Gene a ion o Localized Modes in an Elec ical La ice Using Subha monic D i ing
L. Q. English,
1
F. Palme o,
2
P. Candiani,
1
J. Cue as,
2
R. Ca e e o-Gonza
´lez,
3
P. G. Ke ekidis,
4
and A. J. Sie e s
5
1
Depa men o Physics and As onomy, Dickinson College, Ca lisle, Pennsyl ania 17013, USA
2
Nonlinea Physics G oup, Escuela Te
´cnica Supe io de Ingenie ı
´a In o ma
´ ica, Depa amen o de Fı
´sica Aplicada I,
Uni e sidad de Se illa, A enida Reina Me cedes, s/n, 41012-Se illa, Spain
3
Nonlinea Dynamical Sys ems G oup, Depa men o Ma hema ics and S a is ics, and Compu a ional Science Resea ch Cen e ,
San Diego S a e Uni e si y, San Diego, Cali o nia 92182-7720, USA
4
Depa men o Ma hema ics and S a is ics, Uni e si y o Massachuse s, Amhe s , Massachuse s 01003-4515, USA
5
Labo a o y o A omic and Solid S a e Physics, Co nell Uni e si y, I haca, New Yo k 14853, USA
(Recei ed 10 June 2011; published 22 Feb ua y 2012)
We show expe imen ally and nume ically ha an in insic localized mode (ILM) can be s ably p oduced
(and expe imen ally obse ed) ia subha monic, spa ially homogeneous d i ing in he con ex o a
nonlinea elec ical la ice. The p ecise nonlinea spa ial esponse o he sys em has been seen o depend
on he ela i e loca ion in equency be ween he d i e equency, !d, and he bo om o he linea
dispe sion cu e, !0.I !d=2lies jus below !0, hen a single ILM can be gene a ed in a 32-node la ice,
whe eas, when !d=2lies wi hin he dispe sion band, a spa ially ex ended wa e o m esembling a ain o
ILMs esul s. To ou knowledge, and despi e i s appa en ly b oad ele ance, such an expe imen al
obse a ion o subha monically d i en ILMs has no been p e iously epo ed.
DOI: 10.1103/PhysRe Le .108.084101 PACS numbe s: 05.45.Y , 63.20.Pw, 63.20.Ry
I is well known ha a damped nonlinea oscilla o can
espond a i s in insic esonance equency when i is
d i en a a mul iple o ha equency. A di ec example
o such subha monic d i ing is p o ided by he d i en
Van de Pol oscilla o , whe e he a io o esponse o d i e
equency is exac ly 1=3[1,2]. Many o he nonlinea os-
cilla o s exhibi simila subha monic esonances ( he
Du ing oscilla o being ano he ex ensi ely s udied ex-
ample). In ac , subha monic esponse mus be seen as a
ai ly gene ic p ope y o nonlinea oscilla o s.
Al e na i ely, a nonlinea oscilla o wi h a pa ame e
modula ed a a pa icula equency can also espond a a
ac ion o ha equency in wha is called pa ame ic
exci a ion.
Wha happens when such nonlinea oscilla o s a e con-
nec ed o one ano he in a egula la ice? In nonlinea
la ices, an impo an gene ic phenomenon is he exis ence
o sel - apped localized modes, known as in insic local-
ized modes (ILMs) o disc e e b ea he s. Such a mode
ep esen s an exci a ion which is ( ypically exponen ially)
spa ially localized o e a limi ed ange o la ice nodes and
decays o ze o a om hese, and i is empo ally pe iodic.
In his ega d, i can be hough o as an analog o he
soli ons o con inuous media. Howe e , he disc e eness o
he la ice in oduces in e es ing a ia ions o he p oblem,
including, o ins ance, he ac ha ILMs may be dynami-
cally s able in any dimension. This has made ILMs ele an
exci a ions o a wide a ay o applica ions, including
supe conduc ing Josephson junc ions [3], pho onic c ys als
[4], biopolyme s [5], cha ge- ans e solids [6], an i e o-
magne s [7], and mic omechanical can ile e a ays [8],
among o he s [9].
He e, we blend hese wo b oadly signi ican aspec s o
nonlinea sys ems by add essing he ollowing ques ion:
can subha monic o pa ame ic exci a ions, which igu e so
p ominen ly in isola ed nonlinea oscilla o s, ca y o e o
he la ice se ing? Tha is, we examine whe he ILMs can
be gene a ed and, especially, s abilized by subha monic
and/o pa ame ic d i ing which is homogeneous in space.
So a , his ype o ques ion seems o ha e been conside ed
chie ly in he con ex o con inuous media [10], o o
pa ame ic d i ing [11–13], and has been p incipally heo-
e ical in na u e. In his Le e , we demons a e expe imen-
ally and co obo a e h ough heo e ical modeling and
nume ical compu a ion, and, when possible, in using ana-
ly ical insigh s, ha ILMs can indeed be gene a ed and
s abilized ia subha monic o cing.
The expe imen al sys em, shown in Fig. 1, is he bi-
induc ance elec ical band-pass il e o Re s. [14,15], and
he basic geome y and coupling o an ex e nal d i e is
gi en in Re s. [16–18]. This elec ical la ice becomes non-
linea by i ue o a diode (np junc ion) eplacing a adi-
ional capaci o in he uni cell. The ol age a each la ice
node is moni o ed a 0:4sin e als using a mul ichannel
analog- o-digi al con e e . The bounda y condi ions
a e pe iodic, and he main esul o subha monic ILM
L1
R
C
L2
Vn-1 VnVn+1
V( )
R
CL
2
V( )
V
FIG. 1. Le : Schema ic ci cui diag am o he elec ical ans-
mission line. Righ : Schema ic o a single elemen .
PRL 108, 084101 (2012) PHYSICAL REVIEW LETTERS week ending
24 FEBRUARY 2012
0031-9007=12=108(8)=084101(5) 084101-1 Ó2012 Ame ican Physical Socie y
gene a ion is ealized iden ically in la ge la ices han he
one used. We use ‘‘ la ’’—ze o ol age and cu en —ini ial
condi ions along he la ice. Fu he mo e, since he d i ing
is homogeneous ac oss he sys em, his s udy ela es o
Re s. [6–8] and mo e gene ally o nanoscale (e.g., an i e -
omagne s and cha ge- ans e solids) o e en mesoscale
[such as mic oelec omechanical sys em (MEMS) can ile-
e a ays o Josephson junc ions] applica ions whe e ex-
e nal ields appea homogeneous on he scale o he la ice.
Using basic ci cui heo y, he single elemen composed
o he pa allel combina ion o an induc o , L2, and a diode
(d i en ia a esis o ) is app oxima ely desc ibed by [18]
d
d ¼1
cð ÞcosðÞ
RþRl
Rl
þyiD;
dy
d ¼1
L2
; (1)
whe e RC0!0and he ollowing dimensionless a i-
ables ha e been used: ¼!0 ;iD¼ID=ð!0C0VdÞ; ¼
V=Vd, he dimensionless ol age; cð Þ¼CðVÞ=C0;¼
!d=!0; and !0¼1=ffiffiffiffiffiffiffiffiffiffiffi
L2C0
p.y ep esen s he no malized
cu en h ough he induc o , and CðVÞis he capaci ance o
he diode [18]. A phenomenological (and ampli ude-
dependen ) dissipa ion esis o , Rl, was included in he
model o be e app oxima e he expe imen al diode
dynamics.
When Nsuch oscilla o s a e coupled ia a second in-
duc o , Eq. (1) gene alizes o he la ice equa ions
cð nÞd n
d ¼cosðÞ
RþRl
Rl
nþyniDð nÞ;
dyn
d ¼L2
L1ð nþ1þ n12 nÞ n:(2)
The induc o L1is used o couple he uni cells, and L2
e e s o he induc o o he g ound wi hin each oscilla o
[18]. The a io o hese wo induc o s yields he e ec i e
‘‘disc e eness’’ o he sys em; in he limi o L1much
la ge (smalle ) han L2, he sys em can be iewed as
app oaching he con inuum (an icon inuum) limi . In ou
la ice, L1¼0:68 mH and L2¼0:33 mH, so ha we a e
clea ly no in he con inuum limi , al hough he la e is, in
p inciple, expe imen ally app oachable and ma hema i-
cally in e es ing in i s own igh .
In o de o in es iga e he o igin o hese subha monic
b ea he s, we ha e o examine in de ail he esponse o a
single uni cell o he elec ical la ice (i.e., an e ec i e
an icon inuum limi ). As shown in Fig. 2(a), he esponse is
a ypical nonlinea esonance cu e, as expec ed. Howe e ,
o a ange o equencies loca ed a abo e he linea
esonance cu e, he a ac o o he sys em, which oscil-
la es wi h equency d¼!d=2, expe iences a pe iod-
doubling bi u ca ion and a new, la ge (in ampli ude) a -
ac o and appea s wi h ¼ d=2(see he cu e bi u ca -
ing om poin s A and B in he igu e). Thus, o an in e al
o equencies beyond he op o he linea dispe sion cu e
o he ull elec ical la ice, wo di e en a ac o s, one
small wi h a equency ¼ d¼550 kHz and ano he
one, la ge and wi h a equency ¼ d=2¼275 kHz,
coexis , as illus a ed in Fig. 2(b). When we dec ease he
ol age ampli ude, Vd, he wo bi u ca ion poin s
(labeled A and B in he igu e) ge close and, o a ol age
Vd6:4Vin he model, collide and disappea , and no
subha monic esonance akes place. Expe imen ally, he
cu o ol age is ound a ound 6.2 V; o he wise, he nu-
me ical p edic ions ma ch expe imen al obse a ions
easonably well, especially gi en he model’s phenomeno-
logical ea men o he diodes.
Fu he mo e, in o de o ob ain an app oxima e subha -
monic solu ion co esponding o small ol ages, we can
(Taylo ) app oxima e Eq. (1)as
€
xþRRl
Rlð1þxþx2Þ_
xþxþx2
2¼VdsinðÞ
;
200 300 400 500 600 700 800
−1
0
1
2
3
(kHz)
V (Vol s)
(a)
AB
0246810
0
1
2
(µs)
V (Vol s)
(b)
FIG. 2 (colo online). Response o a uni cell a a d i e
ampli ude o Vd¼8V(in nume ical simula ions, we conside
a small equency shi o 25 kHz o quan i a i ely compa e wi h
he expe imen al cu es). Top: Nonlinea esonance cu es,
whe e g ey ( ed) do s co espond o expe imen al da a while
he solid and dashed black lines co espond, espec i ely, o
s able and uns able nume ical solu ions. Black ci cles show
pe iod-doubling bi u ca ion poin s. The inse zooms in on he
subha monic esponse, whe e he analy ical app oxima ion is
included (dash-do ed blue line), wi h a equency shi o 5 kHz.
Bo om: Coexis ing la ge and small a ac o s co esponding o
d¼!d=2¼550 kHz ob ained nume ically (black lines) and
expe imen ally [g ey ( ed) lines].
PRL 108, 084101 (2012) PHYSICAL REVIEW LETTERS week ending
24 FEBRUARY 2012
084101-2
whe e V¼ðxþx2=2Þ= (in ol s) and is a pa ame e
ela ed o nonlinea capaci ance [19,20]. Using he ha -
monic balance me hod o app oxima e a solu ion o he
subha monic esponse [21], such a solu ion assumes he
o m xð Þ¼A1sinðTþÞþA1=2sinðT=2ÞþB1=2cosðT=2Þ,
whe e A1¼1=ð1!2Þand A1=2and B1=2can be ob ained
by sol ing wo nonlinea algeb aic equa ions (no shown
he e). This app oxima e solu ion is displayed in he inse o
Fig. 2(a). We no e ha his analy ical app oach p edic s he
ange o equencies whe e he subha monic esonance
akes place and he esul ing solu ions in he small ampli-
ude egime.
Le us now u n o he la ice o nonlinea oscilla o s.
Figu e 3shows he s eady-s a e con igu a ions upon uni-
o m d i ing a equencies (a) d¼!d=2¼550 kHz
and (b) d¼590 kHz and an ampli ude o 7.5 V. No e ha
he d i e ’s equency is a de uned om he sys em’s
linea eigenmodes, so ha , in he linea case, we would
expec no ene gy ans e om he d i e . The uni o m
mode equency (k¼0) a he bo om o he linea dispe -
sion cu e occu s a a ound 315 kHz; he op o he
dispe sion cu e (k¼) is a a ound 520 kHz.
Ne e heless, in his nonlinea sys em, a a ound ¼
75 sa e he d i e is i s u ned on, ene gy s a s o
build up a ound he 10 h node, and soon we obse e a
s able ILM cen e ed he e, wi h i s wings expanding abou
h ee nodes in ei he di ec ion. The ILM oscilla es a i s
cen e a 275 kHz [Fig. 3(a)] and 295 kHz [Fig. 3(b)] bu is
d i en a wice he co esponding equency. I is wo h
men ioning ha he pa icula loca ion whe e he ILM is
o med is pa ly due o ( e y sligh ) con igu a ional asym-
me ies (i.e., e y weak de ec s), whe e he d i ing p e e -
en ially exci es a pa icula si e o he la ice and he
esul ing b ea he s a e eme ges spon aneously as a esul
o his ea u e. I should be men ioned ha modula ional
ins abili y o he (subha monically exci ed) uni o m mode
may also a ise and has been obse ed o gi e ise o mul i-
b ea he s a es. The p o iles o he ILM depic ed in Fig. 3
co espond o he imes a which he ILM eaches i s mos
posi i e and nega i e ol ages o bo h he expe imen al
da a (ci cles) and o he heo e ical model esul s (solid
line), sugges ing an excellen ag eemen be ween he wo
app oaches. The inse s in (a) and (b) show he nume ical
linea iza ion spec um o Floque mul iplie s (¼ þ
ii) co esponding o his ime-pe iodic solu ion, indica -
ing dynamical s abili y o he ILM. I is in e es ing o no e
ha he e exis s a na ow equency in e al, whe e he
heo e ical model p edic s he des abiliza ion o he ILM in
a o o as able quasipe iodic ILM h ough a Hop loop
( o wa d Hop and e e se Hop ) bi u ca ion. De ailed
analysis o he expe imen al esul s ( equency spec a)
also e eals he co esponding window in he expe imen s.
Fu he s udies o his in e es ing bi u ca ion will be e-
po ed elsewhe e.
Figu e 4illus a es he ILM dynamics in mo e de ail.
The op panel depic s he expe imen al ime aces a
a ious nodes. The mos p ominen ace co esponds o
he ILM cen e ; he o he aces co espond o i s -,
second-, and hi d-neighbo dynamics ( he expe imen al
aces su e om a mo e limi ed ime esolu ion). In
Fig. 4(b), he nume ical aces gi e a smoo he pic u e in
e y good ag eemen wi h he expe imen ; bo h panels
demons a e ha he equency o oscilla ion a he ILM’s
cen e is hal o he d i ing equency, d. Fu he mo e—as
e idenced by he equency spec a depic ed in Figs. 4(c)–
4( )—as we mo e away om he cen e o neighbo ing
la ice nodes, a second oscilla ion cycle g adually appea s
and we ansi om dominance o he d=2 equency o
he e en ual dominance o he undamen al equency d.
Thus, spa ially, mo ing om he wings o he cen e , a
pe iod-doubling ansi ion occu s.
We now explo e he dependence on he d i e equency.
Single-peak ILMs (shown in Fig. 3) a e ound be ween 525
and 617 kHz (a 7.5 V ampli ude). The lowe bound is
dic a ed by an eme ging o e lap wi h he zone-bounda y
linea mode, and he uppe bound is dic a ed by he coin-
cidence o he ILM wi h he uni o m linea mode. As he
d i e equency is aised beyond 617 kHz, he subha -
monic will s a o in e sec he dispe sion cu e. Wha is
in e es ing is ha , e en inside he linea dispe sion band,
localized s uc u es can be d i en subha monically, as we
will now show.
10 20 30
−1
0
1
2
3
si e
V ( ol s)
(a)
−1 0 1
−1
0
1
λ
λi
10 20 30
si e
(b)
−1 0 1
−1
0
1
λ
λi
FIG. 3 (colo online). Compa ison be ween he heo e ical
(solid line) and expe imen al (ci cles) ILM p o iles, wi h e-
quencies (a) 275 kHz and (b) 295 kHz, gene a ed by a homoge-
neous o cing o 7.5 Va (a) 550 kHz and (b) 590 kHz. The inse s
show he Floque mul iplie nume ical linea iza ion spec um,
con i ming (since all mul iplie s a e inside he uni ci cle) he
s abili y o hese ime-pe iodic solu ions.
0
1
2
V (Vol s)
(a)
0 5 10 15
0
1
2
(µs)
V (Vol s)
(b)
0
1
105
105S( )
(c)
0
0.5
1(d)
0 500
0
0.5
(kHz)
S( )
(e)
0 500
0
0.5
(kHz)
( )
FIG. 4 (colo online). (a) Expe imen al and (b) nume ical
aces o he oscilla ion a ou di e en nodes— he ILM cen e ,
i s neighbo , second neighbo , and hi d neighbo . (c)–( ) The
equency spec um co esponding o he expe imen al ime
aces.
PRL 108, 084101 (2012) PHYSICAL REVIEW LETTERS week ending
24 FEBRUARY 2012
084101-3
Figu e 5(a) cap u es he sys em’s expe imen ally mea-
su ed esponse in ecip ocal space o a d i e a a equency
o 750 kHz and an ampli ude o 7.92 V. We clea ly obse e
ene gy concen a ion a he disc e e alues in kspace
whe e he dispe sion cu e (do ed ed line) in e sec s he
d=2line. Mo eo e , his ene gy concen a ion esul s om
a buildup o e ime. Figu e 5(b) plo s he Fou ie ampli-
ude a dand d=2, o h ee dis inc imes in i s e olu ion.
The bo om ace co esponds o an ea ly ime in e al, ¼
0 o 400 s, wi h only a weak subha monic esponse; he
middle ace indica es he subha monic esponse eme -
gence, om ¼1:2ms o 1.6 ms; while he op one e eals
i s e en ual dominance a la e imes, om ¼2:8ms o
3.2 ms. In he spa ial domain, he pa e n ha esul s in his
si ua ion is shown in Fig. 5(c). A mul ipeaked localized
pa e n (o ILM ain) is obse ed, he pe iodici y o which
is se up by he wa e numbe kon he dispe sion cu e
associa ed wi h !d=2.
Pa e ns esembling ILM ains, as shown in Fig. 5(c),do
no appea a all d i e equencies equally. This is indi-
ca ed in Fig. 6, whe e he linea dispe sion cu e (solid
line) is shown wi h he ( ed) do s indica ing he no mal
modes o a 32-node la ice. Supe imposed on hese linea
no mal modes a e ho izon al lines depic ing equencies
whe e subha monic esponse is he mos di icul o ac-
complish expe imen ally. Namely, a d i e equencies
equal o wice hose indica ed, he mul ipeak pa e ns
anish i s as he ampli ude o d i ing is educed. These
equencies coincide well wi h he linea no mal modes.
This co ela ion sugges s ha such pa e ns a oid o e lap
wi h he linea spec um and hus p e e en ially eside in
he gaps inhe en in small la ices [22,23]. Abo e ¼
420 kHz ( d>840 kHz), no pa e n can be induced e en
a he maximum d i ing ampli ude.
In conclusion, we ha e demons a ed expe imen ally
and ha e suppo ed heo e ically h ough bo h analysis
and nume ical compu a ion he ac ha he subha monic
esponse o coupled, d i en nonlinea oscilla o s in ol es
he o ma ion o in insic localized modes h ough a spa ial
pe iod-doubling sequence building up o e ime. We an-
icipa e ha such conclusions may ha e b oad applicabili y
o mechanical (pendula, g anula chains), supe conduc ing
(Josephson junc ion), and op ical sys ems, among o he s.
This esea ch was suppo ed by he Minis e io de
Ciencia e Inno acio
´n o Spain (FIS2008-04848). A. J. S
was suppo ed by NSF-DMR-0906491. P. G. K is sup-
po ed by NSF-CMMI-1000337 and he A. S. Onassis
Public Bene i Founda ion.
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1200
1000
800
600
400
200
0
equency [kHz]
12x10
3
10
8
6
4
2
0
ampli ude [a.u.]
-0.4 -0.2
0.0 0.2 0.4
k [
π/
a]
750 kHz
375 kHz
x 5
(a)
(b)
10 20 30
0
1
si e
V ( ol s)
(c)
−1 0 1
−1
0
1
λ
λi
FIG. 5 (colo online). (a) Two-dimensional as Fou ie ans-
o m (FFT) o he expe imen al la ice dynamics. (b) Spa ial
componen o he FFT (see ex o desc ip ion) in (a) co e-
sponding o 750 kHz (solid lines) and 375 kHz (do ed lines).
(c) A spa ial snapsho o he esul ing spa ially ex ended s uc-
u e using he same layou as in Figs. 3(a) and 3(b).
500
450
400
350
300
[kHz]
1.00.80.60.40.20.0
k / kZB
FIG. 6 (colo online). Linea dispe sion cu e and no mal
modes ( ed do s) in ela ion o equencies a which nonlinea
subha monic esponse is supp essed (ho izon al lines) in he
expe imen .
PRL 108, 084101 (2012) PHYSICAL REVIEW LETTERS week ending
24 FEBRUARY 2012
084101-4
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PRL 108, 084101 (2012) PHYSICAL REVIEW LETTERS week ending
24 FEBRUARY 2012
084101-5