Generic Hopf bifurcation from linesof equilibria without parameters:III. Binary oscillations �Bernold Fiedler, Stefan LiebscherJ.C. AlexanderyPreprint 42/98Institut f�ur Mathematik I, Freie Universit�at BerlinArnimallee 2-6, 14195 Berlin, Germanyy Dept. of Mathematics, Case Western Reserve University10900 Euclid Avenue, Cleveland, OH 44106-7058, USAJune 22, 1998�This work was supported by the DFG-Schwerpunkt \Analysis und Numerikvon Erhaltungsgleichungen" and with funds provided by the \National ScienceFoundation" at IMA, University of Minnesota, Minneapolis.

1 IntroductionBinary oscillations have been observed, both numerically and analytically, incertain discretizations of systems of nonlinear hyperbolic conservation laws;see [LL96]. More speci�cally, we consider systems of hyperbolic balance lawsof the form ut + f(u)x = g(u):(1.1)Semidiscretization of x 2 IR with step size �=2 > 0 can, for example, beperformed by a central di�erence scheme_uk + ��1(f(uk+1)� f(uk�1)) = g(uk)(1.2)As a cautioning remark we hasten to add that we do not recommend this par-ticular discretization for numerical purposes. Rather, it is our goal to inves-tigate peculiar short range oscillation phenomena of system (1.2). Rescalingtime, we obtain the equivalent system_uk = �g(uk)� f(uk+1) + f(uk�1)(1.3)Note how this system decouples into a direct product ow, if ui+2 = ui, forall k 2 ZZ . Indeed, any solution u0(t) of _u0 = �g(u0) gives rise to a solutionof (1.3) satisfying u1(t) = u0(t+ �)uk+2(t) = uk(t); for all k(1.4)for any �xed choice of � 2 IR.It is the goal of our present paper to investigate loss of stability of thisdecoupling phenomenon. In general, u2k and u2k�1 can de�ne consistentlysmooth, but di�erent pro�les x 7! u(t; x). We consider only the simplestcase k (mod4)(1.5) 1

of uk de�ning the corners of a square. In this case, decoupling phenomenaas above have been discovered by [AA86] in the slightly di�erent context ofperiodic orbits of linearly coupled oscillators. For more intricate, nonplanargraphs of coupled oscillators supporting such decoupling e�ects see [AF89].For simplicity, we consider systems of two balance laws, ui 2 IR2. To facilitateour calculations further, we impose an S1 equivariance conditionf(R'u) = R'f(u);g(R'u) = R'g(u)(1.6)under all rotation matricesR' = 0@ cos' � sin'sin' cos' 1A :(1.7)Denoting the Euclidean norm by juj, we can therefore writef(u) = a(juj2)ug(u) = b(juj2)u:(1.8)where the values of a; b are scalar multiples of rotation matrices.We assume that the decoupled vector �eld _u = g(u) of the reaction termalone possesses an exponentially stable periodic orbit. Assumingb(juj2) = 0@ 1 � juj2 � 1 � juj2 1A ;(1.9)we normalize the periodic orbit to juj = 1, its frequency to , and its ex-ponential rate of attraction to �2. On the slow time scale _u = �g(u) theselatter values become � and �2�, of course.On the square (1.3), (1.5) decoupling produces an invariant 2-torus foliatedby these periodic orbits; see (1.4). Normalizing the time shift �, these solu-2

tions are given explicitly byT 2 : u0(t) := R�te1u1(t) := u0(t+ (�)�1�) = R�u0(t)u2(t) := u0(t)u3(t) := u1(t) = R�u2(t)(1.10)Here � 2 S1 = IR=2�ZZ , due to S1-equivariance, and e1 2 IR2 denotes the�rst unit vector. Our investigation of binary oscillations will focus on thedetailed dynamics near the decoupled 2-torus (1.10).Our assumption (1.6) on S1-equivariance allows us to eliminate one variablefrom our eight-dimensional vector �eld (1.3), (1.5). Indeed, the ow of (1.3),(1.5) maps S1-orbits onto S1-orbits, by equivariance under the S1-action(R'u)i := R'ui(1.11)on u = (u0; : : : ; u3) 2 IR8. The induced ow on the space of group orbitscan be computed in explicit coordinates, representing a cross section to thegroup orbits; see (1.12) below. For spec�c calculations, we will use polarcoordinates (rk; 'k) for uk 2 IR2; see sections 3 and 4.Relating back to dynamics, consider the Poincar�e return map to any Poincar�ecross section X through any of the periodic orbits on our 2-torus T 2. Inparticular, the section X is also transverse to the S1-action (1.11) which isfree near the 2-torus. We can therefore rewrite the Poincar�e map as the timet = 2�(�)�1 map of a suitable associated ow_x = F (x)(1.12)representing the induced ow on the seven-dimensional Poincar�e cross sectionX. The �xed Poincar�e return time can in fact be achieved by incorporating ascalar Euler multiplier into the induced ow on X. We will make an explicitchoice for X later, based on polar coordinates.3

In the coordinates x 2 X, the periodic 2-torus T 2 from (1.10) becomes a one-dimensional curve of equilibria. Indeed, time action and S1-action coincideon T 2. Therefore the �xed points of the Poincar�e return map given by T 2\Xcoincide with equilibria of the induced ow _x = F (x) on X. In other words,the relative equilibria on T 2, relative to the S1-action, become equilibria onthe local space X of S1-orbits.As long as the curve of equilibria remains normally hyperbolic, the localdynamics has been clari�ed by [Sho75], [Fen77], [HPS77], and others; seealso [Shu87], [Wig94]. Locally, the dynamics is �bered into invariant leavesover each equilibrium, with dynamics in each leaf governed by hyperboliclinearization.Bifurcations from lines of equilibria in absence of parameter have been in-vestigated in [FLA98] from a theoretical view point. We brie y recall thepertinent result, for convenience. As in (1.12), now consider general C5 vec-tor �elds _x = F (x) with x 2 X = IRn. We assume a one parameter curve ofequilibria 0 = F (x(�))(1.13)tangent to x0(�0) 6= 0 at � = �0; x(�0) = x0. At � = �0, we assumethe Jacobi matrix F 0(x0) to be hyperbolic, except for a trivial kernel vectoralong the direction of x0(�0) and a complex conjugate pair of simple purelyimaginary eigenvalues �(�); �(�) crossing the imaginary axis transversely as� increases through � = �0 :�(�0) = i!0; !0 > 0Re�0(�0) 6= 0(1.14)Let E be the two-dimensional real eigenspace of F 0(x0) associated to �i!0.Coordinates in E are chosen as coe�cients of the real and imaginary parts ofthe complex eigenvector associated to i!0. Note that the linearization acts4

as a rotation with respect to these not necessarily orthogonal coordinates.Let P0 be the one-dimensional eigenprojection onto the trivial kernel alongthe direction x0(�0). Our �nal nondegeneracy assumption then reads�EP0F (x0) 6= 0:(1.15)Here the Laplacian �E is evaluated with respect to the above eigenvectorcoordinates in the eigenspace E of �i!0. Fixing positive �-orientation, wecan consider �EP0F (x0) as a real number. Depending on the sign� := sign(Re�0(�0)) � sign(�EP0F (x0))(1.16)we call the \bifurcation" point x = x0 elliptic if � = �1, and hyperbolic for� = +1.The following result from [FLA98] investigates the qualitative behavior ofsolutions in a normally hyperbolic three-dimensional center manifold to x =x0.Theorem 1.1Let assumptions (1.13)-(1.15) hold for the C5 vector �eld _x = F (x) alongthe curve x(�) of equilibria. Then the following holds true in a neighborhoodU of x = x0 within a three-dimensional center manifold to x = x0.In the hyperbolic case, � = +1, all nonequilibrium trajectories leave theneighborhood U in positive or negative time direction (possibly both). Thestable and unstable sets of x = x0, respectively, form cones around the pos-itive/negative direction of x0(�0), with asymptotically elliptic cross sectionnear their tips at x = x0. These cones separate regions with di�erent con-vergence behavior. See �g. 1.1a).In the elliptic case, � = �1, all nonequiblibrium trajectories starting in Uare heteroclinic between equilibria x� = x(��) on opposite sides of � = �0.5

Case a) hyperbolic, � = +1. Case b) elliptic, � = �1.Figure 1.1: Dynamics near Hopf bifurcation from lines of equilibria.If F (x) is real analytic near x = x0, then the two-dimensional strong stableand strong unstable manifolds of x� within the center manifold intersect atan angle which possesses an exponentially small upper bound in terms ofjx� � x0j. See �g. 1.1b).Our main result is theorem 2.1 in section 2. It provides speci�c examples of ux functions f and reaction terms g which realize the elliptic variant � = �1of theorem 1.1 in binary oscillation systems (1.3), (1.5), only. The resultholds in the limit � & 0 of small discretization steps. Both the hyperbolicand the elliptic variants occur in viscous pro�les of systems of hyperbolicbalance laws, see [FL98]. For applications to binary oscillations in Ginzburg-Landau and nonlinear Schr�odinger equations as well as a more global studyof decoupling in squares of additively compled oscillators, see [AF98].The proof of theorem 2.1 is spread over the remaining two sections. In sec-tion 3 we present a detailed analysis of the linearization near the curve x(�)of equilibria on X. In particular, we verify assumptions (1.14) on transverve6

crossing of simple, purely imaginary eigenvalues. The nondegeneracy condi-tion (1.15) is veri�ed in section 4, completing the proof of theorem 2.1.AcknowledgmentFor the initiating idea applying bifurcation from curves of equilibria in ab-sence of parameters to binary oscillations problems we are much indebted toBob Pego.This work was supported by the Deutsche Forschungsgemeinschaft, Schw-erpunkt \Analysis und Numerik von Erhaltungsgleichungen" and was com-pleted during a fruitful research visit of one of the authors at the Instituteof Mathematics and its Applications, University of Minnesota, Minneapolis.The visit was supported by the Institute for Mathematics and its Applica-tions with funds provided by the National Science Foundation.2 ResultSetting up for our main result, theorem 2.1, we further specify our squarebinary oscillation system_uk = �g(uk) + f(uk�1)� f(uk+1); k(mod4):(2.1)In the S1-equivariant formulation (1.8), we have already completely speci�edg(u) = b(juj2)u = 0@ 1� juj2 1� juj2 1A u;(2.2)see (1.9). To avoid formula overkill by chain and product rules, we specifythe derivative A(u) = f 0(u) = (a(juj2)u)0(2.3) 7

at u = e1 = (1; 0) to be given by the symmetric inde�nite matrixA(e1) = 0@ c �1�1 0 1A(2.4)In terms of a(juj2); this choice corresponds toa(1) = 0@ 0 �11 0 1A ; a0(1) = 0@ 12c 1�1 12c 1A(2.5)To ensure transverse crossing (1.14) of purely imaginary eigenvalues in thelimit �& 0, we assume jcj < 1 is nonzero:jcj < 1; c 6= 0(2.6)The nondegeneracy condition (1.15) on �EP0P (0) will hold due to the sameassumption (2.6).Theorem 2.1Consider the square binary oscillation system (2.1) with speci�c nonlineari-ties (2.2)-(2.6). Then, for 0 < � < �0 small enough, Hopf points of elliptictype occur at the periodic solution through u = u(�) given by u0 = u2 = e1,u1 = u3 = R�(�)e1.Here � = �(�) 2 (0; �=4) satis�escos(2�(�)) = 2� �22 + c2(2.7)More precisely, the induced ow _x = F (x) on the space of S1-orbits, rep-resented by a Poincar�e cross section X to the above periodic orbit, satis�esassumptions and conclusions of theorem 1.1 for the hyperbolic / elliptic types� = �1 in a neigborhood U = U� of u = u(�(�)): The type determining sign� is given by � = �1;(2.8) 8

independently of the choice of c in (2.6). In particular the stability regionof decoupling into separate, phase-related periodic solutions on odd-labeled /even-labeled discretization points includes a full neighborhood U� of u(�(�))with a prefered sign � � �(�) for the phase shift � of decoupling periodicsolutions u(�).The proof of this theorem consists of checking the transverse crossing as-sumption (1.14) and the nondegeneracy condition (1.15) of theorem 1.1. Inthe limit �& 0, these two conditions are checked in sections 3 and 4, respec-tively.3 Eigenvalue crossingIn this section we provide the linear analysis for Hopf points of purelyimaginary eigenvalues along the 2-torus T 2 of decoupled periodic orbitsu(t) = (u0(t); :::; u3(t)) given byu2(t) = u0(t) = R�te1u3(t) = u1(t) = R�u0(t)(3.1)see (1.10). Passing to polar coordinatesuk = rkR'ke1(3.2)we explicitly factor out the S1�action(R'u)k = rkR'k+'e1(3.3)This explicitly converts the 2-torus T 2 into a line of equilibria x(�) of_x = F (x) in a Poincar�e cross section X; see (1.12), (3.9). We compute thelinearization along the equilibria x(�) and determine the location of Hopf9

points at � = �(�); in the limit �& 0:We then determine an explicit expres-sion for the crossing direction Re �0(�(�))(3.4)of the Hopf eigenvalues. For later use, we also determine ��expansions forthe Hopf eigenvalues � = �(�(�)) = i!(�) and the Hopf eigenvectors v = v(�):Calculations in this and the following section were performed with Mathe-matica and Maple; any other symbolic calculation package should also do.We begin with transformation to polar coordinates (3.2). In variables(rk; 'k); k(mod4); equations (2.1) for binary oscillations mod 4 read_rk = �rk(1 � r2k) +�(rk�1) cos('k�1 � 'k + (rk�1))��(rk+1) cos('k+1 � 'k + (rk+1));_'k = � +r�1k �(rk�1) sin('k�1 � 'k + (rk�1))�r�1k �(rk+1) sin('k+1 � 'k + (rk+1)):(3.5)Here the new nonlinearities �(r); (r) are related to the ux function f(u) =a(juj2)u by a(r2) = r�1�(r)R (r). In fact assumptions (2.5) on a(1); a0(1)translate as �(1) = 1; �0(1) = �1; (1) = �=2; 0(1) = �c:(3.6)The 2-torus (3.1) of decoupled binary oscillations becomesrk � 1;'1 � '0 + �;'k+2 � 'k(3.7)in polar coordinates, with ow _'k = �:(3.8) 10

Note that the right hand side of (3.8) is proportional to the in�nitesimalgenerator of the S1{action (3.3). We choose an orthogonal sectionX = f(r;') j'0 + :::+ '3 = 0g = < e >?(3.9)in coordinates r = (r0; :::; r3), ' = ('0; :::; '3), e = (0; :::; 0; 1; :::; 1). Thevector �eld _x = F (x)(3.10)of the induced ow on S1{orbits, represented by x 2 X, is then given byorthogonal projection of (3.5) onto the section X: In particular, the term �disappears in this projection and the line of equilibria is given byx(�) : rk = 1; 'k = (�1)k+1�=2:(3.11)To simplify our calculations, we restrict the time shift � to the interval� 2 [0; 12�]:(3.12)This can be done without loss of generality due to the D2{symmetry of thesquare ring coupling of our system (1.3),(1.5). Indeed, the D2{symmetry isgenerated by the rotation � and re ection �� : (0; 1; 2; 3) 7! (1; 2; 3; 0)� : (0; 1; 2; 3) 7! (2; 1; 0; 3)(3.13)of the indices. This leads to an equivariance of the system (1.3),(1.5) under� : uk 7! uk+1; k (mod4)� : u0 7! �u2; u2 7! �u0; u1 7! u1; u3 7! u3;(3.14)when we recall that f and g are odd by (1.6). In terms of the time shift �these transformations are� : � 7! 2� � � and� : � 7! �+ �:(3.15) 11

This immediately gives the fundamental domain (3.12).The linearization of F at x(�) is given by restriction and projection of the8 � 8 linearization L(�) of the original polar coordinate vector �eld (3.5)at the relative equilibrium x(�): Rather than writing L(�) out explicity, werecall equivariance of (3.5) and invariance of x(�) under the action k 7! k+2of shifting indices k by 2: The four-dimensional representation subspaces V �under this action are given byV � := frk+2 = �rk; 'k+2 = �'kg:(3.16)By equivariance, these are invariant subspaces of the linearization L(�): Dueto decoupling, V + is in fact also invariant under the nonlinear ow.Let L�(�) denote the respective restrictions of L(�); explicitlyL+(�) = 2 0BBBBBB@ �� 00 0 �� 00 0 1CCCCCCA ;L�(�) = 2 0BBBBBB@ �� 0 �c cos�� sin� cos�0 0 �c sin�+ cos� sin�c cos�� sin� � cos� �� 0�c sin�� cos� sin� 0 0 1CCCCCCA(3.17)with respect to coordinates (r0; '0; r1; '1) in V �:Obviously only L�(�) can carry purely imaginary eigenvalues. In the follow-ing, we therefore restrict our attention to V �. The characteristic polynomialp of L�(�) on V � is given by0 = p(�; �; ) :== �4 + 4��3 + 2((c2 + 4) + c2 + 2�2)�2++16� � + 8(2 + �2( � 1))(3.18) 12

with the abbreviation = cos(2�):(3.19)Decomposing into real and imaginary parts, we immediately see that imagi-nary eigenvalues � = i! satisfy !2 = 4 > 0(3.20)and only occur for �; satisfying = (�) = 2� �22 + c2 :(3.21)This proves (2.7) of theorem 2.1. Before computing the associated eigenspace,we address the transverse crossing condition (3.4) for the eigenvalues � =�(�; ) near Hopf points. Note thatRe �0(�) = �2q1 � 2 Re @ �(�; )(3.22)at = (�), by (3.19) and the chain rule. At � = 0; = (0); � = i!(0) wehave @�p = 8p2ic2(4 + c2)(2 + c2)�3=2 6= 0:(3.23)Hence the implicit function theorem applies:@ �(�; ) = �@ p=@�p(3.24)At � = 0 we obtain @ � = �p2ic�2p2 + c2(3.25)with vanishing real part. Totally di�erentiating (3.24) with respect to � alongthe path � � 0; = (�); � = i!(�); we obtaindd� ������=0 @ � = �@�(@ p=@�p) = �4 (c2 + 2)2c4(c2 + 4) :(3.26) 13

This yields the expansionRe �0(�) = 8 c2 + 2jcj3(c2 + 4)1=2 �+O(�2):(3.27)In particular sign Re �0(�) = +1(3.28)for small � > 0:For later use, we also provide an expansionv(�) = v0 + �v1 +O(�2)(3.29)for the complex eigenvector v(�) associated to the imaginary eigenvalue� = i!(�) = i!0 +O(�2):(3.30)Normalization of v(�) will not be necessary. We decomposeL�� (�(�)) = L0 + �L1 + :::(3.31)where in fact L0 = L�0 (�(0)); �L1 = L+: Comparing coe�cients of �0; �1 in(L0 + �L1 + :::)(v0+ �v1+ :::) = (i!0 + :::)(v0+ �v1 + :::)(3.32)we immediately see (L0 � i!0)v0 = 0;(L0 � i!0)v1 = �L1v0:(3.33)With the abbreviation ~c := pc2 + 4 and some substitutions c2 = ~c2 � 4,14

explicit solutions are given byv0 = 0BBBBBB@ �i(c2~c+ cjcj)�i(c3~c+ c~c� 2jcj)c3 � jcj~c+ 2cc2(c2 + 3) 1CCCCCCAv1 = p24cpc2+2(c2+4) 0BBBBBB@ �c6jcj+ 3c5~c� 6c4jcj+ 6c3~c� 20c2jcj+ 8c~c� 16jcj�c7jcj � 3c5jcj � 2c4~c+ 6c3jcj+ 24cjcji(c5jcj~c+ 3c6 + 4c3jcj~c+ 10c4 + 8cjcj~c+ 8c2)i(c6jcj~c+ c4jcj~c� 4c5 � 6c2jcj~c� 4c3 � 8jcj~c+ 16c) 1CCCCCCA(3.34)Note that v0, v1 are complex orthogonal.4 NondegeneracyIn this section we check the nondegeneracy condition�EP0R(x) 6= 0(4.1)in the limit � & 0; see (1.15). Here the Hopf point x = x�, given in polarcoordinates (r�k; '�k), lies in the section X = hei? = f'0 + : : :+ '3 = 0g andsatis�es r�k = 1'�k = 12(�1)k+1�� � = cos 2�� = 2��22+c2(4.2)see (3.9), (3.11), (3.21). The projection P0 is the eigenprojection onto theone-dimensional kernel of the linearization in X. By our V �� decomposition(3.16), (3.17), the full linearization L(��) possesses kernel only in V +. Indeed,the characteristic polynomial p = p(�; �; �) on V � does not possess zero15

eigenvalues; see (3.18). The two-dimensional kernel of L+(��) in V + is givenby rk = 0; 'k+2 = 'k(4.3)for small � > 0; see (3.17) again. By restriction and orthogonal projection toX, we see that P0F (x) is given byP0F (x) = 12(�F '0 + F '1 � F '2 + F '3 ) � 0BBBBBB@ �1=21=2�1=21=2 1CCCCCCA(4.4)Here we have written the polar coordinate components of the orginal vector�eld (3.5) in the form _'k = F 'k(4.5)Note that the unit vector in (4.4) points along the '-components of the linex(�) of equilibria in positive �-direction. Moreover P0 does not depend on�. We can therefore consider �EP0F (x) to be given by the real number�� := 12�E�(�F '0 + F '1 � F '2 + F '3 )(x�);(4.6)with only the Hopf point x� and the Hopf eigenspace E� depending on �.Equivariance with respect to index change k 7! k + 2 further simpli�es ex-pression (4.6) for ��. Indeed E� � V �, because the Hopf eigenspace E�results from L�(��); see (3.17). Restricted to V �, the quadratic Hessianforms of F 'k and F 'k+2 at x� coincide. Therefore (4.6) simpli�es to�� = �E�(�F '0 + F '1 )(x�)(4.7)Expanding �� to including �rst order terms in �, we immediately notice that�� = �E�(�F '0 + F '1 )(x0) +O(�2)(4.8) 16

Indeed x� = x(��) depends only to second order on �, as does �� itself; see(3.19), (3.21). Therefore we only have to consider dependence of the Hopfeigenspace E� = span fRe v(�); Im v(�)g(4.9)on �, to �rst order. We recall that a �rst order expansionv(�) = v0 + �v1 + : : :(4.10)of complex eigenvectors v(�) to the simple eigenvalue �(�) = i!(�) near i!(0)was derived in section 3; see (3.34). Also recall that v(�) are not normalized.Denoting second derivatives by D2, we abbreviate the Hessian byH0 = �D2F '0 (x0) +D2F '1 (x0)(4.11)and expand �� = H0[v(�); �v(�)] +O(�2)= H0[v0; �v0] + 2Re (H0[v1; �v0])�+O(�2)(4.12)It is worth noting here that indeed �E� has to be evaluated with respect tothe eigenbasis Re v(�), Im v(�) of E� and not with respect to an orthonormalbasis. This follows from the proof of theorem 1.1 in [FLA98]. Indeed, theterm �� arises in the normal form process after a linear transfromation ofthe linearization to pure rotation in the Hopf eigenspace. The length of v(�)is irrelevant in that analysis: only the sign of �� enters into the �nal result.After these preparations we �ndH0[v0; �v0] � 0:(4.13)The term of order � can be considerably simpli�ed to2Re (H0[v1; �v0]) = �8c2((c2 + 1)pc2 + 4jcj � 2c) < 0:(4.14) 17

From (4.12) - (4.14) we �nally obtainsign�� = �1;(4.15)for small � > 0.Proof of theorem 2.1:Theorem 2.1 follows from theorem 1.1, proved in [FLA98]. The location(2.7) of Hopf points was derived in (3.21). Transverse crossing (1.16) ofpurely imaginary eigenvalues has been established in (3.27), (3.28) withsignRe�0(�) = +1;(4.16)for small � > 0. Nondegeneracy condition (1.15) has been veri�ed in (4.15)with sign�E�P 0F (x�) = sign�� = �1;(4.17)again for small � > 0. We therefore have shown that the assumptions oftheorem 1.1 and the conclusions of theorem 2.1 hold with elliptic type �determined by � = �1:(4.18)This completes the proof of theorem 2.1. ./As a concluding remark, we note that the existence of hyperbolic Hopf pointsin central di�erence schemes (1.2) has not been established yet. Further in-vestigations of more general nonlinearities are necessary to clarify the possi-bility of such bifurcation points.References[AA86] J.C. Alexander and G. Auchmuty. Global bifurcation of phase-locked oscillators. Arch. Rat. Mech. Analysis, 93:253{270, (1986).18

[AF89] J.C. Alexander and B. Fiedler. Global decoupling of coupled sym-metric oscillators. In C.M. Dafermos, G. Ladas, and G.C. Papanico-laou, editors, Di�erential Equations, Lect. Notes Pure Appl. Math.118, New York, 1989. Marcel Dekker Inc.[AF98] J.C. Alexander and B. Fiedler. Stable and unstable decoupling insquares of additively coupled oscillators. In preparation, 1998.[Fen77] N. Fenichel. Asymptotic stability with rate conditions, II. IndianaUniv. Math. J., 26:81{93, (1977).[FL98] B. Fiedler and S. Liebscher. Generic Hopf bifurcation from linesof equilibria without parameters: II. Systems of viscous hyperbolicbalance laws. Preprint, FU Berlin, 1998.[FLA98] B. Fiedler, S. Liebscher, and J.C. Alexander. Generic Hopf bi-furcation from lines of equilibria without parameters: I. Theory.Preprint, FU Berlin, 1998.[HPS77] M.W. Hirsch, C.C. Pugh, and M. Shub. Invariant Manifolds.Lect. Notes Math. 583. Springer-Verlag, Berlin, 1977.[LL96] C.D. Levermore and J.-G. Liu. Large oscillations arising in a dis-persive numerical scheme. Physica D, 99:191{216, (1996).[Sho75] A.N. Shoshitaishvili. Bifurcations of topological type of a vector�eld near a singular point. Tr. Sem. Petrovskogo, 1:279{309, (1975).[Shu87] M. Shub. Global Stability of Dynamical Systems. Springer-Verlag,New York, 1987.[Wig94] S. Wiggins. Normally Hyperbolic Invariant Manifolds in Dynami-cal Systems. Applied Mathematical Sciences 105. Springer-Verlag,New York, 1994. 19