Author(s) Kimura, Masayuki; Matsushita, Yasuo.

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1 A study o bifurcatios ad Titlecocerig itrisic localized structu mode mechaical lattice Author(s) Kimura, Masayuki; Matsushita, Yasuo Citatio AIP Coferece Proceedigs: NONLINE State-of-the-Art ad Perspectives ( Issue Date -9- URL America Istitute of Physic Right dowloaded for persoal use oly. A permissio of the author ad the Am Type Joural Article Textversio publisher Kyoto Uiversity

2 A study o bifurcatios ad structure of phase space cocerig itrisic localized modes i a oliear mageto-mechaical lattice Masayuki Kimura, Yasuo Matsushita ad Takashi Hikihara School of Egieerig, The Uiversity of Shiga Prefecture, Hassaka-cho, Hikoe, Shiga -, Japa Departmet of Electrical Egieerig, Kyoto Uiversity, Katsura, Nishikyo, Kyoto -, Japa Abstract. A mageto-mechaical lattice proposed i this paper is oe of the oliear lattices i which itrisic localized mode(ilm) exists. A bifurcatio diagram cocerig ILMs is ivestigated with respect to the magitude of oliear couplig force. I additio, the possibility of the existece of movig ILM is discussed based o the phase structure aroud a ustable ILM which is umerically examied by computig ustable maifolds of the ustable ILM. Keywords: itrisic localized mode, discrete breather, catilever array, movig ILM PACS:..-a,..Pw,..Ry INTRODUCTION Itrisic localized mode(ilm) is a eergy localized pheomeo appeared i oliear lattices []. I this decade, ILM was idetified ot oly i atural structures such as atiferromaget [] but also i artificial structures, for istace, micro-mechaical catilever array [], electroic circuits [, ], ad mageto-mechaical lattice []. The experimets i the micro-mechaical catilever arrays allow us to expect the realizatio of applicatios usig the eergy localizatio i micro/ao-egieerig because ILM ca move without decayig its eergy cocetratio ad ca be maipulated by a extraeous stimulus []. However, it is eeded to realize such applicatio that the mechaism of how ILM moves should be clarified ad the cotrol scheme has to be established. We have proposed the mageto-mechaical lattice to ivestigate the dyamics of ILM []. Although stadig ILMs were successfully observed ad maipulated i the lattice, ay movig oes could ot be geerated. I this paper, a improved magetomechaical lattice which have a oliearity i couplig force is first itroduced. The, bifurcatios ad the phase space aroud ILMs are ivestigated umerically. The possibility of the existece of movig ILM is fially discussed based o the phase structure. MAGNETO-MECHANICAL LATTICE A catilever which behaves as a liear oscillator for small deflectio is used as a oscillator of the mageto-mechaical lattice. A small maget is attached at the tip of NONLINEAR ACOUSTICS State-of-the-Art ad Perspectives AIP Cof. Proc., - (); doi:./.99 America Istitute of Physics 9----/$.

3 Support Strai gauge Catilever Permaet Maget Electromaget ` ` Support Frot view Side view FIGURE. Cofiguratio of mageto-mechaical lattice the catilever ad a electromaget is placed beeath the tip to cause a oliearity i the restorig force of the catilever. For the couplig force betwee adjacet catilevers, aother maget is sticked at the middle of each catilever. The cofiguratio of these magets are show i Fig.. By usig the magetic charge approximatio for the magetic iteractios [], the motio of equatio for the mageto-mechaical lattice is obtaied as follows: ü = ω u u χ O (u + d χ u u + )/ I { (u u + ) + d /κ} / χ u u I { (u u ) + d /κ} /, () where u is the displacemet of the tip of th catilever. Gaps betwee o-site ad iter-site magets are deoted by d (=.mm) ad d (=.mm), respectively. χ O ad χ I are coefficiets of magetic force. Sice the curret flowig i electromaget determies the magitude of magetic flux o the surface, χ O ca be tued dyamically. The resoat frequecy of catilever is represeted by ω = π f (= π.rad/s). The ratio of the positio of iter-site maget to the legth of catilever is deoted by κ = l /l (= mm/mm = /). For simplicity ad geerality, Eq.() is odimesioalized by substitutig t tt ad u U. ẍ = (π) χ O ( ) x + d / + χ I { } ( + ) + d / χ I { } ( ) + d /, () where = u /U, χ O = χ O/U f, χ I = χ I/U f, d = d /U, d = d /U κ. I this paper, the scale parameters are set to T = / f.ms ad U = mm. BIFURCATIONS OF INTRINSIC LOCALIZED MODES I this sectio, bifurcatios of stadig ILMs whose locatio of the ceter of eergy cocetratio is fixed are ivestigated. Sice Eq. () has o dampig term ad o forcig term, the total eergy(e tot ) is coserved ad is a bifurcatio parameter. The magitude of magetic couplig betwee adjacet catilevers χ I is also a bifurcatio parameter. I this paper, we ivestigate bifurcatios with respect to χ I istead of E tot. I Fig. (a), amplitude distributio is show for coexistig ILMs at χ I =, E tot =. These

4 O E x O. E.... (a) Coexistig ILMs. O..... E... x Odd Eve Uiform SN E E E SN Â I = O O T O (b) Bifurcatio diagram SN FIGURE. Amplitude distributio of ILM ad bifurcatio diagram ILMs ca be classified ito two kids by spatial symmetry of the amplitude distributio. The odd-symmetric ILMs are labeled O,O, ad O while the eve-symmetric oes are show by E,E, ad E. The bifurcatio diagram for the ILMs is show i Fig. (b). The ope squares correspod to saddle-ode bifurcatio poits. ILMs havig the same symmetry appear or disappear at the bifurcatio poits. O the other had, a oddsymmetric ILM ad a eve-symmetric ILM coalesce with the uiform vibratio at the taget bifurcatio poit [9] idicated by the ope triagle. The taget bifurcatio poit is oly oe bifurcatio poit related to the two differet kids of ILM i the bifurcatio diagram. I the ext sectio, the phase structure aroud coexistig ILMs ear the taget bifurcatio poit is discussed. PHASE STRUCTURE AND MOVING ILM To ivestigate the phase structure aroud coexistig ILMs, a oe-dimesioal ustable maifold of ustable ILM is computed. I the upper pael of Fig. (a), the ustable maifold of the ustable eve-symmetric ILM(E ) is show for the case of weak couplig regime χ I =. The ivariat maifold shows a homocliic-like structure, but does ot reach to the eighborig odd-symmetric ILMs(O ). O the other had, the ivariat maifold of E surrouds the eighborig odd-symmetric ILMs(E ) for the case of ear the taget bifurcatio poit χ I =. The tedecy of the behavior of the solutio started from ear the ustable ILM ca be estimated because the structure of the ivariat maifold i the phase space reflects the flow of Eq. []. Therefore, if the structure shows coectios betwee two differet ILMs, the existece of movig ILM ca be expected. I fact, as show i the lower pael of Fig. (b), the time developmet of the eergy distributio of a solutio whose iitial coditio is crated by slightly perturbig E shows a waderig locus of the eergy cocetratio, amely, a movig ILM. CONCLUSION The mageto-mechaical lattice which has oliearities i both o-site ad iter-site potetials was itroduced ad modeled as the coupled ordiary differetial equatios.

5 v  I = O E 9  I = O e(, t) x x 9 t v  I = 9  I =. O E e(, t). x..... O.. x.. 9 t - - (a) Ustable maifold (b) Time developmet of eergy distributio FIGURE. Ustable maifolds ad time developmet of eergy distributio of the perturbed ILMs For the odimesioalized equatio, bifurcatios of coexistig ILMs were ivestigated with respect to the couplig force. The saddle-ode bifurcatios ad the taget bifurcatio was observed i the bifurcatio diagram. The phase structure ear ILMs was examied by computig the ustable maifold of the ustable ILM for two cases, oe was i the weak couplig regime ad the other was ear the taget bifurcatio. It was revealed that the ustable maifold surrouds the eighborig ILMs for the latter case. The fact that the movig ILM was easily created implies that the existece of movig ILM ca be expected ear a bifurcatio poit related to differet symmetric ILMs such as the taget bifurcatio poit. ACKNOWLEDGMENTS This work was supported by the Miistry of Educatio, Culture, Sports, Sciece ad Techology i Japa, Grat-i-Aid for Youg Scietist (B) No.. REFERENCES. S. Takeo, ad A. J. Sievers, Solid State Commu., (9).. M. Sato, ad A. J. Sievers, Nature, ().. M. Sato, B. E. Hubbard, A. J. Sievers, B. Ilic, D. A. Czaplewski, ad H. G. Craighead, Phys. Rev. Lett. 9, ().. L. Q. Eglish, F. Palmero, A. J. Sievers, P. G. Kevrekidis, ad D. H. Barak, Phys. Rev. E, ().. M. Sato, S. Yasui, M. Kimura, T. Hikihara, ad A. J. Sievers, Europhys. Lett., ().. M. Kimura, ad T. Hikihara, Phys. Lett. A, (9).. M. Sato, B. E. Hubbard, ad A. J. Sievers, Rev. Mod. Phys., ().. M. Kimura, ad T. Hikihara, Noliear Theory ad Its Applicatios, IEICE, (). 9. S. Flach, ad A. V. Gorbach, Phys. Rep., ().. T. Hikihara, K. Torii, ad Y. Ueda, It. J. Bifurcat. ad Chaos, 999 ().

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