Bifurcation from Relative Equilibria

of Noncompact Group Actions:

Skew Products, Meanders, and Drifts

Bernold Fiedler, Björn Sandstede, Arnd Scheel and Claudia Wulff

Received: October 4, 1996

Communicated by Alfred K. Louis

Abstract. We consider a finite-dimensional, typically noncompact Riemannian manifold M with a differentiable proper action of a possibly noncompact Lie group G. We describe G-equivariant flows in a tubular neighborhood U of a relative equilibrium G·u_{0}, u_{0} ∈ M, with compact isotropy H of u_{0}, by a skew product flow ġ = ga(v), v̇ = φ(v). Here g ∈ G, a ∈ alg(G). The vector v is in a linear slice V to the group action. The induced local flow on G×V is equivariant under the action of (g_{0}, h) ∈ G×H on (g, v) ∈ G×V, given by (g_{0}, h)(g, v) = (g_{0}gh^{−1}, hv). The original flow on U is equivalent to the induced flow on {id} × H-orbits in G × V.

Applications to relative equivariant Hopf bifurcation in V are presented, clarifying phenomena like periodicity, meandering, and drifting. Specific illustrations involving Euclidean groups G are meandering spirals, in the plane, and drifting twisted scroll rings, in three-dimensional Belousov-Zhabotinsky media.

1991 Mathematics Subject Classification: 58F35, 57S20, 55R91

1 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 ∈ IR^{2}. The partial differential equations, which model the dynamics of the solutions u(t, x), are equivariant with respect to the standard affine 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′, S′) ∈ O(N) × IR^{N} is defined by

(R, S) ∘ (R′, S′) := (RR′, S + RS′);  (1.1)

this rule is compatible with the standard affine representation

(R, S)x := Rx + S  (1.2)

on x ∈ 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-profiles 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 fixed shift of x-coordinates, it can be written as

u(t, ·) = (R(t), 0)u(0, ·).  (1.5)

The rotations R(t) ∈ 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.

Bifurcation from Relative Equilibria … 481

The first mathematically rigorous analysis of these phenomena has recently been achieved by Wulff, see [Wul96]. Her result is based on a careful Lyapunov-Schmidt reduction in a scale of Banach spaces. This resolves the difficulties of nondifferentiability and, in some cases, non-continuity of the group action (1.3) on the infinite-dimensional Banach space u(t, ·) ∈ X. For technically related earlier results, restricted to compact group actions, see [Ren82] and [Ran82]. It has recently been shown, for the first time, that a center manifold reduction to a finite-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 differential 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 differentiable 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 flow near relative equilibria inside a finite-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 flow as a skew product flow 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 defined in (1.15), (2.6) below.

Also, we will allow for general compact isotropies H, rather than requiring H to be finite or even trivial. In the following, the reader may find 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}-diffeomorphism u ↦ gu on the finite-dimensional Riemannian C^{k+1}-manifold M, such that the map

\begin{matrix}ρ : G × M & → & M \\ (g, u) & ↦ & gu = ρ(g, u) \end{matrix}  (1.8)

is C^{k+1}. Of course, we assume that G acts on M, that is (gg′)u = g(g′u) for all g, g′ ∈ G and u ∈ M. We also require the action to be proper, that is, the map ρ̃(g, u) := (gu, u) ∈ M ×M is closed (mapping closed sets to closed sets) with compact preimages ρ̃^{−1}(u_{1}, u_{2}), for any u_{1}, u_{2} ∈ M. As a caveat, we note that G = IR activing by shift on BC_{unif}(IR, IR), for example, does not define 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}) := {g ∈ G | gu_{0} = u_{0}}  (1.9)

is compact, for any u_{0} ∈ M. Indeed, H × {u_{0}} = ρ̃^{−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 fix 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 first assume the given Riemannian metric on M to be H-invariant, without loss of generality; see [Bre72], section VI.2. In particular, any h ∈ H acts linearly and orthogonally on the tangent space T_{u_{0}}M to M in u_{0}, by the derivative of u ↦ ρ(h, u) at u = u_{0}. Similarly, ρ induces a C^{k}-action of G on the C^{k} tangent bundle T M; 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} := {gu_{0} | g ∈ G}  (1.10)

of u_{0} as follows. Let alg(G) = T_{id} G denote the Lie algebra of G and

T_{u_{0}}(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 ∈ M by the derivative of g ↦ ρ(g, u) at g = id. Now let the desired disk V of the bundle G×V be defined as the open ϵ_{0}-ball, centered at u_{0}, inside the orthogonal complement

V ⊂ (T_{u_{0}}(G · u_{0}))^{⊥} ⊆ T_{u_{0}}M  (1.12)

to the orbit tangent space T_{u_{0}}(G · u_{0}) in T_{u_{0}}M. Note that the isotropy H of u_{0} acts linearly and orthogonally on V, as it does on T_{u_{0}}M and T_{u_{0}}(G · u_{0}).

To define the slice to the G-action and the G-invariant tube U around G · u_{0}, let ψ : (T_{u_{0}}M)_{loc} → M denote a local C^{k+1}-chart of M which is H-equivariant, that is

ψ(hv) = hψ(v),  (1.13)

for all v ∈ (T_{u_{0}}M)_{loc} and h ∈ H. Here (T_{u_{0}}M)_{loc} denotes an ϵ_{0}-ball in T_{u_{0}}M. In fact we construct ψ^{−1}, first, such that ψ^{−1}(u_{0}) = u_{0}, and then achieve H-equivariance, by Haar measure, preserving the property that ψ^{−1} is a diffeomorphism; see for example [tD91], section I.5. Then ψ(V) ⊂ M is a slice to the G-action at u_{0} ∈ ψ(V), and

U := G · ψ(V)  (1.14)

is an open G-invariant tube around the G-orbit G · u_{0}. For convenience, we also call the ϵ_{0}-disk V ⊂ T_{u_{0}}M around u_{0} a (linear) slice. We will take license to identify u_{0} ∈ V with the origin in IR^{l} = T_{u_{0}}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

\begin{matrix}τ̄ : G × V & → & U ⊃ G · u_{0} \\ (g, v) & ↦ & gψ(v) \end{matrix}  (1.16)

is a submersion for small radius ϵ_{0} of the disk V. In fact, the triple (G × V, U; τ̄) identifies the trivial product G × V as a (generally nontrivial) C^{k+1} principal fiber bundle over U with fiber, alias structure group, H. For more details, we refer to section 2.

Returning to dynamics, consider a G-equivariant C^{k} vector field f on the “center” manifold M, that is

gf(u) = f(gu),  (1.17)

for all u ∈ M, g ∈ G. Of course, here we define gf(u) by the induced (differential) C^{k}-action of G on the tangent space T M. We seek a representation of the (local) G-equivariant flow

u̇ = f(u)  (1.18)

on M near the G-orbit G · u_{0} by the skew product flow

\begin{aligned}ġ & = & ga(v) \\ v̇ & = & φ(v) \end{aligned}  (1.19)

on G × V. Here the maps a : V → alg(G) and φ : V → T_{u_{0}}V are requested to be of class C^{k} and H-equivariant in the following sense:

\begin{aligned}a(hv) & = & Ad(h)a(v) = ha(v)h^{−1}, \\ φ(hv) & = & hφ(v), \end{aligned}  (1.20)

for all h ∈ H and all v ∈ V. Here Ad(h) denotes the standard adjoint representation on the Lie algebra, and hφ is again understood to be differential on the linear ball V ⊆ T_{u_{0}}M. Theorem 1.1 Let f be a G-equivariant C^{k} vector field on the Riemannian C^{k+1}- manifold M, k ≥ 1, with proper C^{k+1}-action of G on M. Let u_{0} ∈ 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, φ, as in (1.20), such that the projection u := τ̄(g, v) ∈ U of any solution (g, v) of the skew product system (1.19) satisfies the original differential equation (1.18) in U. The projection τ̄ is defined in (1.16).

Conversely, for the local G×H-equivariant flow defined on (g, v) ∈ G×V by any C^{k} vector field (1.19), which is H-equivariant in the sense of (1.20), the projection u := τ̄(g, v) ∈ U induces a G-equivariant C^{k} vector field 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 {id} × 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 fix nomenclature. Definition 1.2 Consider u_{0} ∈ M with isotropy H and a G-equivariant vector field f on M, as in the theorem, with lifted skew product ġ = ga(v), v̇ = φ(v) as in (1.19), (1.20).

We call u_{0} a relative equilibrium, if

φ(u_{0}) = 0  (1.21)

In other words, u_{0} ∈ M is a relative equilibrium if, and only if, the time orbit of u_{0} remains inside the group orbit G · u_{0} :

{(u(t)|t ∈ IR} ⊆ G · u_{0}.  (1.22)

Equivalently, G · u_{0} is a flow invariant manifold.

Next, take any solution u(t) ∈ U. Suppose that u(t) is neither stationary nor periodic. Then, we call u(t) meandering if

{(g(t), v(t)) | t ∈ IR} ⊂ G × V  (1.23)

is globally defined and relatively compact. If, in contrast, the G-component

{g(t) | t ∈ IR}  (1.24)

is globally defined but not relatively compact, then we call u(t) drifting.

Equilibria, as well as rotating waves (spirals) are examples of relative equilibria. The reference point u_{0} ∈ 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 flow-defined, the definition (1.21) refers to a specific 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 ũ_{0} ∈ U other than u_{0}, the vector field (1.19) has to be constructed with respect to ũ_{0} instead of u_{0}. This subtlety, however, is irrelevant for small tubular neighborhoods U, as long as H is finite.

In the very special case G = {id} 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 filaments of scroll waves, so-called twisted scroll rings, in N = 3 dimensions. Acknowledgment. We gratefully acknowledge helpful advice by Christian Leis. We are also much indebted to Marty Golubitsky for sharing his insight into the behavior of multi-armed spirals, as well as for some inciting competitiveness. Regina Löhr helped us with careful and very patient typesetting.

2 Tubes, slices, and skew products

In this section we prove theorem 1.1. So, let its assumptions hold. We specifically recall that

u̇ = f(u)  (2.1)

is a G-equivariant C^{k} vector field on the Riemannian C^{k+1}-manifold M with proper C^{k+1}-action of the Lie group G on M. Given u_{0} ∈ M with isotropy H, tube U, and linear slice V, we will relate (2.1) to a G × H-equivariant C^{k} skew product flow

\begin{aligned}ġ & = & ga(v) \\ v̇ & = & φ(v) \end{aligned}  (2.2)

on G × V. The G × H-action on G × V is defined 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 {id} × H. Similarly, G-action refers to G × {id}.

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

τ̄(g, v) = gψ(v),  (2.4)

defined in (1.16), to obtain a G-equivariant C^{k} vector field (2.1) on U from the skew product (2.2). To complete the proof, we finally lift a given G-equivariant C^{k} vector field 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 τ̄.

Checking G × H-equivariance of the skew product (2.2) on G × V is easy: fix (g_{0}, h) ∈ G × H and (g, v) ∈ G × V. Then (2.3), (1.20) imply

\begin{matrix}(g_{0}, h)(ga(v), φ(v)) = (g_{0}ga(v)h^{−1}, hφ(v)) = \\ = (g_{0}gh^{−1}ha(v)h^{−1}, hφ(v)) = ((g_{0}gh^{−1})a(hv), φ(hv)). \end{matrix}  (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 field on G×V. Indeed, (left) G-equivariance forces the ġ component to be of the form ga(v) with a(v) ∈ alg(G). Moreover, the v̇ component must be independent of g. Then H-equivariance provides the equivariance conditions (1.20).

We briefly digress now, to clarify the structure of G × V as an H principal C^{k+1} bundle over U with fiber, 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

G ×_{H} V := G × V / {id} × H.  (2.6)

It turns out that the G(×{id})-equivariant C^{k+1}-submersion τ̄ factorizes over the H orbit space G ×_{H} V, such that

τ̄ : G × V →^{p} G ×_{H} V →^{τ} U.  (2.7)

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 fiber bundle with compact fiber, alias structure group, H. The G-equivariant map τ, called tube map, is a C^{k+1} diffeomorphism 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 τ̄, aiming at the second part of our theorem. Let u ∈ M. Since the C^{k+1}-submersion τ̄ : G × V → U is surjective, there exists (g, v) such that τ̄(g, v) = u. By the bundle digression, any other (g′, v′) in τ̄^{−1}(u) is on the same H-orbit: there exists h ∈ H such that

(g′, v′) = (gh^{−1}, hv).  (2.8)

We define f(u) via the differential Dτ̄(g, v) of τ̄ with respect to g and v,

f(u) := Dτ̄(g, v) · (ga(v), φ(v)).  (2.9)

To show that f is well-defined, 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

τ̄(gh^{−1}, hv) = τ̄(g, v),  (2.10)

for all h ∈ H. Differentiating with respect to g and v, we obtain

Dτ̄(gh^{−1}, hv) · (gah^{−1}, hφ) = Dτ̄(g, v) · (ga, φ),  (2.11)

for any a ∈ alg(G), φ ∈ T_{u_{0}}V. Therefore, f(u) does not depend on the choice of (g′, v′) ∈ τ̄^{−1}(u), because (2.8)–(2.11) and equivariance (1.20) imply

\begin{matrix}Dτ̄(g′, v′) · (g′a(v′), φ(v′)) = \\ = Dτ̄(gh^{−1}, hv) · (gh^{−1}a(hv), φ(hv)) = \\ = Dτ̄(gh^{−1}, hv) · (gh^{−1}ha(v)h^{−1}, hφ(v)) = \\ = Dτ̄(gh^{−1}, hv) · (ga(v)h^{−1}, hφ(v)) = \\ = Dτ̄(g, v) · (ga(v), φ(v)) = f(u). \end{matrix}  (2.12)

This proves that f(u) is indeed well-defined on u ∈ U, by (2.9).

Because τ̄ ∈ C^{k+1} and a, φ ∈ C^{k}, it is obvious that f is a C^{k} vector field on the tube U. It remains to check G-equivariance (1.17) of f. Fixing (g, v) ∈ τ̄^{−1}(u), this follows directly from G-equivariance of τ̄ and of the skew product (ga(v), φ(v)). Explicitly, we have τ̄(g_{0}g, v) = g_{0}τ̄(g, v) = g_{0}u, and hence

\begin{aligned}f(g_{0}u) & = & Dτ̄(g_{0}g, v) · (g_{0}ga(v), φ(v)) = \\ & = & g_{0}Dτ̄(g, v) · (ga(v), φ(v)) = \\ & = & g_{0}f(u). \end{aligned}  (2.13)

This proves the second part of our theorem: the submersion τ̄ projects any G × Hequivariant C^{k} vector field (2.2) on G × V down to a G-equivariant C^{k} vector field 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 τ̄. Since the fiber is the isotropy group H, this is trivial if H happens to be discrete, that is, finite. Then we can simply lift the flow in U, and f, back to any sheet h_{0} of the covering space G × V of U, by the local diffeomorphism τ̄_{h_{0}}^{−1}. Lifting f back to any other sheet h_{1}, locally near u_{0}, where h_{1} = h^{−1}h_{0} for some h ∈ H, we see that

(gh^{−1}, hv) = (τ̄_{h_{0}}^{−1} ∘ τ̄_{h_{1}})(g, v)  (2.14)

induces, by linearization with respect to (g, v), the claimed H-equivariance of the lifted vector fields, where h ∈ H acts freely as a permutation of the sheets in the covering space G × V. The trivial case H = id was first presented by one of the authors, explaining meandering and drifting spirals [Fie95]; for a recent version see also [BHN96].

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) ∈ 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 field F on the total space W of our principal H bundle

τ̄ : G × V → U,  (2.15)

such that F projects down to f by τ̄, that is,

Dτ̄(w)F (w) = f(τ̄(w))  (2.16)

for all w ∈ W.

We first define F on the linear slice w ∈ {id} × 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 define the lifted vector field F at w = (id, v) as

F (w) := P_{v}(Dτ̄(w))^{−1}f(τ̄(w)).  (2.18)

Note that F is now well defined on {id} × V. Indeed τ̄^{−1}(u) = wH, for u = τ̄(w), and

Dτ̄(w) : T_{w}W → T_{τ̄(w)}U  (2.19)

is surjective. Hence the kernel of Dτ̄(w) is given by

ker Dτ̄(w) = T_{w}(w · H)  (2.20)

in the H principal fiber bundle τ̄ : G × V → U, and P_{v} annihilates that kernel. Thus (2.18) defines F (w) properly on w ∈ {id} × V. Moreover, F is of class C^{k} on {id} × V, as are P_{v}, Dτ̄, and f.

We extend F to W = G × V by the left action of G on W, defining

F (g, v) := gF (id, v) ∈ T_{(g,v)}W,  (2.21)

for all g ∈ G. The vector field F is still C^{k}, by smoothness of the free G-action. By construction, F is G-equivariant.

We verify the projection property (2.16) next. Because τ̄, F, and f are all Gequivariant, it is sufficient to verify (2.16) at w = (id, v), that is,

Dτ̄(id, v)F (id, v) = f(τ̄(id, v)).  (2.22)

This follows trivially from definition (2.18) of F at w = (id, v), because P_{v} projects onto a complement of ker Dτ̄(w).

To complete the proof of theorem 1.1, it remains to show equivariance of F under the right action of H, that is

F (gh^{−1}, hv) = F (g, v) · h,  (2.23)

for all g ∈ G, h ∈ H, v ∈ V. By left G-equivariance of F, this is equivalent to showing

F (hw · h) − hF (w) · h = 0  (2.24)

for any w = (id, v) ∈ {id}×V, h ∈ H. To show (2.24), we first differentiate the relation τ̄(gw · h) = gτ̄(w) with respect to w to obtain

Dτ̄(gw · h)(gw̃ · h) = gDτ̄(w)w̃,  (2.25)

for any g ∈ G, h ∈ H, and (w, w̃) ∈ T W. Putting w = (id, v), g := h, and w̃ := F (w), this implies

\begin{matrix}Dτ̄(hw · h)(hF (w) · h) = hDτ̄(w)F (w) = \\ = hf(τ̄(w)) = f(hτ̄(w)) = f(τ̄(hw)) = \\ = f(τ̄(hw · h)), \end{matrix}  (2.26)

so that hF (w) · h is indeed a candidate for F (hw · h) in (2.24): the difference lies in the kernel of Dτ̄(hw · h).

To complete the proof of (2.24), and of theorem 1.1, we finally show that hF (w)·h is orthogonal to ker Dτ̄(hw·h), as is F (hw·h) by definition (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 ∈ h(ker Dτ̄(w))^{⊥} · h =
= (h(ker Dτ̄(w)) · h)^{⊥} =
= (ker Dτ̄(hw · h))^{⊥}.

(2.27)

This completes the proof of G × H-equivariance of F, and of theorem 1.1. □

We note that our orthogonality condition in (2.18) at w ∈ {id}×V determines the lifted vector field F uniquely. We formalize this statement for F (id, v) = (a(v), φ(v)). Corollary 2.1 Let the assumptions of theorem 1.1 hold. Let ⟨·, ·⟩_{alg(G)} denote an invariant scalar product on alg(G) under the adjoint action Ad(h) of h ∈ H, and let (·, ·)_{V} denote an H-invariant scalar product on the linear slice space V.

Then the lifted vector field F (id, v) = (a(v), φ(v)) can be chosen such that

(φ(v), ηv)_{V} = ⟨a(v), η⟩_{alg(G)},  (2.28)

for any v ∈ V, η ∈ alg(H). The above conditions, together with the vector field f on the base U, determine F uniquely.

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

\begin{matrix}ġ = ga(v), \\ v̇ = φ(v). \end{matrix}  (3.1)

To interpret results in terms of u = τ̄(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 ∈ G×_{H} V. In this section, we investigate some elementary consequences of our decomposition (3.1) in case the H-equivariant v̇ 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 v̇ equation; for a detailed background using compactness of H see [GSS88] or [Fie88].

The spatio-temporal symmetry of any periodic solution v(t) of v̇ = φ(v), with minimal period normalized to 1, can be described by a triple (L, K, Θ) as follows. Let L denote the set of h ∈ H mapping some point v(t_{1}) to any point v(t_{2}) on the periodic orbit. Denoting Θ(h) := t_{2} − t_{1}, equivariance of φ then implies

hv(t) = v(t + Θ(h)),  (3.2)

for all real t. Moreover

Θ : L → S^{1} := IR/ Z Z  (3.3)

is a homomorphism into the additively written circle group. Letting K := ker Θ, we have a normal subgroup of L, and L/K ≅ image(Θ). Note that the groups L, K, image(Θ) are closed. The kernel K is the isotropy of some, and hence all, v(t) with t ∈ IR. Following [Fie88], we call v(·) a discrete wave, if image(Θ) = Z Z_{n} = {0, 1/n, ⋯, (n − 1)/n} is finite. A rotating wave has image(Θ) = S^{1}.

The periodic solution v(t) gives rise to solutions g(t) of ġ = 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

\begin{aligned}ġ_{∗}(t) & = & g_{∗}(t)a(t) \\ g_{∗}(0) & = & id \end{aligned}  (3.5)

with the abbreviation a(t) := a(v(t)). Theorem 3.1 Let v(t) be a rotating wave solution of v̇ = φ(v) in (3.1).

Then there exist η ∈ alg(H), a_{0} ∈ alg(G) such that

v(t) = exp(ηt)v_{0}
g_{∗}(t) = exp((a_{0} + η)t) exp(−ηt). (3.6)

The projected solution u(t) = τ̄(g_{∗}(t), v(t)) near the relative equilibrium u_{0} is again a relative equilibrium and can be represented as

u(t) = exp((a_{0} + η)t)u(0).  (3.7)

Proof: To construct η, just note that v_{0} := v(0) is a relative equilibrium to the action of H on V because v(t) is a rotating wave. In particular

v̇(0) ∈ T_{v_{0}}(Hv_{0}) = alg(H) · v_{0}.  (3.8)

Pick η ∈ alg(H) such that ηv_{0} = v̇(0) = φ(v_{0}). Let v_{∗}(t) := exp(ηt)v_{0}. Then v_{∗}(0) = v_{0} and H-equivariance (1.20) of φ implies

v̇_{∗}(t) = exp(ηt)ηv_{0} = exp(ηt)φ(v_{0}) =
= φ(exp(ηt)v_{0}) = φ(v_{∗}(t)), (3.9)

for all t. Therefore v(t) = v_{∗}(t), for all t.

Let a_{0} := a(v(0)) and define g_{∗}(t) as in (3.6). We have to show that g_{∗}(t) solves (3.5). Trivially g_{∗}(0) = id. Using H-equivariance (1.20) of a(v(t)) = a(t) yields

ġ_{∗}(t) = exp((a_{0} + η)t) (a_{0} + η) exp(−ηt)−
− exp((a_{0} + η)t) η exp(−ηt) =
= exp((a_{0} + η)t) exp(−ηt) exp(ηt) a(v(0)) exp(−ηt) =
= g_{∗}(t) a(exp(ηt)v(0)) =
= g_{∗}(t) a(v(t)) =
= g_{∗}(t) a(t).

(3.10)

This proves (3.6). To prove (3.7), we remember that τ̄ is left G equivariant and collapses right H-orbits. Therefore (3.6) implies

u(t) := τ̄(g_{∗}(t), v(t)) = τ̄(exp(a_{0} + η)t)(id, v_{0}) · exp(ηt)) =
= exp((a_{0} + η)t)τ̄(id, v_{0}) = exp((a_{0} + η)t)u(0),

and the theorem is proved. □

Note that the relative equilibrium u(t) above can be stationary, periodic, meandering, or drifting, depending on the values of the infinitesimal generator a_{0} + η ∈ 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} + η vary smoothly with parameters, and phase locking does not occur.

Next let v(t) be a discrete wave with symmetry (L, K, Θ), L/K ≅ Z Z_{n}. We describe the spatio-temporal symmetry of the associated not necessarily periodic solution u(t) = τ̄(g_{∗}(t), v(t)) by a triple (L̃, K̃, Θ̃) similarly to the periodic case. Let L̃ denote the set of g ∈ G such that gu(t_{1}) = u(t_{2}), for some t_{1}, t_{2}. Letting Θ̃(g) := t_{2}−t_{1}, we obtain

gu(t) = u(t + Θ̃(g)),  (3.11)

for all real t and g ∈ L̃, similarly to (3.2). Let Σ := IR/p Z Z if u is periodic with minimal period p > 0, and Σ := IR if u is nonperiodic (p = ∞). Then

Θ̃ : L̃ → Σ  (3.12)

is a homomorphism with kernel, alias isotropy of any u(t), denoted by K̃.

Theorem 3.2 Let v be a discrete wave solution of v̇ = φ(v) in (3.1) with symmetry (L, K, Θ), image (Θ) = 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.

Then, for any k ∈ Z Z , h_{0} ∈ K, h ∈ L, t ∈ IR, we have

\begin{aligned}g_{∗}(t) & = & h_{0}g_{∗}(t)h_{0}^{−1} \\ g_{∗}(t + Θ(h) + k) & = & g_{∗}(k)g_{∗}(Θ(h))hg_{∗}(t)h^{−1}. \end{aligned}  (3.13)

For the stroboscope map g_{∗}(1) of (3.5) we obtain

\begin{aligned}g_{∗}(1) & = & (g_{∗}(1/n)h_{∗})^{n}h_{∗}^{−n} \\ g_{∗}(k) & = & g_{∗}(1)^{k} \end{aligned}  (3.14)

where h_{∗} ∈ L can be chosen arbitrarily such that Θ(h_{∗}) = 1/n generates image(Θ) = Z Z_{n} = {0, 1/n, ⋯, (n−1)/n}. In particular, the stroboscope maps g_{∗}(k) commute with g_{∗}(Θ(h))h, for all k ∈ Z Z and h ∈ L.

For g_{∗}(Θ(h)), Θ(h) = k/n, k = 0, 1, ⋯, n−1, we have the more explicit expression

g_{∗}(k/n) = (g_{∗}(1/n)h_{∗})^{k} h_{∗}^{−k}.  (3.15)

The symmetry (L̃, K̃, Θ̃) of the projected solution u(t) = τ̄(g_{∗}(t), v(t)) with minimal “period” 0 < p ≤ ∞ satisfies

K̃ = K,
L̃ = {g_{∗}(k)g_{∗}(Θ(h))h | k ∈ Z Z , h ∈ L}, and
Θ̃(g_{∗}(k)g_{∗}(Θ(h))h) = Θ(h) + k  (mod p),

(3.16)

where we fix representatives 0 ≤ Θ(h) < 1. In particular we obtain for k := 0, h := h_{∗}

u(1/n) = g_{∗}(1/n)h_{∗}u(0).  (3.17)

Proof: To prove (3.13), we first claim that

g_{♯}(t) := h^{−1}g_{∗}(Θ(h))^{−1}g_{∗}(k)^{−1}g_{∗}(t + Θ(h) + k)h  (3.18)

solves the same initial value problem (3.5) as g_{∗}(t) does. Then g_{♯} ≡ g_{∗}, of course. To prove the claim, differentiate (3.18) with an eye on H-equivariance (1.20) and 1-periodicity of a(t) :

\begin{aligned}ġ_{♯}(t) & = & h^{−1}g_{∗}(Θ(h))^{−1}g_{∗}(k)^{−1}ġ_{∗}(t + Θ(h) + k)h = \\ & = & g_{♯}(t)h^{−1}a(t + Θ(h) + k)h = \\ & = & g_{♯}(t)h^{−1}a(v(t + Θ(h)))h = \\ & = & g_{♯}(t)a(t). \end{aligned}  (3.19)

We now show that g_{♯}(t) satisfies the same initial condition g_{♯}(0) = id as g_{∗}(t), for any choice of h ∈ L. Consider the special case h = id, first. Then Θ(h) = 0 and g_{♯}(0) = id is trivial. In particular g_{♯}(t) = g_{∗}(t), in that case, proving

g_{∗}(t) = g_{∗}(k)^{−1}g_{∗}(t + k)  (3.20)

for all t ∈ IR, k ∈ Z Z . Now (3.20) with t := Θ(h) implies g_{♯}(0) = id, for all choices of h ∈ L. This proves g_{♯}(t) = g_{∗}(t) in (3.18).

Inserting h := h_{0} ∈ K = ker Θ and k := 0 into (3.18) with g_{♯} = g_{∗}, g_{∗}(0) = id, we immediately see that g_{∗}(t) and h_{0} commute. Together with (3.18), g_{♯} = g_{∗}, this proves (3.13).

The choice t = 1 in (3.20) yields

g_{∗}(1 + k) = g_{∗}(k)g_{∗}(1),  (3.21)

whence g_{∗}(k) = g_{∗}(1)^{k}, for all k ∈ Z Z . This also follows directly, because multiplication by g_{∗}(1) is the time 1 stroboscope map for the nonautonomous equation ġ = ga(t) with time period 1.

Inserting k := 0, h := h_{∗} with Θ(h_{∗}) = 1/n in (3.13) yields

g_{∗}(t + 1/n) = g_{∗}(1/n)h_{∗}g_{∗}(t)h_{∗}^{−1}.  (3.22)

An n-fold iteration of (3.22), evaluated at t = 0, yields the expression for g_{∗}(1) in (3.14). Together with (3.21), this proves (3.14).

Similarly, a k-fold iteration of (3.22) for k = 0, 1, ⋯, n − 1 proves (3.15). Since h_{∗}^{−n} ∈ 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_{∗}(Θ(h))h, h ∈ L.

To prove (3.16), let g ∈ L̃. Then gu(t) = u(t + ϑ) for some real ϑ and all t ∈ IR. Upstairs, there exists h ∈ H such that

\begin{aligned}(g_{∗}(t + ϑ), v(t + ϑ)) & = & g(g_{∗}(t), v(t)) · h = \\ & = & (gg_{∗}(t)h^{−1}, hv(t)) \end{aligned}  (3.23)

for some, and hence all, real t. Comparing the second components we see that h ∈ L and there exists a unique k ∈ Z Z such that

ϑ = Θ(h) + k,  (3.24)

if we fix representatives 0 ≤ Θ(h) < 1. Comparing the first components, in view of (3.13), (3.24), we find

g_{∗}(k)g_{∗}(Θ(h))h = g,  (3.25)

after cancellation of g_{∗}(t)h^{−1}. Conversely, any such g lies in L̃, by (3.13), (3.23), (3.24). Letting Θ̃(g) = ϑ (mod p), it only remains to prove K̃ = K.

Note that g ∈ K̃ = ker Θ̃ if, and only if, (3.23) holds with ϑ = 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 ∈ K. Hence K̃ = K, and the proof is complete. □

The simple fact K̃ = K, in our notation, implies that the isotropy groups occurring in the tube U are precisely the conjugates gKg^{−1}, g ∈ 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 L̃ of u(t), given in (3.16), is a group, and Θ̃ : L̃ → Σ is a group homomorphism. For suitable H-equivariant 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 significant restriction on a(t), 0 ≤ t < 1/n, thus leaving sufficient freedom to prescribe a path g_{∗}(t) ∈ 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, Θ) in equivariant Hopf bifurcation certainly deserve further investigation.

4 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 affine action on IR^{N}; see (1.1)–(1.4).

For computations involving the Lie algebras se(N) it is convenient to represent (R, S) ∈ SE(N) = SO(N) × IR^{N} isomorphically as an element in SL(N + 1),

(R, S) ↦ ( \begin{matrix}R & S \\ 0 & 1 \end{matrix} ) ,  (4.1)

in block matrix notation. With this identification, an element (r, s) of the Lie algebra se(N) becomes the (N + 1) × (N × 1) matrix

(r, s) ↦ ( \begin{matrix}r & s \\ 0 & 0 \end{matrix} ) .  (4.2)

In particular, conjugation, iterates, the exponential map exp, the adjoint representation Ad of E(N) on se(N), and the commutator [·, ·] are given by

(R, S)(R′, S′)(R, S)^{−1} = (RR′R^{−1}, (id − RR′R^{−1})S + RS′);
(R, S)^{n} = (R^{n}, (id + R + ⋯ + R^{n−1})S);
(r, s)^{n} = (r^{n}, r^{n−1}s);
exp (r, s) = (exp (r), r^{−1}(exp (r) − id)s);
(R, S)(r, s) = (Rr, Rs);
(R, S)(r, s)(R, S)^{−1} = (RrR^{−1}, −RrR^{−1}S + Rs);
[(r, s), (r′, s′)] = ([r, r′], rs′ − r′s).

(4.3)

The notation in (4.3) is concise, but somewhat tricky. The first/last two relations hold in the group/algebra, respectively. Similarly, exp(r, s) is in the group. The expressions for (r, s)^{n}, (R, S)(r, s) are neither in the group nor in the algebra, in general, and are evaluated in SL(N + 1) in the sense of (4.1), (4.2). Similarly, all equations of (4.3) are easily checked in SL(N + 1).

In the settings of theorems 1.1, 3.1, 3.2, the compact isotropy subgroup H of G was playing a central role. We determine the compact subgroups H of E(N) next. We use the equivariant projection

p : E(N) = O(N) × IR^{N} → O(N)  (4.4)

onto the first 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 fixed translation S_{0} ∈ IR^{N} :

H = (id, S_{0}) p(H) (id, −S_{0})  (4.5)

Proof: We will first prove that there exists a map

σ : p(H) → IR^{N}  (4.6)

such that H has the form

H = p(H)^{σ} := {(R, σ(R)) | R ∈ p(H)}.  (4.7)

In a second step, we identify a fixed S_{0} ∈ IR^{N} such that

σ(R) = (id − R)S_{0},  (4.8)

for all R ∈ p(H). Then (4.5) is proved.

To construct σ, let (R, S), (R, S′) ∈ H possess the same projection R ∈ p(H). Then, for any integer n,

H ∋ ((R, S)(R, S′)^{−1})^{n} = (id, n(S − S′)).  (4.9)

Since H is compact, this implies S′ = S and σ := S is well-defined. This proves (4.6), (4.7).

For the second step note that σ is at least continuous. Indeed, H is compact and the bijection p : H → p(H) is continuous, with inverse determined by σ. Therefore p is a homeomorphism, and σ is continuous.

Multiplying (R, S), (R′, S′) in H yields the functional equation

σ(RR′) = σ(R) + Rσ(R′).  (4.10)

Note continuous dependence on R′. We integrate (4.10) over R′ with respect to the left invariant Haar measure on the compact Lie group p(H). With the abbreviation

S_{0} := ∫_{p(H)} σ(R′)dR′,  (4.11)

we obtain

\begin{aligned}σ(R) & = & ∫ σ(R)dR′ = ∫ σ(RR′)dR′ − ∫ Rσ(R′)dR′ = \\ & = & (id − R)S_{0}. \end{aligned}  (4.12)

This proves the lemma. □

The lemma holds, more generally, for any compact subgroup H of the general affine 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 v̇ = φ(v) in the skew product, with period 1, and with associated fundamental solution g_{∗}(t) of (3.5). We derive a criterion to decide whether the projected solution u(t) = τ̄(g_{∗}(t), v(t)) is meandering or drifting, in the sense of definition 1.2. Lemma 4.2 Let G = SE(N) or E(N). Consider v of period 1, and g_{∗}, u as above. Assume u(t) is neither stationary nor periodic. Let

(R_{∗}, S_{∗}) := g_{∗}(1).  (4.13)

Then u(t) is meandering, if S_{∗} is orthogonal to the fix space of the rotation R_{∗} ∈ SO(N) in IR^{N}, that is,

S_{∗} ⊥ ker(id − R_{∗}) =: (IR^{N})^{R_{∗}}.  (4.14)

If, on the other hand, (4.14) does not hold, then u(t) is drifting. Proof: By definition 1.2, the nonstationary, nonperiodic solution u(t) is meandering if the orbit g_{∗}(t), t ∈ IR, is relatively compact, and drifting otherwise. By theorem 3.2 and the differential equation (3.5) for g_{∗}(t), this orbit is relatively compact if, and only if,

H′ := clos{g_{∗}(k) | k ∈ Z Z }
= clos{g_{∗}(1)^{k} | k ∈ Z Z } (4.15)

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 ∈ IN. By lemma 4.1, the group H′ is compact if, and only if, it can be conjugated to its projection p(H′) ⊆ (S)O(N), by a pure translation S_{0} ∈ 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})  (4.16)

vanishes. Using orthogonality of R_{∗}, this is equivalent to

S_{∗} ∈ image(id − R_{∗}) = ker(id − R_{∗})^{⊥},  (4.17)

proving claim (4.14), and the lemma. □

We note a dichotomy with respect to dimension N, here, which was also observed by [AM96]. For even N, we have (IR^{N})^{R_{∗}} = {0}, 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) ∈ V possesses spatio-temporal symmetry (L, K, Θ) 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 fixed under K, that is

S_{∗} ∈ (IR^{N})^{K}.  (4.18)

In particular, drifting is excluded if

(IR^{N})^{K}⊥(IR^{N})^{R_{∗}}.  (4.19)

Most trivially, of course, condition (4.19) holds if (IR^{N})^{K} = {0} or (IR^{N})^{R_{∗}} = {0}. Proof: Lemma 4.2 and (4.18) imply claim (4.19). To prove (4.18), we let h_{0} ∈ K ≤ O(N). Since h_{0} and g_{∗}(1) commute, by theorem 3.2, (3.13), this implies

\begin{aligned}(R_{∗}, S_{∗}) & = & g_{∗}(1) = h_{0}g_{∗}(1)h_{0}^{−1} = \\ & = & (h_{0}R_{∗}h_{0}^{−1}, h_{0}S_{∗}). \end{aligned}  (4.20)

Therefore h_{0}S_{∗} = S_{∗}, and the lemma is proved. □

The projected solution u(t) satisfies

\begin{aligned}u(k) & = & τ̄(g_{∗}(k), v(k)) = (g_{∗}(1))^{k}τ̄(id, v(0)) = \\ & = & g_{∗}(1)^{k}u(0) \end{aligned}  (4.21)

for all stroboscope times k ∈ 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_{∗})^{†} 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_{∗}). Define

S_{†} := (id − R_{∗})^{†}S_{∗}.  (4.22)

Lemma 4.4 As in the above setting, let S_{∗}⊥ ker(id − R_{∗}). Then g_{∗}(1)^{k}, k ∈ Z Z , are all conjugate to the rotations (R_{∗}^{k}, 0) around the origin, by the fixed translation S_{†} :

\begin{aligned}g_{∗}(1)^{k} & = & (id, S_{†})(R_{∗}^{k}, 0)(id, −S_{†}) \\ & = & (R_{∗}^{k}, S_{†} − R_{∗}^{k}S_{†}). \end{aligned}  (4.23)

In particular, the translation components S_{∗}^{k} ∈ IR^{N} of (g_{∗}(1))^{k} all lie on a sphere around S_{†} ∈ IR^{N} with radius |S_{†}|_{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_{∗})^{†}S_{∗} = (id − R_{∗}^{k})S_{†}
= S_{†} − R_{∗}^{k}S_{†}.

(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. □

The radius |S_{†}|_{2} defined 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_{†} with radius |S_{†}|_{2}, we can call the Euclidean length |S_{†}|_{2} the stroboscope radius of u(t). In section 5, (5.7) we will see how |S_{†}|_{2} → ∞, when a planar meandering spiral passes through a drift resonance, for which S_{∗} ≠ 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 |S_{†}|_{2} depends on our choice of t = 0 as a reference point within the period of v. Indeed, other choices lead to expressions

S̃_{†} = R_{∗}(t)^{−1}(S_{†} − P_{∗}S_{∗}(t)),  (4.25)

0 ≤ t ≤ 1, replacing S_{†}, with correspondingly modified radii |S̃_{†}|_{2}. Here P_{∗} projects onto ker(id−R_{∗}), orthogonally. Note that (4.25) has period 1 in t, by definition (4.22) of S_{†}. Bounded modifications as in (4.25), however, do not affect the asymptotics of |S_{†}|_{2} → ∞, when passage through a drift resonance occurs.

□

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 fixed translation, without loss of generality. As in section 3, we consider H-equivariant Hopf bifurcation for v̇ = φ(λ, v) in the slice v ∈ V of our skew product (3.1). Let (L, K, Θ) 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) := τ̄(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 reflection.

Then u(t) cannot drift, in the sense of definition 1.2. Proof: Suppose K ≤ O(2) contains some nontrivial rotation. Then K fixes only the origin, in IR^{2}, that is (IR^{2})^{K} = {0}. By lemma 4.3, this excludes drifting. □

We look at meandering and drifting for spatio-temporal symmetries (H, K, Θ) of v(·) next. Throughout, we identify IR^{2} = C I and write (R, S) ∈ SE(2) in complex notation:

R = e^{2πiα}, α ∈ IR/ Z Z , S ∈ C I .  (5.1)

We consider solutions v(·) with spatio-temporal symmetry (H, K, Θ) given by

H = Z Z_{n} = {e^{2πik/n} | k = 0, …, n − 1}, n ≥ 2,
K = {1},
Θ(e^{2πik/n}) = mk/n ∈ S^{1} = IR/ Z Z , k = 0, …, n − 1.

(5.2)

We require K = {1}, to give drifting a chance. Note that this is equivalent to requiring the integer m ∈ {1, …, n − 1} to be relatively prime to n.

Meandering, meander radii, drifting, and drift resonance will follow from theorem 3.2 and lemma 4.4. We will express all these effects in terms of the fractional stroboscope map

g_{∗}(1/n) = (exp(2πiα_{1/n}), S_{1/n}) .  (5.3)

Also, we have to choose h_{∗} ∈ H such that Θ(h_{∗}) = 1/n generates image(Θ). Of course, we have to choose

h_{∗} = exp(2πim′/n), where
m′m ≡ 1  (mod n). (5.4)

In other words, m′ is the unique multiplicative inverse of m, mod n. Corollary 5.2 With the above notation, the stroboscope map g_{∗}(1) is given explicitly by

g^{∗}(1) = (exp(2πinα_{1/n}), (∑_{k=0}^{n−1} exp(2πik(α_{1/n} + m′/n))) S_{1/n}) .  (5.5)

The solution u(t) = τ̄(g_{∗}(t), v(t)) satisfies

u (1/n) = (exp(2πi(α_{1/n} + m′/n)), S_{1/n})u(0).  (5.6)

In particular, the solution u(t) is

(i) periodic,  if S_{1/n} = 0 and α_{1/n} ∈ Q I ;
(ii) periodic,  if α_{1/n} + m′/n ∉ Z Z and α_{1/n} ∈ Q I ;
(iii) meandering, if (α_{1/n} + m′/n) ∉ Z Z and α_{1/n} ∉ Q I ;
(iv) drifting,  if α_{1/n} + m′/n ∈ Z Z and S_{1/n} ≠ 0.

In case (iii), the meandering stroboscope radius r is given explicitly by

r = \frac{1}{2} | sin((α_{1/n} + m′/n)π)|^{−1} · |S_{1/n}|_{2}.  (5.7)

Proof: By theorem 3.2, (3.14), we compute the stroboscope map g_{∗}(1) as

\begin{matrix}(exp(2πiα_{∗}), S_{∗}) := g_{∗}(1) = (g_{∗}(1/n)h_{∗})^{n}h_{∗}^{−n} = \\ = (exp(2πi(α_{1/n} + m′/n)), S_{1/n})^{n}. \end{matrix}  (5.8)

In particular (4.3) implies

\begin{aligned}α_{∗} & = & nα_{1/n}  (mod 1) \\ S_{∗} & = & (∑_{k=0}^{n−1} exp(2πik(α_{1/n} + m′/n))) S_{1/n}. \end{aligned}  (5.9)

This proves (5.5) and case (i).

We now have to distinguish two cases. If α_{1/n} + m′/n is integer, then

\begin{aligned}α_{∗} & = & 0  (mod 1), \\ S_{∗} & = & nS_{1/n}, \end{aligned}  (5.10)

proving case (iv). If, on the other hand, α_{1/n} +m′/n ∉ Z Z , then we can easily compute

S_{∗} = \frac{exp(2πiα_{∗}) − 1}{exp(2πi(α_{1/n} + m′/n)) − 1} S_{1/n}.  (5.11)

In view of formula (4.3) for the iterates g_{∗}(n) = (g_{∗}(1))^{n} = (exp(2πiα_{∗}), S_{∗})^{n}, this proves cases (ii), (iii).

In case (iii), we can apply lemma 4.4 to compute the meandering radius r, because ker(id − R_{∗}) = {0} for α_{∗} = nα_{1/n} ∉ Z Z . Therefore (5.11) implies

r = |S_{†}|_{2} = |(id − R_{∗})^{−1}S_{∗}|_{2} =
= |S_{1/n}|_{2}/| exp(2πi(α_{1/n} + m′/n)) − 1| =
= \frac{1}{2} |S_{1/n}|_{2}/| sin((α_{1/n} + m′/n)π)|.

(5.12)

This proves the corollary. □

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(iω_{rot}t)u_{0}(0) with isotropy H = Z Z_{n} and minimal period T_{rot} = 2π/(nω_{rot}). Assume an additional pair ±2πi of imaginary eigenvalues of the linearization (in rotating coordinates). Then we can parameterize

v(t) = ϵe^{2πit} + O(ϵ^{2})  (5.13)

at parameter λ = λ_{0} + λ_{2}ϵ^{2} + O(ϵ^{3}). The equation for g_{∗}(t) becomes

ġ_{∗} = 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(2πiα(t)) of g_{∗}(t). Inserting the v-expansion (5.13) we obtain

2πα̇(t) = ω_{rot} + ⋯, α(0) = 0,  (5.15)

omitting time dependent terms of order ϵ. Note that, indeed, ω_{rot} is the rotation frequency of the rotating spiral u_{0}(t). Solving (5.15), up to terms of order ϵ, we get for g_{∗}(1/n) = (α_{1/n}, S_{1/n})

α_{1/n} = α(1/n) = \frac{ω_{rot}}{2πn} + · · · .  (5.16)

Letting 2π = ω_{Hopf} denote the (normalized) frequency of the nontrivial Hopf eigenvalues, the transition to the drift case (iv) occurs, for example, at

nα_{1/n} + m′ = \frac{ω_{rot}}{ω_{Hopf}} + m′ ≡ 0 (mod n).  (5.17)

From (5.7) we see how the meandering stroboscope radius blows up, at this resonance, provided S_{1/n} ≠ 0.

To analyze S_{1/n} in more detail, we write the differential equation for the component S(t) of g_{∗}(t) as

Ṡ(t) = e^{2πiα(t)} · ξ(v(t)).  (5.18)

In view of the v-expansion (5.13), we can restrict our attention to the case v ∈ C I . Note that the spatio-temporal symmetry

v(t + Θ(h)) = h · v(t)  (5.19)

then forces h ∈ H = Z Z_{n} ⊆ C I to act on v as complex multiplication by h^{m}, to be consistent with (5.2) and (5.13). Writing the H-action in this complex notation, equivariance condition (1.20) becomes

ξ(h^{m}v) = hξ(v).  (5.20)

Expanding, as far as necessary, by

ξ(v) = ∑_{k,l=0}^{∞} ξ_{kl} v^{k} v̄^{l},  (5.21)

we see that ξ_{kl} = 0, unless

(k − l)m ≡ 1  (mod n).  (5.22)

Requiring m coprime to n, still, this yields

k ≡ l + m′  (mod n)  (5.23)

with the mod n multiplicative inverse m′ of m. The terms of leading order are ξ_{m′,0}v^{m′}, if 0 < m′ ≤ n/2, and ξ_{0,n−m′}v̄^{n−m′}, in case n/2 ≤ m′ < n. Integrating the Ṡ equation, up to higher order in ϵ, yields

S(1/n) = ϵ^{m′} \frac{ξ_{m′,0}}{ω_{rot} + 2πm′} (e^{i(ω_{rot}+2πm′)/n} − 1) ≠ 0.  (5.24)

for ϵ, ξ_{m′,0} ≠ 0, in case 0 < m′ < n/2. The case n/2 < m′ < n reads

S(1/n) = ϵ^{n−m′} \frac{ξ_{0,n−m′}}{ω_{rot} + 2π(m′ − n)} (e^{i(ω_{rot}+2πm′)/n} − 1) ≠ 0.  (5.25)

for ϵ, ξ_{0,n−m′} ≠ 0. For m′ = n/2, the coefficients of ξ_{m′,0} and ξ_{0,n−m′} add. Most notably, we see a stroboscopic radius of meandering r proportional to higher powers of ϵ, in these cases; see (5.7). A similar calculation for H = {id}, n = 1, yields r proportional to ϵ.

6 Meandering and drifting in three dimensions: twisted scroll rings

Let G = SE(3), in this section. We first consider a primary wave u_{0}(t) with trivial isotropy H = {id}. 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 filament 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 difference along the family of normal planes. For simplicity, we assume that phase difference to equal the angle difference 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} = ( \begin{matrix}iω_{0} & 0 \\ 0 & 0 \end{matrix} ) , s_{0} = ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} ) .  (6.2)

We use complex notation in the horizontal (x, y)-plane, here, writing IR^{3} = C I × IR. We assume ω_{0} ≠ 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

\begin{aligned}ġ_{∗} & = & g_{∗}a(v), g_{∗}(0) = id, \\ v̇ & = & φ(v), \end{aligned}  (6.3)

describes the flow in a neighborhood U of G · u_{0}. We consider a family of periodic solutions v = v(ϵ, t) of period normalized to 1, bifurcating from the trivial solution v ≡ u_{0}. The parameter λ, so necessary for such a Hopf bifurcation, is suppressed. Instead, we represent dependence of a(v) on v = v(ϵ, t) by a differentiable function

a(ϵ, t) := a(v(ϵ, 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(ϵ, ·) has (normalized) period 1 for ϵ > 0.

As in any differential equation, we can differentiate the solution g_{∗} = g_{∗}(ϵ, t) with respect to ϵ. Writing

\begin{aligned}γ(ϵ, t) & := & (∂_{ϵ}g_{∗})g_{∗}^{−1} \\ η(ϵ, t) & := & g_{∗}^{−1}∂_{ϵ}g_{∗} \end{aligned}  (6.6)

with γ, η ∈ alg(G), the differential equations for γ, η, respectively, are

\begin{aligned}γ̇ & = & g_{∗}(∂_{ϵ}a)g_{∗}^{−1}, \\ η̇ & = & [η, a] + ∂_{ϵ}a, \end{aligned}  (6.7)

with initial conditions γ = η = 0 at t = 0 and ˙ = ∂_{t}. For example, at ϵ = 0 and t = 1, the derivative of the stroboscope map g_{∗} with respect to ϵ becomes

∂_{ϵ}g_{∗} = γ · g_{∗} = ∫_{0}^{1} exp(a_{0}t′)∂_{ϵ}a(t′) exp(−a_{0}t′)dt′ g_{∗},  (6.8)

because g_{∗}(0, t) = exp(a_{0}t).

What are the effects of this ϵ-expansion on the dynamics, alias on the iterates of the stroboscope map g_{∗}(ϵ, 1) = (R(ϵ), S(ϵ))? At ϵ = 0, we have

\begin{aligned}R(0) & = & ( \begin{matrix}exp(iω_{0}) & 0 \\ 0 & 1 \end{matrix} ) , \\ S(0) & = & ( \begin{matrix}0 \\ 1 \end{matrix} ) ∈ C I × IR. \end{aligned}  (6.9)

For small positive ϵ, by (6.8), we get a rotation axis of R(ϵ) near the z-axis, tilted by an angle proportionally to ϵ. Conjugating by a small rotation around a horizontal axis orthogonal to that angle, we can assume

R(ϵ) = ( \begin{matrix}exp(iω) & 0 \\ 0 & 1 \end{matrix} ) ,  (6.10)

with ω = ω(ϵ) near ω_{0}. Conjugating by yet another rotation around the z-axis, afterwards, we can assume

S(ϵ) = ( \begin{matrix}σ(ϵ) \\ 1 + s(ϵ) \end{matrix} ) ,  (6.11)

with small complex σ and small real s. Now we can iterate the stroboscope map g_{∗}(ϵ, 1) = (R, S). Using (3.14) and (4.3),

g_{∗}(ϵ, n) = (R_{n}, S_{n}) = (R, S)^{n} = (R^{n}, ∑_{k=0}^{n−1} R^{k}S).  (6.12)

With (6.10) we obtain the rotation

R_{n} = ( \begin{matrix}exp(iωn) & 0 \\ 0 & 1 \end{matrix} ) .  (6.13)

Similarly, the translation S_{n} = (σ_{n}, n + sn) is given by

σ_{n} = (∑_{k=0}^{n−1} e^{iωk})σ.  (6.14)

Summarizing, the propagation speed of our original twisted scroll ring u_{0} experiences periodic fluctuations, due to v(t). The period near 1 has been scaled to 1, 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 modified 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 = \frac{1}{2}| sin(πω_{rot}/ω_{Hopf})|^{−1} · |σ|,  (6.15)

as has been investigated in section 5. (We have returned to the notation ω_{rot} = ω, ω_{Hopf} ≈ 2π used there). Typically, |σ| will be of order ϵ. Note the horizontal drift resonance which occurs at integer values

ω_{rot}/ω_{Hopf} ∈ 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. Specifically, let (H, K, Θ) 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_{∗} ∈ SO(3) \ {id} commutes with K, by (3.13) and lemma 4.3, and hence R_{∗} and K fix 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 = {id} 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} = ( \begin{matrix}exp(iα_{1/n}) & 0 \\ 0 & 1 \end{matrix} ) .  (6.17)

Then S_{1/n} possesses a rather irrelevant vertical component, which only modifies the vertical propagation speed. The important horizontal component, however, produces periodicity, meandering, and drifting phenomena transversely to the propagation direction. Note how the ϵ-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 flow could lead to destabilization of the straight flight 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 fixes 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).

References

[Bar94] D. Barkley. Euclidean symmetry and the dynamics of rotating spiral waves. Phys. Rev. Lett., 72, 164–167, 1994.

[BE93] M. Braune and H. Engel. Compound rotation of spiral waves in a lightsensitive Belousov-Zhabotinsky medium. Chem. Phys. Lett., 204, 257– 264, 1993.

[BHN96] V. A. Biktashev, A. V. Holden, and E. V. Nikolaev. Spiral wave meander and symmetry of the plane. Preprint, University of Leeds, 1996.

[Bre72] G. E. Bredon. Introduction to Compact Transformation Groups. Academic Press, New York, 1972.

[BtD85] T. Bröcker and T. tom Dieck. Representations of Compact Lie Groups. Springer-Verlag, New York, 1985.

[Con78] C. Conley. Isolated Invariant Sets and the Morse Index. Conf. Board Math. Sci. 38, AMS, Providence, 1978.

[Die72] J. Dieudonné. Treatise on Analysis 3. Academic Press, New York, 1972.

[Fie88] B. Fiedler. Global Bifurcation of Periodic Solutions with Symmetry. Lect. Notes Math. 1309, Springer-Verlag, Berlin, 1988.

[Fie95] B. Fiedler. Meandering Spirals. Lecture at Workshop Dynamics and Symmetry, D. Chillingworth (org.), Isaac Newton Institute, Cambridge, 1995.

[GLM96] M. Golubitsky, V. LeBlanc, and I. Melbourne. Meandering of the spiral tip – an alternative approach. Preliminary version, Houston, 1996.

[GSS88] M. Golubitsky, I. Stewart, and D. Schaeffer. Singularities and Groups in Bifurcation Theory 2. Springer-Verlag, New York, 1988.

[Hel62] S. Helgason. Differential Geometry and Symmetric Spaces. Academic Press, New York, 1962.

[JSW89] W. Jahnke, W.E. Skaggs, and A.T. Winfree. Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and in the two-variable Oregonator model. J. Chem. Phys., 93, 740–749, 1989.

[Kru90] M. Krupa. Bifurcations of relative equilibria. SIAM J. Math. Analysis, 21, 1453–1486, 1990.

[LM96] N. E. Leonard and J. E. Marsden. Stability and drift of underwater vehicle dynamics: Mechanical systems with rigid motion symmetry. Preprint, 1996.

[NvORE93] S. Nettesheim, A. von Oertzen, H.H. Rotermund, and G. Ertl. Reaction diffusion patterns in the catalytic CO-oxidation on Pt(110) – front propagation and spiral waves. J. Chem. Phys., 98, 9977–9985, 1993.

[Pal61] R. S. Palais. On the existence of slices for actions of non-compact Lie groups. Ann. of Math., 73, 295–323, 1961.

[PW85] A.V. Panfilov and A. T. Winfree. Dynamical simulations of twisted scroll rings in three dimensional excitable media. Physica D, 17, 323–330, 1985.

[Pos92] G. Pospiech. Eigenschaften, Existenz und Stabilität von travelling wave Lösungen zu einem System von Reaktions-Diffusions-Gleichungen. Dissertation, Heidelberg, 1992.

[Ran82] D. Rand. Dynamics and symmetry: Predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Analysis, 79, 1–38, 1982.

[Ren82] M. Renardy. Bifurcation from rotating waves. Arch. Rat. Mech. Analysis, 79, 49–84, 1982.

[SSW96a] B. Sandstede, A. Scheel, and C. Wulff. Center-manifold reduction for spiral waves. C.R. Acad. Sc., Série I, to appear.

[SSW96b] B. Sandstede, A. Scheel, and C. Wulff. Dynamics of spiral waves in unbounded domains using center-manifold reductions. In preparation, 1996.

[tD91] T. tom Dieck. Topologie. Walter de Gruyter, Berlin, 1991.

[UNUM93] Zs. Ungvarai-Nagy, J. Ungvarai, and S.C. Müller. Complexity in spiral wave dynamics. Chaos, 3(1), 15–19, 1993.

[Win73] A. T. Winfree. Scroll-shaped waves of chemical activity in three dimensions. Science, 181, 937–939, 1973.

[Wul96] C. Wulff. Theory of meandering and drifting spiral waves in reactiondiffusion systems. Dissertation, Berlin, 1996.

[Reference added in proof:]

[AM96] P. Ashwin and I. Melbourne. Noncompact drift for relative equilibria and relative periodic orbits. Preprint, University of Houston, 1996.

Bernold Fiedler

Institut für Mathematik I Freie Universität Berlin Arnimallee 2-6 D-14195 Berlin, Germany fiedler@fu-berlin.de Björn Sandstede Div. of Applied Mathematics Brown University Providence, RI 02912, USA

Arnd Scheel

Institut für Mathematik I Freie Universität Berlin Arnimallee 2-6 D-14195 Berlin, Germany Claudia Wulff Institut für Mathematik I Freie Universität Berlin Arnimallee 2-6 D-14195 Berlin, Germany 506

Documenta Mathematica 1 (1996)