay special a en ion o he case o wo mu ually
pe pendicula o ces wi h sinusoidal ime dependence, one
wi h wice he pe iod o he o he . In pa icula , we show ha ,
al hough nei he o hese wo o ces induces di ec ed mo ion
when ac ing sepa a ely, he esul an o ce ob ained by adding
hem oge he causes a ne mo ion o he sphe e along he
di ec ion o he o ce wi h he sho es pe iod [22].
The ou line o he emainde o his pape is as ollows.
In Sec. II, we in oduce he sys em unde conside a ion and
2470-0045/2018/97(3)/032219(8) 032219-1 ©2018 Ame ican Physical Socie y
JESÚS CASADO-PASCUAL PHYSICAL REVIEW E 97, 032219 (2018)
de ine he quan i ies o in e es , namely, he ime-dependen
e minal eloci y and he a e age e minal eloci y. We also
p o ide necessa y condi ions o he appea ance o di ec ed
mo ion. The con e gence o he solu ions o he equa ion o
mo ion o he ime-dependen e minal eloci y is analyzed
in he Appendix. In Sec. III, we de i e an exp ession o he
a e age e minal eloci y wi hin he adiaba ic app oxima ion.
In Sec. IV, he heo e ical esul s o he p e ious sec ions a e
illus a ed by nume ical simula ions. Finally, in Sec. V,we
p esen conclusions o he main indings o ou wo k.
II. PROBLEM FORMULATION
We conside he mo ion o a sphe e o mass mand adius ,
imme sed in a s eady luid o densi y ρ and iscosi y η, and
subjec ed o a ime-pe iodic o ce F( )o pe iodTand ze o
ime a e age, i.e., wi h T
0d F( )/T =0. Since he objec i e
is o s udy how di ec ed mo ion eme ges om he combined
ac iono hed ag o ceandF( ), o exposi ionalcla i ywewill
assume ha hese a e he only o ces ac ing on he sphe e. This
implies ha he sys em is in a mic og a i y en i onmen and
ha he size o he sphe e is la ge enough o neglec he e ec s
o he B ownian o ces. The ime e olu ion o he eloci y o
he sphe e ela i e o he luid, ( ), is go e ned by he equa ion
o mo ion
m˙
( )=Fd( )+F( ),(1)
whe e he o e do indica es de i a i e wi h espec o ime and
Fd( ) is he hyd odynamic d ag o ce exe ed on he sphe e a
ime .
He eina e , we will also assume ha he hyd odynamic
d ag o ce can be exp essed in e ms o he s eady d ag
coe icien Cd(Re)[17]as
Fd( )=−π η
4Re( )Cd[Re( )] ( ),(2)
whe e Re( )=2 ρ | ( )|/η is he Reynolds numbe o he
sphe e a ime ( h oughou his pape , a pai o e ical ba s
indica es hemagni udeo heenclosed ec o ).Ino he wo ds,
we will assume ha he hyd odynamic d ag o ce a any gi en
ins an is wha i would be i he sphe e we e mo ing uni o mly
wi h i s ins an aneous eloci y ( ). This assump ion is s ic ly
alid in he low- equency limi de ined by he condi ion
2ρ ω/η 1, whe e ω=2π/T is he undamen al angula
equency o he d i ing o ce (see, e.g., §24 o Re . [17]). In
his limi , he eloci y a ies so slowly in ime ha he low
can be ega ded as s eady a any gi en ins an . In p ac ice, he
low- equency limi can be achie ed by su icien ly dec easing
he d i ing equency and/o he size o he sphe e (bu keeping
i la ge han he B ownian size). In addi ion, i is mo e
easily achie ed in luids wi h high kinema ic iscosi ies η/ρ .
By way o example, in he case o a sphe e o adius =
10−4m mo ing in wa e , e hanol, o ai a no mal empe a u e
and p essu e (i.e., 20◦C and 1 a m), he low- equency limi
is alid o ω1.0×102 ad/s, ω1.5×102 ad/s, and
ω1.5×103 ad/s, espec i ely [23].
The use o Eqs. (1) and (2) implies ha he e ec s
o he i ual mass o ce and he Basse his o y o ce a e
negligible [24,25]. In addi ion, i he densi y o he sphe e,
ρs=3m/(4π 3), is less han o o he o de o ρ , he ine ial
e m m˙
( ) is also negligible, since i is p opo ional o
he i ual mass o ce −2π 3ρ ˙
( )/3 wi h p opo ionali y
cons an −2ρs/ρ . In ac , i can be shown (see Sec. III) ha ,
in dimensionless uni s, he ine ial e m is p opo ional o
he dimensionless pa ame e 2ρsω/η and, consequen ly, i is
negligible in he low- equency limi 2ρ ω/η 1i ρs/ρ
is less han o o he o de o uni y. Thus, o ine ial e ec s
o be non-negligible, i is necessa y o assume ha ρ /ρs1
(hea y-pa icle limi ).
Acco ding o Eq. (2), he d ag o ce is a nonlinea unc ion
o ( ). Mo e speci ically, he a io |Fd( )|/| ( )|inc eases wi h
inc easing | ( )|i , as will be assumed hence o h, | ( )|<
105η/( ρ ). This is so gi en ha ReCd(Re) is an inc easing
unc ion o Reun il he onse o he “d ag c isis,” which occu s
o Re≈2×105[25]. I is only in he limi o anishing
Reynoldsnumbe ha ReCd(Re) ends o24andEq.(2) educes
o he linea exp ession Fd,S ( )=−6π η ( ) (S okes’ law)
[17]. In p ac ice, he depa u e om S okes’ law is al eady
qui e signi ican o Reynolds numbe s o he o de o uni y o
e ensmalle (see, o ins ance,Fig.3.9inRe .[24]).The e o e,
o he nonlinea na u e o he d ag o ce o become appa en , i
is su icien ha | ( )|η/(2 ρ ); his is p ecisely he egime
o in e es in his wo k. Fo example, in he case conside ed
abo e, he nonlinea i y becomes signi ican when | ( )|
5.0×10−3m/s(wa e ),| ( )|7.6×10−3m/s (e hanol),
and | ( )|7.6×10−2m/s (ai )— eloci ies ha can be ea-
sonably achie ed in he labo a o y.
Gi en an ini ial condi ion o he eloci y a some ini ial
ins an 0, he alue o he eloci y a any la e ime >
0
can be calcula ed by in eg a ing he equa ion o mo ion (1). In
he Appendix i is shown ha , as happens in he linea case,
as he ime in e al − 0inc eases, he solu ions o Eq. (1)
become independen o he ini ial condi ions and con e ge
exponen ially o a single ime-dependen e minal eloci y,
which will be deno ed by V( ). Mo eo e , i is also shown
ha he elaxa ion ime o each his e minal eloci y is less
han, o o he same o de as, he cha ac e is ic imescale
τ=m/(6π η).
The ime-dependen e minal eloci y V( ) is uniquely
de e mined by he equa ion o mo ion (1). Consequen ly, i
Eq. (1) is in a ian unde some ans o ma ion, so will be
V( ). Fo example, using Eq. (2) and he ac ha F( )is
pe iodic, i is easy o e i y ha Eq. (1) is in a ian unde
he ans o ma ion ( )→ ( )= ( +T), in he sense ha
i emains unchanged i ( ) is eplaced by ( )= ( +T).
I hen ollows ha V( ) mus also be in a ian unde he same
ans o ma ion, i.e., V( )=V( )=V( +T). The e o e, he
ime-dependen e minal eloci y is pe iodic in ime wi h
he same pe iod Tas F( ). We a e in e es ed in s udying he
a e age e minal eloci y
V=1
TT
0
d V( )(3)
and, mo e speci ically, he condi ions o V o be nonze o.
Fi s ly, i should be poin ed ou ha he exis ence o
nonze o a e age e minal eloci ies is an unequi ocal signal
ha S okes’ law is no longe applicable. Indeed, i S okes’
law we eapplicable, hen he ime-dependen e minal eloci y
032219-2
DIRECTED MOTION OF SPHERES INDUCED BY … PHYSICAL REVIEW E 97, 032219 (2018)
would sa is y he linea di e en ial equa ion
˙
V( )=−V( )
τ+F( )
m.(4)
By in eg a ing he abo e equa ion om 0 o T, and aking in o
accoun he pe iodici y o V( ) and ha F( ) has ze o ime
a e age, i would necessa ily ollow ha V=0.
In addi ion o he iola ion o S okes’ law, he e is ano he
necessa y condi ion o he exis ence o nonze o a e age
e minal eloci ies. Le us assume ha he o ce F( ) ul ills
he ime-shi symme y
F( )=−F( +T/2).(5)
In his case, i can be easily e i ied ha he equa ion o
mo ion(1)isin a ian unde he ans o ma ion ( )→ ( )=
− ( +T/2), so V( +T/2) =−V( ). This las esul , o-
ge he wi h Eq. (3), implies ha V=0. Consequen ly, nonze o
a e age e minal eloci ies a e only possible i he ime-shi
symme y (5) is b oken.
As a pa icula ly illus a i e example, le us conside he
o ce F( )=F1( )e1+F2( )e2, whe e F1( ) and F2( )a e
pe iodic unc ions wi h pe iods T1=Tand T2=T/2, e-
spec i ely, and e1and e2a e wo mu ually pe pendicula uni
ec o s [22]. The pe iod o F( ) is, hus, equal o T. Le us
assume ha bo h F1( ) and F2( ) sa is y he a o emen ioned
ime-shi symme y, i.e., Fj( +Tj/2) =−Fj( ) o j=1,2.
Consequen ly, i each componen o F( ) we e conside ed
sepa a ely, he esul ing a e age e minal eloci y would be
ze o. By con as , i bo h componen s a e added oge he ,
a nonze o alue o Vis possible gi en ha F( +T/2) =
−F1( )e1+F2( )e2=−F( ). Mo eo e , in his case, he a -
e age e minal eloci y is necessa ily pa allel o e2. Indeed,
o he conside ed F( ), he equa ion o mo ion (1)isclea ly
in a ian unde he ans o ma ion ( )→ ( )= 1( )e1+
2( )e2− 3( )e3, whe e e3is a uni ec o pe pendicula o
bo h e1and e2. As a consequence, V3( )=−V3( )=0. Since
Eq. (1) is also in a ian unde he ans o ma ion ( )→
( )=− 1( +T/2)e1+ 2( +T/2)e2+ 3( +T/2)e3,i
hen ollows ha V( +T/2) =−V1( )e1+V2( )e2.Using
his las exp ession in Eq. (3), we ob ain ha , i V= 0, hen
i is necessa ily pa allel o e2[22]. In addi ion, we ha e also
p o ed ha he pe iod o V2( )isT/2.
III. ADIABATIC LIMIT
An explici exp ession o he a e age e minal eloci y can
be ob ained in he adiaba ic limi ωτ 1. No ice ha , in he
hea y-pa icle limi , he adiaba ic limi is mo e es ic i e han
he low- equency limi men ioned in he p e ious sec ion,
since 2ρ ω/η =9ρ ωτ/(2ρs)ωτ i ρ /ρs1. To s udy
he adiaba ic limi , we i s in oduce he dimensionless quan-
i ies θ=ω ,ν(θ)=2 ρ (θ/ω)/η, and (θ)=F(θ/ω)/F0,
wi h F0being a ypical alue o |F( )|such ha | (θ)|⩽1 o
all θ. The alue o he Reynolds numbe a he dimensionless
ime θis hus gi en by |ν(θ)|. In dimensionless a iables, he
equa ion o mo ion (1) becomes
ωτ dν(θ)
dθ =− 1
24 Cd[|ν(θ)|]|ν(θ)|ν(θ)+ 0 (θ),(6)
whe e 0=ρ F0/(3πη2) is a dimensionless pa ame e cha -
ac e izing he s eng h o he d i ing o ce.
The adiaba ic limi o ν(θ), deno ed he ea e as νad(θ), can
be ound by sol ing he equa ion
Cd[|νad(θ)|]|νad(θ)|νad(θ)=24 0 (θ),(7)
which is ob ained by aking he limi ωτ →0inEq.(6). Once
Eq. (7) is sol ed and νad(θ) is known, we can immedia ely
de e mine he adiaba ic limi o he a e age e minal eloci y,
Vad, using he exp ession
Vad =η
4π ρ 2π
0
dθ νad(θ),(8)
which is jus Eq. (3) ew i en in e ms o dimensionless
quan i ies.
In o de o sol e Eq. (7) and ob ain νad(θ), an explici
exp ession o he s eady d ag coe icien Cd(Re) is equi ed.
Fo he p esen pu poses, we will use he semiempi ical
exp ession
Cd(Re)=24
Re1+√Re
δ02
,(9)
wi h δ0=9.06 (see Re . [26] o a heu is ic de i a ion). This
exp ession is in ema kable ag eemen wi h he expe imen s
o Re5×103[26]. Consequen ly, in he case unde con-
side a ion, i s use is jus i ied p o ided ha |νad(θ)|5×103
o all θ.
A e inse ing Eq. (9) in o Eq. (7), one ob ains
1+√|νad(θ)|
δ02
νad(θ)= 0 (θ).(10)
I ollows om he abo e equa ion ha he ec o νad(θ)
poin s in he same di ec ion as he uni ec o (θ)/| (θ)|.
Fu he mo e,i also ollows ha hemagni udeo νad(θ)isaso-
lu iono he algeb aic equa ion [1 +|νad(θ)|1/2/δ0]2|νad(θ)|=
0| (θ)|. Thus, aking in o accoun ha |νad(θ)|mus be eal
and non-nega i e, i is s aigh o wa d o show ha he only
physically meaning ul solu ion o Eq. (10)is
νad(θ)=δ2
0
4⎡
⎣−1+1+4√ 0| (θ)|
δ0⎤
⎦
2
(θ)
| (θ)|.(11)
This las exp ession also allows us o es ima e he maximum
alue o 0consis en wi h he condi ion |νad(θ)|5×103.
Indeed, since | (θ)|⩽1 o all θ, i is hen clea ha |νad(θ)|⩽
δ2
0[−1+(1 +4 1/2
0/δ0)1/2]2/4. F om his inequali y i can
be eadily e i ied ha he condi ion |νad(θ)|5×103is
au oma ically ul illed o all θi 03.87 ×105. Fo la ge
0 alues he use o Eq. (9) is no longe jus i ied, and an
al e na i e exp ession o he s eady d ag coe icien mus be
used (see, e.g., Re . [24] o a discussion o possible choices).
Finally, acco ding o Eq. (8), he adiaba ic limi o he
a e age e minal eloci y, Vad, can be calcula ed by e alua ing
he in eg al
Vad =δ2
0η
16π ρ 2π
0
dθ⎡
⎣−1+1+4√ 0| (θ)|
δ0⎤
⎦
2
(θ)
| (θ)|.
(12)
032219-3
JESÚS CASADO-PASCUAL PHYSICAL REVIEW E 97, 032219 (2018)
As shown in he ollowing sec ion, e en hough 2π
0dθ (θ)=
0, he alueo Vad ob ained om he abo e exp ession is in
gene al nonze o.
IV. RESULTS
In his sec ion, we illus a e ou esul s using he pa icula
case o a dimensionless biha monic o ce o he o m
(θ)=ζcos (θ)e1+(1 −ζ) cos (2θ+ϕ)e2,(13)
whe e ζis a pa ame e ha allows us o simul aneously a y
he ampli udes o he wo ha monic componen s, and ϕis
he phase di e ence be ween hem. The pa ame e ζ akes
alues be ween 0 and 1, wi h ζ=0 and ζ=1 co esponding,
espec i ely, o monoch oma ic o ces along he di ec ions o
he uni ec o s e2and e1.
In o de o de e mine he ime-dependen e minal eloc-
i y, we ha e nume ically in eg a ed he di e en ial equa ion
ob ained by subs i u ing Eqs. (9) and (13) in o Eq. (6), wi h
he ini ial condi ion ν(θ0)=0. The pa ame e θ0has been
chosen o be nega i e and much la ge in magni ude han he
dimensionless elaxa ion ime ωτ, so as o ensu e ha he
asymp o ic ime-pe iodic egime has been eached o θ⩾
0. Once he ime-dependen e minal eloci y is known, he
a e age e minal eloci y can be easily compu ed by e alua ing
nume ically he in eg al appea ing in Eq. (3).
In Fig. 1, he me hod desc ibed abo e has been used o de-
e mine he dependence o he dimensionless e minal eloci y
2 ρ V/η on he dimensionless ime θ o h ee alues o ωτ,
namely, ωτ =0.1 (do ed lines), ωτ =1.1 (dashed lines), and
ωτ =2.1 (do -dashed lines). Only he i s wo componen s o
2 ρ V/η ha e been plo ed, since he hi d one is iden ically
ze o. The esul s ob ained by using he adiaba ic exp ession
in Eq. (11) a e indica ed wi h solid lines. The alues o he
emaining pa ame e s a e 0=100, ζ=0.5, and ϕ=π.We
ha e chosen a ela i ely la ge alue o 0in o de o highligh
he e ec o nonlinea i y. Howe e , his does no necessa ily
mean ha he magni ude o he d i ing o ce is also la ge. Fo
-20
-10
0
10
20
0
-20
-10
0
10
20
θ
π/2π3π/22π
2 ρ V2/η2 ρ V1/η
Adiaba ic limi
ωτ =0.1
ωτ =1.1
ωτ =2.1
FIG. 1. Dependence o he componen s o he dimensionless
ime-dependen e minal eloci y 2 ρ V/η on he dimensionless ime
θ o ωτ =0.1, 1.1, and 2.1. The esul s ob ained by using he
adiaba ic exp ession in Eq. (11) a e depic ed wi h solid lines. The
emaining pa ame e alues a e 0=100, ζ=0.5, and ϕ=π.
example, i he luid is ai a no mal empe a u e and p essu e,
i can be easily e i ied om he de ini ion o 0 ha he
alue 0=100 co esponds o a alue o F0o app oxima ely
2.6×10−7N. In he case o a solid i on sphe e o adius
=10−4m, his is abou 80.4% o i s weigh on Ea h.
The esul s in Fig. 1con i m ha he i s componen o
2 ρ V/η (bo om panel) e e ses sign e e y hal pe iod and
ha he second componen ( op panel) has hal he pe iod o
he i s , as was shown a he end o Sec. II. Fu he mo e, i
is obse ed ha he ag eemen be ween he analy ical esul s
ob ained om Eq. (11) and he nume ical esul s is qui e
good o he lowes dimensionless equency, ωτ =0.1, bu
becomes p og essi ely wo se as ωτ inc eases, as is o be
expec ed in an adiaba ic app oxima ion. Mo e speci ically,
wi h inc easing he alue o ωτ, he cu es shi o he igh
and he ampli udes o he oscilla ions dec ease.
A his poin , i is con enien o ecall ha , as poin ed ou
in Sec. III, he adiaba ic limi is mo e es ic i e han he
low- equency limi i ρ /ρs1; as a esul , in he hea y-
pa icle limi , he low- equency ange ex ends beyond he
adiaba ic egime.Toillus a e his ac ,le usconside again he
example o a solid i on sphe e o adius =10−4m imme sed
in ai a no mal empe a u e and p essu e. In his case, i
is easy o e i y ha ρ /ρs≈1.5×10−4and 2ρ ω/η =
9ρ ωτ/(2ρs)≈6.9×10−4ωτ. F om his las exp ession, i is
clea ha he h ee alues o ωτ conside ed in Fig. 1a e in he
low- equency ange. Howe e , as can be seen in Fig. 1, he
adiaba ic app oxima ion is no longe alid o ωτ =1.1 and
ωτ =2.1.
Figu e 2depic s he dependence o he second componen
o 2 ρ V/η on he ela i e phase ϕ o he same alues o ωτ,
0, and ζas in Fig. 1. In his and he ollowing igu es, he
i s componen o 2 ρ V/η is no shown as i is iden ically
ze o (see he p oo in Sec. II). We ha e es ic ed he alues o
ϕ o he in e al [0,2π] since he a e age e minal eloci y is
2πpe iodic in ϕ. This pe iodici y ollows om he in a iance
o he dynamics unde he ans o ma ion ϕ→ ϕ+2π[see
Eqs. (6) and (13)]. Again, he adiaba ic app oxima ion (solid
0
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
π/2π3π/22π
2 ρ V2/η
ϕ
Adiaba ic limi
ωτ =0.1
ωτ =1.1
ωτ =2.1
FIG. 2. Dependence o he second componen o he dimension-
less a e age e minal eloci y 2 ρ V/η on he ela i e phase ϕ o
ωτ =0.1, 1.1, and 2.1. The esul ob ained in he adiaba ic limi by
using Eq. (12) is depic ed wi h a solid line. The emaining pa ame e
alues a e 0=100 and ζ=0.5.
032219-4
DIRECTED MOTION OF SPHERES INDUCED BY … PHYSICAL REVIEW E 97, 032219 (2018)
line) p o ides a e y sa is ac o y desc ip ion o he nume ical
esul s o he lowes dimensionless equency (do ed line).
A close examina ion o Fig. 2 e eals ha he unc ion
V2(ϕ), as well as i s adiaba ic limi Vad,2(ϕ), sa is ies he
phase-shi symme y
V2(ϕ+π)=−
V2(ϕ).(14)
To elucida e he o igin o his symme y, obse e ha eplacing
ϕby ϕ+πin Eq. (13) is equi alen o changing he sign o
he second componen o (θ). Using his ac , i is easy o
e i y ha he equa ion o mo ion (1) is in a ian unde he
ans o ma ion ( ,ϕ)→ ( ,ϕ)= 1( ,ϕ +π)e1− 2( ,ϕ +
π)e2+ 3( ,ϕ +π)e3, whe e he dependence o he eloci y
on he phase di e ence ϕhas been explici ly indica ed. Since
he ime-dependen e minal eloci y is uniquely de e mined
by Eq. (1), i is clea ha V( ,ϕ)=V1( ,ϕ +π)e1−V2( ,ϕ +
π)e2. This, oge he wi h Eq. (3), leads o Eq. (14). No e ha ,
acco ding o his easoning, Eq. (14) is alid independen ly o
he alue o ωτ and, in pa icula , in he limi ωτ →0 [i.e.,
when V2(ϕ) is eplaced by Vad,2(ϕ)].
The esul s in Fig. 2also e eal ha he adiaba ic limi o
V2(ϕ) sa is ies he symme y ela ion
Vad,2(ϕ)=Vad,2(−ϕ),(15)
whe eas ou side he adiaba ic egime, gene ally V2(ϕ)=
V2(−ϕ).Tounde s andwhy hisisso,obse e ha he unc ion
(θ)inEq.(13) is unchanged i he sign o bo h θand ϕis
swi ched. F om his, i eadily ollows ha Eq. (7) is in a ian
unde he ime-and-phase- e e sal ans o ma ion νad(θ,ϕ)→
ν
ad(θ,ϕ)=νad(−θ, −ϕ) and, consequen ly, ha νad(θ,ϕ)=
νad(−θ, −ϕ). A e an app op ia e change o a iables, his
las exp ession, oge he wi h Eq. (8),leads o Eq.(15). Ou side
he adiaba ic egime, howe e , his a gumen ails because he
in a iance o Eq. (6) unde ime-and-phase e e sal is b oken
by he p esence o he i s -o de ime de i a i e.
Thesymme y ela ion(15) is bu a special case o he mo e
gene al ela ion
Vad,2(nπ/2+ϕ)=(−1)nVad,2(nπ/2−ϕ),(16)
alid o any in ege n.Top o eEq.(16), i su ices o
no e ha , acco ding o Eq. (14), he igh -hand side o he
equali y Vad,2(nπ/2+ϕ)=Vad,2(−nπ/2−ϕ) is equal o
(−1)nVad,2(nπ/2−ϕ). By se ing ϕ=0inEq.(16), we
conclude ha
Vad,2(nπ/2)=0 (17)
i nis odd. The p ope ies (16) and (17) a e isible in Fig. 2.
As a inal commen on Fig. 2, i is wo h men ioning ha he
cu es shown can be e y well i ed by an exp ession o he
o m A1cos(ϕ+χ1)+A3cos(3ϕ+χ3), whe e A1,χ1,A3,
and χ3a e i ing pa ame e s which depend on he alue o ωτ;
he i ingcu esa eno shownin he igu e,as heya e isually
indis inguishable om he o iginal ones. The unc ional o m
o his i ing unc ion is a di ec consequence o he sys em
symme ies, being independen o he de ails o he dynamics
(see, e.g., Re s. [27,28]). In pa icula , in he adiaba ic limi ,
i is only necessa y o calcula e wo i ing pa ame e s since,
acco ding o Eq. (15), χ1and χ3can be chosen o be ze o.
In Fig. 3, he dependence o 2 ρ V2/η on he pa ame e ζ
is shown o he same alues o ωτ, 0, and ϕas in Fig. 1.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
ζ
2 ρ V2/η
Adiaba ic limi
ωτ =0.1
ωτ =1.1
ωτ =2.1
FIG. 3. Dependence o he second componen o he dimen-
sionless a e age e minal eloci y 2 ρ V/η on he pa ame e ζ o
ωτ =0.1, 1.1, and 2.1. The esul ob ained in he adiaba ic limi by
using Eq. (12) is depic ed wi h a solid line. The emaining pa ame e
alues a e 0=100 and ϕ=π.
The nume ical esul s ob ained o ωτ =0.1 (do ed line) a e
indis inguishable om hose p o ided by he adiaba ic exp es-
sion (12) (solid line). No ice ha , independen ly o he alue
o ωτ, he a e age e minal eloci y anishes o ζ=0 and
ζ=1. This is so because he dimensionless o ce in Eq. (13)
sa is ies he ime-shi symme ies (θ+π/2) =− (θ), i ζ=
0, and (θ+π)=− (θ), i ζ=1 (see discussion a he end
o Sec. II). The cu es in Fig. 3also e eal ha , o ixed
alues o he o he pa ame e s, he e exis s an op imal alue
o ζwhich maximizes he second componen o he a e age
e minal eloci y. Fu he mo e, as ωτ inc eases, he maximum
eloci y dec eases and i s loca ion shi s owa d lowe alues
o ζ.
I should be no ed he e ha , in he lowes o de , he gene al
o malism de eloped in Re s. [27,28] leads o he app oxima e
exp ession V2(ζ)≈Cζ2(1 −ζ), whe e Cis independen o
ζ. This exp ession anishes a ζ=0 and ζ=1, and displays
a maximum a ζ=2/3, hus quali a i ely esembling he
beha io seen in Fig. 3. Howe e , i is unable o accoun o
he dependence o he loca ion o he maximum eloci y on
ωτ. This de iciency is no su p ising, gi en ha he abo e
app oxima ion is expec ed o be accu a e only o small alues
o 0and, in Fig. 3, we ha e aken 0=100.
In Fig. 4, we plo he dimensionless a e age e minal
eloci y 2 ρ V2/η as a unc ion o he dimensionless d i ing
s eng h 0 o he same alues o ωτ,ϕ, and ζas in Fig. 1.We
ha e limi ed he ange o 0 o alues well below 3.87 ×105,
so as o ensu e he applicabili y o Eq. (9) (see Sec. III).
In pa icula , he la ge panel shows he esul s ob ained o
alues o 0 a ying om 0 o 104, while he inse zooms
in on he ange 0 ⩽ 0⩽400. Again he nume ical esul s
ob ained o ωτ =0.1 (do ed line) a e indis inguishable om
hose p o ided by he adiaba ic exp ession (12) (solid line). A
glance a he la ge panel migh emp one o conclude ha , o
he pa ame e alues conside ed, an inc ease in ωτ causes an
inc ease in 2 ρ V2/η. Howe e , his conclusion is e u ed by
he da a shown in he inse .
032219-5

JESÚS CASADO-PASCUAL PHYSICAL REVIEW E 97, 032219 (2018)
0 2000 4000 6000 8000 10000
0
5
10
15
20
25
30
0 100 200 300 400
0
0.5
1
1.5
2
0
0
2 ρ V2/η
2 ρ V2/η
Adiaba ic limi
ωτ =0.1
ωτ =1.1
ωτ =2.1
FIG. 4. Dependence o he second componen o he dimension-
less a e age e minal eloci y 2 ρ V/η on he pa ame e 0 o
ωτ =0.1, 1.1, and 2.1. The esul ob ained in he adiaba ic limi by
using Eq. (12) is depic ed wi h solid lines. The emaining pa ame e
alues a e ζ=0.5andϕ=π. The inse shows a zoomed-in iew o
he cu es in he ange 0 ⩽ 0⩽400.
In he case shown in Fig. 4 he a e age e minal eloci y
is an inc easing unc ion o 0.F omEq.(14) i is e iden
ha , i we had used ϕ=0 ins ead o ϕ=π, we would
ha e obse ed ha he a e age e minal eloci y dec eases
mono onically wi h 0. The ques ion hen a ises as o whe he
he e a e pa ame e alues o which he a e age e minal
eloci y exhibi s a nonmono onic dependence on 0.The
answe o his ques ion is a i ma i e, as can be seen in Fig. 5
o ϕ=π/2 and ωτ =1.1 and ωτ =2.1. No ice ha he
nonmono onic beha io is accompanied by he appea ance o
a cu en e e sal as a unc ion o he dimensionless d i ing
s eng h 0. As he alue o he dimensionless equency ωτ
dec eases, he posi ion o he cu en e e sal shi s owa d
0=0, becoming indis inguishable om ze o o ωτ =0.1.
This ype o cu en e e sal is qui e common in unde damped
ocking a che s (see, e.g., Re s. [10,29]). The abo e esul s
clea ly show ha , when nonlinea ic ion is p esen , cu en
0 2000 4000 6000 8000 10000
-5
-4
-3
-2
-1
0
0 100 200 300 400
-0.2
0
0.2
0.4
0
0
2 ρ V2/η
2 ρ V2/η
Adiaba ic limi
ωτ =0.1
ωτ =1.1
ωτ =2.1
FIG. 5. The same as in Fig. 4bu now ϕ=π/2.
e e sals can be obse ed e en in he absence o any pe iodic
subs a e po en ial.
V. CONCLUSIONS
In his pape , a heo e ical s udy o he mo ion o a sphe e
imme sed in a iscous luid and subjec ed o a ime-pe iodic
o ce o ze o a e age has been p esen ed. Ou ocus has been
on si ua ions in which S okes’ law is no applicable, so he d ag
o ce depends nonlinea ly on he eloci y o he sphe e ela i e
o he luid. Le us summa ize he main esul s o his wo k.
(i) I has been shown ha , when he ime-shi symme y
(5) is b oken, he combined ac ion o he ze o-mean oscilla ing
o ce and he nonlinea d ag o ce is able o induce a di ec ed
mo ion o he sphe e. Unlike in he adi ional ocking a che ,
in his case he di ec ed mo ion eme ges in he absence o any
pe iodic subs a e po en ial.
(ii) Explici exp essions o he e minal eloci y and he a -
e age e minal eloci y ha e been de i ed wi hin he adiaba ic
app oxima ion. A compa ison be ween he p edic ions o hese
exp essionsand he esul sob ainedbynume icallysol ing he
equa ion o mo ion has been ca ied ou . As expec ed, i has
been ound ha he lowe he equency o he d i ing o ce,
he mo e accu a e he adiaba ic app oxima ion becomes.
(iii) By way o example, he case o wo mu ually pe -
pendicula o ces wi h sinusoidal ime dependence has been
conside ed. Al hough nei he o hese wo o ces induces
di ec ed mo ion when ac ing sepa a ely, i has been shown ha
he esul an o ce ob ained by adding hem oge he causes a
ne mo ion o he sphe e along he di ec ion o he o ce wi h
he sho es pe iod [22].
(i ) A de ailed analysis o he dependence o he a e age
e minal eloci y on he sys em pa ame e s has been made
and some aspec s o he obse ed phenomenology, such as he
supp essiono anspo o pa icula alueso he pa ame e s,
ha e been a ionalized using symme y a gumen s.
( ) A ema kable inding o his analysis is ha , o
app op ia e pa ame e alues, he a e age e minal eloci y
exhibi s a nonmono onic beha io as a unc ion o he o cing
s eng h, esul ing in he appea ance o cu en e e sal. This
kind o beha io esembles ha obse ed in unde damped
ocking a che s [10,29].
A na u al ex ension o his wo k would be o empi ically
e i y he heo e ical esul s epo ed he e. I is hoped ha
he p esen pape will p o ide he s imulus o do expe imen al
esea ch in his a ea.
ACKNOWLEDGMENTS
I hank he Jun a de Andalucía o unding suppo . I also
wan o hank M. L. Oli e a-A encio, N. R. Quin e o, R.
Al a ez-Noda se, J. A. Cues a, and A. T. Pé ez o hei c i ical
eading o he manusc ip and ui ul discussions.
APPENDIX: CONVERGENCE OF THE SOLUTIONS OF
EQ. (1) TO THE TIME-DEPENDENT
TERMINAL VELOCITY
Le ( ) and ( ) be wo solu ions o he equa ion o
mo ion (1) co esponding o he ini ial condi ions ( 0)= 0
032219-6
DIRECTED MOTION OF SPHERES INDUCED BY … PHYSICAL REVIEW E 97, 032219 (2018)
and ( 0)= 
0= 0. By using Eqs. (1) and (2), i is no ha d
o e i y ha
d
d | ( )− ( )|2=−( )| ( )− ( )|2−( ),(A1)
whe e
( )=π η
4m{Re( )Cd[Re( )] +R
e( )Cd[R
e( )]}(A2)
and
( )=3πη3
2m ρ2
{[Re( )]2−[R
e( )]2}
×{Re( )Cd[Re( )] −R
e( )Cd[R
e( )]},(A3)
wi h Re( )=2 ρ | ( )|/η and R
e( )=2 ρ | ( )|/η.
As men ioned in Sec. II, in he ange o Reynolds numbe s
conside ed in his wo k, ReCd(Re) is an inc easing unc ion o
Re. Thus, om Eq. (A3) i ollows ha ( )⩾0. In addi ion,
since limRe→0ReCd(Re)=24, i is clea om Eq. (A2) ha
( )⩾12π η/m. Using hese esul s, Eq. (A1) leads o he
inequali y
d
d | ( )− ( )|2⩽−2
τ| ( )− ( )|2,(A4)
wi h τ=m/(6π η). I hen ollows om G onwall’s inequal-
i y (see, e.g., Re . [30]) ha
| ( )− ( )|2⩽| 0− 
0|2e−2( − 0)/τ .(A5)
The e o e, as he ime in e al − 0inc eases, he solu ions
o Eq. (1) become independen o he ini ial condi ions and
con e ge exponen ially o a single ime-dependen e minal
eloci y. Fu he mo e, acco ding o Eq. (A5), he elaxa ion
ime o each his e minal eloci y is less han, o o he same
o de as, he cha ac e is ic imescale τ=m/(6π η).
[1] R. D. As umian, The modynamics and kine ics o a B ownian
mo o , Science 276,917 (1997); R. D. As umian and P. Hänggi,
B ownian mo o s, Phys. Today 55(11), 33 (2002); P. Hänggi, F.
Ma chesoni,andF.No i, B ownianmo o s,Ann.Phys.(Leipzig)
14,51 (2005); D. Cube o and F. Renzoni, B ownian a che s:
F om s a is ical physics o bio and nano-mo o s (Camb idge
Uni e si y P ess, Camb idge, 2016).
[2] P. Reimann, B ownian mo o s: Noisy anspo a om equilib-
ium, Phys. Rep. 361,57 (2002).
[3] P. Hänggi and F. Ma chesoni, A i icial B ownian mo o s:
Con olling anspo on he nanoscale, Re . Mod. Phys. 81,387
(2009).
[4] L. Mahade an and P. Ma sudai a, Mo ili y powe ed by
sup amolecula sp ings and a che s, Science 288,95 (2000);
R. Di Leona do, L. Angelani, D. Dell’A cip e e, G. Ruocco, V.
Iebba,S.Schippa,M.P.Con e, F. Meca ini,F. DeAngelis, andE.
DiFab izio,Bac e ial a che mo o s,P oc.Na l. Acad.Sci. USA
107,9541 (2010); D. Chowdhu y, S ochas ic mechano-chemical
kine ics o molecula mo o s: A mul idisciplina y en e p ise
om a physicis ’s pe spec i e, Phys. Rep. 529,1(2013).
[5] A. an Oudenaa den and S. G. Boxe , B ownian a che s:
Molecula sepa a ions in lipid bilaye s suppo ed on pa e ned
a ays, Science 285,1046 (1999); H. Linke, T. E. Humph ey,
A. Lo g en, A. O. Sushko , R. Newbu y, R. P. Taylo , and
P. Omling, Expe imen al unneling a che s, ibid. 286,2314
(1999); V. Se eli, C. F. Lee, E. R. Kay, and D. A. Leigh,
A molecula in o ma ion a che , Na u e (London) 445,523
(2007); C. D exle , S. A. Ta asenko, P. Olb ich, J. Ka ch, M.
Hi me , F. Mülle , M. Gmi a, J. Fabian, R. Yakimo a, S.
La a-A ila, S. Kuba kin, M. Wang, R. Vaj ai, P. M. Ajayan, J.
Kono, and S. D. Ganiche , Magne ic quan um a che e ec in
g aphene, Na . Nano echnol. 8,104 (2013).
[6] D. C. Rapapo , The wonde ul wo ld o g anula a che s,
Compu . Phys. Commun. 147,141 (2002); G. Cos an ini, U.
Ma ini Be olo Ma coni, and A. Puglisi, G anula B ownian
a che model, Phys.Re .E75,061124 (2007); V. Be a di, J.
Lydon, P. G. Ke ekidis, C. Da aio, and R. Ca e e o-González,
Di ec ed a che anspo in g anula chains, ibid. 88,052202
(2013).
[7] F.Ma chesoni,The malRa che sin1+1Dimensions,Phys.Re .
Le . 77,2364 (1996); M. Sale no and N. R. Quin e o, Soli on
a che s, Phys. Re . E 65,025602 (2002); L. Mo ales-Molina,
F. G. Me ens, and A. Sánchez, Soli on a che s ou o poin -
like inhomogenei ies, Eu .Phys.J.B37,79 (2004); Ra che
beha io innonlinea Klein-Go donsys emswi hpoin likeinho-
mogenei ies, Phys. Re . E 72,016612 (2005); B. Sánchez-Rey,
J. Casado-Pascual, and N. R. Quin e o, Kink a che induced
by a ime-dependen symme ic ield po en ial, ibid. 94,012221
(2016).
[8] M. Sale no and Y. Zolo a yuk, Soli on a che like dynamics by
ac o ces wi hha monic mixing,Phys.Re .E65,056603(2002).
[9] R. Ba ussek, P. Hänggi, and J. G. Kissne , Pe iodically ocked
he mal a che s, Eu ophys. Le . 28,459 (1994).
[10] P. Jung, J. G. Kissne , and P. Hänggi, Regula and Chao ic
T anspo in Asymme ic Pe iodic Po en ials: Ine ia Ra che s,
Phys.Re .Le .76,3436 (1996);J. L.Ma eos,Chao icT anspo
and Cu en Re e sal in De e minis ic Ra che s, ibid. 84,258
(2000).
[11] A. Sa mien o and H. La alde, De e minis ic anspo in a ch-
e s, Phys. Re . E 59,4878 (1999); D. Cube o, J. Casado-
Pascual, A. Al a ez, M Mo illo, and P. Hänggi, O e damped
de e minis ic a che s d i en by mul i equency o ces, Ac a
Phys. Pol. B 37, 1467 (2006); J. Casado-Pascual, D. Cube o,
and F. Renzoni, Uni e sal asymp o ic beha io in nonlinea
sys ems d i en by a wo- equency o cing, Phys. Re . E 88,
062919 (2013); D. Cube o, J. Casado-Pascual, and F. Renzoni,
I a ionali y and Quasipe iodici y in D i en Nonlinea Sys ems,
Phys. Re . Le . 112,174102 (2014).
[12] F. Ma chesoni, Ha monic mixing signal: Doubly di he ed ing
lase gy oscope, Phys. Le . A 119,221 (1986).
[13] S. Flach, O. Ye ushenko, and Y. Zolo a yuk, Di ec ed Cu en
due o B oken Time-Space Symme y, Phys. Re . Le . 84,
2358 (2000); S. Deniso , S. Flach, A. A. O chinniko , O.
Ye ushenko, and Y. Zolo a yuk, B oken space- ime symme-
ies and mechanisms o ec i ica ion o ac ields by nonlinea
(non)adiaba ic esponse, Phys.Re .E66,041104 (2002).
[14] P. Reimann, Supe symme ic Ra che s, Phys.Re .Le .86,4992
(2001).
032219-7
JESÚS CASADO-PASCUAL PHYSICAL REVIEW E 97, 032219 (2018)
[15] D. Cube o and F. Renzoni, Hidden Symme ies, Ins abili ies,
and Cu en Supp ession in B ownian Ra che s, Phys.Re .Le .
116,010602 (2016).
[16] H. Hagman, M. Zelan, and C. M. Dion, B eaking he symme y
o a B ownian mo o wi h symme ic po en ials, J. Phys. A:
Ma h. Theo . 44,155002 (2011).
[17] L. D. Landau and E. M. Li shi z, Cou se o heo e ical physics.
Fluid mechanics, 2nd ed. (Pe gamon, Ox o d, 2003).
[18] F. Schwei ze , W. Ebeling, and B. Tilch, Complex Mo ion o
B ownian Pa icles wi h Ene gy Depo s, Phys. Re . Le . 80,
5044 (1998); A. Fiascona o, W. Ebeling, and E. Gudowska-
Nowak, Ac i e B ownian mo ion models and applica ions o
a che s, Eu . Phys. J. B 65,403 (2008); P. Romanczuk and L.
Schimansky-Geie , B ownian Mo ion wi h Ac i e Fluc ua ions,
Phys. Re . Le . 106,230601 (2011).
[19] J.Dunkeland P. Hänggi,Theo yo ela i is ic B ownianmo ion:
The (1+1)-dimensional case, Phys. Re . E 71,016124 (2005);
Theo y o ela i is ic B ownian mo ion: The (1+3)-dimensional
case, 72,036106 (2005); Rela i is ic B ownian mo ion, Phys.
Rep. 471,1(2009).
[20] P. G.de Gennes,B ownianmo ionwi hd y ic ion,J.S a . Phys.
119,953 (2005); A. Po o sky and F. Ma chesoni, Pe iodically
d i en B ownian mo ion wi h d y ic ion and ul a ela i is ic
Lange in equa ions, Phys.Re .E87,032132 (2013).
[21] Y. L. Bolo in, A. V. Tu , and V. V. Yano sky, Nonlinea ic ion
as a mechanism o di ec ed mo ion, Tech. Phys. 47,803 (2002).
[22] S. Deniso , Y. Zolo a yuk, S. Flach, and O. Ye ushenko,
Vo ex and T ansla ional Cu en s due o B oken Time-Space
Symme ies, Phys.Re .Le .100,224102 (2008); V. Lebede
andF. Renzoni,Two-dimensional ocking a che o colda oms,
Phys.Re .A80,023422 (2009).
[23] The iscosi y and densi y da a used in he p esen pape we e
ob ained om Wol am|Alpha.
[24] R. Cli , J. R. G ace, and M. E. Webe , Bubbles, d ops, and
pa icles (Academic, New Yo k, 1978).
[25] C. T. C owe, T. R. T ou , and J. N. Chung, Pa icle in e ac ion
wi h o ices, in Fluid o ices ( luid mechanics and i s applica-
ions), edi ed by Sheldon I. G een (Sp inge -Science+Business
MediaB. V.,Do d ech , 1995),Vol. 30,Chap. XIX,pp. 829–861.
[26] Fa id F. Ab aham, Func ional dependence o d ag coe icien o
a sphe e on Reynolds numbe , Phys. Fluids 13,2194 (1970).
[27] J. A. Cues a, N. R. Quin e o, and R. Al a ez-Noda se, Time-
Shi In a iance De e mines he Func ional Shape o he Cu en
in Dissipa i e Rocking Ra che s, Phys.Re .X3,041014 (2013).
[28] J. Casado-Pascual, J. A. Cues a, N. R. Quin e o, and R. Al a ez-
Noda se, Gene al app oach o dealing wi h dynamical sys ems
wi h spa io empo al pe iodici ies, Phys. Re . E 91,022905
(2015).
[29] D. Cube o, V. Lebede , and F. Renzoni, Cu en e e sals in a
ocking a che : Dynamical e sus symme y-b eaking mecha-
nisms, Phys. Re . E 82,041116 (2010); A. Wickenb ock, D.
Cube o, N. A. Abdul Wahab, P. Phoon hong, and F. Renzoni,
Cu en e e sals in a ocking a che : The equency domain,
ibid. 84,021127 (2011).
[30] N. A. A. Wahab, In ini e-dimensional dynamical sys ems in
mechanics and physics (Sp inge -Ve lag, New Yo k, 1988).
032219-8