Interaction of moving discrete breathers with interstitial defects

a Xi :0907.4499 1 [nlin.PS] 26 Jul 2009 Manusc ip submi ed o Websi e: h p://AIMsciences.o g AIMS’ Jou nals Volume X, Numbe 0X, XX 200X pp. X–XX INTERACTION OF MOVING DISCRETE BREATHERS WITH INTERSTITIAL DEFECTS J. Cue as, B. S´ anchez–Rey G upo de F´ısica No Lineal. Depa amen o de F´ısica Aplicada I. Escuela Uni e si a ia Poli ´ecnica. Uni e sidad de Se illa. C/ Vi gen de ´ A ica, 7. 41011 Se illa, Spain J.C. Eilbeck and F.M. Russell Depa men o Ma hema ics and he Maxwell Ins i u e o Ma hema ical Sciences He io -Wa Uni e si y - Ricca on, Edinbu gh, EH14 4AS, UK Abs ac . In his pape , in e s i ial mig a ion gene a ed by sca e ing wi h a mobile b ea he is in es iga ed nume ically in a F enkel-Kon o o a one- dimensional la ice. Consis en wi h expe imen al esul s i is shown ha in e s i ial diffusion is mo e likely and as e han acancy diffusion. Ou sim- ula ions suppo he hypo hesis ha a long- ange ene gy anspo mechanism in ol ing mo ing nonlinea ib a ional exci a ions may significan ly enhance he mobili y o poin de ec s in a c ys al la ice. 1. In oduc ion. The F enkel–Kon o o a (FK) model, in oduced almos 70 yea s ago [1], is one o he mos pa adigma ic nonlinea sys ems, whose dynamics has been widely s udied du ing he las decades (see [2,3,4,5] and e e ences he ein). F om he poin o iew o condensed ma e physics, i s pa amoun impo ance elies on he abili y o desc ibe a as numbe o phenomena, including di e en kinds o de ec s such as acancies (Scho ky de ec s) and, o some ex en , in e s i ials (F enkel de ec s), which can play an impo an ole in he design o new ma e ials [6]. As he FK model is basically a one-dimensional la ice o pa icles subjec ed o a nonlinea pe iodic subs a e po en ial and a nea es -neighbou in e ac ion, i con ains he basic ing edien s o sus ain localized exci a ions such as opological soli ons (kinks o an ikinks) o b ea he s. Disc e e b ea he s (DBs), also called in insic localized modes ( o a e y ecen e iew abou hei p ope ies, exis ence p oo s, compu a ional me hods and applica ions see [7]), a e exac solu ions o he dynamical equa ions whose ene gy, in con as wi h no mal ex ended wa e exci a- ions, is no sha ed among la ice componen s bu ex ends only o e a ew la ice si es. In his sense, hei spa ial p o iles esembles localized ib a ional modes in- duced by a de ec si e in a ha monic la ice [8]. Howe e DBs a ise only hanks o he in e play be ween nonlinea i y and disc e eness and, o ha eason, hey may occu anywhe e in he la ice gi en su icien ib a ional ampli ude. They a e also a he uni e sal since hey a e no speci ic o Hamil onians wi h a pa icula o m and can be ound in la ices o a bi a y dimensions. Mo eo e , heo e ical s udies 2000 Ma hema ics Subjec Classi ica ion. P ima y: 70K75, 74J30; Seconda y: Key wo ds and ph ases. Mo ing b ea he s, kinks, de ec s, F enkel–Kon o o a model. 1 2 CUEVAS, S ´ ANCHEZ-REY, EILBECK AND RUSSELL ha e shown DBs a e linea ly s able [9], which implies hey can pe sis o e e y long imes on op o a he malized backg ound [10]. Thei in es iga ion is no es ic ed o simple oy models. Apa om indi ec spec oscopic obse a ions [11], DBs ha e been de ec ed and s udied expe imen ally in such di e en mac oscopic sys- ems as wa eguide a ays [12], mic omechanical can ile e s [13], an i e omagne ic s uc u es [14] and Josephson-junc ions [15]. In his con ex , an in e es ing p oblem ha has a ac ed much a en ion in ecen yea s is he in e ac ion be ween a mo ing localized exci a ion and a la ice de ec . The p oblem has been add essed wi hin di e en amewo ks: impu i ies [16,17], la ice junc ions [18,19], bending poin s o a polyme chain [20,21,22], bu mos s udies assume ha he posi ion o he de ec is ixed and is no able o mo e along he la ice. O cu en in e es is he in e ac ion be ween la ice de ec s and mo ing localized exci a ions, which migh esul in mo emen o he de ec . This is especially ue in he case o in e ac ions a ising du ing i adia ion o solids by swi pa icles, which usually in ol e he c ea ion o DBs o ei he longi udinal o ans e se op ical mode ype. The possibili y o such in e ac ions a ose in he s udy o high ene gy cha ged pa icles passing h ough c ys als o musco i e, when sca e ing e en s we e pos u- la ed o c ea e many mo ing highly ene gy DBs. I was sugges ed ha when such DBs ( he e called quodons) eached he end o a chain, which ep esen s a de ec in a chain, i migh cause he las a om o be ejec ed om he su ace [23]. This p e- dic ion was suppo ed by s udies using bo h mechanical and nume ical models [24]. Subsequen ly, i was e i ied by expe imen using a na u al c ys al o musco i e [25]. In he expe imen one edge o a c ys al was bomba ded wi h alpha pa icles a nea g azing incidence o c ea e mo ing DBs. These p opaga ed in chain di ec ions in he laye ed c ys al and caused a p opo iona e ejec ion o a oms om a emo e edge o he c ys al ha was >107uni cells dis ance in a chain di ec ion om he si e o bomba dmen . As his expe imen was pe o med a 300K i no only e i ied he p edic ion bu also demons a ed he s abili y o hese mobile DBs agains he mal mo ion. O he i adia ion s udies ha e p o ided mo e empi ical signs o he in e ac ion o DBs wi h de ec s. Fo ins ance, in e . [26] he au ho s p o ide e idence ha , a e i adia ing a silicon c ys al wi h sil e ions, a pileup o la ice de ec s is accomplished a loca ions spa ially sepa a ed om he i adia ion si e. The e idence indica ed ha de ec s could be swep by up o abou 1 mic on om he i adia ed egion. This e ec was asc ibed o he p opaga ion o highly localized packe s o ib a ional ene gy, o DBs, c ea ed by he bomba dmen o hea y ions. Ano he ion-induced, a he mal anspo p ocess was epo ed in e . [27]. In his case in e s i ial N di usion in aus eni ic s ainless s eel unde A ion bomba dmen was in es iga ed. I was ound ha N mobili y inc eases in dep hs se e al o de s o magni ude la ge han he ion pene a ion dep h. This i adia ion-induced en- hancemen o N di usion is consis en wi h p e ious obse a ions which show a dependence o he ni iding dep h on ion ene gy [28] and also on he c ys alline o ien a ion [29], bu no con en ional mechanism o di usion can explain hem. Fo his eason i was sugges ed ha di usion o in e s i ial a oms migh be assis ed by highly anha monic localized exci a ions which p opaga e dis ances well beyond he ion pene a ion dep h. In e s i ial a oms eside in po en ial wells be ween he la ice a oms. When a b ea he p opaga es i s ongly dis u bs he la ice locally. I i passes nea an BREATHER–INTERSTITIAL INTERACTION 3 in e s i ial hese oscilla o y mo ions will dis o he po en ial well con ining he in e s i ial and will a ec signi ican ly i s mobili y. In e s i ial mo ion consis s o jumps om one po en ial well o he nex . Since expe imen al measu es deal wi h concen a ion dep h p o iles, in e s i ial di usion p ocess can be analyzed in e ms o an e ec i e mo emen along a one-dimensional chain o po en ial wells. Mo eo e he p esence o an in e s i ial modi ies po en ials in adjacen a omic chains, causing he spacing be ween he wo nea es a oms in a chain o he in e s i ial o inc ease. The e o e, in a i s app oxima ion, an in e s i ial can be modelled in oducing and addi ional pa icle in a one-dimensional sys em and his p o ides he link o he FK model. In his pape , using a FK model wi h nonlinea nea es -neighbou in e ac ion, i is shown ha mig a ion o he dis u bance in a chain caused by an in e s i ial can be induced by sca e ing wi h a mobile longi udinal mode b ea he . Compa ison wi h p e ious wo k on acancies mig a ion [30,31] also sugges s ha , acco ding o expe imen al esul s, in e s i ial mobili y is mo e likely and as e han ha o acancy de ec s. O cou se, he speci ic cons ain s o a one-dimensional sys em im- plies ha ca e is needed when a emp ing o ca y o e esul s o highe dimensional la ices. Ne e heless we hink ha a one-dimensional s udy is a necessa y and use- ul i s s ep be o e app oaching he p oblem wi h a mo e ealis ic and complex wo o h ee-dimensional model. 2. The model. As desc ibed in he in oduc ion, he F-K model consis s o a chain o in e ac ing pa icles subjec o a pe iodic subs a e po en ial. This sys em is desc ibed by he ollowing Hamil onian: H= N X n=1 1 2m˙x2 n+V(xn) + W(xn−xn−1),(1) whe e xnis he absolu e coo dina e o he n- h pa icle. The co esponding dy- namical equa ions a e m¨xn+V′(xn) + [W′(xn−xn−1)−W′(xn+1 −xn)] = 0, n ∈Z.(2) In o de o in es iga e in e s i ial mobili y we ha e chosen a cosine po en ial wi h he la ice pe iod a V(x) = a2 4π2[1 −cos(2πx/a)] ,(3) as he simples , pe iodic subs a e po en ial, wi h he linea equency no malized o uni y ω0=pV′′ (0) = 1. Fo he in e ac ion be ween pa icles, we ha e selec ed he Mo se po en ial W(x) = C 2b2[e−b(x−a)−1]2, x > 0.(4) which has a minimum a he la ice pe iod aand a ha d pa ha p e en s pa icles om c ossing. The well dep h o his po en ial is C/2b2while b−1is a measu e o he well wid h. I s cu a u e a he bo om is gi en by C=W′′ (a), so ha we can modula e he s eng h o he in e ac ion wi hou changing i s cu a u e by a ying pa ame e b. In his sys em, an in e s i ial a om is ep esen ed by a doubly occupied well o he pe iodic po en ial (see he s able equilib ium con igu a ion in panel (a) o Fig.1). The ela i e coo dina e o each pa icle wi h espec o i s equilib ium posi ion can be w i en as un=xn−na. Using hese ela i e coo dina es, he in e s i ial 4 CUEVAS, S ´ ANCHEZ-REY, EILBECK AND RUSSELL −10 −5 0 5 10 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 un n 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10−3 EPN b Figu e 1. (a) Scheme o he s able equilib ium s a e o he F enkel–Kon o o a model wi h cosine subs a e po en ial and Mo se nea es neighbo in e ac ion. The doubly-occupied well ep- esen s an in e s i ial. (b) An ikink co esponding o he s able equilib ium con igu a ion in ela i e coo dina es o b= 1 and C= 0.5. (c) Uns able equilib ium con igu a ion. (d) Peie ls- Naba o ba ie o he an ikink. can be isualized as an an ikink [2,3]1as i is shown in panel (b) o Fig. 1. I is well-known ha an an ikink can be pu in o mo emen as soon as an ene gy ba ie , he so-called Peie ls-Naba o ba ie (PNB), is o e come. The PNB can be calcula ed as he ene gy di e ence be ween he uns able and s able an ikink equilib ium con igu a ions (panels (c) and (a) o Fig. 1 espec i ely) and dec eases mono onically wi h b(see panel (d)). I is wo h no ing ha a acancy can be isualized as a kink in ela i e coo di- na es. I s PNB inc eases wi h he pa ame e band is always highe han he PNB o an in e s i ial, excep o b= 0 whe e bo h ac i a ion ene gies coincide. This is in acco dance wi h he expe imen al ac ha di usion o in e s i ials is as e han ha o acancies, and suppo he idea ha i is necessa y o conside a nonlinea in e ac ion po en ial in o de o s udy di usion o de ec s, since b= 0 ep esen s he linea limi o he Mo se po en ial. 1No ice ha in Re . [3] he e ms kink and an ikink a e in e changed. BREATHER–INTERSTITIAL INTERACTION 5 In ou F-K chain, s a iona y disc e e b ea he s can be nume ically ob ained using he s anda d me hod o con inua ion om he an icon inuous limi [32,33]. T ansla ional mo ion o disc e e b ea he s can be induced [34,35] by adding a pe u ba ion ~ =λ(..., 0,−1/√2,0,1/√2,0, ...) o he eloci ies o he s a iona y b ea he , wi h he nonze o alues a he neighbo ing si es o he ini ial b ea he cen e . The esul ing DB kine ics is e y smoo h and esembles ha o a classical ee pa icle. The e o e, he o al ene gy o a mo ing disc e e b ea he can be es ima ed as he sum o i s ib a ional in e nal ene gy, equal o ha o he s a iona y b ea he , plus i s ansla ional ene gy, which is equal o he ene gy o he added pe u ba ion K=λ2/2. 3. Nume ical s udy. In o de o in es iga e in e s i ial mobili y, we ha e gene - a ed a b ea he cen e ed a si e n=−25, ela i ely a om an in e s i ial whose le mos pa icle is loca ed a n= 0, and hen launched ha b ea he owa ds i ollowing he depinning me hod men ioned abo e. Th oughou he pape , we ha e no malized he la ice pe iod aand masses o uni y and ha e aken C= 0.5 so ha mo ing b ea he s (MBs) exis in he sys em o a b ea he equency ωb= 0.9. As a esul o he sca e ing he de ec can be pu in o mo emen leading o long– ange anspo . We ha e ound h ee well-di e en ia ed egimes depending on he s eng h o he in e ac ion po en ial. Below a c i ical alue b≈0.83 he esul o he sca e ing is unp edic able. The dynamics is ex emely sensi i e o ini ial condi ions ( alue o he pe u ba ion λand ini ial posi ion o he b ea he ) and he in e s i ial can a el o make andom jumps (backwa d o o wa d) o e en emain a es . Howe e , a ne backwa d mo emen o he de ec is only possible i he in e ac ion po en ial is s ong enough. In ac we ha e obse ed i only o alues o b.0.69. An example o a backwa ds a elling in e s i ial is shown in Fig. 2, whe eas Figs. 3and 4show a backwa ds and o wa ds, espec i ely, hopping in e s i ial. In his case, he in e s i ial, a e se e al andom jumps, emains pinned on he la ice. These h ee igu es display h ee panels. Le panel co esponds o an ene gy densi y plo whe e lines join poin s wi h he same ene gy in ime while da ke colo indica es la ge ene gy. Cen al panel displays he ime e olu ion o he an ikink (in e s i ial) cen e o mass. This g aph helps o isualize mo e clea ly he jumps o he in e s i ial pa icle and he inal oscilla o y s a e a ound an equilib ium con igu a ion. Finally, igh panel shows a s eak plo wi h he ime e olu ion o he b ea he and he in e s i ial. I is no ewo hy ha in ou nume ical expe imen s smalle alues o benhance backwa d mo emen and hopping beha iou o he in e s i ial pa icle. This la es beha iou is he only obse ed in he ha monic limi o he in e ac ion po en ial (b= 0). No ice ha he complexi y o he dynamics is linked o he disc e eness o he F-K model conside ed [36,37]. In he con inuous limi wi h b= 0 he b ea he - an ikink in e ac ion is an in eg able and well-known case, and he esul ing scena io is qui e simple: he b ea he always c osses he an ikink which mo es backwa ds du ing a b ie lapse o ime [38]. Due o he exis ence o an ac i a ion ene gy o mo e an an ikink in he disc e e case, in e s i ial mo ion is only ound abo e a h eshold alue, Kc, o he kine ic ene gy o he inciden b ea he . In he chao ic egime, b.0.83, his h eshold alue, plo ed in igu e 5, inc eases mono onically in con as wi h he PNB beha io ound in he p e ious sec ion. On he con a y, o b&0.87 we ind he opposi e endency: Kcdec eases wi h bindica ing a deep change in he dynamics. Indeed in 6 CUEVAS, S ´ ANCHEZ-REY, EILBECK AND RUSSELL B ea he pe iods n −50 0 50 0 10 20 30 40 50 60 0 10 20 30 40 50 60 −35 −30 −25 −20 −15 −10 −5 0 Xins B ea he pe iods 10 20 30 40 50 60 −25 −20 −15 −10 −5 0 5 10 xn B ea he pe iods Figu e 2. (Le panel) Ene gy densi y plo , showing a backwa d mo emen o he in e s i ial de ec a e b ea he sca e ing. The lines join poin s wi h he same ene gy in ime. The da ke colou he la ge ene gy. (Cen al panel) Time e olu ion o he an ikink ene gy cen e . (Righ panel) S eak plo . Pa ame e s: K= 0.0220 and b= 0.5. B ea he pe iods n −50 0 50 0 20 40 60 80 100 120 140 160 180 200 0 50 100 150 200 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 Xins B ea he pe iods 20 40 60 80 100 120 140 −25 −20 −15 −10 −5 0 5 10 xn B ea he pe iods Figu e 3. Same as Fig. 2bu o a hopping in e s i ial wi h ne backwa ds displacemen . Pa ame e s: K= 0.0050 and b= 0.2. B ea he pe iods n −50 0 50 0 20 40 60 80 100 120 140 160 180 200 0 50 100 150 200 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Xins B ea he pe iods 20 40 60 80 100 120 140 −25 −20 −15 −10 −5 0 5 10 xn B ea he pe iods Figu e 4. Same as Fig. 2bu o a hopping in e s i ial wi h ne o wa d displacemen . Pa ame e s: K= 0.00605 and b= 0.1. his pa ame e egime, o K > Kc, he in e s i ial always mo es o wa d a e he sca e ing and, ema kably, i always mo es wi h app oxima ely cons an eloci y. In his egime, he Mo se po en ial becomes essen ially “ la ” wi h a ha d co e and he dynamics is domina ed by he epulsi e pa o he in e ac ion po en ial. An example can be obse ed in Fig. 6. A e he collision wi h he b ea he , in e s i ial mo ion is clea ly linea in ime. I s eloci y has been compu ed i ing he poin s o he cen al panel wi h linea eg ession. In he ansi ion be ween bo h egimes, i.e. o 0.83 .b.0.87, he in e s i ial always emains pinned on he la ice, a leas o hose alues o λ o which he b ea he p opaga es wi hou signi ican dis o ion. BREATHER–INTERSTITIAL INTERACTION 7 0 0.5 1 1.5 2 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Kc b Figu e 5. Minimum ansla ional ene gy (Kc) o he incoming b ea he needed o mo e an in e s i ial. In he band 0.83 &b. 0.87 he in e s i ial always emains pinned on he la ice. B ea he pe iods n −50 0 50 0 10 20 30 40 50 60 0 10 20 30 40 50 60 5 10 15 20 25 30 Xins B ea he pe iods 10 20 30 40 50 60 −15 −10 −5 0 5 10 15 20 xn B ea he pe iods Figu e 6. Same as Fig. 2bu o a b ea he wi h K= 0.020 and b= 1.5. The in e s i ial always mo es o wa d wi h cons an eloci y in he pa ame e egion b&0.87, K > Kc Fig. 7shows he e olu ion o a pinned in e s i ial o b= 1. In his case he inciden b ea he possesses a ansla ional ene gy smalle han he c i ical alue Kc and, consequen ly, he in e s i ial ac s as a wall which o ally e lec s he b ea he . I is obse ed ha , a e he collision, pa o he b ea he ene gy is employed in exci ing an in e nal mode o he in e s i ial wi h a equency smalle han ha o he inciden b ea he . This linea localized mode co esponds o he line below he phonon spec um shown in Fig. 8 o he in e s i ial s able equilib ium con igu a ion. No e ha nonlinea localized modes do no exis close o he in e s i ial since he in e ac ion po en ial is so , and he equency o he linea localized mode is always below ωb= 0.9. As men ioned abo e, o b&0.87 a s able in e s i ial p opaga ing mode appea s i he kine ic ene gy o he inciden b ea he is highe han he h eshold alue Kc. In his pa ame e egion he Mo se po en ial becomes essen ially a epulsi e po en ial. In ac , in he limi b→ ∞ i becomes a ha d-sphe e po en ial. Fo his eason, in his dynamical egime in e s i ial pa icles mo e oughly like ha d sphe es on a wa y ene gy landscape. A e he b ea he sca e ing, he in e s i ial pa icle su moun s he ene gy ba ie o he on-si e po en ial well and collides wi h he pa icle ha occupies he ollowing well ans e ing i s ene gy and momen um o i . In his way he de ec p opaga es a cons an eloci y o e e . In igu e 9we 8 CUEVAS, S ´ ANCHEZ-REY, EILBECK AND RUSSELL B ea he pe iods n −50 0 50 0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 50 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Xins B ea he pe iods 10 15 20 25 30 35 40 45 50 −25 −20 −15 −10 −5 0 5 xn B ea he pe iods Figu e 7. Same as Fig. 2bu o a b ea he wi h K= 0.0162 < Kcand b= 1. As he kine ic ene gy o he inciden b ea he is below he h eshold alue Kc, he in e s i ial emains pinned on he la ice. 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 b Linea modes equencies Figu e 8. Linea modes spec um o he s able equilib ium con- igu a ion o C= 0.5. ωband 2ωba e depic ed h ough dashed lines. ha e plo ed he dependence o he eloci y o his p opaga ing mode on he kine ic ene gy o he inciden b ea he . One can obse e ha jus abo e he h eshold ene gy, in e s i ial eloci y inc eases wi h he kine ic ene gy, K, o he incoming b ea he . Howe e o highe alues o K he in e s i ial eloci y ends o sa u a e a ound a alue 0.14, wha means ha he in e s i ial pa icle mo es app oxima ely 0.14 2π ωb≈1 si e on he chain pe b ea he pe iod, independen ly o he coupling s eng h. This phenomenon is con i med in igu e 10 whe e we ha e plo ed he in e s i- ial eloci y e sus pa ame e b o a ixed alue o K. Indeed, o K= 0.045 (dashed line) well abo e he ene gy h eshold, in e s i ial eloci y akes oughly he sa u a ion alue 0.14 independen ly o he coupling s eng h. In e media e alues o kine ic ene gy as K= 0.02 (con inuous line) also leads o sa u a ion eloci ies independen ly o bbu wi h alues lowe han 0.14 and less luc ua ions. 4. Conclusions. We ha e p esen ed nume ical esul s a ising om he in e ac ion be ween a mo ing disc e e b ea he and an in e s i ial de ec in a FK chain. The main esul is he exis ence o h ee di e en ia ed egimes depending on he s eng h o he in e ac ion po en ial. When he in e ac ion be ween neighbo s is s ong he BREATHER–INTERSTITIAL INTERACTION 9 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 In e s i ial eloci y K b=1.25 b=1.5 b=2 Figu e 9. In e s i ial eloci y as a unc ion o he ansla ional ene gy Ko he inciden b ea he o h ee di e en alues o he coupling s eng h in he egime (b&0.87). In his egime he in e s i ial always mo es o wa d wi h cons an eloci y because he in e ac ion po en ial educes essen ially o a epulsi e ha d co e. 0.8 1 1.2 1.4 1.6 1.8 2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 In e s i ial eloci y b K=0.02 K=0.045 Figu e 10. In e s i ial eloci y e sus coupling s eng h o a ixed ansla ional ene gy o he inciden b ea he . dynamics is chao ic and he beha io o he in e s i ial pa icle is unp edic able: i can jump backwa ds, o wa ds o emains a es . Howe e , i he in e ac ion po en ial is weak enough, he de ec mo es o wa ds along he la ice wi h cons an eloci y. This s able p opaga ing mode had no been obse ed o ou knowledge in p e ious nume ical s udies conce ning he in e ac ion be ween mo ing b ea he s and poin de ec s. The e ec is asc ibed o he ac ha he in e ac ion po en ial educes essen ially o a epulsi e ha d co e. Be ween hese wo dynamical egimes he e is an na ow in e media e ange o he coupling s eng h in which he in e s i ial always emains pinned. Ou o ha pinned egime, he kine ic ene gy o he incoming b ea he s mus su pass a h eshold in o de o mo e he in e s i ial. This ene gy h eshold has a non-mono onic beha io . I g ows wi h pa ame e bin he chao ic egime, bu dec eases wi h bwhen he sys em losses sensi i i y o ini ial condi ions and he