Generation of localized modes in an electrical lattice using subharmonic driving

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. 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Phys. 78, 137 (2006). 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 [9] S. Flach and A. V. Go bach, Phys. Rep. 467, 1 (2008);S. Aub y, Physica (Ams e dam) 103D, 201 (1997). [10] I. V. 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