Bifurcation from Relative Equilibria of Noncompact Group Actions: Skew Products, Meanders, and Drifts

We consider a nite-dimensional, typically noncompact Rie-mannian manifold M with a diierentiable proper action of a possibly non-compact Lie group G: We describe G-equivariant ows in a tubular neigh


Introduction
Going beyond rigidly rotating spirals, meandering and drifting spiral wave patterns have been observed in Belousov-Zhabotinsky media UNUM93], JSW89], BE93] and in low pressure CO-oxidation on platinum monocrystals NvORE93].Mathematically speaking, the wave patterns are described by concentration vectors u = u(t; x) depending on time t and location x 2 IR 2 : The partial di erential equations, which model the dynamics of the solutions u(t; x); are equivariant with respect to the standard a ne action of the planar Euclidean group E(2): The Euclidean group E(N); N = 2; 3; : : : ; is a semidirect product E(N) = O(N) IR N of the orthogonal group O(N) with the Abelian translation group IR N : The composition for (R; S); (R 0 ; S 0 ) 2 O(N) IR N is de ned by (R; S) (R 0 ; S 0 ) := (RR 0 ; S + RS 0 ); (1.1) this rule is compatible with the standard a ne representation (R; S)x := Rx + S (1.2) on x 2 IR N : Equivariance of our dynamical system means that u(t; ) is a solution if, and only if, (R; S)u(t; ) is a solution for any (R; S): Here the linear representation of (R; S) in the state space X of solution x-pro les u(t; ) is given by ((R; S)u(t; )) (x) := u(t; (R; S) 1 x): (1.3) The inverse (R; S) 1 x is, of course, given explicitly by (R; S) 1 = (R 1 ; R 1 S): (1.4) A spiral wave u(t; ) is a special time periodic solution, for which the time orbit is contained in a single group orbit.After a xed shift of x-coordinates, it can be written as u(t; ) = (R(t); 0)u(0; ): (1.5) The rotations R(t) 2 SO(N) are given as a periodic one-parameter subgroup R(t) = exp(r 0 t) (1.6) generated by r 0 in the Lie algebra so(N) of anti-symmetric matrices.In the terminology of Fie88], non-stationary spiral waves are called rotating waves; see also section 3. The term \spiral" arises from the above applied context, where the concentration patterns largely follow Archimedean spirals.Quite analogously, a meandering wave u(t; ) is a special solution of the form u(t; ) = (R(t); S(t)) v(t; ); (1.7) where this time v(t; ) is a nonstationary time periodic solution and the shifts S(t) remain bounded.If the shifts S(t) are unbounded, we call the solution u(t; ) drifting.Numerically, meandering and drifting one-armed spirals have been observed in planar (N = 2) models by Barkley Bar94].Emphasizing the lack of a theoretical framework, based on Euclidean E(2) equivariance, he also presented an ad-hoc heuristic ODE model exhibiting meandering and drifting solutions.
The rst mathematically rigorous analysis of these phenomena has recently been achieved by Wul ,see Wul96].Her result is based on a careful Lyapunov-Schmidt reduction in a scale of Banach spaces.This resolves the di culties of nondi erentiability and, in some cases, non-continuity of the group action (1.3) on the in nite-dimensional Banach space u(t; ) 2 X: For technically related earlier results, restricted to compact group actions, see Ren82] and Ran82].It has recently been shown, for the rst time, that a center manifold reduction to a nite-dimensional globally group-invariant and locally time-invariant C k+1 manifold M X can also be achieved in an E(2)-equivariant context, if the nonlinearity of the di erential equation governing the dynamics of the spiral waves is smooth; see SSW96a], SSW96b].The reduction is based on the assumption that the linearization at the spiral wave does not exhibit continuous spectrum near the imaginary axis.Most notably, the group action becomes di erentiable on M, albeit its possible noncontinuity on X.
Communicated by one of the present authors, this idea is already being used successfully to investigate meandering of multi-armed spirals GLM96].The method of center bundles, there, is similar in spirit to a previous approach to bifurcation from relative equilibria of compact group actions Kru90].
In the present paper we give an alternative, new description of the ow near relative equilibria inside a nite-dimensional Riemannian C k+1 -manifold M; typically noncompact, with a C k+1 -smooth action of a possibly noncompact Lie group G: Our principal aim is to represent the ow as a skew product ow on a trivial disk bundle G V over G, see (1.19).The alternative approach by GLM96], instead, works on a center bundle over the coset space G=H with respect to some discrete isotropy subgroup H: In our approach this amounts to working in the space G H V of H-orbits on G V; as de ned in (1.15), (2.6) below.Also, we will allow for general compact isotropies H, rather than requiring H to be nite or even trivial.In the following, the reader may nd some background in Lie groups helpful; see for example Bre72], BtD85], tD91], Hel62], Pal61], or Die72].
To set up, we assume g in the Lie group G to act as a C k+1 -di eomorphism u 7 !gu on the nite-dimensional Riemannian C k+1 -manifold M, such that the map : G M !M (g; u) 7 !gu = (g; u) (1.8) is C k+1 .Of course, we assume that G acts on M, that is (gg 0 )u = g(g 0 u) for all g; g 0 2 G and u 2 M: We also require the action to be proper, that is, the map ~ (g; u) := (gu; u) 2 M M is closed (mapping closed sets to closed sets) with compact preimages ~ 1 (u 1 ; u 2 ); for any u 1 ; u 2 2 M: As a caveat, we note that G = IR activing by shift on BC unif (IR; IR); for example, does not de ne a proper IR action.Still, the action of G = SE(2) on a center manifold M is proper SSW96b].Picking u 1 = u 2 = u 0 ; in particular, we observe that the isotropy subgroup H = H(u 0 ) := fg 2 G j gu 0 = u 0 g (1.9) is compact, for any u 0 2 M: Indeed, H fu 0 g = ~ 1 (u 0 ; u 0 ) is compact.Although M; G are allowed to be compact, in principle, we note here that the interesting new cases arise for noncompact M and G: We x u 0 and its isotropy H, henceforth.We construct the disk V of the trivial bundle G V as a geometric cross section to the action of G near u 0 : Using the Haar measure on the compact Lie group H; we may rst assume the given Riemannian metric on M to be H-invariant, without loss of generality; see Bre72], section VI.2.
In particular, any h 2 H acts linearly and orthogonally on the tangent space T u0 M to M in u 0 ; by the derivative of u 7 !(h; u) at u = u 0 : Similarly, induces a C k -action of G on the C k tangent bundle TM; we cannot assume G to act as an isometry on tangent spaces in general, if G is non-compact.It should be noted, however, that the special action (1.3) of the Euclidean group, arising in spiral wave motion, is an isometry in the usual L p and W k;p spaces.In that case, G would automatically act as an isometry on a center manifold M; see SSW96a], SSW96b].
We will construct V as a linear version of a slice to the action of G in an arbitrarily small G-invariant neighborhood U, called a tube, around the G-orbit G u 0 := fgu 0 j g 2 Gg (1.10) of u 0 as follows.Let alg(G) = T id G denote the Lie algebra of G and T u0 (Gu 0 ) = alg(G) u 0 (1.11) the tangent space to the group orbit G u 0 at u 0 : The Lie algebra of G acts on u 2 M by the derivative of g 7 !(g; u) at g = id: Now let the desired disk V of the bundle G V be de ned as the open 0 -ball, centered at u 0 ; inside the orthogonal complement V (T u0 (G u 0 )) ?T u0 M (1.12) to the orbit tangent space T u0 (G u 0 ) in T u0 M: Note that the isotropy H of u 0 acts linearly and orthogonally on V; as it does on T u0 M and T u0 (G u 0 ): To de ne the slice to the G-action and the G-invariant tube U around G u 0 ; let (1.13) for all v 2 (T u0 M) loc and h 2 H: Here (T u0 M) loc denotes an 0 -ball in T u0 M: In fact we construct 1 ; rst, such that 1 (u 0 ) = u 0 ; and then achieve H-equivariance, by Haar measure, preserving the property that 1 is a di eomorphism; see for example tD91], section I.5.Then (V ) M is a slice to the G-action at u 0 2 (V ); and is an open G-invariant tube around the G-orbit G u 0 : For convenience, we also call the 0 -disk V T u0 M around u 0 a (linear) slice.We will take license to identify u 0 2 V with the origin in IR l = T u0 V sometimes.
To describe the dynamics in the tube U well, we consider the C k -action of the direct product Lie group G H on the Cartesian product G V; given by (g 0 ; h)(g; v) := (g 0 gh 1 ; hv): (1.15)Because the derivative of this action at (id; u 0 ) is surjective, by the choice (1.12) of V , the G-equivariant map is a submersion for small radius 0 of the disk V .In fact, the triple (G V; U; ) identi es the trivial product G V as a (generally nontrivial) C k+1 principal ber bundle over U with ber, alias structure group, H: For more details, we refer to section 2.
Returning to dynamics, consider a G-equivariant C k vector eld f on the \center" manifold M, that is gf(u) = f(gu); (1.17) for all u 2 M; g 2 G: Of course, here we de ne gf(u) by the induced (di erential) C k -action of G on the tangent space TM: We seek a representation of the (local) G-equivariant ow _ u = f(u) (1.18) on M near the G-orbit G u 0 by the skew product ow on G V: Here the maps a : V !alg(G) and ' : V !T u0 V are requested to be of class C k and H-equivariant in the following sense: (1.20) for all h 2 H and all v 2 V: Here Ad(h) denotes the standard adjoint representation on the Lie algebra, and h' is again understood to be di erential on the linear ball V T u0 M: Theorem 1.1 Let f be a G-equivariant C k vector eld on the Riemannian C k+1manifold M; k 1, with proper C k+1 -action of G on M. Let u 0 2 M: Then the isotropy H of u 0 is compact.Moreover, there exists a disk slice V , an open G-invariant tube U around the group orbit G u 0 ; and H-equivariant C k -maps a; '; as in (1.20), such that the projection u := (g; v) 2 U of any solution (g; v) of the skew product system (1.19) satis es the original di erential equation (1.18) in U.
The projection is de ned in (1.16).
Conversely, for the local G H-equivariant ow de ned on (g; v) 2 G V by any C k vector eld (1.19), which is H-equivariant in the sense of (1.20), the projection u := (g; v) 2 U induces a G-equivariant C k vector eld f on U such that (1.17), (1.18) hold.
We do not think that this theorem is particularly surprising: our proof, given in section 2, is essentially based on a coordinatization of U by the space G H V of the orbits in G V under the action of the group fidg H: This point of view is due to Pal61] and is concisely presented in the beautiful topology textbook tD91], section I.5.We do think, however, that our theorem is particularly useful: in the present paper, it enables us to analyze drifting and meandering solutions on the \center manifold" M. To be precise, we x nomenclature.The reference point u 0 2 M is not required to be a relative equilibrium in theorem 1.1, although it will typically be in applications, and may be forced to be, by nontrivial H-equivariance of the skew product.While the notion (1.22) of a relative equilibrium u 0 is intrinsically ow-de ned, the de nition (1.21) refers to a speci c G V lifting with respect to the isotropy H of u 0 , as stated.For example, to apply condition (1.21) to any given point ũ0 2 U other than u 0 , the vector eld (1.19) has to be constructed with respect to ũ0 instead of u 0 : This subtlety, however, is irrelevant for small tubular neighborhoods U, as long as H is nite.
In the very special case G = fidg the maximal isolated invariant set, in the sense of Con78], of an isolating neighborhood V = U of u 0 consists precisely of the equilibria, the periodic solutions, and the meanders in U = V: A similar statement holds for the case of compact G: As mentioned above, we prove our theorem in section 2. In section 3 we discuss H-equivariant Hopf bifurcation in V , in general.Section 4 collects some useful facts on actions of the Euclidean groups SE(N) before we proceed sorting out drifts and meanders for N = 2, in section 5. We conclude, in section 6, with a slow-fast analysis of drifting circular laments of scroll waves, so-called twisted scroll rings, in N = 3 dimensions.linear slice V , we will relate (2.1) to a G H-equivariant C k skew product ow on G V: The G H-action on G V is de ned as (g 0 ; h)(g; v) = (g 0 gh 1 ; hv); (2.3) see in particular (1.15){(1.20).Talking about the H-action on G V; below, we will mean the action of fidg H: Similarly, G-action refers to G fidg.
Our proof can be outlined as follows.First, we check G H-equivariance of (2.2).After a brief digression clarifying the structure of G V over U as a principal H bundle, we project (2.2) from G V down to the tube U M by the submersion (2.4) de ned in (1.16), to obtain a G-equivariant C k vector eld (2.1) on U from the skew product (2.2).To complete the proof, we nally lift a given G-equivariant C k vector eld f on U back to a G H-equivariant C k skew product (2.2) on G V; such that the skew product projects down to the prescribed f; by : Checking G H-equivariance of the skew product (2.2) on G V is easy: x )a(hv); '(hv)): (2.5) In other words, (g 0 ; h)(g(t); v(t)) is a solution of the skew product (2.2) if, and only if, (g(t); v(t)) is.This proves G H-equivariance of the skew product on G V: In passing, we note that the skew product (2.2) with equivariance condition (1.20) is the most general form of a G H-equivariant C k vector eld on G V: Indeed, (left) G-equivariance forces the _ g component to be of the form ga(v) with a(v) 2 alg(G).
Moreover, the _ v component must be independent of g: Then H-equivariance provides the equivariance conditions (1.20).
We brie y digress now, to clarify the structure of G V as an H principal C k+1 bundle over U with ber, alias structure group, H.Our presentation essentially follows Pal61] and the textbook tD91].
Identifying H-orbits of the free H-action on G V; we obtain the H orbit space Here p is the canonical G-equivariant C k+1 -submersion which projects (g; v) onto its H-orbit; it induces the structure of a C k+1 -manifold on G H V because the free H-action on G V is C k+1 : In fact, (G V; G H V; p) is a G-equivariant C k+1 principal ber bundle with compact ber, alias structure group, H.The G-equivariant map ; called tube map, is a C k+1 di eomorphism onto the open tube U around the group orbit G u 0 : We emphasize that these results are by no means original.They are essentially due to Pal61], and are concisely summarized in the textbook tD91], sections I.5, II.6.
After our bundle digression, we now project the skew product (2.2) down to M with the submersion , aiming at the second part of our theorem.Let u 2 M: Since the C k+1 -submersion : G V !U is surjective, there exists (g; v) such that (g; v) = u: By the bundle digression, any other (g 0 ; v 0 ) in 1 (u) is on the same H-orbit: there exists h 2 H such that (g 0 ; v 0 ) = (gh 1 ; hv): (2.8) We de ne f(u) via the di erential D (g; v) of with respect to g and v, f(u) := D (g; v) (ga(v); '(v)): (2.9) To show that f is well-de ned, we use the action of H on G V: In fact, we prefer an explicit calculation even though we could also argue \elegantly" with the H bundle structure.We start from (gh 1 ; hv) = (g; v); (2.10) for all h 2 H: Di erentiating with respect to g and v, we obtain D (gh 1 ; hv) (gah 1 ; h') = D (g; v) (ga; '); (2.11) for any a 2 alg(G); ' 2 T u0 V: Therefore, f(u) does not depend on the choice of (g 0 ; v 0 ) 2 1 (u); because (2.8){(2.11)and equivariance (1.20) imply D (g 0 ; v 0 ) (g 0 a(v 0 ); '(v 0 )) = = D (gh 1 ; hv) (gh 1 a(hv); '(hv) (2.12) This proves that f(u) is indeed well-de ned on u 2 U; by (2.9).Because 2 C k+1 and a; ' 2 C k ; it is obvious that f is a C k vector eld on the tube U: It remains to check G-equivariance (1.17) of f: Fixing (g; v) 2 1 (u); this follows directly from G-equivariance of and of the skew product (ga(v); '(v)): Explicitly, we have (g 0 g; v) = g 0 (g; v) = g 0 u; and hence f( This proves the second part of our theorem: the submersion projects any G Hequivariant C k vector eld (2.2) on G V down to a G-equivariant C k vector eld f on U. It remains to, conversely, lift f from U M up to a skew product on G V; such that the lift projects back onto the prescribed f, by : Since the ber is the isotropy group H; this is trivial if H happens to be discrete, that is, nite.Then we can simply lift the ow in U, and f; back to any sheet h 0 of the covering space G V of U, by the local di eomorphism 1 h0 : Lifting f back to any other sheet h 1 ; locally near u 0 ; where h 1 = h 1 h 0 for some h 2 H; we see that (gh 1 ; hv) = ( 1 h0 h1 )(g; v) ( for all w 2 W: We rst de ne F on the linear slice w 2 fidg V G V: Let P v denote the orthogonal projection, with respect to the H H-invariant Riemannian metric on W; in the tangent space T (id;v) W = alg(G) V onto the orthogonal complement T (id;v) ((id; v) H) ? of the right H-action.So P v projects onto the second summand of the orthogonal decomposition T (id;v) W = T (id;v) ((id; v) H) T (id;v) (id; v) H) ?: (2.17) Then we de ne the lifted vector eld F at w = (id; v) as F(w) := P v (D (w)) 1 f( (w)): (2.18) Note that F is now well de ned on fidg V: Indeed 1 (u) = wH; for u = (w); and D (w) : T w W !
(2.26) so that hF(w) h is indeed a candidate for F(hw h) in (2.24): the di erence lies in the kernel of D (hw h): To complete the proof of (2.24), and of theorem 1.1, we nally show that hF(w) h is orthogonal to ker D (hw h); as is F(hw h) by de nition (2.18), at hw h = (id; hv): Indeed, by invariance of the Riemannian metric on W with respect to the action of the compact group H H; we conclude from (2.18) at w and (2.25) that hF(w) h 2 h(kerD (w)) ?h = = (h(ker D (w)) h) ?= = (ker D (hw h)) ?: (2.27)This completes the proof of G H-equivariance of F, and of theorem 1.1.2 We note that our orthogonality condition in (2.18) at w 2 fidg V determines the lifted vector eld F uniquely.We formalize this statement for F(id; v) = (a(v); '(v)): Corollary 2.1 Let the assumptions of theorem 1.1 hold.Let h ; i alg(G) denote an invariant scalar product on alg(G) under the adjoint action Ad(h) of h 2 H; and let ( ; ) V denote an H-invariant scalar product on the linear slice space V: Then the lifted vector eld F(id; v) = (a(v); '(v)) can be chosen such that ('(v); v) V = ha(v); i alg(G) ; (2.28) for any v 2 V; 2 alg(H): The above conditions, together with the vector eld f on the base U, determine F uniquely.
Documenta Mathematica 1 (1996) 479{505 3 Equivariant periodic orbits in a slice By theorem 1.1 we can discuss any local bifurcation from a relative equilibrium u 0 with isotropy H in the associated G H-equivariant skew product system _ g = ga(v); _ v = '(v): (3.1) To interpret results in terms of u = (g; v) in the tube U around G u 0 ; we just have to identify points w = (g; v) on the same right H-orbit w H 2 G H V: In this section, we investigate some elementary consequences of our decomposition (3.1) in case the H-equivariant _ v equation possesses a periodic orbit.Such periodic orbits may arise by H-equivariant Hopf bifurcation from the H-invariant equilibrium v = u 0 of the _ v equation; for a detailed background using compactness of H see GSS88] or Fie88].
The spatio-temporal symmetry of any periodic solution v(t) of _ v = '(v); with minimal period normalized to 1, can be described by a triple (L; K; ) as follows.
Let L denote the set of h 2 H mapping some point v(t 1 ) to any point v(t 2 ) on the periodic orbit.Denoting (h) := t 2 t 1 ; equivariance of ' then implies hv(t) = v(t + (h)); (3.2) for all real t: Moreover is a homomorphism into the additively written circle group.Letting K := ker ; we have a normal subgroup of L, and L=K = image( ): Note that the groups L; K; image( ) are closed.The kernel K is the isotropy of some, and hence all, v(t) with t 2 IR: Following Fie88], we call v( ) a discrete wave, if image( ) = Z Z n = f0; 1=n; ; (n 1)=ng is nite.A rotating wave has image( ) = S 1 : The periodic solution v(t) gives rise to solutions g(t) of _ g = ga(v): By left Gequivariance, any solution g(t) with initial condition g(0) = g 0 is given by g(t) = g 0 g (t); (3.4) where g (t) denotes the fundamental solution _ g (t) = g (t)a(t) g (0) = id with the abbreviation a(t) := a(v(t)): Theorem 3.1 Let v(t) be a rotating wave solution of _ v = '(v) in (3.1).

2
Note that the relative equilibrium u(t) above can be stationary, periodic, meandering, or drifting, depending on the values of the in nitesimal generator a 0 + 2 alg(G): In particular, the closure of the orbit u( ) can have large dimension, for example if G contains large dimensional tori.Although the motion of u( ) can then be quasiperiodic in time, the associated rotation numbers given by a 0 + vary smoothly with parameters, and phase locking does not occur.
Next let v(t) be a discrete wave with symmetry (L; K; ); L=K = Z Z n : We describe the spatio-temporal symmetry of the associated not necessarily periodic solution u(t) = (g (t); v(t)) by a triple ( L; K; ~ ) similarly to the periodic case.Let L denote the set of g 2 G such that gu(t 1 ) = u(t 2 ); for some t 1 ; t 2 : Letting ~ (g) := t 2 t 1 ; we obtain gu(t) = u(t + ~ (g)); (3.11) for all real t and g 2 L; similarly to (3.2).Let := IR=p Z Z if u is periodic with minimal period p > 0; and := IR if u is nonperiodic (p = 1): Then ~ : L ! (3.12) is a homomorphism with kernel, alias isotropy of any u(t); denoted by K: Documenta Mathematica 1 (1996) 479{505 Theorem 3.2 Let v be a discrete wave solution of _ v = '(v) in (3.1) with symmetry (L; K; ); image ( ) = Z Z n ; and minimal period 1, as above.Let g (t) denote the associated solution of (3.5), a nonautonomous, 1-periodic, G-equivariant equation.
Similarly, a k-fold iteration of (3.22) for k = 0; 1; ; n 1 proves (3.15).Since h n 2 K commutes with h and, by (3.13), with g (1=n); the stroboscope map g (1) in (3.14) also commutes with g (1=n)h : Since g (1) also commutes with K; and because h generates H=K; the stroboscope g (1) and its iterates g (k) also commute with all g ( (h))h; h 2 L: To prove (3.16), let g 2 L: Then gu(t) = u(t + #) for some real # and all t 2 IR: Upstairs, there exists h 2 H such that (g (t + #); v(t + #)) = g(g (t); v(t)) h = = (gg (t)h 1 ; hv(t)) (3.23) for some, and hence all, real t: Comparing the second components we see that h 2 L and there exists a unique k 2 Z Z such that # = (h) + k; (3.24) if we x representatives 0 (h) < 1: Comparing the rst components, in view of (3.13), (3.24), we nd g (k)g ( (h))h = g; (3.25) after cancellation of g (t)h 1 : Conversely, any such g lies in L; by (3.13), (3.23), (3.24).Letting ~ (g) = # (mod p); it only remains to prove K = K: Note that g 2 K = ker ~ if, and only if, (3.23) holds with # = 0 and for some (hence all) t, say t = 0: Comparing components and using g (0) = id; this is equivalent to gh 1 = id with h 2 K: Hence K = K; and the proof is complete.2 The simple fact K = K; in our notation, implies that the isotropy groups occurring in the tube U are precisely the conjugates gKg 1 ; g 2 G; of isotropy groups K occurring in the (linear) slice V: Concisely: the isotropy types in U and V coincide.We emphasize that the spatio-temporal symmetry L of u(t); given in (3.16), is a group, and ~ : L ! is a group homomorphism.For suitable H-equivariant Documenta Mathematica 1 (1996) 479{505 choices of a(v); the element g (1=n) can be thought of as an arbitrary element of the connected component G 0 of the identity in G: Indeed, H-equivariance does not impose any signi cant restriction on a(t); 0 t < 1=n; thus leaving su cient freedom to prescribe a path g (t) 2 G from g (0) = id to g (1=n): However, the skew product consequences of the interplay of the various spatio-temporal symmetries (L; K; ) in equivariant Hopf bifurcation certainly deserve further investigation.

Basic facts on Euclidean groups
We collect some background material concerning G = E(N) (or SE(N)), the (special) Euclidean groups on IR N : In lemma 4.1 below, we identify the compact subgroups of G as translation conjugates of purely orthogonal groups.In lemma 4.2 this is applied to distinguish meandering from drifting solutions.We recall the semidirect product structure (S)E(N) = (S)O(N) IR N and the composition rule, coming from the standard a ne action on IR N ; see (1.1){(1.4).
For the second step note that is at least continuous.Indeed, H is compact and the bijection p : H ! p(H) is continuous, with inverse determined by : Therefore p is a homeomorphism, and is continuous.
This proves the lemma.

2
The lemma holds, more generally, for any compact subgroup H of the general a ne group GL(N) R N : The proof is the same, and the compact group p(H) GL(N) may in fact be assumed to act orthogonally.
Using the notation of section 3, we now consider a periodic solution v(t) of _ v = '(v) in the skew product, with period 1, and with associated fundamental solution Documenta Mathematica 1 (1996) 479{505 g (t) of (3.5).We derive a criterion to decide whether the projected solution u(t) = (g (t); v(t)) is meandering or drifting, in the sense of de nition 1.2.
is a compact subgroup of SE(2): (Note here that theorem 3.2 also applies to rotating waves v(t); viewed as discrete waves with arbitrary n 2 IN: By lemma 4.1, the group H 0 is compact if, and only if, it can be conjugated to its projection p(H 0 ) (S)O(N); by a pure translation S 0 2 IR N : This is possible if, and only if, the translation component of (id; S 0 )(R ; S )(id; S 0 ) = (R ; S 0 + S + R S 0 ) ( vanishes.Using orthogonality of R ; this is equivalent to S 2 image(id R ) = ker(id R ) ? ; (4.17) proving claim (4.14), and the lemma.

2
We note a dichotomy with respect to dimension N, here, which was also observed by AM96].For even N; we have (IR N ) R = f0g; for generic rotations R ; and hence generic meandering.For odd N, in contrast, dim(IR N ) R = 1; generically, which implies generic drifting.
If the 1-periodic solution v(t) 2 V possesses spatio-temporal symmetry (L; K; ) with non-trivial pointwise isotropy K, we obtain a particularly simple criterion excluding drifts.
Lemma 4.3 Let G = SE(N) or E(N), consider v; u; g as above, and let g (1) = (R ; S ): Assume the compact isotropy group K of v(t) to be contained in O(N); after conjugation by a translation as in lemma 4.1.
Then the translation component S of the stroboscope map g (1) is xed under K, that is S 2 (IR N ) K : (4.18)In particular, drifting is excluded if Most trivially, of course, condition (4.19) holds if (IR N ) K = f0g or (IR N ) R = f0g: Proof: Lemma 4.2 and (4.18) imply claim (4.19).To prove (4.18), we let h 0 2 K O(N): Since h 0 and g (1) commute, by theorem 3.2, (3.13), this implies (R ; S ) = g (1) = h 0 g (1)h 1 0 = = (h 0 R h 1 0 ; h 0 S ): (4.20) Therefore h 0 S = S ; and the lemma is proved.2 The projected solution u(t) satis es u for all stroboscope times k 2 Z Z : Let g (1) k = (R k ; S k ): Aside from a compact part, due to R k ; and possibly the isotropy H of u 0 ; the displacement of u(0) is therefore given by the translation component S k of the k-fold iterated stroboscope g (1) k : From (4.3), we recall S k = (id + R + + R k 1 )S and R k = (R ) k : To analyze S k ; we consider the meandering case S ?ker(id R ) next, for the stroboscope map g (1) = (R ; S ): Let (id R ) y denote the pseudo-inverse of (id R ); that is, the isomorphism inverting (id R ) within the R -invariant subspace (ker(id R )) ?= image(id R ): De ne S y := (id R ) y S : ( Lemma 4.4 As in the above setting, let S ?ker(id R ): Then g (1) k ; k 2 Z Z ; are all conjugate to the rotations (R k ; 0) around the origin, by the xed translation S y : g (1) k = (id; S y )(R k ; 0)(id; S y ) = (R k ; S y R k S y ): (4.23) In particular, the translation components S k 2 IR N of (g (1)) k all lie on a sphere around S y 2 IR N with radius jS y j 2 : Proof: By (4.3), applied to (R k ; S k ) = (g (1)) k ; k > 0; and geometric summation, we have S k = (id + R + + R k 1 )S = = (id R k )(id R ) y S = (id R k )S y = S y R k S y : (4.24) In case k < 0; the same formula holds, by (g (1)) k = ((g (1)) 1 ) 1 and (1.4).This proves (4.23) and, by orthogonality of R k ; the lemma. 2 The radius jS y j 2 de ned in (4.22) and lemma 4.4 relates to the \radius" of a meandering solution u(t) = g (t)v(t) as follows.Let u 0 (t) = (exp(r 0 t); 0)u 0 be a primary rotating wave solution, as in the introduction (1.5), (1.6).Then u 0 (t) rotates around its core point centered at zero.For v(0) near u 0 ; we can consider zero also as the core point of u(0) = id v(0): Then S k ; the translation component of g (k) = g (1) k ; is the core position of u(k) = g (k)v(0); by 1-periodicity of v( ): Since S k all lie on a sphere around S y with radius jS y j 2 ; we can call the Euclidean length jS y j 2 Documenta Mathematica 1 (1996) 479{505 the stroboscope radius of u(t): In section 5, (5.7) we will see how jS y j 2 !1; when a planar meandering spiral passes through a drift resonance, for which S 6 = 0 and R = id: We caution our reader that our notion of a stroboscope radius requires u 0 (t) to rotate around the origin.Moreover, the precise value of jS y j 2 depends on our choice of t = 0 as a reference point within the period of v: Indeed, other choices lead to expressions Sy = R (t) 1 (S y P S (t)); (4.25) 0 t 1; replacing S y ; with correspondingly modi ed radii j Sy j 2 : Here P projects onto ker(id R ); orthogonally.Note that (4.25) has period 1 in t, by de nition (4.22) of S y : Bounded modi cations as in (4.25), however, do not a ect the asymptotics of jS y j 2 !1; when passage through a drift resonance occurs. 2 5 The planar case E(2): meandering and drifting multi-armed spirals First rigorous results on meandering and drifting one-armed spirals in the plane were obtained by Wul96], using a Lyapunov-Schmidt procedure in scales of Banach spaces.First formal results on meandering and drifting multi-armed spirals in the plane were obtained by GLM96], using a formal center bundle reduction in the spirit of Kru90].
Using the rigorous center manifold reduction due to SSW96a], SSW96b], the skew product structure developed in the present paper applies.We recover results of GLM96], and investigate the behavior of meander radii at drift resonance.
Throughout this section, G = E(2); and H is a compact subgroup which we may consider to be a subgroup of O(2); after conjugation by a xed translation, without loss of generality.As in section 3, we consider H-equivariant Hopf bifurcation for _ v = '( ; v) in the slice v 2 V of our skew product (3.1).Let (L; K; ) denote the spatio-temporal symmetry of our periodic solution v(t); with minimal period normalized to 1.We also normalize the primary relative equilibrium u 0 to become v = 0; without loss of generality.The case of a rigidly rotating \primary" spiral wave with n identical arms, in the setting of the introduction, now corresponds to a rotating wave u 0 with H = Z Z n SO(2): We begin with a simple criterion excluding drifting solutions u(t) := (g (t); v(t)) for general H O(2): Corollary 5.1 In the above planar setting, assume the isotropy group K of v(t) contains some nontrivial rotation, that is, K O(2) is neither trivial nor generated by a single re ection.

2
In a Hopf bifurcation situation, it is easy to derive expansions for the various cases of the previous corollary.Indeed, consider a primary n-armed spiral u 0 (t) = exp(i!rot t)u 0 (0) with isotropy H = Z Z n and minimal period T rot = 2 =(n! rot ): Assume an additional pair 2 i of imaginary eigenvalues of the linearization (in rotating coordinates).Then we can parameterize v(t) = e 2 it + O( 2 ) (5.13) at parameter = 0 + 2 2 + O( 3 ): The equation for g (t) becomes _ g = g (a 0 + a 1 v(t) + ); (5.14) where a 0 = a(v = 0) and a 1 = Da(v = 0): For simplicity of presentation, we focus on the rotational component R (t) = exp(2 i (t)) of g (t): Inserting the v-expansion (5.13) we obtain 2 _ (t) = !rot + ; (0) = 0; (5.15) omitting time dependent terms of order : Note that, indeed, !rot is the rotation frequency of the rotating spiral u 0 (t): Solving (5.15), up to terms of order ; we get for g (1=n) = ( 1=n ; S 1=n ) 1=n = (1=n) = !rot 2 n + : (5.16) Letting 2 = !Hopf denote the (normalized) frequency of the nontrivial Hopf eigenvalues, the transition to the drift case (iv) occurs, for example, at n 1=n + m 0 = !rot !Hopf + m 0 0 (mod n): 6 Meandering and drifting in three dimensions: twisted scroll rings Let G = SE(3); in this section.We rst consider a primary wave u 0 (t) with trivial isotropy H = fidg: At the end of this section, we comment on the case H = Z Z n : Pictorially, we think of u 0 as a hypothetical one parameter family of one-armed spirals with a core lament aligned along a unit circle parallel to the (x; y)-plane.The spiral patterns occur, locally, in the bundle of normal planes to the core circle.Such patterns have been called scroll waves by Win73].Moreover, assume the spirals to possess a phase di erence along the family of normal planes.For simplicity, we assume that phase di erence to equal the angle di erence of the core points on the unit circle (rather than equaling an integer multiple of that angle.)While that pattern rotates, horizontally, around the vertical z-axis, as a rotating wave, it also propagates, vertically, along the z-axis, at constant speed.We call such a hypothetical pattern (if it exists) a twisted scroll ring PW85].The so inclined reader may also visualize smoke rings, with an inner rotating structure.For another recent example involving rigid body motion (of submarines) with SE(3) symmetry see LM96].More mathematically, we require u 0 (t) = exp(a 0 t)u 0 (0); (6.1)where u 0 has trivial isotropy H, and a 0 = (r 0 ; s 0 ) in the Lie algebra of SE(3) has the special form r 0 = i! 0 0 0 0 ; s 0 = 0 @ 0 0 1 1 A : (6.2) We use complex notation in the horizontal (x; y)-plane, here, writing IR 3 = C I IR: We assume !0 6 = 0 for the horizontal rotation frequency.Technically speaking, we might call u 0 (t) a drifting and rotating relative equilibrium.Lemma 4.2 explains why we choose the translation s 0 to be vertical to the rotation plane.
Because the isotropy H is trivial, the skew product _ g = g a(v); g (0) = id; _ v = '(v); (6.3) describes the ow in a neighborhood U of G u 0 : We consider a family of periodic solutions v = v( ; t) of period normalized to 1, bifurcating from the trivial solution v u 0 : The parameter ; so necessary for such a Hopf bifurcation, is suppressed.Instead, we represent dependence of a(v) on v = v( ; t) by a di erentiable function a( ; t) := a(v( ; t)) (6.4) in the Lie algebra, directly.Note that a 0 := a(0; t) = a(u 0 ) (6.5) does not depend on time, while a( ; ) has (normalized) period 1 for > 0: As in any di erential equation, we can di erentiate the solution g = g ( ; t) with respect to : Writing ( ; t) := (@ g )g 1 ( ; t) := g 1 @ g (6.6) Documenta Mathematica 1 (1996) 479{505 here.Lighted with a stroboscope at (normalized) integer times t = n; we observe identical shapes of the twisted scroll ring.It propagates along the (slightly tilted) z-axis at a slightly modi ed average speed 1 + s: This oscillating propagation is a three-dimensional analogue of Hopf bifurcation from a traveling wave in one space dimension; for the latter see Pos92].In a plane perpendicular to the vertical propagation direction, our scroll ring performs a planar meandering motion of stroboscopic radius r = 1 2 j sin( !rot =! Hopf )j 1 j j; (6.15) as has been investigated in section 5. (We have returned to the notation !rot = !; !Hopf 2 used there).Typically, j j will be of order : Note the horizontal drift resonance which occurs at integer values !rot =! Hopf 2 Z Z : (6.16)At these values, the meandering propagation along a spiral around the z-axis becomes a slow sidewards drift, away from the z-axis.
Additional isotropies H = Z Z n ; commuting with the primary rotation exp(r 0 t) of u 0 (t) in (6.1), (6.2), can be incorporated.Note that H rotates around the vertical z-axis.For the horizontal planar meandering, the results of section 5 will reappear.Speci cally, let (H; K; ) be the spatio-temporal symmetry of a bifurcating periodic solution v(t) in the skew product.According to lemma 4.3, nontrivial rotations in K will force translations S in the stroboscope map g (1) = (R ; S ) to point along the z-axis.Likewise, R near exp r 0 will rotate around the z-axis, unless R = id: Indeed R 2 SO(3) n fidg commutes with K; by (3.13) and lemma 4.3, and hence R and K x the same axis of rotation.Therefore horizontal meandering is impossible, if K contains a nontrivial rotation.Pure drifts g (1) = (id; S ) can only point along the z-axis.
If K = fidg is trivial, transverse meandering perpendicular to the direction of propagation becomes possible.Indeed, let g (1=n) = (R 1=n ; S 1=n ): Again we conjugate the axis of R 1=n to be vertical, so that R 1=n = exp(i 1=n ) 0 0 1 : (6.17)Then S 1=n possesses a rather irrelevant vertical component, which only modi es the vertical propagation speed.The important horizontal component, however, produces periodicity, meandering, and drifting phenomena transversely to the propagation direction.Note how the -expansions (5.24), (5.25) force the transverse drifting to be of small radius, or the transverse drifting to be slow.
Arrows by American Indians and other early, even neolithic civilizations are a practical visualization of some of the results discussed here.In fact, elastic vibrations and interaction with the air ow could lead to destabilization of the straight ight path.However, the feathers can provide an isotropy K, if they prevent rotation around the axis of the arrow.This isotropy, in turn, prevents transverse drifting and xes the direction of propagation to be, quite literally, \straight as an arrow".Even in the case of a rotating feathered arrow, transverse deviations caused by symmetry breaking bifurcations from the straight path will be slow, due to (5.24), (5.25).
We use the equivariant projection p : E(N) = O(N) IR N !O(N) (4.4) onto the rst component.
Lemma 4.1 Let H be a compact subgroup of E(N): Then H is conjugate to its projection p(H) O(N) by a xed translation S 0 2 IR N : H = (id; S 0 ) p(H) (id; S 0 ) (4.5) Proof: We will rst prove that there exists a map : p(H) !IR N (4.6) Multiplying (R; S); (R 0 ; S 0 ) in H yields the functional equation(RR 0 ) = (R) + R (R 0 ): (4.10)Note continuous dependence on R 0 .We integrate (4.10) over R 0 with respect to the left invariant Haar measure on the compact Lie group p(H).With the abbreviation S We now return to the general case of compact isotropy H: It is convenient to describe the lift of f in slightly more abstract notation.Let w = (g 0 ; v) 2 W := G V with left action gw := (gg 0 ; v) of G and right action wh := (g 0 h 1 ; hv) of H describe the action of the direct product G H on W: Note that G; H act freely, separately.It remains to construct a G H-equivariant C k vector eld F on the total space W of our principal H bundle : G V !U;(2.15) such that F projects down to f by , that is, The vector eld F is still C k ; by smoothness of the free G-action.By construction, F is G-equivariant.