Subcritical bifurcation to innitely many rotating waves. Arnd Scheel. Freie Universitat Berlin. Arnimallee Berlin, Germany

Transcription

1 Sbcritical bifrcation to innitely many rotating waves Arnd Scheel Institt fr Mathematik I Freie Universitat Berlin Arnimallee Berlin, Germany 1

2 Abstract We consider the eqation r 0 k2 r 2 = + ajj 2 on r 2 R + with k 2 N, a; 2 C, Re > 0 > Re a, and j Im j + j Im aj << 1. Bonded soltions possess an interesting interpretation as rotating wave soltions to reaction-dision systems in the plane. Or main reslts claim that there are contably many soltions which are decaying to zero at innity. The proofs rely on nodal properties of the eqation and a Melnikov analysis. 2

3 1 Introdction The problem of nding spiral wave soltions in reaction-dision systems has been stdied intensively throghot the last fteen years. In order to be able to address this problem, many athors assmed the reaction term to be in a specic form which allows for a decopling of Forier modes. These reaction-dision eqations t = D4 + g(jj); 2 C (1.1) were called -! systems and many interesting reslts on the existence of nonlinear waves nder varios assmptions on the particlar strctre of the reaction term have been derived [1, 2, 3, 4, 5, 6]. Recently, a systematic and mathematically rigoros procedre has been fond, which allows to prove the approximation property of -! systems for general reaction-dision eqations [8]. The crcial assmption is that a homogeneos steady state is close to a Hopf bifrcation point in the pre reaction system. The ODE describing the shape of spiral wave soltions is the same as the one which can be derived from -! systems. In a typical example, it is shown in [8] that the eqations are of the form r 0 k2 r 2 = + ajj2 (1.2) with 0 = d, a 2 C and the complex parameter being close to zero. The spiral wave dr soltion to the original reaction dision eqation is then approximately given by an expression of the form U(r; '; t) = (r)e ik' e ict ; see [8]. The speed of rotation c of the spiral wave { whose vale mst be fond as a part of the problem { determines the imaginary part of the parameter. Indeed d I =dc 6= 0, which allows s to control the imaginary part I by the wave speed c. The eqation (1.2) has been stdied for small imaginary parts of the parameters and a in the case when R = 1 and a R = 1; see [2, 3, 6]. As R < 0 corresponds to an nstable zero state in the reaction-dision system (1.1), this case can be interpreted as a spercritical bifrcation. Here we address or attention to the case of a sbcritical bifrcation, that is we sppose throghot this work that a R = 1 and R = 1: Moreover we assme that the imaginary part of a is small, qite as in the qoted references on the spercritical bifrcation. Soltions to (1.2) which are bonded at r = 0 actally satisfy the expansion (r) = r k +O(r k+1 ). We are interested in localized soltions: we reqire that (r) decays to zero as r! 1. The next propositions state that in the limit a I = I = 0, there are contably many soltions of this type. Proposition 1 Sppose a I = I = 0. Then for all k 2 N, (1.2) possesses a soltion 0 (r) sch that 0 (r) > 0 for all r 2 (0; 1) and 0 (0) = 0 (1) = 0: Proposition 2 Sppose a I = I = 0. Then for all k; n 2 N, (1.2) possesses a soltion n (r) sch that n (0) = n (1) = 0 and n (r) possesses exactly n simple zeroes in (0; 1). 3

4 The proofs are carried ot in the next section exploiting the nodal strctre of eqation (1.2) and of its linearization v r v0 k2 r v = v 2 32 v; (1.3) restricted on the real sbspace I = 0 = 0, In section 5 we give a completely dierent I proof sing variational argments. In order to be able to consider a I nonzero we need more detailed information on the soltion: Proposition 3 The soltions j (r); j = 0; 1; 2; : : :, are transverse in the real sbspace. The variational eqation (1.3) with = j (r) does not possess a bonded soltion on (0; 1). Or last reslt concerns the existence of localized soltions when a I is close to zero. Proposition 4 Fix k; n 2 N and a neighborhood of the soltion n (r). Then there is a smooth fnction I (a I ) with I (0) = 0 sch that eq. (1.2) possesses a bonded soltion n (r; a I ) with n (0; a I ) = n (1; a I ) = 0. Moreover this soltion is niqe in the xed neighborhoods of n (r) p to mltiplication with e i' ; ' 2 R. In the next two sections we proof or propositions for the positive soltion 0 (r). We then otline the necessary modications for the soltions j (r). In section 5 we then give an alternative proof sing variational methods. We conclde with a brief discssion on the implication of the reslts presented here. 2 Proof of Propositions 1 and 3 The real system is given after a sitable rescaling by r 0 k2 r 2 = 3 : (2.1) Any soltion which is bonded at r = 0 is of the form (r) = r k + O(r k+1 ). We stdy in detail the variational eqation of (2.1) v r v0 k2 r 2 v = v 32 v: (2.2) Any soltion of this eqation which is bonded at r = 0 is proportional to the and (r; ) is the niqe bonded soltion of (2.1) growing like r k for small r. For large, soltions of (2.1) are approximated by an atonomos eqation as follows. We set ~ =, r = s and obtain which is close to the homogeneos eqation ~ ss + 1 s ~ s k2 s 2 ~ = 1 2 ~ ~3 ; ~ ss + 1 s ~ s k2 s 2 ~ = ~3 : Rescaling time s = e t and setting e t ~ = ^ yields ~ tt k 2 ~ = e 2t ~ 3 and ^ tt 2^ t (1 + k 2 )^ = ^ 3 : (2.3) 4

5 Withot the negative damping 2^ t the phase portrait of this system is the well-known doble homoclinic loop to the origin, lled and srronded by periodic orbits. If (r) is bonded as r! 0, then r k, ~ r k = (1=) k s k, ^ (1=) k e (k+1)t! 0 as t! 1. Therefore (^; ^ 0 ) belongs to the nstable manifold of the origin in (2.3) and dierent parameter vales now correspond jst to a time shift. Lemma 2.1 Let be sciently large. Then there exists R () > 0 sch that (r) > 0 on (0; R ()) and (R ()) = 0. Frthermore there is R v () < R () sch that the soltion of the variational eqation (2.2) satises v(r) > 0 on (0; R v ()); v(r) < 0 on (R v (); R ()] and v 0 < 0 on [R v (); R ()]: Moreover, this implies that dr =d < 0. Proof. We solve the scaled atonomos eqation (2.3). Global existence is ensred as the fnction 2^ 2 + t ^4 can grow at most linearly with time. Level lines of the atonomos system, given by ^ 2 t (1 + k 2 )^ ^4 const are always crossed otwards. One can easily check that for positive energy vales the soltion ^ cannot stay positive (otherwise ^ wold have to get nbonded bt for large vales of ^ the rotational component ^ 3 of the vector eld becomes dominant). Shooting with the nstable manifold yields the desired positive soltion ^ with some R (), where ^(t(r ())) = 0 and ^ t () < 0. The transverse intersection with the axis ^ = 0 persists for nite when adding the pertrbation term ~= 2. The claim on the sign of v is a claim on the sign of ^ t in the limit = 1 and an immediate conseqence of the phase portrait. The derivative of R is calclated from 0 = d(r ()) d and v < 0; 0 < 0, becase ^ t < 0 at R (R (R ()) dr () = v + 0 R 0 d For close to zero, we can show that the soltion (r) does not possess any zero. Lemma 2.2 Sppose is sciently small. Then the soltion (r) is strictly positive for all r > 0. Proof. We se a shooting argment. Let M R 2 R + denote the manifold of soltions (; 0 )(r; ) bonded at r = 0. Then the tangent space of M along = 0 = 0 is calclated from the linear eqation v r v0 ( k2 + 1)v = 0; v(0) = 0: r2 The soltions are mltiples of the modied Bessel fnctions of the rst kind I k (r); see [10]. Asymptotic expansions for these fnctions yield v0 (r) v(r) = 1 1 2r + O( 1 ). In particlar, for r2 large r, we see that v 0 =v % 1. Next we constrct a forward invariant region close to r = 1 where 0 and we show that a part of M gets trapped in this region, exclding zeroes of the corresponding soltions. 5

6 We consider the original eqation, extended by the eqation 0 = 2 with = 1=r to make ot of it an atonomos eqation. At = 0; = 0 = 0, we have an eqilibrim with a niqely dened two-dimensional center-stable manifold. The intersection of this manifold with the plane = 0 is the homoclinic crve q(r) > 0 (and its symmetric). The tangent space of the center-stable manifold along this soltion evolves nder the linearized eqation q 2 (r) + q 0 (r) = 0 0 = 0: The positive damping term q 0 (r) forces soltions to the eqation for xed > 0, which are bonded for r! 1, to ct the axis = 0 at 0 > 0 at a nite time r = R(). The invariant region S we were looking for is now constrcted as being bonded by: the interior of the homoclinic (q; q 0 ) in = 0, the plane = 0, the center-stable manifold and the plane = 0 > 0 sciently small. All bondaries are ow invariant, except the planes = 0 and = 0, where the vector eld is pointing strictly inwards. By the calclations on the tangent spaces of the centerstable manifold and the shooting manifold M, there are bonded trajectories (r) entering S. These trajectories do not possess any zeroes of in S. Making sciently small, we can garantee that sch a soltion is close to I k (r), the modied Bessel fnction of the rst kind [10] as long as it stays otside of S and thereby does not possess any zeroes at all. Now we want to decrease, preserving the sign strctre from Lemma 2.1. Sppose that v wold achieve its minimm on (0; R ()) and sppose it wold be negative. For sciently large v does not achieve its minimm in the interior of the interval by the previos lemma. A minimm cold appear in the interior of the interval if either at a point r = R 0 we had v 0 = 0 and v 00 = 0 { which is exclded becase then necessarily v 0 { or, a minimm cold become negative { bt then again v 0 = v = 0 wold imply v 0 { or, alternatively, a minimm cold enter throgh the bondary, at R (). Bt then at R () we wold have v 00 0; v 0 = 0; v < 0 and = 0. Using the eqation for v this wold imply 0 v r v0 = ( k2 r + 1)v 2 32 v = ( k2 + 1)v < 0; r2 a contradiction. Thereby v is strictly decreasing on the interval (R v (); R ()) as long as R v () < R (). Lemma 2.3 For all > 0 we have R v () < R () and R v () R for all with R () < 1. In particlar there is 0 > 0 sch that R ( 0 ) = 1 and R v ( 0 ) < 1. Proof. We arge by contradiction. Sppose R v () = R () = R. Then ; v > 0 on (0; R) and = v = 0 on f0; Rg. As both ; v r k at 0 and 0 ; v 0 < 0 at R, there is a sch that v > on (0; R). We dene = inffj v > on (0; R)g: 6

7 This implies v 0 for all r 2 (0; R) and at some R 0 2 [0; R] we have v(r 0 ) (R 0 ) = 0, v 0 (R 0 ) 0 (R 0 ) = 0 and v 00 (R 0 ) 00 (R 0 ) 0. Bt from (2.1) and (2.2) we can dedce that ( v )(R 0 ) = 2 3 (R 0 ) 0 with strict ineqality, in case R 0 2 (0; R), which is thereby rled ot. Now sppose that R 0 = R. From the eqation for w = v w r w0 = ( k2 r )w 2 2 v; and from the condition w = w 0 = 0 at r = R we can get expansions of w at R, eqating the lowest order terms w 00 = 2 2 v; then w (5) = 12( 0 ) 3 > 0 and, for r < R bt r close to R, w is negative which contradicts v 0. If R = 0, we can conclde that both, v and are at leading order given by r k for some > 0. Then w 00 = 2 3 r 3k < 0 at leading order and thereby again w(r) < 0 for small enogh r. This proves dr ()=d < 0 for all 0 0, and R ( 0 ) = 1. By Lemma 2.2, we know that 0 > 0. In order to prove the lemma, we have to exclde that R v ( 0 ) = 1. This follows for the same reasons as for the nite interval above. We can similarly dene 1 = inffj v > on (0; 1)g becase and v possess at leading order the same exponential decay property e r as r! 1. Then again 6= 1v on [0; 1) by the same argments as above. The last possibility we have to rle ot is that R 0 = 1. Then actally w(r) is given by the variation of constants formla w(r) = Z 1 with the linear evoltion operator given by r (r; s)2 2 (s)v(s)ds (r; s) = I k(s)k k (r) I k (r)k k (s) I k (s)y 0(s) k I0 (s)y k k(s) : Here the I k and K k are the modied Bessel fnctions of the rst and second kind. The expression in the nmerator is the Wronski determinant and strictly positive, whereas the expression in the denominator is negative for large r, as I k (r) e r =r 1=2 and K k (r) e r =r 1=2 ; see [10]. Thereby again w(r) < 0, for large r, and we have reached a contradiction. This proves the lemma. Together with the previos lemmata the proof of Proposition 1 and Proposition 3 for 0 (r) is now easy. We dene 0 (r) = (r; 0 ) with = 0 > 0 from the previos lemma. This soltion is bonded, converges to zero at innity and is transverse, again by Lemma Proof of Proposition 2 We mimic the proof for n = 0. As in Lemma 2.1, we can garantee that for large there are soltions bonded at r = 0 with innitely many zeroes, winding arond the two homoclinic crves of the atonomos problem (r = 1). They possess a seqence of non-degenerate zeroes R n (). Similarly the soltion to the variational eqation (1.3) possesses a seqence of non-degenerate zeroes R n v () and we have R 0 v() < R 0 () < R 1 v() < R 1 () < : : :. We have to contine this pattern for decreasing. As in the previos section, we can conclde that drn < 0 if we can ensre that d Rn v () < R n () < R n+1 v (). Arging as in the case 7

8 n = 0, we can show that v(r; ) cannot achieve its local minimm on (R n v (); R n ()). It is therefore scient to prove an analoge of Lemma 2.3. Proceeding by indction on n, we compare and v on (R n 1 (); (R n ()). By the indction hypothesis, sign v(r n 1 ()) = ( 1) n+1. Now assme that v(r n ()) = 0 and, to x signs, v 0 on (Rn 1 (); (R n()) (n is spposed to be even, the case of n odd being similar). Then there is a sch that w = v 0 on (R n 1 (); (R n ()) and w(r) = 0 for some R 2 (Rn 1 (); (R n()]. Then w achieves its local maximm in R which is however forbidden from (1.2) and (1.3), becase w 00 > 0 where w 0 = w = 0 and v < 0. Ths we have reached a contradiction showing that there is n > 0 sch that R n ( n ) = 1. Arging as above and in Lemma 2.3, it is easy to see that R n v ( n ) < 1, which shows that the soltions are transverse. This proves Proposition 2 and Proposition 3 for n (r). 4 Proof of Proposition 4 The proof reqires a Melnikov type calclation. In the phase space extended by the eqation for = 1=r, or soltions are transverse intersections of a shooting manifold M (the set of soltions bonded at r = 0) and the center-stable manifold W cs (0) of the origin = 0 = = 0. The intersection is transverse only when restricted to the real sbspace = 0 = 0. The complex problem, a I 6= 0, possesses an additional S 1 -symmetry (; 0 )! (e i' ; e i' 0 ) for ' 2 R. De to this symmetry, there is one direction orthogonal to the sm of the tangent spaces of M and W cs (0) at the heteroclinic orbits n (r). This direction is orthogonal to the generator of the rotational symmetry at the heteroclinics i( n (r); 0 n(r), and therefore given as i( 0 (r); n n(r)). The derivative of the vector eld with respect to I, or pertrbation parameter, points in the direction (0; i n ). The scalar prodct with the direction orthogonal to the sm of the tangent spaces has a denite sign for all r and gives a nonzero contribtion to the Melnikov integral. In other words, the two manifolds intersect transversely if the phase space is extended by the eqation 0 = 0. This transverse intersection persists at a I point I (a I ) for small pertrbations a I. This proves Proposition 4. 5 Variational approach Proposition 1 might be proved sing variational methods. We have to consider r 0 k2 r 2 = 3 : (5.1) We want to apply montain-pass lemma to a variational formlation of the eqation I() = Z [ 2 + r (k2 R+ r 2 + 1) ]r dr; 2 H 1;2 (R + ): Of corse, any critical point of the fnctional I gives a soltion of the above eqation on R +. We next apply the montain pass lemma to or fnctional. First of all zero is a nondegenerate local minimm: the kernel of the linearization are the Bessel fnctions of the rst kind which however do not lie in H 1;2 (R + ) as they grow exponentially at r! 1. On the other hand, the fnctional decays to 1 along any ray s, 2 H 1;2 (R + ) xed, s 2 R +. It remains to establish convergence of a Palais-Smale seqence, a non-trivial task de to non-compactness at 0 and +1. We do not carry ot details here. 8

9 This wold then establish the existence of a heteroclinic orbit as claimed in Lemma 1. Transversality does not follow from this constrction. We sspect that one cold prove as well the existence of innitely many critical points, sing the Z 2 -symmetry of the fnctional,!, see for example [9, Chapter II, Theorem 6.5 and 6.6]. 6 Discssion As already pointed ot in the introdction, the soltions proved to exist in Proposition 4 have an interesting interpretation as localized rotating wave soltions of reaction-dision eqations. A particlar eqation ndergoing a Hopf bifrcation and exhibiting sch spatiotemporal phenomena was given in [8]. The soltions are, in contrast to the ones fond for spercritical bifrcations, localized, that is, along rays emanating from the origin, the amplitde and derivative of the phase of the soltions decay exponentially to zero. In particlar for the soltions 0 (r), regions of constant phase form arcs which rn from the origin to innity, asymptotic to a straight line throgh the origin. The soltions with zeroes of the amplitde form more complicated patterns: there are n circles, where the amplitde gets close to zero. The phase changes sign, when crossing these circles. We sspect that the localized soltions of Proposition 1 and Proposition 2 are niqe as localized soltions with a prescribed nmber of zeroes. We did not try to prove stability or instability of the soltions for the fll reaction-dision system. The considerations on a variational approach in Section 5 sggest that all waves are nstable, with Morse index increasing with n (which is well dened becase the continos spectrm of the linearization is bonded away from the imaginary axis, see [7, Lemma 5.4]). References [1] D.S. Cohen, J.C. Ne, and R.R. Rosales, Rotating spiral wave soltions of reactiondision eqations, SIAM J. Appl. Math. 35 (1978), 536{547. [2] J.M. Greenberg, Spiral waves for! systems, SIAM J. Appl. Math. 39 (1980), 301{309. [3] J.M. Greenberg, Spiral waves for! systems, II, Adv. Appl. Math. 2 (1989), 450{455. [4] P.S. Hagan, Spiral waves in reaction-dision eqations, SIAM J. Appl. Math. 42 (1982), 762{786. [5] N. Kopell and L.N. Howard, Plane wave soltions to reaction-dision eqations, Stdies in Appl. Math. 52 (1973), 291{328. [6] N. Kopell and L.N. Howard, Target patterns and spiral soltions to reactiondision eqations with more than one space dimension, Adv. Appl. Math. 2 (1981), 417{449. [7] B. Sandstede, A. Scheel, and C. Wl, Dynamics of Spiral Waves on Unbonded Domains Using Center-Manifold Redctions, to appear in J. Di. Eq.. 9

10 [8] A. Scheel, Bifrcation to spiral waves in reaction-dision systems, Preprint FU Berlin. [9] M. Strwe, \Variational Methods," Springer, New York, [10] G.N. Watson, \Theory of Bessel fnctions," Cambridge Univ. Press,