Bifurcation of homoclinic orbits to a. saddle{focus in reversible systems. December 19, Abstract

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1 Bifurcation of homoclinic orbits to a saddle{focus in reversible systems with SO(2){symmetry Andrei Afendikov y and Alexander Mielke z December 9, 997 Abstract Bifurcation of homoclinic orbits of reversible SO(2){invariant vector elds in R 4 in a vicinity of the primary homoclinic orbit H 0 is studied. Under transversality conditions on H 0 with respect to the bifurcation parameter the existence of n-homoclinic solutions is demonstrated together with the study of their parity with respect to involution. The existence of cascades of 2 k 3 m -homoclinic solutions is established by showing their transversality and then using induction. The method relies on the construction of an SO(2)-invariant Poincare map which, after factorization, consists of a composition of a logarithmic twist map and a smooth global map. As an application we treat the steady complex Ginzburg{Landau equation for which a primary homoclinic solution is known explicitly. Introduction We study the existence of homoclinic orbits as well as their bifurcation in reversible ordinary dierential equations in R 4 with SO(2){symmetry. The analysis is motivated by the steady complex Ginzburg{Landau equation (cgl) 0 = A + aa + bjaj 2 A; a =? 2 ; b = ( + i")( + ); (.) which, for " = 0, is known to have the explicit homoclinic solution A(x) = (cosh x)?. Throughout we will use the normalization = +i! with! > 0, such that the origin is a reversible saddle{focus. Our aim is to study the existence of homoclinic orbits for all small ". These homoclinic orbits will then be n{homoclinic with respect to the primary homoclinic orbit H 0, which means that they wind around n times close to H 0 before they return to the origin. Research partially supported by VW-Stiftung under I/7 06 and by DFG under Mi 459/2-2. ykeldysh Institute of Applied Mathematics, Miusskaya sq. 4, Moscow, RUSSIA zinstitut fur Angewandte Mathematik, Universitat Hannover, Welfengarten, D{3067 Hannover, GERMANY

2 The important role of cgl for bifurcation problems on spatially unbounded domains (see [Doe93, AfM95]) led to several previous studies of its homoclinic orbits, see [Hol86, Doe96, KaMP96], where however only the case of small! was treated. In the present work we develop a general theory for reversible SO(2)-invariant vector elds and thereby improve the known results for cgl, in particular by generalizing them to the case of large!. This is important since the applications in 3D Poiseuille ow (see [AfM95, AfM98]) need! in the range between 8 and 8. Numerical work for cgl can be found in [Lan87] where n-homoclinic solutions for several n as well as the sequences of periodic solutions close to n-homoclinic solutions were detected. Homoclinic bifurcations in generic four{ dimensional reversible systems with the same linear part of saddle{focus type (hence without SO(2){symmetry or Hamiltonian structure) are considered in [H97, Cha97]. In such systems one typically has homoclinic bifurcations only on one side of the parameter value where the primary homoclinic orbit exists. Our methods are closely related to [MiHO92] where homoclinic orbits to a saddle{ center (xed point with eigenvalues, i!) in a reversible Hamiltonian system were studied. There the conserved Hamiltonian was used to reduce the dimension of Poincare sections from 3 to 2 whereas in the present setting the SO(2){invariance gives the desired reduction. However, the reduced Poincare map has exactly the same features. Our main starting point is a xed point of saddle{focus type which is taken to be the origin. Because of the reversibility the eigenvalues of the linearization form the quadruple ;?; ;? and by choosing appropriate coordinates the system can be written in the form _y = y + F ("; Y; Y ); _y 2 =?y 2 + F 2 ("; Y; Y ); (.2) where Y = (y ; y 2 ) 2 C 2, and the equations for the complex conjugates Y are suppressed. The nonlinearities F j are assumed to be real analytic with F j ("; Y; Y ) = O(jY j 2 ). (However, vector elds with nite dierentiability can be handled similarly, see [AfM98].) The SO(2){action T and the reverser R (involution with R 2 = I) are given via T (y ; y 2 ) = (e i y ; e i y 2 ); R(y ; y 2 ) = (y 2 ; y ): For " = 0 we assume the existence of the primary homoclinic H 0. Our method relies in constructing a Poincare map for this orbit which will allow us to nd n{homoclinic orbits for small " 6= 0. From the SO(2){invariance it is clear that a homoclinic orbit exists if and only if the two{dimensional stable and unstable manifolds coincide completely and are given by rotations T H " of a homoclinic orbit H ". Moreover, from the reversibility follows that there is an 2 R such that H " (x) = " R H " (?x) for all x 2 R with " 2 f?; +g: We call " = par(h " ) the parity of the homoclinic orbit H ". The Poincare map P is a composition of a local map e S of Shilnikov type (see [Sh67]) and a global map e G along the primary homoclinic H0. It is possible to factor out the SO(2){symmetry before these mappings are constructed as was done in [KaMP96] for the cgl. However, this leads to a three{dimensional system with a singularity at the xed point which makes the analysis dicult. In contrast to this we rst construct mappings 2

3 and then factorize. We use the Poincare section K in f (y ; y 2 ) 2 C : jy j r; jy 2 j = g with > 0 and K out = R K in which are sections at the local stable and unstable manifold of 0, respectively. Thus, the local map e S : Kin! K out, the global map e G : Kout! K in, and the Poincare map P = e G e S : Kin! K in are obtained. In fact, it is more convenient to use S = R e S and G = e G R, which both map Kin into itself, and take the form S( ; w) = ( + (w); s(w)) and G("; ; w) = ( + ("; w); g("; w)) after choosing suitable coordinates ( ; w) 2 S K r with (y ; y 2 ) (we i ; e i ). The local map S is a logarithmic twist map given exactly via (w) = arg w + b (jwj), s(w) = e?2ib (jwj) w and b (jwj) =! log(=jwj). To show this we use the local normal form theory in [Brj7] for analytic vector elds to simplify the ow of (.2) near (y ; y 2 ) = 0. The case of vector elds with nite dierentiability can be treated similarly with the results in [Sam82]. Note that the reverser R is involved in the denition of S and G such that P = e G e S = G S and the reversibility amounts into the fact that G and S are involutions, i.e., G G = S S = id. The global mapping G is smooth and satises G(0; ; 0) = ( + arg 0 ; 0) with 0 = par(h 0 ). Together with G G = id it follows that g has the expansion g("; w) = "a +?w + O(" 2 +jwj 2 ) with?a + a = 0;? 2 = I; det? =?: (.3) We say that the primary orbit H 0 is transversal (with respect to the perturbation in ") if a 6= 0. The set of parameters " 2 (?" 0 ; " 0 ) decomposes into the disjoint union of E n where E n is the set of all " such that (.2) has an n{homoclinic orbit. The main results (see Section 3) can be summarized as follows: If H 0 is transversal, then for suciently small " 0 > 0 we have: () For each n 2 the set E n is innite and has 0 in its closure. (2) Both, the set E 2 and E 3 consist of one monotone decreasing and one monotone increasing sequence converging to " = 0. All the associated homoclinic orbits are again transversal. (3) In each sequence in E 2 the parity of the associated homoclinic orbits alternates. All " 2 E 3 have the parity par(h 0 ). (4) In E 2 [ E 3 the elements of E 2 and E 3 separate each other. An essential new feature of our work is the transversality of the 2- and 3-homoclinic orbits existing for " 2 E 2 [ E 3. It is this result which allows us to use induction to produce n{homoclinic orbits which are transversal for any n = 2 k 3 m. In Section 4 we present an analytical method which enables us to calculate the coef- cients a and? in (.3) explicitly from solving the variational equation for (.2) around H 0. For cgl in the form (.) we are thus able to nd a and? as a functions of the constant! > 0. Since?(!) always has eigenvalues with eigenvectors (!) with j (!)j = we have a(!) = c(!)? (!) with c : [0; )! R. The analysis in [KaMP96] shows c(!) 6= 0 for small! 6= 0. Since c is an analytic function we have transversality for all! > 0 except for an exceptional set f! ;! 2 ; : : :g which may be nite or innite, but it has no nite points of accumulation. Our numerical calculations give! 8:032 and! 2 9:5. 3

4 2 Construction of the Poincare map 2. Setup of the problem We consider homoclinic orbits in a real four-dimensional SO(2){invariant reversible system depending on the small parameter ". Without loss of generality we assume that the xed point is the origin. The eigenvalue constellation of the linearization is a nontrivial quadruple ;?; ;? with = + i! where! > 0. After a linear coordinate transformation we may assume that the system is in the complex form _Y = JY + F ("; Y; Y ); Y = (y ; y 2 ) 2 C 2 ; F = (F ; F 2 ) 2 C 2 (2.) with the diagonal linear part J = diag ( ; 2 ) = diag (;?; ) and F j ("; Y; Y ) = O(jY j 2 ). Throughout we will assume that the vector eld is (real) analytic in ("; Y; Y ), but this is just for convenience since the case with nite dierentiability can be treated similarly, see [AfM98]. The action of SO(2) and of the reverser is as follows: T (y ; y 2 ) = (e i y ; e i y 2 ); R(y ; y 2 ) = (y 2 ; y ): Then, SO(2){invariance of (2.) means F T = T F, and reversibility means F R =?RF. (Of course the linear part JY has this symmetries as well.) In fact, having chosen the coordinates according to the eigenvalues as above the action is uniquely determined up to conjugacies under the additional assumption T R = RT. It is also interesting to note that there is no Hamiltonian system of the form (2.) with the desired symmetries. Since we want to switch between real and complex notation frequently we use the realication functor R( ) and the complexication functor C ( ). For instance, R(y ; y 2 ) = (Re y ; Imy ; Re y 2 ; Imy 2 ) and RC n = R 2n denotes the realication of the complex space and RB 2 R 2n2n is the realication of B 2 C nn. In the real coordinates u = R(y ; y 2 ) T 2 R 4 equation (2.) takes the form _u = f("; u); u 2 R 4 ; " 2 R: (2.2) The induced actions on R 4 are RT and RR. Problem (2.) is obviously reversible under the family T R of transformations. But only two of them are involutions, namely those with = 0; =. Thus, problem (2.2) must be treated as a bireversible system with respect to R and?r. An important observation is that any orbit homoclinic to 0 is in fact reversible. Lemma 2. Suppose that H " (x) is a homoclinic orbit of (2.) " then there are unique " 2 f?; g and 2 R such that H " (x) = " R H " (? x) for all x 2 R: The sign " is called the parity of H " and written as par(h " ) = ". 4

5 Proof: Let W s (0) and W u (0) be stable and unstable manifolds of 0 and dimw s (0) = dim W u (0) = 2. Let y s (x) be any solution in W s (0), then by SO(2){invariance we have y(x) = T y s (x + x s ) for each solution y in W s (0). By reversibility, for a solution y in W u (0) we have y(x) = T Ry s (x u? x). Since the homoclinic orbit lies in W s (0) \ W u (0) we nd y s (x) = T? Ry s (? x) for all x 2 R where = x s +x u. Evaluating at x = =2 gives (I?T? R)y s (=2) = 0. Since y s (=2) 6= 0 and since T? R has the eigenvalues ie i(?) and their complex conjugates it follows that 2(? ) = 0 2 S. If? = 0 2 S we let = and if? = 2 S we let =?, then we have T? = I; and thus H(x) = R H(? x) as desired. QED Simple equations of the above type with an explicit homoclinic orbit can be constructed as follows. Let r(x) be a real homoclinic orbit to the two-dimensional system r? r[g(r 2 ) + r 2 G 0 (r 2 )] = 0; (such that ( _r) 2 = r 2 G(r 2 ) ) where G is a smooth function with G(0) =. Then, the complex function A(x) = r(x) = r(x)e i! log r(x) solves the complex second order equation A? A[ 2 G(jAj 2 ) + jaj 2 G 0 (jaj 2 )] = 0: With G() =? we obtain the important example of the time-independent complex Ginzburg-Landau equation A? 2 A + ( + i")( + )AjAj 2 = 0; = + i!;! > 0; A 2 C ; (2.3) which has, for " = 0, the explicit homoclinic solution A (x) = (cosh x)?. To our knowledge this observation appears rst in [HoS72]. This example is worth keeping in mind throughout the paper. These problems, treated as a complex system in coordinates (A; _ A) are invariant with respect to T = diag (e i ; e i ) and reversible with the involution R A = diag (;?). Transforming (2.3) to the eigenbasis of the linear part we nd system (2.) together with the specied symmetries. It is immediate that the homoclinic orbits constructed above have parity = +. From now on we consider the case that (2.) has a homoclinic orbit H 0 for " = 0. Our goal is to study the existence of other homoclinic orbits for all small ". By SO(2)-invariance and reversibility the existence of reversible orbits is a phenomenon of codimension, such that we expect a discrete set E (?" 0 ; " 0 ) for which homoclinic orbits exist. This point will become clearer through the following analysis. The approach we take is that of constructing a Poincare section and a suitable Poincare return map which allows us to study the existence of homoclinic orbits which wind around the original homoclinic orbit n times. To this end we construct a local map which takes care of the passage of orbits near the xed point Y = 0 and a global map which contains information about the ow near the homoclinic orbit. In [KaMP96] (2.3) was rst reduced by SO(2)-symmetry to a real three-dimensional ODE and then Poincare maps 5

6 were constructed. To avoid the diculties arising from the singularity at the origin, we rst construct a Poincare map and then study its symmetry properties with respect to SO(2). 2.2 The local map Consider the analytical ODE _Z = JZ + ("; Z; Z); = ( ; 2 ) with Z = (z ; z 2 ) 2 C 2 : (2.4) It is said to be in normal form if j("; Z; Z) = z j X h;qi=0 g jq (")(Z; Z) Q ; = ( ; 2 ; 3 ; 4 ) = (;?; ;?) (2.5) where Q = (q ; : : : ; q 4 ) 2 Z 4 ; q j?, q k 0 for k 6= j, and (Z; Z) Q = z q z q 2 2 z q 3 z q 4 2. Monomials z j (Z; Z) Q with h; Qi := q q 4 = 0 are called resonant monomials. Lemma 2.2 There exist a neighborhood U C 2 of 0 and an analytical transformation Z = Y + ("; Y; Y ); ("; Y; Y ) = ( ("; Y; Y ); 2 ("; Y; Y )); j ("; ) : U! C (2.6) such that j ("; Y; Y ) contains no resonant monomials and that (2.) is transformed into the normal form _z = z [ + ("; jz z 2 j 2 )]; _z 2 =?z 2 [ + ("; jz z 2 j 2 )]; (2.7) with ("; 0) = 0. Moreover, the transformation is SO(2)- and R-equivariant. Proof: Observe that the problem (2.) is analytical and the set of eigenvalues f j g; j = ; : : : ; 4 does not belong to the simple Poincare domain, but to the more dicult Siegel domain (see [Arn83]) where analyticity of normal form transformations only holds if further strong conditions are satised [Brj7]. For this purpose we write (2.) as 4 complex equations with X = (Y; Y ) 2 C 4 by adding the complex conjugate equations, i.e., _X = diag ()X + G("; X) where G = (F ; F 2 ; F ; F 2 ). Of course, SO(2)-symmetry and reversibility are preserved. We rst derive the general form of the normal form under the given symmetries. Since = =2 R the resonance condition q? q 2 + q 3? q 4 = 0 yields q = q 2 and q 3 = q 4. From SO(2)-invariance we conclude q + q 2? q 3? q 4 = 0 which implies q = q 2 = q 3 = q 4. Since the normal form transformation (2.6) with no resonant terms preserves symmetries and reversibility (see [Brj7, Arn83, IoA92]) we arrive at _x j = x j [ j + j ("; x x 2 x 3 x 4 )] for j = ; : : : ; 4: (2.8) Using (x 3 ; x 4 ) = (x ; x 2 ) and reversibility we nd ( 2 ; 3 ; 4 ) = (? ; ;? ), and thus (2.7) is established. 6

7 For the analyticity of the normal form transformation we use Theorem 2 of [Brj7]. There are no small denominators in this problem; Siegel's and hence Brjuno's small denominator condition is obvious, as for all Q 2 Z 4 with h; Qi 6= 0 we have jh; Qij = [(q?q 2 +q 3?q 4 ) 2 +! 2 (q?q 2?q 3 +q 4 ) 2 ] =2 minf;!g: It is left to check the condition A 2 from [Brj7]. It reads as follows: There exist two power series ("; X); ("; X) such that in the normal form (2.5) j ("; X) = j ("; X) + j ("; X); j = ; : : : ; 4: For our special normal form (2.8) this condition is fullled with real =? 2? 2 and =?. This completes the proof.? QED We describe the local mapping in the normal form coordinates (z ; z 2 ) 2 C 2. For a homoclinic solution H " (x) of (2.2) in Y -coordinates we know that H " (x) 2 U for jxj x 0. Hence, in Z-coordinates we have e H" (x) = H " (x) + ("; H " (x)) for jxj x 0. Z(x) = e H" (x) = (c + e?x ; 0) for x x 0 ; Z(x) = e H" (x) = (0; par(h " )c + e (+x) ) for x?x 0 ; (2.9) with 2 R and c + 2 C. To see this, we use the fact that the stable manifold of (2.7) is given by z 0 while the unstable manifold is given by z 2 0. The result on the parity is a consequence of Lemma 2.. We construct a local map e S between the hypersurfaces K in = f(z ; z 2 ) 2 C 2 : jz j r; jz 2 j = g and K out = f(z ; z 2 ) 2 C 2 : jz j = ; jz 2 j rg; where and r are taken small enough. Clearly, K out is transversal to the unstable manifold of Z = 0 and K in is transversal to the stable manifold. The local (Shilnikov) map induced by the ow of (2.7) is given by where es("; ) : ( K in n f(0; z 2 ) : jz 2 j = g! K out ; (z ; e i 2 ) 7! ( z jz j ei(";jz j) ; e i( 2?(";jz j) jz j); ("; jz j) =! + Im("; 2 jz j 2 ) + Re("; 2 jz j 2 ) log jz j : (2.0) This special form of e S follows easily from (2.7) since jz z 2 j 2 is constant along solutions. The reversibility of the problem is such that R maps K in into K out as well as K out into K in. Moreover the mapping e S("; ) satises the conditions for 2 S = R= 2Z. es("; )? = Re S("; R ) and e S("; T ) = T e S("; ); (2.) 7

8 2.3 The global and the Poincare map The global map, which will take care of the ow along the homoclinic excursion, has to be constructed in the Y -coordinates since the validity of the Z-coordinates is necessarily restricted to a small neighborhood of Z = 0. Thus, denote the normal-form transformation of Lemma 2.2 for (z ; z 2 ) 2 C 2 by Z = M("; Y; Y ) and dene the Poincare sections in the Y -coordinates. bk in = M? ("; K in ) C 2 and b Kout = M? ("; K out ) C 2 Since the homoclinic orbit H 0 (x) of (2.2) exists for " = 0 we 7! Kin b, where as conclude the existence of a smooth Poincare mapping G("; b ) : Kout b usual, G b may only be dened in a smaller set. In order to be able to combine this map with the local map S e we transform it back to Z-coordinates and let eg("; ) : Kout! K in ; (e i ; z 2) 7! M("; b G("; M? ("; (e i ; z 2)))): This map preserves the SO(2)-symmetry and the reversibility as follows: eg? ("; ) = R e G("; R ); e G("; T ) = T e G("; ); (2.2) for 2 S. The assumption that (2.2) has a homoclinic orbit H 0 for " = 0 now means that eg(0; (e i ; 0)) = (0; ( 0 e i ; 0)) with 0 = par(h 0 ): (2.3) This follows easily from (2.9) by shifting x! x? =2. We are now in the position to dene the total Poincare map P ("; ) = e G("; ) e S("; ) : Kin! K in : In fact, we can avoid the use of the two sections K in and K out by using the fact RK out = K in and dening the maps S : K in! K in and G : K in! K in via S("; ) = Re S("; ) and G("; ) = e G("; R ); which implies P = G S. As a consequence of the symmetries of e S and e G in (2.) and (2.2), respectively, we have the following result. Lemma 2.3 Both maps S; G : K in! K in are involutions on their domains of denition: S S = I and G G = I. Moreover both maps are SO(2)-equivariant, that is S T = T S and G T = T G for 2 S. Note that the mappings S and G are reversible in the sense that S? = S and G? = G and there are no other reversibilities left over from R. 8

9 2.4 The factorization with respect to SO(2)-symmetry The next step is to use the SO(2)-symmetry to reduce the dimension of the problem from 3 to 2. For this purpose we introduce in K in local coordinates = 2 2 S and w = z e?i 2 2 K r = f w 2 C : jwj r g; such that (z ; z 2 ) = (we i ; e i ) 2 K in. The SO(2)-action in these coordinates is T ( ; w) = ( + ; w) and the maps S and G take the form S("; ; w) = ( + ("; w); s("; w)) and G("; ; w) = ( + ("; w); g("; w)): (2.4) For the local map we have the following explicit expressions ("; w) = arg w + ("; jwj); s("; w) = e?2i(";jwj) w (2.5) The complex conjugation in s("; ) arises from R in the denition S = Re S. Notice that for w = 0 there is the unique continuation s("; 0) = 0. In contrast ("; w) : R C 7! R has a discontinuity in the origin and cannot be extended continuously. The functions and g can be obtained from e G via (g("; w)e i(";w) ; e i(";w) ) = e G("; (; w)) = T? e G("; (e i ; e i w)): (2.6) The properties of the maps G and S in ( ; w)-coordinates can be summarized as follows. Lemma 2.4 Let E 0 = (?" 0 ; " 0 ) for suciently small positive " 0. Then, we have: A. The functions : E 0 K r! S and g : E 0 K r! C are real analytic and satisfy on their domain of denition ("; g("; w)) + ("; w) = 0 2 S and g g = I: (2.7) B. The functions : E 0 K r n f0g! S and s : E 0 K r n f0g! K r are real analytic and s("; ) is a Lipschitz map on K r. Moreover we have ("; s("; w)) + ("; w) = 0 2 S and s s = I: (2.8) C. Let p("; w) = g("; s("; w)) then p? = s p s = g p g. The proof is straightforward from the above construction. We note that whenever g("; w) = w we obtain from (2.7) the relation 2("; w) = 0 2 S. Assuming that for " = 0 there exists a homoclinic orbit H 0 with 0 = par (H 0 ) we conclude with (2.3) that 0 = e i(0;0). Since is a smooth function we conclude ("; w) = arg 0 2 S whenever g("; w) = w: (2.9) (Note that arg = arg e i0 = 0 2 S and arg(?) = arg e i = 2 S.) Finally the total Poincare map P = G S is written in ( ; w)-coordinates as P ("; ; w) = ( + e("; w); p("; w)) where e("; w) = ("; s("; w)) + ("; w) and p("; w) = g("; s("; w)). 9

10 2.5 The canonical form of S and G Notice rst that the local map s("; w) can be simplied with the use of an analytical coordinate transform to es("; v) = e?2i (jvj) b v with (jvj) b =! log(=jvj): (2.20) The needed transformation is given by v = w exp?!+im (";2 jwj 2 )![+Re("; 2 jwj = w + O(jwj 3 ). For 2 )] brevity we rename v 7! w and es 7! s to keep old notations for simplied transformation and use S( ; w) = ( + arg w + (jwj); b e?2ib (jwj) w) as the nal form of our local map from K in! K in. The main advantage is that S is now independent of ". In these new coordinate w the global map G still is an involution with g(0; 0) = 0 such that we have the expansion Rg("; Rw) = "a +?Rw + O(" 2 +jwj 2 ) with? 2 = I and?a + a = 0: (2.2) This implies that eigenvalues of the matrix? lie in f?; g. If det? = then either? = I or? =?I; however in our situation only the case det? =? is relevant. This is seen as follows. The Poincare map P = G S from K in into itself is orientation preserving and hence the same is true for the factorized map p = g s, that is det D w p(w) = whenever it is dened. Hence det D w s(w) =? implies det D w g(w) =? as desired. Having? 2 = I and det? =? there is a rotation w 7! e i w such that Re?i?Re i = 0 =b b 0 with b > 0. For? = q v u?q take = 2 u+v arcctg(? ). Note that such a rotation q of w 2 K r does change the local map s simply by b 7! b + 2 which amounts into the same as changing slightly. The canonical form of the global map in this case is?b 0 b Rg("; Rw) = c" + =b 0 Rw + O(" 2 + jwj 2 ) with c 2 = jaj2 + b : (2.22) 2 Despite the fact that we only need the case det? =? in this paper the following result is kept general for possible future reference. Lemma 2.5 Suppose that g is the real analytical involution with the expansion (2.2). Then the xed point set Fix g("; ) = f w 2 K r : g("; w) = w g has the following characterization. A. If? = I, then a = 0 and Fix g("; ) = K r. B. If? =?I, then Fix g("; ) = f(")g K r with (") =?"a=2 + O(" 2 ). C. If det? =? then Fix g("; ) = f (x ; x 2 ) 2 RK r : ("; x ; x 2 ) = 0 g (2.23) where ("; x ; x 2 ) = bx 2? x? bc" + O(" 2 +jxj 2 ). 0

11 Proof: The real analytical involution Z("; w) = ("; Rg("; Rw)) on (?" 0 ; " 0 ) K r has the expansion Z(v) = Nv + O(v 2 ) with v = ("; Rw) near the xed point v = 0. It is analytically conjugate to its linear part around the xed point. This is the contents of Bochner theorem (see [Bre72]) for Z 2 -actions. Thus, statements A, B, and C follow since they are obviously true in the linear case. We give a simple and explicit proof as we want be able to conclude that the conjugation of Bochner's theorem does not change ". In fact, the conjugating transformation is Q(v) = v + NZ(v) since NQ = (NZ + Z 2 )Z = QZ and hence Z = Q? N Q. QED 3 n-homoclinic orbits We return to system (2.2) for which we assumed the existence of a primary homoclinic orbit H 0 () for " = 0. To this orbit we x a small tubular neighborhood N = f Y 2 C 2 : 9 x 2 R : jy?h 0 (x)j g, i.e. > 0 is small. For small " 6= 0 there might again exist homoclinic orbits. Denition 3. An orbit H " () of (2.2) " which is homoclinic to 0 is called an n-homoclinic orbit with respect to the primary orbit H 0 if it winds around n times in N. From our construction of the Poincare section we conclude that an n-homoclinic orbit has to intersect K in in n dierent points ( j ; w j ) 2 K in such that ( j+ ; w j+ ) = P ("; j ; w j ) with w j 6= 0 for j = ; : : : ; n?; and additionally w = g("; 0) and w n = 0. Here the condition w j 6= 0 guarantees that we do not have a homoclinic orbit which is m-homoclinic with m < n. Hence, the set of all parameter values " 2 E 0 = (?" 0 ; " 0 ) where (2.2) has a n- homoclinic orbit is given by E n = f " 2 E 0 : p n ("; 0) = 0 and p j ("; 0) 6= 0 for j = ; : : : ; n? g: 3. Characterization of E n We introduce the xed point sets for s and g, respectively, Fix s = f w 2 K r : s(w) = w g; Fix g("; ) = f w 2 K r : g("; w) = w g: From the reversibility properties of S and G stated in Lemma 2.4 we know that 2("; w) = 0 on Fix s and 2("; w) = 0 on Fix g("; ). From (2.9) we already know that ("; w) = arg 0 on Fix g("; ), and Lemma 2.5 states that Fix g("; ) is a smooth curve in K r. The set Fix s decomposes into Fix s = Fix s + [ Fix s? [ f0g with Fix s = f w 2 K r : arg w + b (jwj) = arg g: These two sets are logarithmic spirals due to the special form of b given in (2.20).

12 Theorem 3.2 We have E 2m = E 2m + [ E2m? where E 2m = f " 2 E 0 : p m ("; 0) 2 Fix s ; p j ("; 0) 6= 0 for j = ; : : : ; m g: For " 2 E 2m the associated 2m-homoclinic orbit H " satises par(h " ) =. Moreover, E 2m+ = f " 2 E 0 : p m+ ("; 0) 2 Fix g("; ); p j ("; 0) 6= 0 for j = ; : : : ; m + g and for " 2 E 2m+ we have par(h " ) = par(h 0 ). Another useful characterization is E 2m+ = f " 2 E 0 : p?m ("; 0) 2 Fix g("; ); p?j ("; 0) 6= 0 for j = ; : : : ; m g: Proof: If p n (0) = 0 then part C of Lemma 2.4 gives p n?k (0) = p?k (0) = g p k p(0) = s p k (0) where we used s(0) = 0. If n = 2m this gives p m (0) = s p m (0) and thus p m (0) 2 Fix s. If n = 2m+ we take k = m and nd p m+ (0) = g p m g(0) = g p m+ (0) since 0 = s(0). Thus, we nd p m+ (0) 2 Fix g("; ) as desired. Since s(0) = 0 and g g = I this is equivalent to p?m (0) 2 Fix g("; ). It remains to consider the parities of the associated homoclinic orbits H " for " 2 E n. This can only be decided in the mapping without factorization. Consider the case n = 2m + rst, and denote by ( k ; w k ), k = ; : : : 2m the intersection points of H " with K in. In Z-coordinates we have (z k; zk) = 2 (wk e i k ; e i k ) 2 Kin and (ez k ;ezk ) = e 2 G? ("; (z k; zk)) 2 2 K out. Since w m+ 2 Fix g("; ) we have m+2 = m+ + ("; w m+ ) = m+ + arg 0. With eg? = RG? we conclude (ez m+ ;ez m+ 2 ) = RG? ("; (w m+ e i m+ ; e i m+ )) = R(w m+ 0 e i m+ ; 0 e m+) i = 0 (e i m+ ; w m+ e m+) i = 0 R(z m+ ; z m+ 2 ) Thus, we know that there exist and 2 such that H " ( ) 2 K out and H " ( 2 ) 2 K in with H " ( ) = 0 RH " ( 2 ) and Lemma 4.2 shows that par(h " ) = 0. The arguments for " 2 E 2m are analogous. QED 3.2 Sequences of 2- and 3-homoclinic orbits In order to give precise statements on the secondary homoclinic solutions we have to introduce some genericity condition on the primary homoclinic solution H 0 (x). Denition 3.3 The primary homoclinic solution H 0 (x) is called transversal with respect to the perturbation in " if a in (2.2) satises a 6= 0. For the transversal homoclinic orbits it is possible to use the characterization given in Theorem 3.2 for proving the existence of n-homoclinic orbits. We starts with 2- and 3-homoclinic orbits. 2

13 Theorem 3.4 Suppose that the primary homoclinic orbit H 0 with parity 0 is transversal. Then for suciently small " 0 > 0 we have E 2 = f " (2) >;k ; "(2) <;k : k 2 N g and E3 = f " (3) >;k ; "(3) <;k : k 2 N g; where the sequences are ordered as follows " (2) <;k < "(3) <;k < "(2) <;k+ < 0 < "(2) >;k+ < "(3) >;k < "(2) >;k ; (3.) and for the vector of quotients we have " (2) ; >;k+ "(3) ; >;k "(2) ; <;k "(3) <;k " (2) >;k?! e?2=! ; 2 ; 3 ; 4 for k! : (3.2) For " 2 E 3 we have par(h " ) = 0 = par(h 0 ) and for " = " (2) >;<;k we have par(h ") = (?) k. Proof: We are looking for " 2 (?" 0 ; " 0 ) such that g("; 0) 2 Fix s. Since the primary homoclinic orbit is transversal g("; 0) = "c(?b; ) T +O(" 2 ) is a smooth curve in K r passing through the origin with nonzero speed with respect to ". Hence, there are innitely many intersections with the two logarithmic spirals Fix s given by 2 arcsin(b= b b) + O(") + n = b (j"cj b b+o(" 2 )); n 2 N; (3.3) whereb b = (+b2 ) =2. For all n 2 N with n N 0 there are unique solutions " (2) <;n < 0 < " (2) >;n of (3.3). Obviously, they lie in (?" 0 ; " 0 ) for N 0 large enough and, according to Theorem 3.2, the parities are equal to (?) n. Renumbering the sequences gives the result for E 2. To study 3-homoclinic orbits we have to check the condition p? ("; 0) 2 Fix g("; ) or (p? ("; 0)) = 0. This gives " c [?(+b 2 ) sin 2b (j"cj b b+o("2 )) + b] + O(" 2 ) = 0. This is equivalent to solving b(j"cjb b+o(" 2 )) = 2 arcsin b + n + O("); n 2 N; (3.4) + b2 which gives sequences " (3) <;n < 0 < " (3) >;n. According to Theorem 3.2 the parity of all these solutions is 0 = par (H 0 ). Moreover, comparing (3.3) and (3.4) and using jbj=(+b 2 ) jbj=b b proves the result on the ordering of the sequences. The explicit formula for b gives nice expansions of the solutions such that the statement of the limits in (3.2) is easily deduced. QED Our aim is to use induction to generate n-homoclinic orbits with n 4 by considering H " as a primary homoclinic orbits to which the above theorem can be applied as well. Theorem 3.5 The homoclinic solutions found in Theorem 3.4 are transversal for suciently small " 2 E 2 [ E 3. 3

14 Proof: For " 2 2 E 2 we consider H "2 as a primary homoclinic solution with the local map s and the global map g 2 = g s g where g 2 (" 2 ; 0) = 0. Note that by reparametrization of " we can always suppose c =. In the vicinity of (" 2 ; 0) we have the following expansion Rg 2 ("; Rw) = ("?" 2 )a 2 +? 2 Rw + O(("?" 2 ) 2 + jwj 2 ) with a 2 " Rg(" 2 ; s(" 2 ; g(" 2 ; 0))) + G 2 (" 2 )S (" 2 )@ " g(" 2 ; 0) and? 2 = G 2 S G where G (") = D Rg("; 0); S (") = D Rs("; g("; 0)), G 2 (") = D Rg("; s("; g("; 0))). Since g is smooth, a 2 = a +?S (" 2 )a + O(" 2 ). That is why a 2 6= 0 for suciently small " 2 if and only if 0 6=?(?S (" 2 )a + a) = S (" 2 )a? a. The matrix S (") can be decomposed into S (") = S e (") + S b + O(") with! cos 2b (" b b)? sin 2(" b b b) es (") =? sin 2b (" b b)? cos 2(" b b ; S b = 2! b? b) + b 2 b 2?b Notice S e (" 2 )g(" 2 ; 0) = s(" 2 ; g(" 2 ; 0)) and hence S e (" 2 )a = a. Therefore we arrive at S (" 2 )a? a = S b a = 2! b??b =?2! 6= 0 + b 2 b 2?b b Thus, the transversality of the bifurcating 2-homoclinic solutions is established for " 2 small enough. For " 3 2 E 3 the associated homoclinic orbit H "3 has the same local map s and the global map g 3 = g s g s g with g(" 3 ; 0) = 0. The expansion is Rg 3 ("; Rw) = ("?" 3 )a 3 +? 3 Rw + O(("?" 2 ) 2 + jwj 2 ) where? 3 = G 3 (" 3 )S 2 (" 3 )G 2 (" 3 )S (" 3 )G (" 3 ) and a 3 = a + G 3 (" 3 )S 2 (" 3 )[a + G 2 (" 3 )S (" 3 )], where G 3 (" 3 ) = G? (" 3 ); S 2 (" 3 ) = S? (" 3 ) and G 2 (" 3 ) = G? 2 (" 3 ) since s(" 3 ; g(" 3 ; 0)) 2 Fix g("; ). Since g is smooth, a 3 6= 0 for " 3 small enough if and only if a +?S? (" 3 )[a +?S (" 3 )a] 6= 0 or equivalently (?S (" 3 )? S (" 3 ) + I)a 6= 0 with S (" 3 ) = e S (" 3 ) + b S + O(" 3 ): (3.5) From s(" 3 ; g(" 3 ; 0)) 2 Fix g("; ) we conclude S e (" 3 )a = a +?e S (" 3 )a + O(" 3 ), and hence condition (3.5) is equivalent to! 0 6= S b? cos 2b (?? I)a = ( + b 2 ("3 b b) ) b? cos 2b ("3 b : b) According to Theorem 3.4 for " 3 2 E 3 we have sin 2b ("3 b b) = hence a 3 6= 0 follows and the transversality is established. b + O(" 2 +b 3) 6= ; and 2 QED 3.3 n-homoclinic orbits with n 4 An immediate corollary of Theorems 3.4 and 3.5 is the existence of cascades of transversal 2 k 3 m -homoclinics for k; m 2 N. This follows by induction where Theorem 3.2 is applied in each step. As a consequence we nd that the sets E n with n = 2 k 3 m with k+m 2 4

15 accumulate on each point " which corresponds to a 2 b k 3 bm -homoclinic orbit with b k k; bm m, and b k+bm < k+m. In fact, there is a fractal structure. For n 5 which are not of the above form our results are less precise. However, Lemma 3.4 in [MiHO92] is fully applicable in our situation and we conclude that between each " m 2 E m and each " m+ 2 E m+ there is at least one point " 2m+ 2 E 2m+. By induction it is then easy to conclude that all E n contain innitely many points in each neighborhood of " = 0, see Theorem 3.5 in [MiHO92]. The main argument there is to use the sign of the function dening Fix g("; ). For instance, to prove the existence of sequences of 5-homoclinic solutions we have to check the condition (p 3 ("; 0)) = 0. Notice that ("; 0) =?bc" and (p("; 0)) = bc". That is why if p 3 (" 3 ; 0) = 0 and p 3 (" 2 ; 0) = p(" 2 ; 0) then between " 2 and " 3 there is a point " 5 2 E 5 such that (p 3 (" 5 ; 0)) = 0. Now the existence of the sequences of 2 and 3-homoclinic values of " described in the theorem 3.4 gives the innite sequence of 5-homoclinic values of ". Unfortunately, these arguments leaves the question on transversality unstudied. We remark that all the results do not depend on the analyticity of the vector eld. The transversality argument in Theorem 3.5 just needs taking one derivative. Similarly, the construction of (2m+)-homoclinic orbits involves a monotonicity argument which works for continuous functions. This is contrast to the results in [MiHO92] where analyticity was needed to obtain higher order homoclinic orbits. However, analyticity can be used to show that the number of points in E 2m+ between any to adjacent points in E m and E m+ must be nite. Without analyticity we even may have situations where E n contains a closed interval. 4 Calculation of? and a In this section we want to show how a and? in the canonical form (2.2) of the global map g can be calculated. This will be achieved by studying the linearized equation (2.2) around the primary homoclinic orbit H 0. Thus, we will be able to check the transversality conditions for the complex Ginzburg-Landau equation (2.3) numerically. In particular, we obtain exact numbers for the normal form coecients c and b in (2.22) and to check the transversality condition in Denition 3.3. This part of the analysis will mainly use real coordinates, thus we recall u = R(y ; y 2 ) and set v = R(z ; z 2 ) for the local normal-form coordinates. By x ("; v(0)) = v(x) and x("; u(0)) = u(x) we denote the ow maps x ; x : R 4! R 4 of be the ows of (2.7), rewritten in real coordinates, and of (2.2), respectively. The normal-form transformation (2.6) written in real coordinates is v = M("; u), and it preserves the symmetry and the reversibility. We choose > 0 such that M(0; H 0 ()) 2 K in and M(0; H 0 (?)) 2 K out and dene the map ("; ; v) = (M 2 + M? )("; Rv) which maps a neighborhood of K in into another neighborhood of K in. Via x this map includes the global information of the ow in a tubular neighborhood along H 0 which is not 5

16 contained in the normal form. By the construction has SO(2)-symmetry T = T for 2 S and is reversible with = id. Moreover we have the expansion ("; ; e H0 () + eu) = e H0 () + "l + Leu + _ e H 0 () + O(" jeuj 2 ); (4.) where L 2 R 44 and l 2 R 4 satisfy L 2 = I and Ll + l = 0. Without loss of generality we may assume e H0 () = R(0; ), compare (2.9). We rst show that L and l contain all the information on? and a and then we give analytic expressions for them using innite integrals over x 2 R of Melnikov type which can be calculated numerically. 4. Reduction to the factorized global map g By dierentiation of (0; 2; T e H0 ( + )) = T e H0 ( + ) with respect to and and by using e H0 (+) = R(0; e? ) we nd the relation L R(0; i+) = R(0; i?) for ; 2 R. Writing L = L L 3 L 2 L 4 with L j 2 R 22 gives L 2 = 0 and L 4 =???2! The global map G : K in! K in is obtained from by restriction to K in and by adjusting the travel time 2 + with = b(";eu) such that ("; b(";eu); H 0 ()+eu) 2 K in. From (2.6) we know that g("; w) is obtained by evaluating the rst two components ( ; 2 ) of 2 R 4 in the form Rg("; Rw) = 0 ( ; 2 )("; b("; R(w; 0)); R(w; )) + O(" 2 +jwj 2 ); where 0 = e?i(0;0) is the parity of the primary orbit. At lowest order the travel time correction b does not enter into the rst two components ( ; 2 ). Thus, we immediately conclude by comparison with Rg("; Rw) = "a +?Rw + O(" 2 +jwj 2 ) the identities? = 0 L and a = 0 l ; where l = (l ; l 2 ) 2 R 4 with l ; l 2 2 R Calculation of L This calculation can be done completely with " = 0. Dene (x) = D u x (0; RH 0 (x)) the linearization of (2.2) around RH 0 (x) which satis- es _ = C(x) with (0) = I; C(x) = Du f(0; RH 0 (x)): (4.2) Since e H0 (x?) = R( 0 e x ; 0) for x 0, e H0 (x+) = R(0; e?x ) for x 0, and R e H 0 (x) = M(0; RH 0 (x)) the linearization of (2.7) along e H0 gives the real system _ = C with C = R 0 0? ; (x)()? = R e (x?) 0 : (4.3) 0 e?(x?) Note that we have avoided to use (x) for x 2 (?; ) since there e H0 is not dened. 6

17 Suppose that x; such that M(0; x (RH 0 (0))) = x? M(0; (RH 0 (0))). This implies by dierentiation N(x) (x) = (x)()? N() () with N(x) = D v M(0; RH 0 (x)). It follows that the real matrix B() = ()? N() () (4.4) is in fact independent of. Let us denote it by B + 2 R 44. On the other hand the matrix L in (4.) takes the form N() () ()? N()R which gives by reversibility and the denition of B + the formula L = ()B + RB? + ()? : Since () is explicitly known, it remains to nd B +, which can be managed as follows. We know (x) explicitly by solving the linear system (4.2). The normal-form transformation v = M(0; u) can be written as v = u + R 3 (0; u) + O(juj 5 ) (4.5) where 3 (0; u) is a homogeneous part of (0; u) (polynomial) of the order 3. Thus, N(x) = [I + N 2 (x) + O(e?4x )] with N 2 (x) = D u 3 (0; RH 0 (x)). Finally B + takes the form B + = (x)[i + N 2 (x) + O(e?4x )] (x): (4.6) Since (x) and (x) only grow like e x it is clear that omitting terms of order O(e?4x ) the function on the right-hand side converges to B + with the error bounded by O(e?2x ). Thus, B + can be obtained numerically by evaluating the right-hand side for suciently large x. 4.3 Calculation of l The vector l is obtained using the rst order perturbation in ", m(x) " f(0; RH 0 (x)) and the linear inhomogeneous system _y = C(x)y + m(x). Since " only appears in (2.) through F ("; Y; Y ) = O(jY j 3 ) we have m(x) = O(e?3jxj ), and there is a unique solution which decays at least like e x for x!? q(x) = (x) Z x Thus, the perturbed stable manifold in v-coordinates for x is? ()? m() d: (4.7) V " (x) = M("; RH 0 (x)+"q(x)+o(" 2 )) = R e H 0 (x) + "(N(x)q(x)+@ " M(0; RH 0 (x))) + O(" 2 ): Recall that x ("; v) is the ow of (2.7) and V " () =?x ("; V " (x)). Expanding in powers of " and comparing with (4.) gives the formula l " V "=0 () = ()(x)? [N(x)q(x) " M(0; RH 0 (x))]: (4.8) Since the perturbation in " acts only on terms of the order 3 it " M(0; RH 0 (x)) = O(e?3x ). Thus, with the use of (4.4) and (4.8) the vector l 2 R 4 in (4.) takes the form 7

18 l = ()[B(x) R x? ()? m() d + O(e?2x )]. With x! we nd convergence like e?2x to the limit l = ()B + b l with b l = Z? ()? m() d: By reversibility we have b R R l = R (?)? m(?) d = R R ()? R(?Rm()) d =?Rb l. That is why it suces to calculate b R l+ = ()? m()d and nd b 0 l = b l+? Rb l+. This implies Ll + l = 0 as desired. 4.4 Application to the complex Ginzburg-Landau equation Finally we apply the above arguments to the complex Ginzburg-Landau equation as given in (.) and (2.3). Clearly we can dene (y ; y 2 ) = ( A _ + A; A _? A) and immediately obtain the desired form of (2.). To calculate? and a in (2.2) through (4.) we have to do the normal form transformation shown in (4.5). Then the calculation of B + and b l+ as indicated is straight forward and was done with a standard Runge{Kutta solver. The convergence of B + as given in (4.6) as well as the identity Ll +l = 0 served as good checks of the correctness of the result. The numerical results are done as a parameter study in the regime! 2 [0; 0]. The the coecients c(!) and b(!) in the canonical form (2.22) of g("; w) were plotted. We nd b(0) = and then an exponential decay to 0 with b(8) 0?2. For c(!) we obtain c(!) =! + O(! 3 ) for small! with 80. Then c remains positive until it reaches its rst zero! 8:032 and the second zero appears at! 2 9:5. Since c : [0; )! R is an analytic function which does not vanish identically, it follows that it can only have a discrete set f! ;! 2 ; : : :g of zeros. This set which might be nite or countable, where in the latter case! j! for j!. References [AfM95] A.L. Afendikov, A. Mielke [995]: Bifurcations of Poiseuille ow between parallel plates: three-dimensional solutions with large spanwise wavelength. Arch. Rat. Mech. Anal., 29, [AfM98] A.L. Afendikov, A. Mielke [998]: Multibump solutions for slightly subcritical three-dimensional Poiseuille ow. In preparation. [Arn83] V.I. Arnold [983]: Geometrical methods in the theory of ordinary dierential equations. Springer-Verlag. New York, Heidelberg, Berlin. [Bre72] G.E. Bredon [972]: Press. New York. Introduction to compact transformation group. Academic [Brj7] A.D. Brjuno [97]: The analytical form of dierential equations. Transactions of the Moscow Mathematical Society, 25, [Cha97] A.R. Champneys [997]: Homoclinic bifurcation in reversible systems. Talk given at the \Multibump Workshop, Leiden, October 997". 8

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