Discrete breathers and Anderson modes: two faces of the same phenomenon?

Disrete breathers and Anderson mo des: two faes of the same phenomenon? J.F.R. Arhilla a ; 1 , R.S. MaKay b ; 2 , J.L. Marn  ; 3 a Faulty of Computer Siene, Avda Reina Meredes, 41013-Sevil la, Spain b The Nonlinear Centre, Department of Applied Mathematis and Theoretial Physis, University of Cambridge, Silver Street, Cambridge CB3 9EW, U.K.  Department of Mathematis, Heriot-Watt University, Edinburgh EH14 4AS, U.K. Abstrat Time-p erio di lo alised osillations o ur in a variety of ontexts, in partiular in weakly oupled anharmoni latties and in disordered harmoni networks of osillators, where they are known resp etively as disrete breathers and Anderson mo des. It is shown numerially in some examples of systems whih interp olate b etween these two limits that disrete breathers an b e ontinued to Anderson mo des, mo dulo small jumps asso iated with resonane with Anderson mo des on other parts of the network. Key words: Disrete breathers; Intrinsi lo alized mo des; Anderson lo alization PACS: 63.20.Pw 1 Intro dution Spatially lo alized time-p erio di osillations, or \lo alized osillations" (LO) for short, app ear in various typ es of spatially disrete mo dels. In partiular they o ur in latties of weakly oupled anharmoni osillators and in disordered latties of linear osillators, but their origins in these two ontexts have up till now b een thought of as indep endent. For example, Sott's assessment of the numeris of Feddersen [1℄ on a disordered disrete self-trapping (DST) 1 Corresp onding author. E-mail: arhillaia.es 2 E-mail: r.s.makaydamtp.am.a.uk 3 E-mail: josema.hw.a.uk Preprint submitted to Physia D 1 Marh 2000 equation is that \Anderson lo alization is ompletely dierent from anharmoni lo alization" (setion 5.2 of [2℄), b eause the numeris show Anderson mo des delo alising rapidly as the anharmoniity is turned on. In our pap er, however, it is shown numerially that on moving parameters in the opp osite diretion in a family of mo dels, LOs of anharmoni ordered systems an b e onneted virtually ontinuously to LOs of harmoni disordered systems. First we reall the basi results in the two ontexts. LOs are ommon in anharmoni latties and have reeived onsiderable attention reently (for a review, see [3℄). They have b een given a variety of names, but the one we shall use is \disrete breather", or more simply \breather". In partiular, MaKay and Aubry [4℄ proved analytially the existene of breathers in Hamiltonian latties of weakly oupled osillators of nonlinear Klein-Gordon typ e | those with an anharmoni 4 on-site p otential. The breathers are proved to exist as smo oth ontinuations of the trivial solutions at zero oupling where some osillators are on p erio di orbits and the remainder at rest. This limit has b een alled the anti-integrable or anti-ontinuous limit [7℄, but is most simply referred to as the \unoupled limit". Pro ofs have also b een given for some systems with no on-site p otential, e.g. [8,9℄. On the other hand, lo alization is also a physial phenomenon observed in the ontext of wave propagation through disordered media. By \disorder" for a lassial lattie we mean that the lo al frequenies and/or ouplings b etween osillators are random. When the disorder is suÆiently large, ertain typ es of waves b eome trapp ed and the spreading of any initial wave paket is anomalously slow or imp ossible. Instead, the eigenfuntions are spatially lo alized, so give lo alized mo des that we all \Anderson mo des", following the pioneering work of Anderson for Shrodinger equations with random p otential [10{12℄. In this pap er we address the question raised in [13℄, whether there is a onne- tion b etween these two dierent typ es of LO, i.e. if there is a ontinuous path that transforms a breather in an anharmoni ordered lattie into an Anderson mo de in a harmoni disordered lattie. We study the question numerially, using a variant of the ontinuation metho d develop ed in [14,15℄. Our mo dels are of the form H = m X n =1 1 2 p 2 n + 1 2 ! 2 n u 2 n  s u 3 n + " 1 2 m  1 X n =1 ( u n  u n +1 ) 2 (1) where " is xed at 0 : 05, the \site frequenies" ! n are hosen randomly from a 2-p oint distribution of variable width  , and the \anharmoniity" s is also 4 \Anharmoni" means that the p erio d of osillation varies with amplitude. A b etter term is \non-iso hronous", f. [5℄, b eause the harmoni osillator is by no means the only p otential for whih the p erio d is indep endent of amplitude, e.g. [6℄. 2 variable. We onsider paths in the ( ; s ) plane onneting the extremes (1 ; 0) (random harmoni) to (0 ; 1) (translation-invariant anharmoni). This typ e of mo del diers from that of [1℄ in three ways. Firstly, Feddersen's mo del has global phase rotation symmetry whih gives it sp eial features (e.g. onserved exitation \numb er" and existene of \stationary solutions" { disrete breathers with time-dep endene purely of the form e i ! t ) 5 . Seondly, Feddersen has a disordered oupling matrix in addition to random site frequenies 6 . Thirdly, Feddersen's distribution of site frequenies is ontinuous rather than disrete. Further work will b e required to assess the eets that suh dier- enes an make, but we note that Kopidakis and Aubry [17℄ have reently studied a similar mo del to ours but with a ontinuous distribution of site frequenies, and it will b e interesting to ompare their results. There are some mathematial results on this problem. Continuation of Anderson mo des to weak nonlinearity (equivalently, from innitesimal to small amplitude in a nonlinear system) was studied mathematially by Albanese and Frohlih [18℄. They proved that Anderson mo des have a \ontinuation" to a Cantor set of amplitudes whose omplement has vanishing relative measure in the neighb ourho o d of zero amplitude. Their pro of leaves gaps in the set of allowed amplitudes b eause of the p ossibility of resonane with nearby Anderson mo des. Non-ontinuability aross these gaps is a priori quite likely but we onsidered it interesting to investigate numerially to what extent there is a genuine obstale. In the other diretion, the pro ofs of existene of disrete breathers apply equally well to disordered systems as to translation invariant latties: simply, more are must b e taken to satisfy the ondition of non-resonane b etween the breather frequeny and the linearised frequenies ab out the equilibria [19℄ (see also [13℄ for a sp eial ase). Thus ontinuation of disrete breathers from the ordered nonlinear regime is guaranteed to at least some amount of disorder and weakening of the nonlinearity. The questions we address here numerially are whether one an nd ontinuous paths of LOs from the ordered nonlinear regime to the edge of the Albanese- Frohlih regime and whether there is a genuine obstale to ontinuation within the Albanese-Frohlih regime. The answer we nd in our hosen mo del is that all the disrete breathers we studied ontinue to Anderson mo des exept for some small disontinuities o urring in the Albanese-Frohlih regime. 5 DST mo dels (also known as disrete nonlinear Shrodinger equations) were an imp ortant early soure of LOs in nonlinear latties [16℄, but the reent wave of ativity on LOs is onneted with the realisation that the phenomenon is muh more general. 6 He also kept the disorder xed, but by saling the eld by the square ro ot of the nonlinearity this is equivalent to dereasing the disorder to zero as the nonlinearity go es to innity. 3 Throughout the pap er we use the term LO for Anderson mo des, breathers and any time-p erio di spatially lo alized osillations in b etween. We avoid the terms \mo de", whih suggests a linear regime, and \intrinsi", whih is supp osed to denote an eet not requiring randomness, b eause our message is that lo alized osillation is a more general phenomenon enompassing b oth limits. We b egin by desribing the hosen mo del in Setion 2. Then in Setions 3,4 and 5, resp etively, we desrib e the numerial metho ds used to obtain a breather in the anharmoni ordered ase, to ontinue it towards the linear disordered ase, and to ompute its linear stability. The results are presented in Setions 6 and 7. The pap er onludes with a short summary in Setion 8. 2 The mo del We study an anharmoni Hamiltonian system of the Klein-Gordon typ e, given by the Hamiltonian: H = m X n =1 1 2 p 2 n + 1 2 ! 2 n u 2 n  s u 3 n + " 1 2 m  1 X n =1 ( u n  u n +1 ) 2 (2) where f u n g m n =1 are the o ordinates of the m partiles with resp et to their equilibrium p ositions; f ! n g m n =1 are the frequenies of small amplitude osillation of the partiles, whih an b e dierent as will b e explained shortly;  s u 3 n is the anharmoni part of the on-site p otential for the partile n , therefore a softening p otential; s is a parameter whih desrib es the degree of anharmoniity and takes its value in [0 ; 1℄, s = 0 b eing the harmoni ase; and " is the oupling parameter, " = 0 orresp onding to no oupling. The oupling p otential is harmoni and nearest neighb our, though b oth these assumptions an b e relaxed. The disorder is implemented here by means of the urvatures ! 2 n of the lo al p otentials at their minima. Sine the masses have all b een hosen equal to 1, this gives rise to frequenies ! n . Supp ose they take two values randomly distributed, say: ! n = ! 0 (1 +  r n 2 ) : (3) Here  is the disorder parameter, taking its value in [0 ; 1℄. At  = 0, there is no disorder and all the frequenies are equal to ! 0 , whih is taken to b e 1; f r n g m n =1 are the omp onents of a random vetor of  1. At  = 1, the most disordered ase, the frequenies are 0 : 5 ! 0 or 1 : 5 ! 0 . The parameter  is to b e onsidered a monotoni funtion  =  ( s ) of the nonlinearity s , with  (0) = 1 4 and  (1) = 0, giving a path from the anharmoni ordered ase to the harmoni disordered ase. Examples of these paths are  = 1  s q ; q > 0 : (4) We study paths with q = 1 and q = 1 4 in this pap er (it might have b een more natural to take only C 1 paths, but we were interested to investigate the eet of the q = 1 4 approah to the disordered linear limit to o, b eause of the small numb er of bifurations on it). We ould have hosen random masses or random oupling strengths instead of or as well as random urvatures, or a ontinuous distribution for ! n rather than a disrete one, but we think that our hoie is a go o d rst ase to study. For a study of an example with ontinuous distribution of ! n , see [17℄. Note also how this approah diers from that used in [1℄: in that work, the degree of disorder was kept xed while the nonlinearity was tunable; here we tune b oth. We susp et that this is more relevant than the p ossible dierenes arising from the two underlying mo dels (a Hamiltonian lattie of lassial osillators vs. a disrete self-trapping equation). We onsider only 1D hains, but it would also b e interesting to study 2D or 3D mo dels and mo dels in whih the randomness arises partially or ompletely from a random network of ouplings. The dynamial equations of the system are ( _ p n =  u n =   H = u n ): F n ( u; s; " ) =  u n + ! 2 n u n  3 s u 2 n + " (2 u n  u n  1  u n +1 ) = 0 : (5) 3 Obtaining breathers and Anderson mo des 3.1 Obtaining a breather Numerial metho ds for nding breathers are desrib ed in [14,15,20℄. We work in the spae of time-p erio di, time-symmetri solutions of sp eied frequeny ! b , with ontinuous seond derivative, whih is denoted by E 2 s ( ! b ). Therefore the funtions u n ( t ) an b e approximated by trunated Fourier series (rotating wave approximation) of the form: u n ( t ) = k m X k =  k m z k n e i k ! b t = z 0 + k = k m X k =1 2 z k n os( k ! b t ) : (6) The seond equality holds b eause z k n is real and z k n = z k  n , as a result of the op erator in Eq. (5) b eing real and the time-symmetry of u n ( t ). 5 Using the Newton-Raphson metho d, we rst nd a solution for the isolated osillator with a given frequeny ! b , using as a seed the solution of the harmoni problem. The starting seed for obtaining a breather is a partile with this solution and the others at rest. By varying " from " = 0, with small steps, we are able to obtain a breather from the unoupled limit. 3.2 Obtaining Anderson modes This is a muh simpler task. When the on-site p otential is harmoni the dynamial equations of the system (Eq. (5)) an b e written:  u n =  ! 2 n u n  " (2 u n  u n  1  u n +1 ) ; (7) or in more ompat notation:  u =   u: (8) Then it is simply a question of nding eigenvetors u of the matrix  and heking whether they are spatially lo alized. In one dimension (f. [21℄), for arbitrarily weak disorder the sp etrum of  onsists of m eigenvalues ~ ! 2 k in (0 ; 1 ), whose eigenvetors v k are lo alized near n = k and deay exp onentially as j n j ! 1 . Thus equation (8) has p erio di solutions: u k n ( t ) = os( ~ ! k t ) v k n ; sin( ~ ! k t ) v k n : (9) As we are working in the spae of time-reversible solutions, we are only on- erned with the rst set of solutions of Eq. (9). They are easy to obtain numerially. Two of them are shown in Fig. 1, whih are the LOs at the end of the q = 1 = 4 and q = 1 paths. 4 Continuation of lo alized osillations Here we explain the metho d that we have used for ontinuing the breather in its path towards disorder. Sample results are presented in Setions 6 and 7. The ontinuation annot b e done at onstant frequeny as there is no guarantee that at the harmoni limit there is any Anderson mo de with the breather frequeny (exept for the trivial solution, u n = 0, 8 n ). Instead, we ho ose to maintain onstant the ation of the lo op, I = m X n =1 I p n dq n = m X n =1 I _ u n du n (10) 6 0 10 20 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 w=0.5466 Components of the Anderson mode 0 10 20 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 w=1.5498 Fig. 1. The omp onents of two Anderson mo des, whih are at the end of the paths with (a) q = 1 = 4, and (b) q = 1. (p erhaps b etter alled its \area", as \ation" has many other meanings). Among other advantages, this prevents the ontinuation from moving towards the zero solution. As the p erio di funtions of arbitrary frequeny do not form a vetor spae, we hange the time saling: t !  = ! b t . Denoting dierentiation with resp et to  by prime ( 0 ), we have  u = ! 2 b u 00 , and ! b will app ear in the dynamial funtion as a parameter to b e determined. Therefore, we work in E s (2  )  R , E s (2  ) b eing a suitable Banah spae of time-symmetri 2  -p erio di funtions. The ation is I = ! b P m n =1 R 2  0 ( u 0 n ) 2 d . Substituting the expressions (6) with  = ! b t into the dynamial equations (5), with u 00 = ! 2 b  u , and olleting the terms in the exp onentials exp (i k  ), we obtain m ( k m + 1) dierent algebrai equations G ( z ; s ) = ff G k n ( z ; s ) g k m k =0 g m n =1 for the m ( k m + 1) variables z = ff z k n g k m k =0 g m n =1 and the frequeny of the selflo alized mo de ! b . Using the expressions for u n in Eq. (6), I an b e easily alulated to b e I ( z ; ! b ) =  ! b m X n =1 k m X k =1 ( k 2 z k n ) 2 : (11) Then, we add to the system of equations G the equation: G I ( z ; ! b ) = I ( z ; ! b )  I 0 = 0 : (12) The dynamial equations are redued to a set of m ( k m + 1) + 1 algebrai equations in the m ( k m + 1) + 1 unknowns z k n and ! b . Hereafter we will use 7 the notation ~ G = ( G; G I ) and ~ z = ( z ; ! b ) : (13) The Jaobian of ~ G with resp et of ~ z is of sp eial imp ortane. On the one hand its invertibility is the ondition for the Impliit Funtion Theorem (IFT) to apply and, therefore, to guarantee the existene of a unique branh of solutions ~ z ( s ) in the neighb ourho o d of a solution ~ z 0 at s 0 . On the other hand, it is used within the Newton metho d for ontinuing a solution, whih is the pratial realisation of the IFT. If ( ~ z 0 ; s 0 ) is a solution of ~ G ( ~ z ; s ) = 0, and s 1 = s 0 + Æ s is a nearby value, writing ~ z 1 = ~ z 0 + Æ ~ z , we have: ~ G ( ~ z 0 + Æ ~ z ; s 1 )  ~ G ( ~ z 0 ; s 1 ) +  ~ z ~ G ( ~ z 0 ; s 1 ) Æ ~ z : (14) Therefore, we obtain a rst approximation for the solution at s 1 by making the last expression zero. This approximation is: ~ z 1 = ~ z 0 + Æ ~ z = ~ z 0  [  ~ z ~ G ( ~ z 0 ; s 1 )℄  1 ~ G ( ~ z 0 ; s 1 ) : (15) Using ~ z 1 as a new seed instead of ~ z 0 , and rep eating this pro edure, we are able to obtain a sequene of approximations to the solution at s 1 whih, provided it exists and it is lose enough to s 0 , onverges quadratially to mahine preision. We need an initial breather solution at s 0 = 1 to start our ontinuation pro edure. This an b e obtained by a preliminary ontinuation from the unoupled limit, as disussed ab ove in Setion 3.1. In our numeris we started with the simplest onguration, single breathers (i.e. those whih orresp ond in the unoupled limit to one partile osillating and the others at rest). One the solution is ontinued to the desired value of the oupling " , it b eomes the starting p oint for our ontinuation into Anderson mo des. We have used a mo del with 21 partiles, a numb er whih proves to b e large enough to observe the LOs. For every initial ondition and path in parameter spae that we examined, we found several values of s where the Jaobian  ~ z ~ G has zero eigenvalues, so in priniple a bifuration ould take plae. In spite of that, the Newton metho d generated an almost ontinuous path of LOs. To larify whether an eigenvalue is really zero or only very small, b eause of the inauray of the trunations of the Fourier series and the numerial error, we have plotted the whole set of eigenvalues with resp et to s . Then it is easy 8 to observe the evolution of the dierent eigenvalues, and if one of them hanges sign it is lear that we have a zero eigenvalue. As the eigenvalues f v i g an b e omplex we plot abs ( v i ) sign(Re v i ), whih gives the learest plot. A omplex eigenvalue whose real part hanges sign would app ear as a disontinuity, but there are two of them, and they are easily reognized and heked with the plot of their ounterpart abs ( v i ) sign(Im v i ). Most of the eigenvalues of  ~ z ~ G are easily reognized, due to the fat that most osillators in a LO are osillating with very small amplitudes. An easy way to trae their origin is to analyze the sp etrum from the limit of weak oupling. In suh ase, the dynamial equations orresp onding to osillators with small amplitudes are: F n ( u; s ) = ! 2 b u 00 n + ! 2 n ( s ) u n = 0 : (16) Substitution of the Fourier representation in terms of the mo des z k n os k  , gives:  k 2 ! 2 b z k n + ! 2 n ( s ) z k n = 0 : (17) That is, these osillators would only ontribute with diagonal entries in the Jaobian  ~ z ~ G , of the form: v k n = ! 2 n ( s )  k 2 ! 2 b ( s ) : (18) As ! n an take only two dierent values, !  ( s ) = ! 0  1 2  ( s ), we obtain: v k  = ! 2  ( s )  k 2 ! 2 b ( s ) : (19) Therefore, we obtain 2( k m +1) dierent p ossible values for these eigenvalues. At " = 0, eah of these has a degeneray oiniding with the numb er of osillators at rest with frequeny ! + ( s ) or !  ( s ), resp etively. When the oupling " is swithed on, eah of these will split into a \band", and that degeneray will generially b e raised. Then there will b e other eigenvalues of dierent origin, those asso iated to sites whih are in a high-amplitude osillation (just one site for a single breather). Those will b e typially detahed from the bands disussed ab ove, so they are easily sp otted. Another omplex onjugate pair of isolated eigenvalues omes from the last equation (Eq. 12). It is asso iated with the interhange of amplitude and frequeny that maintains the ation onstant. 9 0 0.2 0.4 0.6 0.8 1 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 s: 0 disorder, linearity; 1 order, nonlinearity |v|sign(real(v)) Evolution of the eigenvalues of the Jacobian. 1/4 path. w0+ w0− w1+ w1− Iw Iw w2+ w2− b2 Fig. 7. Evolution of the eigenvalues of the  ~ z ~ G ( ~ z ; s ) along the 1/4 path. The lab els are the same as in Fig. 5. it is helpful to ompare with Fig. 9 and to notie how the frequenies of the LO and a linearised mo de lo ated mainly on partiles 2{4 ross (Fig. 11). If the frequenies of a (time-symmetri) LO of ation I on one nite hain of osillators and a linearised mo de on another nite hain ross as a parameter s passes through a value s  , then for the joint system of two indep endent hains there is a pithfork bifuration in the set of time-symmetri LOs of ation I at s = s  , b eause one an add any small amount of the linearised mo de, in either phase, and derease the ation of the LO orresp ondingly. Generially, these hanges ause a slight shift in the frequeny ratio, requiring a omp ensating shift in parameter leading to a pithfork. If one now onnets the two hains by oupling their ends to make a single hain, the pithfork an b e exp eted generially to break (unless some spatial symmetry is preserved), making an imp erfet pithfork. This an b e seen in the bifuration diagram in Fig. 10. We have obtained the two outer prongs of the pithfork by the following pro edure: we add to a solution f u n g , not to o lose to the bifuration, the eigenvetor orresp onding to the entre subspae, saled so as b oth have the same ation. We resale again so as to obtain a vetor with the same ation as our solutions. This is used as a seed for the Newton metho d to obtain a branh of solutions that joins to the previous one at the bifuration p oint. Subtrating the eigenvetor, instead of adding, we are able to obtain the other branh. This an b e onrmed easily by omparing the plots of the o ordinates versus 16 0 0.2 0.4 0.6 0.8 1 −0.01 0 0.01 0.02 0.03 0.04 0.05 s: 0 disorder, linearity; 1 order, nonlinearity |v|sign(real(v)) Evolution of the eigenvalues of the Jacobian. 1/4 path. Fig. 8. Enlargement of Fig. 7, showing the eigenvalues in the group w 1  . Apparently only one eigenvalue rosses the zero line; it is related to a hange of phase of a small amplitude mo de on partile 5. The double LO on partiles 11 and 12 evolves ontinuously. Also visible in this piture is the group of eigenvalues w 2+ at very small s , whih rosses the zero line at ab out s = 0 : 01; this is related to the disapp earane of the Fourier omp onents with k 6 = 1 at the linear limit. time b efore and after the bifuration. It shows the hange of phase in a group of six partiles. Moreover, the null eigenvetor is omp osed mainly of the rst Fourier omp onents of these partiles. This is not a standard imp erfet pithfork bifuration, however, as b oth the \horizontal" branhes are observed to b e linearly stable, in ontrast to a standard imp erfet pithfork for p erio di orbits of a Hamiltonian system for whih the \horizontal" branh would hange stability. We onjeture that the bifur- ation diagram should b e ompleted by reation of a pair of unstable timeasymmetri LOs by a \Rimmer bifuration" (pithfork for p erio di orbits of reversible Hamiltonian systems) [24℄ from one of the \vertial branhes" after a short interval of instability. Note that although the standard setting for analyzing bifurations of p erio di orbits in Hamiltonian systems is to onsider variation of the set of p erio di orbits of given energy E with resp et to E and/or external parameters, the same results apply for given ation I , b eause d I = d E = T , the p erio d, whih is neither zero nor innite. 17 5 10 15 20 0 0.05 0.1 0.15 0.2 2 4 6 8 0 5 10 15 20x 10−4 5 10 15 20 0 0.05 0.1 0.15 0.2 2 4 6 8 −20 −15 −10 −5 0 x 10−4 Fig. 9. Evolution of the LO through the zero eigenvalue at s = 0 : 2938 on the 1/4 path. On the right, Fourier omp onents b efore the bifuration, at s = 0 : 294, on the left, after the bifuration, at s = 0 : 293. On the top, the 21 partiles, on the b ottom, partiles 1{6. 2 4 6 FC of the centre subspace 0 0.2 0.4 0.6 0.8 1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 Bifurcation variable Main bifurcation in the 1/4 path Nonlinearity/order parameter, s breather double Anderson mode Fig. 10. Bifuration diagram along the 1 = 4 path. The bifuration variable is the omp onent in Fourier spae in the diretion of the eigenvetor with zero eigenvalue for the op erator  ~ z ~ G ( ~ z ; s ) at s = 0 : 2938, whih is shown in the inset. The outer prongs, omp osed of the entral osillation plus an osillation similar to this eigen- vetor, with the same phase for the upp er prong, and a phase dierene of  for the lower one, lead to non lo alized osillations in the nonlinear limit. 18 0 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 Nonlinearity/order parameter s Frequencies Frequencies of the linearised modes about equilibrium 2 wb wb, frequency of the LO Fig. 11. Frequenies of the LO (also twie its frequeny) and all the linearised mo des ab out equilibrium for the 1/4 path. 7.2 Bifurations in the linear path The bifuration diagram an b e seen in Fig. 12. The bifuration variable is the pro jetion of the solutions on the entre subspae for bif. 2. The breather starts at s = 1 with o de +1 on partile 11 (in group C) and follows the path 1(+), on whih the LOs A and B grow as s dereases. The rst eigenvalue rossing the zero line is not visible in Fig. 12, but orresp onds to a linear mo de on another part of the hain, given a disontinuity like the one desrib ed ab ove. The seond pro dues the rst imp ortant bifuration, i.e. related to interations of LOs of non-negligible amplitude. This is bif. 1, at s 1 = 0 : 322. There the entral mo de on C, disapp ears, pro duing path 2, on whih the o dd Fourier o eÆients are negligible and presumed to b e due to numerial error, i.e. the frequeny is doubled. Continuation of the solution bakwards from the bifuration leads to path 1(-), where the solution is idential to that on path 1(+) exept for a phase hange of  , leading at s = 1 to the single breather o ded -1 on partile 11. Manipulation of the solutions in path 1, b efore and near the bifuration, eliminating the entral LO, and ontinuing it forwards and bakwards, makes it p ossible to nd a third branh, path 5, whih leads at s = 1 to a general osillation of the system. Putting these together, we obtain an inverse period-doubling bifuration , shown in the inset to Fig. 12 by using as bifuration variable a o ordinate on the entre subspae for bif. 1, whih distinguishes b etween the two phases 1(  ). It 19 0.32 0.35 0.32 0.35 −0.1 0 0.1 path 5 A(+),B A(+),B path 1 (+) path 1 (−) bif. 1 bif. 2 0 0.2 0.4 0.6 0.8 1 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 Nonlinearity/order parameter s Bifurcation variable Bifurcations in the linear path path 3 B path 4 A(−),B −> general oscillation path1(+,−) A(+),B,C(+,−) −> breather(+.−) path 5 A(+),B −> general oscillation path 2 path 6 B −> general oscillation Fig. 12. Bifuration diagram along the linear path. The bifuration variable is the omp onent in Fourier spae in the diretion of the eigenvetor with zero eigenvalue for the op erator  ~ z ~ G ( ~ z ; s ) at bif. 2 (bif. 1 in the inset). It is fully explained in the text. is not a standard p erio d doubling bifuration (as desrib ed by [25℄ for generi Hamiltonian systems or [24℄ for reversible Hamiltonian systems), however, b eause numeris show that all branhes are linearly stable. Our onjeture is that there is a short interval of instability on path 5 whih terminates in another p erio d doubling bifuration, generating an unstable LO whih diers from that on path 1 mainly by group C b eing phase-shifted by  = 2. This would still b e a time-symmetri orbit, but ab out t = T = 4 rather than t = 0 ( T b eing the p erio d). Continuation of path 2 leads to bif. 2, at s 2 = 0 : 234, where it jumps onto path 3, where the LO A has disapp eared, and at s = 0, it is the only surviving LO. Using analogous pro edures to the ab ove, we also obtain two other nearby branhes, giving paths 4 and 6. On path 4, the LO A(-) diers from A(+) by a phase  , and on path 6, the LO A is absent. The limits of paths 4 and 5 leading bakwards to general osillations give, approahing to s = 1, dierent osillations. The approximate symmetry b etween paths 4 and 5 o urs only near the bifuration. We identify this bifuration as an imp erfet pithfork, but again sine all the branhes shown are linearly stable, there must b e some unstable branhes to o and we onjeture the same diagram as in the previous subsetion. 20 7.3 General omment In eah ase, we found only a limited numb er of eigenvalues rossing zero, whih are the eigenvalues of the two or three rst linear mo des of the partiles with frequenies ! + or !  near rest, and the omp ound mo des, dep ending on the path followed; their numb er is of the order of the numb er of partiles. Inreasing the numb er of Fourier omp onents do es not hange this fat; only the very few rst omp onents dier appreiably from zero, and the higher mo des are of very small amplitude. Inreasing the numb er of partiles, on the other hand, inreases prop ortionally the numb er of zero rossings. Therefore, we guess that for innite systems there will b e an innite numb er of eigenvalues rossing zero. 8 Conlusion Disrete breathers are time-p erio di lo alised osillations (LO) of weakly oupled networks of anharmoni osillators. Anderson mo des are LOs for spatially random networks with linear dynamis. For a mo del whih interp olates b etween these two limits, we have found numerially that although Anderson mo des typially 7 evolve to a \general osillation" with no spatial lo alisation, on moving parameters in the opp osite diretion, disrete breathers an b e followed almost ontinuously to Anderson mo des, whih are often (though not always) lo alised around the same sites as the breathers. This asymmetry is natural in view of the fat that there are many more time-symmetri p erio di solutions of given p erio d at the ordered anharmoni limit ((3 m  1) = 2 for a system of size m ) than at the disordered harmoni limit ( m ), but only m of them are single-site breathers. The others are \multi-site breathers" [4℄. No paths have b een found, however, that ompletely avoid degeneraies in the linearized ontinuation op erator. We found bifurations at these p oints. They are often fold bifurations, whih stritly sp eaking makes it imp ossible to ontinue further. Nevertheless, at every fold p oint we always found a very nearby path onto whih to jump and ontinue towards the disordered linear limit. For some paths these bifurations are related to hanges in b ehaviour of groups of partiles near rest. This happ ens, for example, in the path alled the 1 = 4 path. The maximum value of the Fourier o eÆients of the partiles involved in the bifuration are ab out 10  2 times the largest of the whole system. Even less signiant are the eigenvalues of the k = 2 mo des rossing zero almost at 7 Not always: in some ases, the omputation ontinued an Anderson mo de onto a disrete breather, but this dep ended on how fast it went through some bifuration p oints where there was a \random" hoie of branhes. 21 the linear limit where they will disapp ear. No hange of stability is observed in these paths, although it an b e found in some others. A detailed study of the main bifurations has b een made, obtaining the dierent branhes and plotting them in numerially determined entre subspaes. We found inverse p erio d doublings and broken pithforks. Bifurations suh as these, destroying LOs as disorder inreases and nonlinearity dereases, should b e exp eted, sine there are many more time-symmetri p erio di orbits of given ation at the ordered anharmoni limit than at the disordered harmoni limit, as already remarked ab ove. A very interesting pro jet for the future is to understand the generi bifurations for suh systems, whih are not standard ones b eause they are unfoldings of bifurations for deoupled systems. A tehnial, but p ossibly imp ortant, p oint is that we ontinue LOs at onstant ation, rather than onstant p erio d. Aknowledgements We thank A. C. Sott for useful omments on the manusript. JLM aknowledges a Marie-Curie TMR fellowship from the EU, no. ERBFMBICT972761. Referenes [1℄ H Feddersen. Lo alization of vibrational energy in globular protein. Phys Lett A , 154:391{5, 1991. [2℄ A Sott. Davydov's soliton. Phys Rpts , 217:1{67, 1992. [3℄ S Flah and CR Willis. Disrete breathers. 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