II. Systems of viscous hyperbolic balance laws. Bernold Fiedler, Stefan Liebscher. Freie Universitat Berlin. Arnimallee 2-6, Berlin, Germany

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1 Generic Hopf bifurcation from lines of equilibria without parameters: II. Systems of viscous hyperbolic balance laws Bernold Fiedler, Stefan Liebscher Institut fur Mathematik I Freie Universitat Berlin rnimallee 2-6, 495 Berlin, Germany June 2, 998 This work was supported by the DFG-Schwerpunkt \nalysis und Numerik von Erhaltungsgleichungen" and with funds provided by the \National Science Foundation" at IM, University of Minnesota, Minneapolis.

2 Introduction Searching for viscous shock proles of the Riemann problem, we consider systems of hyperbolic balance laws of the form (.) u t + f(u) x = g(u)+u xx ; with u =(u ;u ;;u N ) 2 IR N+ ;f 2 C 3 ;g 2 C 2 ; > ; and with real time t and space x: We assume strict hyperbolicity, that is, the Jacobian (u) =f (u) possesses simple real distinct eigenvalues (.2) ; ;; N 2spec (u) The case of conservation laws, g ; has been studied extensively. for example [Smo83] for a background. Viscous proles are traveling wave solutions of the form x st (.3) u = u with wave speed s: Here (.3) provides a solution of (.) if See (.4) s _u + (u)_u= g(u)+u Here (u) =f (u) denotes the Jacobian and = d with =(x st)=: Note d that (.4) is independent of>:ny solution of system (.4) for which (.5) lim u() =u! exist gives rise, for & ; to a solution of the Riemann problem of (.) with values u = u connected by a shock traveling with shock speed s: We call solutions u() of (.4), (.5) viscous proles. We rewrite the viscous prole equation (.4) as a second order system (.6) _u = v _v = g(u)+((u) s)v

3 Note that any viscous prole must satisfy (.7) g(u ) = : In other words, the asymptotic states u must be equilibria of the reaction term g(u): In the conservation law case, g ; this condition does not impose any restraint on the Riemann values u : In the other extreme of a reaction term g with unique equilibrium, we obtain u + = u and traveling shock proles do not exist. In the present paper, we assume that the u component does not contribute to the reaction terms and still all reaction components vanish, say at u= :Specically, we assume throughout this paper that (.8) g = g(u ; ;u N )= is independent ofu and satises B g g. g N C (.9) g()=: This gives rise to a line of equilibria (.) u 2 IR; u = =u N =; v= of our viscous prole system (.6). The asymptotic behavior of viscous proles u() for!depends on the linearization L of (.6) at u = u ; v =:In block matrix notation corresponding to coordinates (u; v) we have (.) L= id g ( s) 2

4 Here = (u); and g = g (u) describes the Jacobi matrix of the reaction term g at u = u : In (.) we write s rather than s id; for brevity. In the case g of pure conservation laws, the linearization L possesses an (N + ) dimensional kernel corresponding to the then arbitrary choice of the equilibrium u 2 IR N + ; v =:Normal hyperbolicity of this family of equilibria, in the sense of dynamical systems [HPS77], [Fen77], [Wig94], is ensured for wave speeds s not in the spectrum of the strictly hyperbolic Jacobian (u): (.2) s 62 spec (u) =f ;; N g: Indeed, (.2) ensures that additional zeros do note arise in the real spectrum (.3) spec L = fg[ spec ((u) s) In the present paper we investigate the failure of normal hyperbolicity ofl along the line of equilibria u =(u ;;;);v = given by (.). lthough our method applies in complete generality, we just present a simple specic example for which purely imaginary eigenvalues of L arise when >is xed small enough. Specically, we consider three-dimensional systems, N = 2; satisfying (u ; ; ) = + u (.4) = B C ; 6= ; symmetric g () = B C ; jj < ; 3

5 with omitted entries being zero. Note that these data can arise from ux functions f which are gradient vector elds, still giving rise to purely imaginary eigenvalues. t the end of this paper we present a specic example where the reaction terms _u = g(u) alone, likewise, do not support even transient oscillatory behavior; see (3.2). The interaction of ux and reaction, in contrast, is able to produce purely imaginary eigenvalues of the linearization L as follows. Proposition. Consider the linearization L = L(u ) along the line u 2 IR;u = u 2 = of equilibria of the viscous prole system (.6) in IR 3 ;see (.). ssume (.4) holds. For & ; and small jsj; ju j; the spectrum of L then decouples into two parts: (i) an unboundedly growing part spec (L) = spec ( s)+o() (ii) a bounded part spec bd (L) =spec(( s) g )+O() Here ; g are evaluated atu=(u ;;): For =;s =;jj<;the bounded part spec bd (L) at u =limits onto simple eigenvalues 2f;i! g;! = p 2 with eigenvectors given by ~v = ~u and ~u = (.5) ~u = B B C ; for = C i! ; for =+i! : ~u ~v 4

6 Proof: Regular perturbation theory applies to the scaled block matrix (.6) L = g s + id which becomes lower triangular for = :This provides us with the unbounded part spec (L) of the spectrum, generated in v u =: Moreover L possesses three-dimensional kernel, at = ;given by space alone with (.7) g u =( s)v: On this kernel, the eigenvalue problem for L reduces to (.8) u = v = ( s) g u The characteristic polynomial of (.8) at u = is given by! 2s p () = 2 2 (.9) s2 + s 2 Direct calculation completes the proof../ For explicit calculations here and below, we have used and recommend assistance by symbolic packages like Mathematica, Maple, etc. Bifurcations from lines of equilibria in absence of parameters have been investigated in [Lie97], [FL98a] from a theoretical view point. recall that result, for convenience. Consider C 2 vector elds We briey (.2) _u = F (u) with u =(u ;u ;;u n )2IR n+ : We assume a line of equilibria (.2) =F(u ;;;) 5

7 along the u axis. t u =;we assume the Jacobi matrix F (u ; ; ;) to be hyperbolic, except for a trivial kernel vector along the u axis and a complex conjugate pair of simple, purely imaginary, nonzero eigenvalues (u ); (u ) crossing the imaginary axis transversely as u increases through u =: () = i!();!() > (.22) Re () 6= Let Z be the two-dimensional real eigenspace of F () associated to i!(): By Z we denote the Laplacian with respect to variations of u in the eigenspace Z: Coordinates in Z are chosen as coecients of the real and imaginary parts of the complex eigenvector associated to i!(): Note that the linearization acts as a rotation with respect to these not necessarily orthogonal coorrdinates. Let P be the one-dimensional eigenprojection onto the trivial kernel along the u axis. Our nal nondegeneracy assumption then reads (.23) Z P F () 6= : Fixing orientation along the positive u a real number. Depending on the sign axis, we can consider Z P F () as (.24) := sign (Re ()) sign ( Z P F ()); we call the \bifurcation" point u =elliptic, if= =+: ;and hyperbolic for The following result from [FL98a] investigates the qualitative behavior of solutions in a normally hyperbolic three-dimensional center manifold to u = : The results for the hyperbolic case = + are based on normal form theory and a spherical blow-up construction inside the center manifold. The elliptic case = is based on Neishtadt's theorem on exponential elimination of 6

8 rapidly rotating phases [Nei84]. For a related application to binary oscillators in discretized systems of hyperbolic balance laws see [FL98b]. For an application to square rings of additively coupled oscillators see [F98]. Theorem.2 Let assumptions (.2) { (.23) hold for the C 5 vector eld _u = F (u): Then the following holds true in a neighborhood U of u =within a threedimensional center manifold to u =: In the hyperbolic case, = +; all nonequilibrium trajectories leave the neighborhood U in positive or negative time direction (possibly both). The stable and unstable sets of u =;respectively, form cones around the positive/negative u axis, with asymptotically elliptic cross section near their tips at u =:These cones separate regions with dierent convergence behavior. See Fig.., a). In the elliptic case all nonequilibrium trajectories starting in U are heteroclinic between equilibria u =(u ;;;) on opposite sides of u =:If F (u) is real analytic near u =;then the two-dimensional strong stable and strong unstable manifolds of u within the center manifold intersect at an angle which possesses an exponentially small upper bound in terms of ju j: See Fig..,b). In the present paper, we apply theorem.2 to the problem of zero speed viscous proles of systems of hyperbolic balance laws near Hopf points as in proposition.. Nonzero shock speeds can be treated completely analogously, absorbing them into the ux term. Theorem.3 Consider the problem (.5){(.7) of nding viscous proles with shock speed s =to hyperbolic balance laws (.). Let assumptions (.8){(.), (.4) 7

9 Case a) hyperbolic, = +. Case b) elliptic, =. Figure.: Dynamics near Hopf bifurcation from lines of equilibria. hold, so that a pair of purely imaginary simple eigenvalues occurs for the linearization L, in the limit! : Then there exist nonlinearities (u) =f (u)and g(u); compatible with the above assumptions, such that the assumptions and conclusions of theorem.2 are valid for the viscous prole system (.6). Both the elliptic and the hyperbolic case occur; see g... Since both conditions are open, the results persist in particular for small nonzero shock speeds s, even when f, g remain xed. Specic choices of ux f(u) and reaction terms g(u) are presented in corollary 3.3; see (3.2). In the elliptic case =, we nevertheless observe (at least) pairs of weak shocks with oscillatory tails, connecting u and u + : In the hyperbolic case = +;viscous proles leave the neighborhood U and thus represent large shocks. t u =;their proles change discontinuously and the role of the u axis switches from left to right asymptotic state with oscillatory tail. 8

10 This paper is organized as follows. In section 2 we check transversality condition (.22) for the purely imaginary eigenvalues. We also compute an expansion in terms of for the eigenprojection P onto the trivial kernel along the u axis. In section 3, we check nondegeneracy condition (.23) for Z P F (); in the limit & ; completing the proof of theorem.3 by reduction to theorem.2. cknowledgment Helpful discussions with Bob Pego and Heinrich Freistuhler concerning the set-up of viscous proles for hyperbolic balance laws are gratefully acknowledged. This work was supported by the Deutsche Forschungsgemeinschaft, Schwerpunkt \nalysis und Numerik von Erhaltungsgleichungen" and was completed during a fruitful and enjoyable research visit of one of the authors at the Institute of Mathematics and its pplications, University of Minnesota, Minneapolis. This research was supported in part by the Institute for Mathematics and its pplications with funds provided by the National Science Foundation. 2 Linearization and transverse eigenvalue crossing In this section we continue our linear analysis of the linearization (2.) L (u )= id g with = (u), g = g (u) evaluated along the line of equilibria u =(u ;;): See (.) with s = and proposition.. In the limit & ; we address 9

11 the issue of transverse crossing of purely imaginary eigenvalues in lemma 2.. In lemma 2.2, we explicitly compute the one-dimensional eigenprojection P onto the trivial kernel of L (): Throughout this section we x the notation (2.2) (u ; ; ) = + u = B C + u (a jk) j;k2 with a = jk a kj symmetric; see (.4). We also assume that the linearized reaction term (2.3) g (u ; ; ) = g () = B C ; jj < ; which is independent of u g component. by assumption (.8), possesses a vanishing By proposition., purely imaginary eigenvalues of L (u ) arise from an O() perturbation of the matrix (2.4) g =( +u ) g () with spectrum spec bd (L): Let (2.5) (u ); (u ) denote the continuation of the simple, purely imaginary eigenvalues () = i! ; () = i! with u derivatives (u ); (u ):

12 Lemma 2. In the above setting and notation, (2.6) Re () = 2 (a + a 22)+a 2 : Proof : Since the unit vector e in u direction is a trivial kernel vector of g () and since the remaining eigenvalues of g remain conjugate complex for small ju j; we have (2.7) Re (u )= 2 trace ( g ): In particular, trace g =:With the u expansion (2.8) =( +u ) = u + we immediately obtain (2.9) Re (u )= 2 trace ( g ): Inserting ; ;g proves the lemma../ By regular perturbation of spec bd (L), the result Re () 6= of lemma 2. extends to small positive : We now turn to an expansion for the eigenprojection P onto the onedimensional kernel of the 66-matrix L (u )at u = ; see (2.), (2.2). ligning the notations of proposition. and of theorem.2, we decompose u =(u; v) 2 IR 6 = IR 3 IR 3 : gain, e T =(;;) denotes the rst unit vector in IR 3 and e T =(e T ;) the rst unit vector in IR 6 :

13 Lemma 2.2 In the above setting and notation (2.) P = e e T ; with e T = ( + ( )2 ) =2 (e T ; et ) Proof : Kernel and co-kernel of L (u ) are one-dimensional, corresponding to the simple zero eigenvalue of L (u ): Obviously (2.) ker L (u )=e ; because g (u ; ; )e =:tu =;the co-kernel of L (u ) is given by (2.2) =e T id g Inserting from (.4) proves the lemma../ 3 Higher order nondegeneracy In this section we complete the proof of theorem.3. In view of theorem.2, we have already checked transverse crossing of purely imaginary eigenvalues, assumption (.22), in lemma 2.. Letting (3.) F (u) =F(u; v)= v g(u)+ (u)v it remains to check the nondegeneracy assumption Z P F 6= ; see (.23). In lemma 3., we check this assumption in the limit & : In corollary 3.2, we provide explicit expressions for the type determining sign = dened in (.24). In particular, we show in corollary 3.3 that both the hyperbolic 2

14 case = + and the elliptic case = can be realized by our nonlinear hyperbolic balance laws, even with gradient ux terms. This then completes the proof of theorem.3. Tocheck nondegeneracy condition (.23) on Z P F in the limit & ; we use the following notation. By transverse eigenvalue crossing at = ;lemma 2., we also obtain purely imaginary eigenvalues i! at equilibria u = (u ; ; ;)=(u ;) on the u axis, for small >:Let Z denote the corresponding eigenspace. We recall our expression for the eigenprojection P onto the trivial kernel, (3.2) P =(+( )2 ) =2 e (e T ; et ) with e =(e T ;)T ; see lemma 2.2. Note that u ;! ;P ; and Z vary dierentiably with : Lemma 3. In the above setting and notation, we have (3.3) Z P F (u )= g (u )[~u ; ~u ] at the Hopf point u =(u ;) with complex eigenvector (~u ;~v ) of i! : Consider in particular quadratic forms g (), which are strictly positive/negative denite on (u ;u 2 ) space, with = indicating the sign of deniteness. Then (3.4) sign Z P F (u )= sign ; for all small >: Proof : By lemma 2.2 we have (3.5) ( + ( )2 ) =2 P = e e T 3 e (;e T )

15 The explicit form (3.) of the nonlinearity F implies (3.6) Z P F (u )=e Z v = on any subspace Z and for any u ; simply because the u is linear. With P instead of P we obtain more generally ( + ( )2 ) =2 e T Z P F (u) = Z (; et )( ( (3.7) = Z ( g (u)+((u)v) ): component off g(u)+(u)v)) Here ((u)v) denotes the zero-componentof(u)v: We treat this term rst, using the notation (3.8) ~u = ~u ~v for the complex eigenvector of the purely imaginary Hopf eigenvalue = i! at u = u ;v =:Then Z = spanfre ~u ; Im ~u g: Denoting by = the standard Laplacian, evaluated at =;and inserting ~v = ~u yields Z ((u)v) = (u + Re ~u + 2 Im ~u )( Re ~v + 2 Im ~v )) = 2(( (u )Re ~u )Re ~v +(( (u )Im ~u )Im ~v ) = (3.9) = 2Re(( (u )~u )~v ) = = 2Re( )(f (u )[~u ; ~u ]) = = 2Re( )f (u )[~u ; ~u ]= all along the Hopf curve u = u ;v =:Here we have used (u) =f (u) for the ux function and the fact that the Hessian matrix f () is symmetric. Therefore, we can conclude from (3.7), (3.9) that (3.) Z P F (u )= Z g (u)= g (u )[~u ; ~u ]: This proves (3.3), and the lemma../ 4

16 Corollary 3.2 Combining lemmata 2. and 3., the sign = distinguishing elliptic from hyperbolic Hopf bifurcation along our line of equilibria is given explicitly by (3.) = sign Re () sign Z P F () = sign( u a 2 2 u (a + a 22 )) sign for >small enough. Here derivatives are evaluated atu=;and are assumed tobe chosen such that 6= : The sign = indicates positive/negative deniteness of g () on (u ;u 2 ) space. Obviously, both signs of can be realized. Corollary 3.3 Theorems.2,.3 hold true for = with the following specic choices of agradient ux term f(u) =r(u) and a reaction term g(u) : (3.2) g(u)= B u 2 + u u 2 u + u 2 2 C (u) =u (u2 u 2 2 )+ 2 u u 2 + u u 2 2 These choices correspond to =2; = 2 ; =+: References [F98] [Fen77] J.C. lexander and B. Fiedler. Stable and unstable decoupling in squares of additively coupled oscillators. In preparation, 998. N. Fenichel. symptotic stability with rate conditions, II. Indiana Univ. Math. J., 26:8{93, (977). 5

17 [FL98a] B. Fiedler, S. Liebscher, and J.C. lexander. Generic Hopf bifurcation from lines of equilibria without parameters: I. Theory. Preprint, FU Berlin, 998. [FL98b] B. Fiedler, S. Liebscher, and J.C. lexander. Generic Hopf bifurcation from lines of equilibria without parameters: III. Binary oscillations. Preprint, FU Berlin, 998. [HPS77] M.W. Hirsch, C.C. Pugh, and M. Shub. Invariant Manifolds. Lect. Notes Math Springer-Verlag, Berlin, 977. [Lie97] S. Liebscher. Stabilitat von Entkopplungsphanomenen in Systemen gekoppelter symmetrischer Oszillatoren. Diploma Thesis, 997. [Nei84]. Neishtadt. On the seperation of motions in systems with rapidly rotating phase. J. ppl. Math. Mech., 48:34{39, (984). [Smo83] J. Smoller. Shock Waves and Reaction-Diusion Equations. Springer-Verlag, New York, 983. [Wig94] S. Wiggins. Normally Hyperbolic Invariant Manifolds in Dynamical Systems. Springer-Verlag, New York,