Transitions from Relative Equilibria

to Relative Periodic Orbits

Claudia Wulff

Received: June 1, 1998 Revised: February 8, 2000

Communicated by Bernold Fiedler

Abstract. We consider G-equivariant semilinear parabolic equations where G is a finite-dimensional possibly non-compact symmetry group. We treat periodic forcing of relative equilibria and resonant periodic forcing of relative periodic orbits as well as Hopf bifurcation from relative equilibria to relative periodic orbits using Lyapunov-Schmidt reduction. Our main interest are drift phenomena caused by resonance. In comparison to a center manifold approach Lyapunov-Schmidt reduction is technically easier. We discuss impacts of our results on spiral wave dynamics.

2000 Mathematics Subject Classification: 35B32, 35K57, 57S20. Keywords and Phrases: spiral waves, equivariant dynamical systems, noncompact groups.

1 Introduction

1.1 Spiral wave dynamics

Relative equilibria and relative periodic solutions are ubiquitous in systems with continuous symmetry. Examples of relative equilibria and relative periodic solutions are spiral waves. Spiral waves have been observed in various chemical and biological systems, for example in the Belousov-Zhabotinsky reaction [5], [26], [35], and in catalysis on platinum surfaces [16]. The spiral tip of a rigidly rotating spiral wave moves on a circle. In mathematical terms rigidly rotating spiral waves are rotating waves. Rotating waves are stationary in a corotating frame and therefore examples of relative equilibria. Meandering spiral waves are modulated rotating waves, i.e., they are periodic in Figure 1: Meandering spiral wave in the Belousov Zhabotinsky reaction, from Steinbock et al. [27], with kind permission of Nature. The tip trajectory is overlaid with a white curve.

a corotating frame. In this case the spiral tip performs a quasiperiodic motion, which is called meandering, see Fig. 1. Meandering spiral waves are generated by external periodic forcing of rigidly rotating spiral waves [16]. Let ω_{ext} be the frequency of the external forcing and let μ_{ext} be its amplitude. If the periodic forcing is resonant, i.e., if the rotation frequency ω_{rot}^{∗} of the rigidly rotating wave at μ_{ext} = 0 is a multiple of the external frequency ω_{ext} of the system then a curve of drifting spiral waves in the (ω_{ext}, μ_{ext})-plane is observed which separates modulated rotating wave states with inward petals and outward petals, cf. [16]. This phenomenon is called resonance drift. Drifting spiral waves, see Fig. 2, are modulated

Figure 2: Drifting Spiral Waves in the CO-Oxidation on Pt(110), courtesy of [16]. The cross is always at the same position. So we see that the spiral wave drifts away from the cross. travelling waves, i.e., they are periodic in a comoving frame. Both, meandering and drifting spiral waves are examples of relative periodic orbits. In experiments also meandering spiral waves have been forced periodically [35]. Here invariant 3-tori are found and frequency locking between the period of the relative periodic orbits and the period of the external forcing occurs. Furthermore for certain external periods modulated travelling waves are generated. Experimentalists call this phenomenon generalized resonance drift [35].

Figure 3: Phase diagram for the spiral wave dynamics depending on the parameters a, b; courtesy of Barkley [4]. Shown are regions containing N: no spiral waves, RW: stable rigidly rotating waves, MRW: modulated rotating waves, MTW: modulated travelling waves (dashed curve). Spiral tip paths illustrate states at 6 points. Small portions of spiral waves are shown for the two rotating wave cases.

Meandering spiral waves can also emanate from rigidly rotating spiral waves by a spontaneous bifurcation in autonomous systems, see [26], [32]. Barkley found in numerical simulations [3], see Fig. 3, that this transition is a Hopf bifurcation in the corotating frame. Hopf-bifurcation in autonomous systems leads to analogous drifting phenomena as periodic forcing of rigidly rotating waves. The media in which spiral waves occur can be modelled by reaction-diffusion systems of the form

\frac{∂u_{i}}{∂t} = δ_{i}Δu_{i} + f_{i}(u, t, μ), i = 1, …, M.  (1.1)

Here u = (u_{1}, …, u_{M}) is a vector of concentrations of chemical species, the functions u_{i}, i = 1, …, M, map the plane ℝ^{2} to ℝ, the constants δ_{i} ≥ 0, i = 1, …, M, are diffusion coefficients, μ ∈ ℝ^{p} is a parameter, and the functions f_{i}, i = 1, …, M, are reaction-terms which are autonomous or time-periodic. Barkley [4] was the first to notice the importance of the Euclidean symmetry for spiral wave dynamics. The Euclidean group E(2) = O(2) ⋉ ℝ^{2} of rotations, translations and reflections on the plane acts on the functions u(x), x ∈ ℝ^{2}, via

(ρ_{(R,a)}u)(x) = u(R^{−1}(x − a)), where R ∈ O(2), a ∈ ℝ^{2}.  (1.2)

System (1.1) is equivariant with respect to the symmetry group E(2). In this article we want to study the transition from rigidly rotating to meandering spiral waves on the infinitely extended plane ℝ^{2}. More generally the aim of the paper is to understand the transition from relative equilibria to relative periodic orbits in equivariant systems. Furthermore we want to explain the drift and resonance effects which we just described for general symmetry groups. We will discuss implications of our results on spiral wave dynamics in the plane and on the sphere (for simulations of spiral waves on the sphere see [36]). Further we want to apply our results to the evolution of scroll-waves in three-dimensional excitable media. Scroll waves have been studied numerically for example in [15], [18].

1.2 Related literature

In the thesis [33] the first results on bifurcations from rotating waves in systems with a non-compact, non-commutative symmetry group have been obtained. This paper is based on the dissertation [33]; but whereas in [33] we restricted attention to the symmetry group E(2) and applications in spiral wave dynamics in this article we treat arbitrary symmetry groups. As in [33] we study the transition from relative equilibria to relative periodic orbits using Lyapunov-Schmidt reduction. Shortly after [33] was finished a whole bunch of papers on spiral wave dynamics and non-compact symmetry groups appeared: Golubitsky et al. [10] used a formal center-bundle construction to derive ordinary differential equations describing bifurcations near ℓ-armed planar spiral waves of autonomous reaction-diffusion systems and derived new conditions for drifting. In [1] the drift of relative equilibria and periodic orbits along their group orbit is analyzed for general non-compact groups. Fiedler et al. [7] clarified the structure of the autonomous ordinary differential equations near relative equilibria with compact isotropy for general non-compact groups and gave conditions for drifting. In [21], [22] we presented a center-manifold reduction near relative equilibria and derived rigorously the ordinary differential equations on the center-manifold which were already guessed in [4] and formally derived in [10]. In [23] we extended these results to relative periodic orbits. In [8] normal forms near relative equilibria of non-compact group actions are computed. In [34] bifurcations from relative periodic orbits are treated.

Scheel [24], [25] proved the existence of rotating waves in unbounded domains. The thesis [33] was inspired by work of Renardy on bifurcations from rotating waves [19]. Renardy also studied bifurcations from rotating waves of semilinear differential equations using Lyapunov-Schmidt reduction and applied his results to the Laser equations [20]. But his results for partial differential equations are restricted to compact symmetry groups.

1.3 Lyapunov-Schmidt-reduction versus center-manifold theory

To analyze bifurcations there are mainly two reduction methods: centermanifold reduction and Lyapunov-Schmidt reduction. Both have advantages and disadvantages. Here we will use Lyapunov-Schmidt reduction as tool for the analysis of bifurcations; for a center-manifold approach see [21], [22]. The advantage of Lyapunov-Schmidt reduction versus center-manifold theory is that we obtain C^{∞}-paths of relative periodic orbits if the nonlinearity in (1.1) is C^{∞} whereas we only obtain a C^{k}-smooth center-manifold, k < ∞. Besides this we do not need the assumptions that the group action is isometric and that the group orbit of the relative equilibrium is an embedded manifold which are necessary for the center-manifold reduction. Finally the proofs are simpler since they do not rely upon the highly developed invariant manifold machinery. On the other hand the Lyapunov-Schmidt method is limited to relative equilibria and relative periodic orbits – we cannot handle more complicated dynamics. But for our purposes this is sufficient.

1.4 Organization of the paper

The paper is organized as follows. First, in subsections 1.5 and 1.6 we study the functional-analytic framework of spiral wave dynamics and show some of the difficulties arising in the mathematical treatment of spiral waves. In subsection 1.7 we define an appropriate abstract setting which covers the reaction-diffusion system (1.1) modelling spiral wave dynamics. In this abstract setting we henceforth work. In section 2 we study periodic forcing of relative equilibria and relative periodic orbits. First, in subsection 2.1 we consider periodic forcing of relative equilibria and resonance drift. In subsection 2.3 we study the scaling of the drift velocity. As example we consider periodic forcing of rotating waves in E(2)-equivariant systems which lead to modulated rotating waves or, in the resonance case, to modulated travelling waves. This explains the experiments described in subsection 1.1. In subsection 2.4 we consider resonant periodic forcing of relative periodic orbits and discuss conditions for generalized resonance drift. The results apply to periodic forcing of meandering spiral waves as investigated experimentally by [35], see also subsection 1.1. In section 3 we discuss Hopf bifurcation from relative equilibria, resonances, scaling of drift velocity and effects of spatial isotropy of the relative equilibrium. As an example we study the Hopf bifurcation from multi-armed spiral waves. Section 4 is devoted to the proof of the main results.

1.5 Functional-analytic framework

To describe spiral wave dynamics we consider reaction-diffusion systems of the form (1.1) on a domain Ω ⊂ ℝ^{3} to ℝ, where Ω is a C^{∞}-manifold without boundary, for example ℝ^{2}, the unit sphere S^{2} in ℝ^{3} or ℝ^{3} itself. The reactionterms f_{i}, i = 1, …, M, are assumed to be C^{k}-smooth functions where k ∈ ℕ. The domain Ω is invariant under some subgroup G of the Euclidean group E(3) of motions in three-dimensional space consisting of rotations, reflections and translations. The group E(3) = O(3) ⋉ ℝ^{3} acts on the functions u(x), x ∈ ℝ^{3}, via (1.2), i.e.,

(ρ_{(R,a)}u)(x) = u(R^{−1}(x − a)), where R ∈ O(3), a ∈ ℝ^{3}.

System (1.1) is equivariant with respect to the group G. If G = E(2) is the Euclidean group of motions in the plane we write (ϕ, a) for (R_{ϕ}, a) where R_{ϕ} is a rotation with angle ϕ and a ∈ ℝ^{2}. We consider (1.1) in the space of bounded uniformly continuous functions X = BC_{unif}(Ω, ℝ^{M}) or in the space X = L^{2}(Ω, ℝ^{M}). In X = BC_{unif} we get a time-evolution Φ_{t,t_{0}} of (1.1) on Y = X; if X = L^{2} we obtain a time-evolution on Y = X^{α}, α > 1/2 without any growth conditions on f provided that f(0, t, μ) = 0 for all t, μ and δ_{i} > 0, i = 1, …, M. If δ_{i} = 0 for some i we still obtain a semiflow on X = H^{2} provided that f(0, t, μ) ≡ 0. Note that the group action is not smooth on the whole function space X. If the domain is Ω = ℝ^{2} and we choose X = BC_{unif}(ℝ^{2}, ℝ^{M}) then the E(2)-action is even not strongly continuous because on the function u(x_{1}, x_{2}) = cos x_{1} the rotation acts discontinuously: For large radius r the term |(ρ_{(ϕ,0)}u)(x) − u(x)| can become equal to 2 even for arbitrarily small ϕ. We encounter the same problem if Ω = ℝ^{3}. Since we want to have a strongly continuous group action on our base space X we consider the reaction-diffusion system (1.1) on a subspace of BC_{unif} which is invariant under the semiflow and where the group acts in a strongly continuous way: We define BC_{Eucl}(ℝ^{N}, ℝ^{M}) as the subspace of BC_{unif}(ℝ^{N}, ℝ^{M}) on which E(N) acts continuously, N = 2, 3. The Laplacian is sectorial on X = BC_{unif} and on L^{2}, see [13]. We will now show that the Laplacian is also sectorial on X = BC_{Eucl}(ℝ^{N}, ℝ^{M}): let Y be any Banach space with a group G acting on it by a (not necessarily strongly continuous) representation ρ_{g}, g ∈ G. Let Y_{0} be the subspace of Y on which G acts strongly continuously. If A is sectorial on Y and Aρ_{g} = ρ_{g}A for all g ∈ G then A is sectorial in Y_{0}: from ρ_{g}e^{−At} = e^{−At}ρ_{g} we deduce that (e^{−At})_{t≥0} is a C^{0}-semigroup from Y_{0} to Y_{0}; furthermore e^{−At}y is complex differentiable in t for y ∈ Y, t > 0, with derivative Ae^{−At}y ∈ Y. Since ρ_{g}Ae^{−At} = Ae^{−At}ρ_{g} and therefore Ae^{−At}Y_{0} ⊂ Y_{0} we conclude that (e^{−At})_{t≥0} is an analytic semigroup on Y_{0}. Since (λ − A)^{−1}u ∈ Y_{0} for u ∈ Y_{0}, λ ∈ ℂ, λ ∉ spec_{Y} (A), the spectrum of A on Y_{0} is contained in the spectrum of A on Y. Especially the Laplacian is sectorial on BC_{Eucl}, and its spectrum is contained in the spectrum of the Laplacian defined on BC_{unif}.

We also get a time-evolution of (1.1) in BC_{Eucl}(ℝ^{N}, ℝ^{M}) because we have ρ_{g}Φ_{t,t_{0}}(u) = Φ_{t,t_{0}}(ρ_{g}u) and therefore Φ_{t,t_{0}} maps Y_{0} into itself. Now we have a C^{0}-group action on X = BC_{Eucl}, but if Ω = ℝ^{2}, ℝ^{3} the semiflow does not smoothen the group-action even if all diffusion coefficients δ_{i} are positive. We demonstrate this for Ω = ℝ^{2} and for the heat equation where the nonlinearity f is zero. We will show that on ℝ^{2} the operator \frac{∂}{∂ϕ} is not bounded w.r.t. the Laplacian Δ and to the semiflow (e^{Δt})_{t≥0}:

Remark 1.1 The operator \frac{∂}{∂ϕ} is not bounded relatively to the Laplacian Δ or relatively to the semiflow e^{Δt}, t ≥ 0, on BC_{unif}(ℝ^{2}, ℝ) and BC_{Eucl}(ℝ^{2}, ℝ).

Proof. The functions w_{ℓ,b}(x) := J_{ℓ}(b|x|)e^{iℓarg(x)} where b ≥ 0 and J_{ℓ} is the ℓ-th Bessel function of the first kind are elements of BC_{Eucl}(ℝ^{2}, ℝ) ⊂ BC_{unif}(ℝ^{2}, ℝ) and they are eigenfunctions of the Laplacian Δ and of the angle derivative \frac{∂}{∂ϕ}:

\frac{∂}{∂ϕ}w_{ℓ,b} = iℓw_{ℓ,b}, Δw_{ℓ,b} = −b^{2}w_{ℓ,b}.

Since iℓ(1 + b^{2})^{−1} and iℓe^{−b^{2}t} are not bounded for arbitrary b ∈ ℝ, ℓ ∈ ℕ_{0}, we conclude that \frac{∂}{∂ϕ} is not bounded relatively to Δ on BC_{Eucl}(ℝ^{2}, ℝ), BC_{unif}(ℝ^{2}, ℝ) and that \frac{∂}{∂ϕ}e^{Δt} is not a bounded operator on BC_{Eucl}(ℝ^{2}, ℝ), BC_{unif}(ℝ^{2}, ℝ) for t ≥ 0.

Remark 1.2 Also on L^{2}(ℝ^{2}, ℝ) the angle-derivative \frac{∂}{∂ϕ} is not bounded relatively to Δ or e^{Δt}, t ≥ 0.

Proof. By direct computation we see that F(\frac{∂}{∂ϕ}u) = \frac{∂}{∂ϕ}F(u). Here F(u) denotes the Fourier transform of u. From this formula and from F(Δu)(x) = −|x|^{2}F(u)(x) we deduce that \frac{∂}{∂ϕ} is not bounded with respect to Δ. Furthermore the operator \frac{∂}{∂ϕ} is not bounded relatively to e^{Δt} in L^{2}(ℝ^{2}, ℝ) since

(F(\frac{∂}{∂ϕ}e^{Δt}u))(x) = \frac{∂}{∂ϕ}e^{−|x|^{2}t}(F(u))(x) is not defined for all u ∈ L^{2}(ℝ^{2}, ℝ). Therefore we cannot simply change coordinates into a corotating frame to deal with the meandering transition.

1.6 Representations of E(N)

The function spaces Y = BC_{Eucl}(ℝ^{N}, ℝ), L^{2}(ℝ^{N}, ℝ), N = 2, 3, do not contain finite-dimensional subspaces which are E(N)-invariant and in which the E(N)- action is non-trivial. Again we will demonstrate this in the case Ω = ℝ^{2}, G = E(2):

Lemma 1.3 Let the action of E(2) on the spaces X = BC_{Eucl}(ℝ^{2}, ℝ), X = L^{2}(ℝ^{2}, ℝ) be given by (1.2). Then the function spaces BC_{Eucl}, L^{2} do not contain finite-dimensional E(2)-invariant subspaces with nontrivial E(2)-action.

In Greenleaf [12] a general theory on the action of topological groups on function spaces is developed. If we allow polynomial growth in our function space then the polynomials of degree ≤ j are finite-dimensional representations of E(2). Proof of Lemma 1.3. Let V_{j} = span(e_{1}, …, e_{j}) be a j-dimensional representation of E(2) in BC_{unif} or L^{2}. Then the translations act as a C^{0}-group of isometries on V_{j} since they act in such a way on BC_{unif}, L^{2}. Since V_{j} is finite-dimensional, we know that ρ_{(0,(a_{1},a_{2}))}e_{i} = ∑_{i=1}^{j}(e^{η_{1}a_{1}+η_{2}a_{2}})_{ij}e_{j} where η_{1} = \frac{∂}{∂x_{1}} |_{V_{j}}, η_{2} = \frac{∂}{∂x_{2}} |_{V_{j}} are (j, j)-matrices. Since ρ_{(0,a)} is an isometry we conclude that Re spec(η_{1}) = Re spec(η_{2}) = 0 and that η_{1}, η_{2} do not contain Jordan blocks. After simultaneous diagonalization of η_{1}, η_{2} (note that [η_{1}, η_{2}] = 0) we see that the eigenfunctions of η_{1}, η_{2} are of the form e^{ibx}, b, x ∈ ℝ^{2}. These functions are not elements of X = L^{2}(ℝ^{2}, ℝ). So the proof is finished for the function space L^{2}. If we choose b = 0 we obtain an E(2)-invariant subspace of X = BC_{unif}(ℝ^{2}, ℝ) which consists of all constant functions. The E(2)-action on this space is trivial. The action of the rotation is not continuous on the functions e^{ibx}, b ≠ 0, with respect to the norm ∥ · ∥_{BC_{unif} (ℝ^{2},ℝ)}. Therefore the functions e^{ibx} do not span a finite-dimensional E(2)-invariant subspace of BC_{Eucl}(ℝ^{2}, ℝ) for b ≠ 0. Of course, the same considerations apply for x ∈ ℝ^{3}, G = E(3) instead of x ∈ ℝ^{2}, G = E(2). Especially for an E(2)-invariant steady state the eigenspace to each eigenvalue is E(2)-invariant and therefore infinite-dimensional. This makes the study of bifurcations from E(2)-invariant equilibria for an abstract equivariant parabolic equation very difficult. We will not attack this problem and rather study bifurcations from relative equilibria where these difficulties do not occur. Bifurcations from homogeneous steady states of reaction diffusion equations have been studied by Scheel [24], [25] using spatial dynamics.

1.7 Abstract Setting

In this paper we study semilinear parabolic equations

\frac{du}{dt} = −Au + f(u, ω_{ext}t, μ)  (1.3)

on some Banach space X which are equivariant under a m-dimensional Lie group G which may be non-compact. We assume that A is sectorial (for a definition see [13]) and that f is C^{k}-smooth from Y ×ℝ×ℝ^{p} to X. Here k ∈ ℕ or k = ∞, μ ∈ ℝ^{p} and Y = X^{α} for 0 ≤ α < 1. By [13] there exists a time-evolution Φ_{t,t_{0}}(·; μ) of (1.3) on Y, and Φ_{t,t_{0}}(u; μ) is C^{k}-smooth in u, μ for t ≥ t_{0} and in u, μ, t, t_{0} for t > t_{0}. We assume that the group G acts on Y by the linear strongly continuous representation ρ_{g} ∈ L(Y), g ∈ G and that (1.3) is G-equivariant, i.e.,

∀ g ∈ G ρ_{g}A = Aρ_{g}, f(ρ_{g}u, t, μ) = ρ_{g}f(u, t, μ)

This implies that ρ_{g}Φ_{t,t_{0}}(·; μ) = Φ_{t,t_{0}}(ρ_{g}·; μ) for all g ∈ G. Assume that f in (1.3) is time-independent. Then a group orbit Gu^{∗} is called a relative equilibrium of (1.3) if Φ_{t}(u^{∗}) = ρ_{exp(ξ^{∗}t)}u^{∗} for some ξ^{∗} ∈ alg(G). Here alg(G) denotes the Lie algebra of G. Sometimes we denote u^{∗} itself as relative equilibrium. A point u^{∗} lies on a relative periodic orbit

O^{∗} = {ρ_{g}Φ_{t,0}(u^{∗}) | g ∈ G, t ∈ ℝ}

if Φ_{T^{∗},0}(u^{∗}) = ρ_{g^{∗}}u^{∗} for some T^{∗} > 0, g^{∗} ∈ G. In this case we suppose that f(u, ω_{ext}t, μ) is independent of time or time-periodic with frequency ω_{ext} = 2πj/T^{∗}, j ∈ ℕ. Sometimes we sloppily denote u^{∗} itself as relative periodic orbit. We call T^{∗} the relative period of the relative periodic orbit. The aim of this article is to study transitions from relative equilibria to relative periodic orbits of (1.3).

2 Periodically forced G-equivariant systems

This section deals with the effects of periodic forcing on relative equilibria and relative periodic orbits. In particular, we will investigate drift phenomena caused by resonant periodic forcing. We will apply our results to spiral wave dynamics. This helps to understand the experiments mentioned in the introduction. Proofs of the main theorems are postponed to section 4. In this section we assume that the nonlinearity f of (1.3) is of the form

f(u, t, μ) = f̂(u, μ̂) + μ_{ext}f_{ext}(u, ω_{ext}t, μ).

Here f_{ext}(u, τ, μ) is 2π-periodic in τ; ω_{ext} is the frequency of the periodic forcing, T_{ext} = \frac{2π}{ω_{ext}} is its period, μ_{ext} is its amplitude and we decompose μ = (μ_{ext}, μ̂), where μ_{ext} ∈ ℝ, μ̂ ∈ ℝ^{p−1}. So we consider the periodically forced differential equation

\frac{du}{dt} = −Au + f̂(u, μ̂) + μ_{ext}f_{ext}(u, ω_{ext}t, μ).  (2.1)

A typical example of the abstract semilinear differential equation (2.1) is a periodically forced reaction-diffusion system on the domain Ω ⊂ ℝ^{N}, N = 2, 3, cf. (1.1):

\frac{∂u_{i}}{∂t} = δ_{i}Δu_{i} + f̂_{i}(u, μ̂) + μ_{ext}f_{ext,i}(u, ω_{ext}t, μ), i = 1, …, M.  (2.2)

2.1 Periodic forcing of relative equilibria

This subsection deals with effects of periodic forcing on relative equilibria. First we state two general theorems, then we study examples in spiral wave dynamics.

Consider system (2.1) without periodic forcing, i.e., at μ_{ext} = 0. Assume that u^{∗} is a relative equilibrium of the unforced system for the parameter μ̂ = μ̂^{∗} = 0. Then u^{∗} satisfies

Φ_{t}(u^{∗}) = ρ_{e^{tξ^{∗}}} u^{∗}

for some ξ^{∗} ∈ alg(G). Since Φ_{t}(·) is equivariant and C^{k}-smooth in t for t > 0 we conclude that e^{tξ^{∗}}u^{∗} is C^{k}-smooth in t for all t ∈ ℝ. We will write ξu for \frac{d}{dt}ρ_{e^{tξ}}u|_{t=0}. Furthermore denote by

Ad_{g} ξ := gξg^{−1} = \frac{d}{dt}(g exp(ξt)g^{−1})|_{t=0} ∈ alg(G)

the adjoint action of G on alg(G) and by

K = {g ∈ G | ρ_{g}u^{∗} = u^{∗}}

the isotropy group of u^{∗}. We assume that K is compact. Let G^{0} denote the identity component of G. We have ξ^{∗} ∈ alg(N(K)) where N(K) is the normalizer of the isotropy group K of u_{∗} because for g ∈ K, t ∈ ℝ,

ρ_{g}ρ_{exp(tξ^{∗})}u^{∗} = ρ_{g}Φ_{t}(u^{∗}) = Φ_{t}(ρ_{g}u^{∗}) = Φ_{t}(u^{∗}) = ρ_{exp(tξ^{∗})}u^{∗}

and therefore g exp(tξ^{∗}) ∈ exp(tξ^{∗})K. Similarly the pull-back element g^{∗} of a relative periodic orbit u^{∗} = ρ_{g^{∗}}^{−1}Φ_{T_{ext},0}(u^{∗}) lies in the normalizer of the isotropy K of u^{∗}. Actually for a relative equilibrium the drift velocity ξ^{∗} lies in the Lie algebra of the centralizer Z(K) of K, which follows from the formula N(K)^{0} = K^{0}Z(K)^{0}, see [9]. Since by periodic forcing isotropy is not changed we assume without loss of generality in the whole section that K = {id}. Otherwise we change the space Y to the fixed point space Fix(K) = {g ∈ G, ρ_{g}u^{∗} = u^{∗}} of K and the symmetry group G to N(K)/K. Let u^{∗} be a relative equilibrium, i.e., −Au^{∗} + f̂(u^{∗}) = ξ^{∗}u^{∗}, and let

L^{∗} = −A + D_{u}f̂(u^{∗}) − ξ^{∗}

be the linearization at the relative equilibrium in the comoving frame. Assume that ρ_{g}u^{∗} is C^{1} in g ∈ G. We compute that for ξ ∈ alg(G)

\begin{aligned}L^{∗}ξu^{∗} & = & (−A + D_{u}f̂(u^{∗}) − ξ^{∗})ξu^{∗} \\ & = & −ξAu^{∗} + D_{u}f̂(u^{∗})ξu^{∗} − ξ^{∗}ξu^{∗} \\ & = & ξ(−A + f̂(u^{∗})) − ξ^{∗}ξu^{∗} \\ & = & (ξξ^{∗} − ξ^{∗}ξ)u^{∗} \\ & = & [ξ, ξ^{∗}]u^{∗} = −ad_{ξ^{∗}}u^{∗}. \end{aligned}  (2.3)

Here [·, ·] denotes the commutator, ad_{ξ^{∗}}(ξ) = [ξ^{∗}, ξ] and we used that gf̂(u) = f̂(gu) and therefore D_{u}f̂(u^{∗})ξ = ξf̂(u). From (2.3) we see that L^{∗} maps T_{u^{∗}}Gu^{∗} = alg(G)u^{∗} into itself.

Example 2.1 Let u^{∗} be a rotating wave of the unforced system (2.1), e.g a rigidly rotating spiral wave of the reaction-diffusion system (2.2) on Ω = ℝ^{2} at μ_{ext} = 0. Then the symmetry group is G = E(2). We write g = (ϕ, a) ∈ SO(2) ⋉ ℝ^{2} = SE(2). Let ξ_{1} denote the generator of the rotation and ξ_{2}, ξ_{3} denote the generators of the translation. Then ξ^{∗} = ω_{rot}^{∗}ξ_{1} where ω_{rot}^{∗} is the rotation frequency of the spiral, and we compute

L^{∗}ξ_{1}u^{∗} = 0, L^{∗}(ξ_{2} + iξ_{3})u^{∗} = ω_{rot}^{∗}[ξ_{2} + iξ_{3}, ξ_{1}]u^{∗} = iω_{rot}^{∗}(ξ_{2} + iξ_{3})u^{∗}.

Therefore the linearization L^{∗} of the rotating wave in the rotating frame has always eigenvalues on the imaginary axis.

For a relative periodic orbit u^{∗} = ρ_{g^{∗}}^{−1}Φ_{T^{∗},0}(u^{∗}) with ρ_{g}u^{∗} C^{1} in g we get

ρ_{g^{∗}}^{−1}DΦ_{T^{∗},0}(u^{∗})ξu^{∗} = (Ad_{g^{∗}}^{−1} ξ)u^{∗}, ξ ∈ alg(G).

If u^{∗} is a relative equilibrium then the linearization of the time-T -map in the comoving frame ξ_{∗} is given by

e^{L^{∗}T} = ρ_{g^{∗}}^{−1}DΦ_{T} (u^{∗})

where g^{∗} = e^{T ξ^{∗}}. For the groups relevant in applications (compact and Euclidean groups) the eigenvalues of the linear maps [ξ, ·], ξ ∈ alg(G), on alg(G) are purely imaginary and similarly the spectrum of the maps Ad_{g}, g ∈ G, on alg(G) lies on the unit circle. We will restrict our attention to these groups in this article. So we make the overall hypothesis

Overall Hypothesis The spectra of the linear maps Ad_{g}, g ∈ G, are subsets of the unit circle {λ ∈ ℂ; |λ| = 1}.

Therefore in the case of continuous symmetry where alg(G) is nontrivial the linearization L^{∗} at a relative equilibrium always has eigenvalues on the imaginary axis and similarly the linearization ρ_{g^{∗}}^{−1}DΦ_{T} (u^{∗}) of a relative periodic orbit u^{∗} = ρ_{g^{∗}}^{−1}Φ_{T} (u^{∗}) of (2.1) has always center-eigenvalues on the unit circle. If u^{∗} is a relative equilibrium fix some T > 0. In the case of a relative periodic orbit take T = T^{∗}. We need the following assumption on the spectrum:

Hypothesis (S) The set {λ ∈ ℂ; |λ| ≥ 1} is a spectral set for the spectrum spec(B^{∗}) of the operator

B^{∗} := ρ_{g^{∗}}^{−1}DΦ_{T} (u^{∗}) ∈ L(Y)  (2.4)

(called center-unstable spectral set) with associated spectral projection P ∈ L(Y) and the corresponding generalized eigenspace E_{cu} := R(P) (the center-unstable eigenspace) is finite-dimensional.

We will show in Section 4 below that Hypothesis (S) implies that ρ_{g}u^{∗} is C^{k} in g. Let Gu^{∗} = {ρ_{g}u^{∗}; g ∈ G} denote the group orbit at u^{∗}. Frequently we employ the following notion:

Definition 2.2 We say that a relative periodic orbit or a relative equilibrium u^{∗} of (2.1) is non-critical if ρ_{g}u^{∗} is C^{1} in g and if the operator B^{∗} from (2.4) satisfies Hypothesis (S) and if the center-eigenspace

E_{c} = T_{u^{∗}}Gu^{∗} + span(∂_{t}Φ_{t}(u^{∗})|_{t=0})

only consists of eigenvectors which are forced by G-symmetry or time-shift symmetry (in the case of relative periodic orbits of autonomous systems).

Denote the dual space of Y by Y^{⋆}, let m = dim(G) and assume that ρ_{g}u^{∗} is C^{1} in g. Choose l_{i} ∈ Y^{⋆}, i = 1, …, m, such that the equations l_{i}(u − u^{∗}) = 0, , i = 1, …, m, define a section S_{l} = u^{∗} + Ŝ_{l} transverse to the group orbit Gu^{∗} of the relative equilibrium at u^{∗}. If u^{∗} is non-critical we can choose the functionals l_{i} as left center-eigenvectors of L^{∗}. The following theorem essentially states that external periodic forcing leads to a transition from relative equilibria to relative periodic orbits.

Theorem 2.3 Let u^{∗} = ρ_{e^{−tξ^{∗}}} Φ_{t}(u^{∗}) be a relative equilibrium of the unforced system (2.1), i.e., for the parameter μ = 0. Compute B^{∗} = e^{T_{ext}^{∗}L^{∗}} as in (2.4) and assume that u^{∗} satisfies assumption (S). Then ρ_{g}u^{∗} is C^{k} in g. If the generalized eigenspace of B^{∗} to the eigenvalue 1 lies in alg(G)u^{∗} then for each small amplitude μ_{ext} of the periodic forcing, each frequency ω_{ext} ≈ ω_{ext}^{∗} of the forcing and each small μ̂ there is exactly one relative periodic orbit u = u(ω_{ext}, μ), of (2.1) satisfying

u = ρ_{g}^{−1}Φ_{T_{ext},0}(u, μ) and u ∈ S_{l},  (2.5)

for some g = g(ω_{ext}, μ). Furthermore ρ_{g}u(ω_{ext}, μ) is C^{k} in g ∈ G, ω_{ext} and μ, g(ω_{ext}, μ) is C^{k} in (ω_{ext}, μ) and u(ω_{ext}, 0) = u^{∗}, g(ω_{ext}, 0) = g^{∗}.

Often we need not use the full symmetry G of (3.1) to prove Theorem 2.3. If L^{∗} does not have eigenvalues ijω_{ext}^{∗}, j ∈ ℤ, forced by symmetry then the symmetry group is discrete and we need not take it into account to prove the theorem. If [·, ξ^{∗}] has eigenvalues in iω_{ext}^{∗}ℤ, then the corresponding (generalized) eigenvectors form a Lie-subalgebra of alg(G) as can be seen from the Jacobi-identity. We call the Lie group generated by the generalized eigenvectors of [·, ξ^{∗}] to the spectral set iω_{ext}^{∗}ℤ the minimal symmetry group for the forcing frequency ω_{ext}^{∗} that we consider.

2.2 Resonance drift

Now we deal with the effects of resonant periodic forcing. We need the following notion:

Definition 2.4 Let g ∈ G. If g^{n} = exp(ξn) for some ξ ∈ alg(G) with Ad_{g} ξ = ξ and n ∈ ℕ then we call ξ average velocity of g.

There may be many average velocities for each group element g; for example if G = SO(2) then for g^{∗} = ϕ^{∗} the set {ξ^{∗} = ϕ^{∗} + j2π | j ∈ ℤ} consists of average velocities for g^{∗}. If u = ρ_{g}^{−1}Φ_{T,0}(u) is a relative periodic orbit of (2.1) and ξ is an average velocity of g then we call ξ/T average velocity of the relative periodic orbit.

Definition 2.5 If exp(·) is not locally surjective near ξ^{∗} ∈ alg(G) then there are elements g ∈ G close to exp(ξ^{∗}) which have (if any) only average velocities ξ which are far away from ξ^{∗}. We call this phenomenon resonance drift.

Similarly, let u^{∗} be a non-critical relative equilibrium of the unperturbed system (2.1) which travels with velocity ξ^{∗}. If the period of the external forcing T_{ext}^{∗} is such that exp(·) is not locally surjective near ξ = ξ_{∗}T_{ext}^{∗} then it may happen that relative periodic orbits of (2.1) which are generated by external periodic forcing, see Theorem 2.3, drift with an average velocity completely different to the drift velocity ξ^{∗} of the relative equilibrium at μ_{ext} = 0. We also call this effect resonance drift. Due to [31, Theorem 2.14.2] we know that the map (D exp(ξ^{∗}))exp(−ξ^{∗}) : alg(G) → alg(G) is given as

(D exp(ξ^{∗}))exp(−ξ^{∗}) = ∑_{n=0}^{∞} \frac{(−1)^{n}}{(n+1)!}(ad_{ξ^{∗}})^{n}
= (−ad_{ξ^{∗}})^{−1}(exp(−ad_{ξ^{∗}}) − id)

(2.6)

where ad_{ξ^{∗}}(ξ) = [ξ^{∗}, ξ]. Hence exp(·) is not locally surjective at ξ^{∗} iff ad_{ξ^{∗}} has eigenvalues in 2πiℤ \ {0}. Consequently, for resonance drift to occur it is necessary that the periodic forcing is resonant, i.e., that the linearization L^{∗} of the relative equilibrium in the comoving frame has a symmetry eigenvalue in iω_{ext}^{∗}ℤ \ {0}. Otherwise exp(·) would be surjective near T_{ext}^{∗}ξ^{∗} and the relative periodic orbits u(μ) generated by periodic forcing would drift with velocity ξ(μ) ≈ ξ^{∗}. As we mentioned in the introduction even a transition from compact to noncompact drift may take place. We will deal with this in the following example:

Example 2.6 Consider Example 2.1 again: Let the symmetry group be G = E(2), write g = (ϕ, a) ∈ SO(2)⋉ℝ^{2} = SE(2) and let u^{∗} be a non-critical rotating wave u^{∗} = ρ_{(−ω_{rot}^{∗}t,0)}Φ_{t}(u^{∗}) of the unforced system (2.1), ie. for μ_{ext} = 0. For example u^{∗} could be a rigidly rotating spiral wave of the reaction-diffusion system (2.2) on Ω = ℝ^{2}. By Theorem 2.3 for each small forcing amplitude μ_{ext} ≈ 0 and each forcing frequency ω_{ext} there is a relative periodic orbit u(ω_{ext}, μ_{ext}) ≈ u^{∗}. If ω_{rot}^{∗}/ω_{ext}^{∗} ∉ ℤ then the forcing is non-resonant and the relative periodic orbits u(μ_{ext}, ω_{ext}) with ω_{ext} ≈ ω_{ext}^{∗} are modulated rotating waves of (2.1) (called meandering spiral waves in the example (2.2)). If ω_{rot}^{∗}/ω_{ext}^{∗} = j ∈ ℤ then we see from (2.6) that D exp(2πξ^{∗}/ω_{ext}^{∗}) has rank defect 2. We talk of a j : 1-resonance. In this case modulated travelling waves (called drifting spiral waves of (2.2)) are generated as the following proposition shows:

Proposition 2.7 If a rotating wave of an E(2)-equivariant system (2.1) is subject to j : 1-resonant periodic forcing then there is a C^{k}-smooth path u(μ_{ext}), a(μ_{ext}), ω_{ext}(μ_{ext}), of modulated travelling waves satisfying

Φ_{2π/ω_{ext}(μ_{ext})}(u(μ_{ext})) = ρ_{(0,a(μ_{ext}))}u(μ_{ext})

such that u(0) = u^{∗}, a(0) = 0, ω_{ext}(0) = ω_{ext}^{∗}.

Proof. By Theorem 2.3 we get a surface u(ω_{ext}, μ_{ext}) of relative periodic orbits satisfying (2.5) where g(ω_{ext}, μ_{ext}) = (ϕ(ω_{ext}, μ_{ext}), a(ω_{ext}, μ_{ext})). To obtain modulated travelling waves we need to solve the equation

ϕ(ω_{ext}, μ_{ext}) = 0 mod 2π.

We have ∂_{ω_{ext}}ϕ(ω_{ext}, μ_{ext})|_{(ω_{ext},μ_{ext})=(ω_{ext}^{∗},0)} ≠ 0. This can be seen as follows: Let ξ_{1} be the generator of the rotation, and ξ_{2}, ξ_{3} be the generators of the translation. Computing the derivative w.r.t. ω_{ext} of (2.5) in (ω_{ext}, μ_{ext}) = (ω_{ext}^{∗}, 0) we get

−\frac{2πω_{rot}^{∗}}{(ω_{ext}^{∗})^{2}} ξ_{1}u^{∗} + (DΦ_{T_{ext}^{∗},0}(u^{∗}) − 1)∂_{ω_{ext}}u(ω_{ext}^{∗}, 0)

= (∂_{ω_{ext}}ϕ(ω_{ext}^{∗}, 0)ξ_{1} + ∂_{ω_{ext}}a_{1}(ω_{ext}^{∗}, 0)ξ_{2} + ∂_{ω_{ext}}a_{2}(ω_{ext}^{∗}, 0)ξ_{3})u^{∗}.

(2.7)

Here we used that

∂_{ω_{ext}}Φ_{2π/ω_{ext}^{∗}}(u^{∗}) = −\frac{2π}{(ω_{ext}^{∗})^{2}} ∂_{t}Φ_{t}(u^{∗})_{t=2π/ω_{ext}^{∗}} = −\frac{2πω_{rot}^{∗}}{(ω_{ext}^{∗})^{2}} ξ_{1}u^{∗}.

If we choose the l_{i} in (2.5) as left center-eigenvectors of L^{∗} then

l_{i}((DΦ_{T_{ext}^{∗},0}(u^{∗}) − 1)∂_{ω_{ext}}u(ω_{ext}^{∗}, 0)) = 0, i = 1, 2, 3.

Applying the functionals l_{i}, i = 1, 2, 3, onto (2.7) we conclude that

∂_{ω_{ext}}ϕ(ω_{ext}^{∗}, 0) = −2πω_{rot}^{∗}/(ω_{ext}^{∗})^{2} ≠ 0.

Hence we can apply the implicit function theorem to get a smooth path μ_{ext}(ω_{ext}) parametrizing modulated travelling waves.

A transition from rotating waves to modulated travelling waves has been observed in experiments [16] in the case of 1 : 1-resonance and 2 : 1-resonance. Ashwin and Melbourne [2] talk of drift bifurcation of relative equilibria if a rotating wave of an E(2)-equivariant system becomes a travelling wave in the limit ω_{rot} → 0. So their drift bifurcation and our resonance drift are related. But in our case the resonance drift is enforced by periodic forcing.

Example 2.8 Consider the reaction-diffusion system (2.2) on the sphere Ω = S^{2}. Then the symmetry group is G = O(3). We will show that a wave u^{∗} rotating around the x_{3}-axis starts meandering around some vector in the (x_{1}, x_{2})- plane if it is subject to resonant periodic forcing.

Let ξ_{i} denote the generators of the rotation around the unit vectors e_{i} ∈ ℝ^{3}, i = 1, 2, 3, and write g ∈ SO(3) as g = exp(∑_{i=1}^{3} ϕ_{i}ξ_{i}). Let u^{∗} = ρ_{exp(−ξ^{∗}t)}Φ_{t}(u^{∗}) be a non-critical wave of the unforced system (2.2), μ_{ext} = 0, rotating around the x_{1}-axis, i.e., ξ^{∗} = ω_{rot}^{∗}ξ_{1}. As in (2.3) we compute

L^{∗}(ξ_{2} + iξ_{3})u^{∗} = iω_{rot}^{∗}(ξ_{2} + iξ_{3})u^{∗}.

If we switch on resonant periodic forcing with ω_{ext}^{∗} = ω_{rot}^{∗}/j, j ∈ ℤ, then there is a smooth path u(μ_{ext}), ω_{ext}(μ_{ext}) of waves meandering around some vector in the (x_{2}, x_{3})-plane:

Φ_{T_{ext}(μ_{ext}),0}(u(μ_{ext})) = ρ_{exp(ϕ_{2}(μ_{ext})ξ_{2}+ϕ_{3}(μ_{ext})ξ_{3})}u(μ_{ext})

where ϕ_{2}(0) = 0, ϕ_{3}(0) = 0, ω_{ext}(0) = ω_{ext}^{∗}, u(0) = u^{∗}. This can be seen as in Example 2.6. For numerical simulations of rotating waves on the sphere S^{2} see [36].

In the last two examples of resonant forcing the relative equilibria were always rotating waves. But also for nonperiodic relative equilibria resonance drift occurs:

Example 2.9 Consider the reaction-diffusion system (2.2) in three space Ω = ℝ^{3}. Then the symmetry group is the Euclidean group E(3). Let u^{∗} be a twisted scroll ring of the unforced system (2.2). Such a wave consists of a circular filament in the (x_{2}, x_{3})-plane along which vertical spiral waves are located and an additional infinitely extended vertical filament [18]. It is a relative equilibrium which translates along its vertical filament and simultaneously rotates around it. Because of the vertical filament only translations a ∈ ℝ^{3} and rotations around the x_{3}-axis act continuously on u_{∗} in the space BC_{unif}. So the effective symmetry group is in this case G = E(2) × ℝ. cf. [23]. We write g = (ϕ, a) for the elements of E(2) × ℝ where ϕ is the rotation angle around the x_{1}-axis and a ∈ ℝ^{3} is a translation vector. The time-evolution of the twisted scroll ring is given by Φ_{t}(u^{∗}) = ρ_{exp(ξ^{∗}t)}u^{∗} where ξ^{∗} = (ω_{rot}^{∗}, v^{∗}e_{1}). If the twisted scroll ring is forced periodically with frequency ω_{ext} it will typically start meandering in the (x_{2}, x_{3})-plane:

Φ_{T_{ext},0}(u(μ_{ext})) = ρ_{(ϕ(μ_{ext}),a(μ_{ext}))}u(μ_{ext}), a(μ_{ext}) = v(μ_{ext})T_{ext}e_{1}.

But by resonant periodic forcing, i.e., if ω_{rot}^{∗}/ω_{ext}^{∗} ∈ ℤ, we can achieve that the scroll ring drifts away in another direction than the x_{1}-axis as the following proposition shows:

Proposition 2.10 If the twisted scroll ring of (2.2) is noncritical and forced periodically such that ω_{rot}^{∗}/ω_{ext}^{∗} ∈ ℤ then there is a C^{k}-smooth path u(μ_{ext}), ω_{ext}(μ_{ext}) of relative periodic orbits satisfying

Φ_{2π/ω_{ext}(μ_{ext}),0}(u(μ_{ext})) = ρ_{(0,a(μ_{ext}))}u(μ_{ext}), a(μ_{ext}) ∈ ℝ^{3}.

The direction of the drift a(μ_{ext}) of the periodically forced twisted scroll rings in the above proposition will typically not point in x_{1}-direction. The proof of the proposition is similar as the proof of Proposition 2.7.

Note again that to the isotropy K of the relative equilibria not all kinds of noncompact drift are possible. As mentioned before the drifts g(ω_{ext}, μ) of the emanating relative periodic orbits have to lie in N(K). Remember that we have chosen G = N(K)/K in the whole section. In a second step we have to interpret our results on periodic forcing for the original group G. In a system with E(2)-symmetry for instance we see that a rotating wave with spatial symmetry K can not start drifting under the influence of the periodic forcing if K contains a non-trivial rotation (ϕ, 0). In this case N(K) = SO(2), see [7]. Similarly if G = E(2) and K only consists of one reflection then the relative equilibrium u^{∗} can not rotate. Hence it is a travelling wave in general. A relative equilibrium in an E(2)-equivariant system with K ⊃ D_{n}, n > 1, even has to be stationary. We can generalize Propositions 2.7, 2.10 as follows: Let g = g̃(χ), χ ∈ ℝ^{n}, |χ| ≤ 1, be a smooth n-dimensional hyper-surface in G such that g(0) = g^{∗} = exp(T_{ext}^{∗}ξ^{∗}). Let {ξ_{i} |i = 1, … m}, m = dim(G), denote a basis of alg(G). Write

g̃(χ) = exp(ζ̃(χ))g^{∗}, ζ̃(χ) = ∑_{i=1}^{dim(G)} ζ̃_{i}(χ)ξ_{i},  (2.8)

ζ̃_{i}(0) = 0, i = 1, …, m, and assume that (∂_{χ_{j}} ζ̃_{i}(0))_{i,j=1,...,n} is an invertible (n, n)-matrix

(∂_{χ_{j}} ζ̃_{i}(0))_{i,j=1,...,n} ∈ GL(n),  (2.9)

and that

∂_{χ}ζ̃_{i}(0) = 0 for i = n + 1, …, m.  (2.10)

Let u^{∗}(μ̂) = ρ_{exp(−t ∑_{i=1}^{m} ζ_{i}^{∗}(μ̂)ξ_{i})}Φ_{t}(u^{∗}(μ̂)) be relative equilibria of (2.1) at μ_{ext} = 0 such that u^{∗}(0) = u^{∗}, ∑_{i=1}^{m} ζ_{i}^{∗}(0)ξ_{i} = ξ^{∗} and u^{∗}(μ̂) ∈ S_{l}. Then the following holds:

Proposition 2.11 Let the assumptions of Theorem 2.3 jold. Then there is a C^{k}-smooth hyper-surface (ω_{ext}(μ_{ext}, ν), μ(μ_{ext}, ν)) of relative periodic orbits u(μ_{ext}, ν) in the (ω_{ext}, μ)-parameter-space with ν ∈ ℝ^{d}, d = p − (m − n) and |ν| small, satisfying

Φ_{2π/ω_{ext}(μ_{ext},ν),0}(u(μ_{ext}, ν); μ(μ_{ext}, ν)) = ρ_{g̃(χ(μ_{ext},ν))}u(μ_{ext}, ν)

and

u(μ_{ext}, ν) ∈ S_{l}, u(0, 0) = u^{∗}, χ(0, 0) = 0,

provided that the (m − n, p)-matrix

(∂_{(ω_{ext},μ̂)}T_{ext}ζ_{i}^{∗}(0))_{i=n+1,...,m}

has full rank.

Proof. We solve the equation

g̃(χ)^{−1}g(ω_{ext}, μ) = id

by the implicit function theorem. In the examples 2.6, 2.8, 2.9 above the hyper-surface g = g̃(χ) consists of elements with average drift velocity far away from the drift velocity ξ^{∗} of the relative equilibrium.

2.3 Scaling of drift velocity

In this section we study the scaling of drifts induced by a harmonic periodic forcing where the forcing term in (2.1) is of the form

f_{ext}(u, ω_{ext}t, μ) = f̃(u) cos(ω_{ext}t, μ).  (2.11)

Such a forcing term is usually used in experiments [16], [35]. Further let μ = μ_{ext} ∈ ℝ. We first state a general proposition, then we apply this result to some examples in spiral wave dynamics explaining scaling laws which were observed in experiments or simulations. In the end we give a mathematical definition of the spiral tip. The motion of the spiral tip is measured in experiments to visualize the drift [5]. We assume that the unforced system (2.1) has a non-critical relative equilibrium u^{∗} and denote again by {ξ_{1}, ξ_{2}, …, ξ_{m}} a basis of alg(G).

Proposition 2.12 Assume that the periodic forcing term in (2.1) is of the form (2.11). Fix a forcing frequency ω_{ext}^{∗}. Let u(μ_{ext}), g(μ_{ext}) be relative periodic orbits for μ_{ext} ≈ 0. Write

g(μ_{ext}) = e^{T_{ext}ζ(μ_{ext})}e^{T_{ext}ξ^{∗}}, ζ(μ_{ext}) = ∑_{i=1}^{m} ζ_{i}(μ_{ext})ξ_{i}.

Assume that the geometric multiplicity of the eigenvalue 0 of the linear map [·, ξ^{∗}] on alg(G) equals its algebraic multiplicity. Then

∂_{μ_{ext}}ζ_{i}(0) = 0 if [ξ_{i}, ξ^{∗}] = 0.

This is also true if f_{ext} is not a harmonic periodic forcing, but the mean value ∫_{0}^{2π} f_{ext}(u, t)dt of f_{ext} is zero. Now assume that the periodic forcing is resonant so that the linear map [·, ξ^{∗}] on alg(G) has eigenvalues ±iω_{G}^{∗} with eigenvectors ξ_{1} ±iξ_{2} such that ω_{G}^{∗}/ω_{ext}^{∗} = j ∈ Z. Assume that the algebraic and the geometric multiplicity of the eigenvalue ±iω_{G}^{∗} of [·, ξ^{∗}] are equal. Then

∂_{μ_{ext}}ζ_{i}(0) = 0 for i = 1, 2 if j > 1.

If u^{∗} is a rotating wave then e^{T_{ext}ξ^{∗}} = id for some T_{ext}. Therefore the (m, m)- matrix [·, ξ^{∗}] is semisimple and has eigenvalues ±iω_{G}^{∗} with ω_{G}^{∗}/ω_{ext}^{∗} = j ∈ Z and the above proposition can be applied, see Example 2.13 below.

Proof of Proposition 2.12. We write a prime for ∂_{μ_{ext}} in the following calculation. We choose the functionals l_{i} in (2.5) defining the section transversal to the group orbit again as left center-eigenvectors of L^{∗}. Differentiating (2.5) with respect to μ_{ext} in μ_{ext} = 0 gives

∑_{i=1}^{m} T_{ext}^{∗}ζ′_{i}(0)ξ_{i}u^{∗} = (e^{T_{ext}L^{∗}} − 1)u′(0)
+ρ_{exp(−T_{ext}ξ^{∗}})∂_{μ}Φ_{T_{ext}^{∗}}(u^{∗}; μ)|_{μ=0}

(2.12)

where

ρ_{exp(−ξ^{∗}T_{ext}})∂_{μ_{ext}}Φ_{T_{ext}}(u^{∗}; μ)|_{μ=0} = ∫_{0}^{2π/ω_{ext}} e^{L^{∗}(2π/ω_{ext}−t)}f̃(u^{∗}) cos(ω_{ext}t)dt.

Let P be the spectral projection of L^{∗} to the eigenvalue 0. Since algebraic and geometric multiplicity of the eigenvalue 0 of [ξ^{∗}, ·] are equal by assumption and the relative equilibrium u^{∗} is noncritical we conclude that P L^{∗} = 0 and therefore

∫_{0}^{2π/ω_{ext}} P e^{L^{∗}(2π/ω_{ext}−t)}f̃(u^{∗}) cos(ω_{ext}t)dt = 0.

Applying P onto (2.12) we therefore get

∑_{i=1}^{m} T_{ext}^{∗}ζ′_{i}(0)P ξ_{i}u^{∗} = ∫_{0}^{2π/ω_{ext}} P e^{L^{∗}(2π/ω_{ext}−t)}f̃(u^{∗}) cos(ω_{ext}t)dt = 0.

This proves that ζ′_{i}(0) = 0 if [ξ_{i}, ξ^{∗}] = 0, and we see that we get the same result if only the time average of f_{ext}(u, t, 0) is zero. Now let Q be the spectral projection to the eigenvalue iω_{G}^{∗}, ω_{G}^{∗}/ω_{ext}^{∗} = j. Applying Q onto (2.12) we get, similarly as above,

∑_{i=1}^{m} T_{ext}^{∗}ζ′_{i}(0)Qξ_{i}u^{∗} = ∫_{0}^{2π/ω_{ext}} Qe^{L^{∗}(2π/ω_{ext}−t)}f̃(u^{∗}) cos(ω_{ext}t)dt.

As above we conclude that ζ′_{i}(0) = 0 for i = 1, 2 if j > 1.

Example 2.13 Again let G = E(2) and let u^{∗} be a non-critical rotating wave of the unforced system (2.1), e.g. a rigidly rotating spiral wave of the reactiondiffusion system (2.2) on the plane Ω = ℝ^{2}. Assume that the periodic forcing is resonant ω_{rot}^{∗} = jω_{ext}^{∗}, j ∈ ℤ. Then according to Example 2.6 there is a path u(μ_{ext}), a(μ_{ext}), ω_{ext}(μ_{ext}) of modulated travelling waves (drifting spiral waves of the reaction-diffusion system (2.2)) in the parameter-plane (ω_{ext}, μ_{ext}) ∈ ℝ^{2}. Assume that the periodic forcing is harmonic. By Proposition 2.12 the drift velocity v(μ_{ext}) = \frac{a(μ_{ext})}{T_{ext}} of the modulated travelling waves satisfies v′(0) = 0 if |j| > 1.

Drift velocities which only grow with the square μ_{ext}^{2} of the amplitude of the external periodic forcing are rather small and apparently difficult to find in experiments. That is why in experiments [35] mainly the 1:1-resonance is observed; however in [16] also a 2 : 1-resonance could be detected experimentally.

Example 2.14 Let G = SO(3) and let u^{∗} be a non-critical wave of the unforced system (2.1) rotating around the x_{1}-axis with speed ω_{rot}^{∗}, for instance, a rigidly rotating spiral wave in the reaction-diffusion system (2.2) on the sphere, see Example 2.8; if the periodic forcing is resonant ω_{rot}^{∗} = jω_{ext}^{∗}, j ∈ ℤ then according to Example 2.8 there is a path u(μ_{ext}), ϕ(μ_{ext}), ω_{ext}(μ_{ext}) of modulated rotating waves meandering around some vector in the (x_{2}, x_{3})-plane. By Proposition 2.12 their drift velocity ω_{rot}(μ_{ext}) = ϕ(μ_{ext})/T_{ext}(μ_{ext}) satisfies ω′_{rot}(0) = 0 if j > 1.

Example 2.15 We again consider a twisted scroll ring, see Example 2.9. In this case the symmetry group is G = E(2) × ℝ and the drift velocity of the scroll ring is given by ξ^{∗} = (ω_{rot}^{∗}, v^{∗}e_{1}). Denote by u(μ_{ext}), g(μ_{ext}) the relative periodic orbits generated by periodic forcing of the twisted scroll with fixed forcing frequency ω_{ext}. We write g(μ_{ext}) = (ϕ(μ_{ext}), a(μ_{ext})) where a(μ_{ext}) ∈ ℝ^{3}, a(0) = a^{∗}e_{1} = T_{ext}v^{∗}e_{1}, ω_{rot}(μ_{ext}) = ϕ(μ_{ext})/T_{ext}, ω_{rot}(0) = ω_{rot}^{∗}. By Proposition 2.12 we have

|ω_{rot}(μ_{ext}) − ω_{rot}^{∗}| = O(μ_{ext}^{2}), |a_{1}(μ_{ext}) − a^{∗}| = O(μ_{ext}^{2}),

but in general |a_{i}(μ_{ext})| = O(μ_{ext}), i = 2, 3. This is also observed in numerical simulations, see [15].

Now we define the tip position x_{tip}(u) for u ∈ Y. It is not clear at all how to define the spiral tip exactly. Experimentalists often determine the tip of a spiral wave in two dimensions visually as point with maximal curvature at the end of the spiral [5], but there are also other more or less precise definitions around [14]. From a symmetry point of view the position x_{tip}(u) ∈ ℝ^{2} of the spiral tip in the case G = E(2) is a function of the spiral wave solution u into ℝ^{2} and has the following property.

Definition 2.16 The tip position x_{tip}(·) is a C^{1}-smooth G-equivariant function which maps an open set of Y into a G-manifold M.

For example in the case G = E(2) we choose π(ϕ, a) = a, π(G) = ℝ^{2} and G acts on π(G) by the natural affine representation [8]; in the case G = SO(3) we choose π(G) = S^{2}; each g ∈ SO(3) can be represented by a vector ϕ ∈ so(3) = ℝ^{3} such that g = exp(ϕ) is a rotation around the unit vector ϕ/|ϕ| by the rotation angle |ϕ|; we set π(exp(ϕ)) = ϕ/|ϕ|. In experiments the drift phenomena we talked about are detected by following the spiral tip x_{tip}(u). For the spiral tip x_{tip}(u(ω_{ext}, μ_{ext})) the same scaling phenomena hold as for the drifts g(ω_{ext}, μ_{ext}).

2.4 Resonant periodic forcing of relative periodic orbits

Now we consider resonant periodic forcing of relative periodic orbits. We still assume that the isotropy K of the relative periodic orbit is trivial, otherwise we choose G = N(K)/K, Y = Fix(K) as before. Experiments on periodic forcing of meandering spiral waves have been carried out e.g. by Müller and Zykov [35]. Here invariant 3-tori were found and frequency locking between the period of the relative periodic orbits and the period of the external forcing was observed. Furthermore for certain periods of the external forcing modulated travelling waves were found in experiments. This phenomenon is called ”generalized resonance drift” [35]. We will only consider frequency locked relative periodic solutions generated by external periodic forcing. Let again T_{ext} = \frac{2π}{ω_{ext}} denote the period of the forcing, let μ_{ext} denote its amplitude and let T^{∗} be the period of the relative periodic orbit for μ = 0. Assume that u^{∗} is a non-critical relative periodic orbit in μ = 0, that is, u^{∗} satisfies Φ_{T^{∗}}(u^{∗}) = ρ_{g^{∗}}u^{∗}, for some T^{∗} > 0, B^{∗} = ρ_{g^{∗}}^{−1}DΦ_{T^{∗}}(u^{∗}) satisfies Hypothesis (S) and the center-eigenspace only consists of eigenvectors forced by G-symmetry or time-shift symmetry:

E_{c} = alg(G)u^{∗} ⊕ span(∂_{t}Φ_{t}(u^{∗})|_{t=0}).

Furthermore suppose that

T_{new} = jT_{ext} = ℓT^{∗} where gcd(j, ℓ) = 1.

Let P_{θ} be the spectral projection corresponding to the center spectral set of ρ_{g^{∗}}^{−1}DΦ_{1}(Φ_{θ}(u^{∗})). The condition P_{θ}(u − Φ_{θ}(u^{∗})) = 0 defines a section S_{θ} transversal to the relative periodic orbit in Φ_{θ}(u^{∗}).

Proposition 2.17 Under the above conditions there is a C^{k}-smooth hypersurface u(θ, μ) of ℓ : j-frequency-locked relative periodic solutions with μ ∈ ℝ^{p}, θ ∈ [0, T^{∗}], satisfying

Φ_{\frac{2πj}{ω_{ext}(θ,μ)} ,0}(u(θ, μ)) = ρ_{g(θ,μ)}u(θ, μ), u(θ, μ) ∈ S_{θ},  (2.13)

and u(θ, 0) = Φ_{θ}(u^{∗}), g(θ, 0) = (g^{∗})^{ℓ}.

This proposition is proved similarly as Theorem 2.3. We refer to section 4 for a proof. Assume for a moment that G is compact. Due to periodic forcing it may happen that a discrete rotating wave, i.e., a relative periodic orbit u^{∗} for which g^{∗} lies in a discrete Cartan subgroup Z_{n}, starts drifting. If gcd(n, ℓ) > 1, then (g^{∗})^{ℓ} may lie in a Cartan subgroup Z_{n/gcd(n,ℓ)} × T^{N}, N > 0 and ℓ : j-frequency locked relative periodic orbits nearby starts drifting. An example is the group G = O(2) where g^{∗} is a reflection. If ℓ = 2 then modulated rotating waves with relative period T_{new} ≈ 2T^{∗} are generated by the resonant periodic forcing of the discrete rotating wave u^{∗}. Such a phenomenon can not occur in the case of relative equilibria.

Another phenomenon that may occur in the case of periodic forcing is resonance drift as we saw in the preceding sections. Let ξ^{∗} be a drift velocity of g^{∗}. By resonance drift we mean that there are group elements g close to (g^{∗})^{ℓ} with all average drift velocities ξ far away from the drift velocity ξ^{∗} of g^{∗}. We first give an example. Then we state a general proposition.

Example 2.18 We consider periodic forcing of meandering spiral waves. In this case the symmetry group is G = E(2), and

u^{∗} = ρ_{(−ϕ^{∗},0)}Φ_{T^{∗}}(u^{∗})

is a modulated rotating wave. Assume that

ℓϕ^{∗} = 0 mod 2π, ℓ ≠ 0,

and that μ̂ ∈ ℝ (p = 2). If ∂μ̂ϕ^{∗}(0) ≠ 0 then there is an ℓ : j-frequency locked modulated travelling wave u(θ, μ_{ext}) to the parameter μ = (μ_{ext}, μ̂(θ, μ_{ext})), ω_{ext}(θ, μ_{ext}) such that u(θ, 0) = u^{∗}. Here ϕ^{∗}(μ̂) is the rotation angle for the modulated rotating wave u^{∗}(μ̂) = ρ_{(−ϕ^{∗}(μ̂),0)}Φ_{T^{∗}(μ̂)}(u^{∗}(μ̂)) for the autonomous system (μ_{ext} = 0) with parameter μ̂. This explains the ”generalized drift resonance” of locked solutions reported by [35].

Let g = g̃(χ) as in section 2.1 be a hyper-surface of dimension n in G such that g(0) = (g^{∗})^{ℓ} and that (2.8), (2.9), (2.10) hold. The hyper-surface g = g̃(χ) may for example consist of the group elements with average velocities far away from the drift velocity ξ^{∗} of g^{∗}. Let u^{∗}(μ̂) = ρ_{exp(∑_{i=1}^{m} ζ_{i}^{∗}(μ̂)T^{∗}(μ̂)ξ_{i})g^{∗}}^{−1}Φ_{T^{∗}(μ̂)}(u^{∗}(μ̂)), P_{0}(u^{∗}(μ̂) − u^{∗}) = 0, be relative periodic orbits of the unforced system (2.1) where μ_{ext} = 0 such that u^{∗}(0) = u^{∗}, T^{∗}(0) = T^{∗}, ζ_{i}(0) = 0, i = 1, …, m. Similarly as in Proposition 2.11 we find:

Proposition 2.19 Under the above assumptions there is a C^{k}-smooth hypersurface of ℓ : j-frequency locked relative periodic orbits near u^{∗} satisfying

Φ_{\frac{j2π}{ω_{ext}(θ,μ_{ext},ν)} ,0}(u(θ, μ_{ext}, ν); μ(θ, μ_{ext}, ν)) = ρ_{g̃(χ(θ,μ_{ext},ν))}u(θ, μ_{ext}, ν),

and u(θ, μ_{ext}, ν) ∈ S_{θ}, where ν ∈ ℝ^{d}, d = p − 1 − (n − dim(G)), |ν| small, provided that the (n − dim(G), p − 1)-matrix

(∂μ̂ζ_{i}^{∗}(0))_{i=n+1,...,dim(G)}

has full rank.

Now we study the scaling behaviour of the drift velocities in the case of harmonic periodic forcing (2.11) which is usually used in experiments [35]. Let μ = μ_{ext} ∈ ℝ.

Proposition 2.20 Let the periodic forcing be harmonic as in (2.11). Fix a frequency ω_{ext} of the periodic forcing and write the pull-back elements g(θ, μ_{ext}) of the ℓ : j-frequency locked periodic orbits, see Proposition 2.17, as

g(θ, μ_{ext}) = exp(∑_{i=1}^{m} jT_{ext}(θ, μ_{ext})ζ_{i}(θ, μ_{ext})ξ_{i})(g^{∗})^{ℓ}.

If ℓ > 1 and if the geometric multiplicity of the eigenvalue 1 of Ad_{g^{∗}} equals its algebraic multiplicity then we have:

∂_{μ_{ext}}ζ_{i}(0) = 0 for all i with Ad_{g^{∗}} ξ_{i} = ξ_{i}.

Moreover the Arnold tongues where the frequency locking occurs grow as |μ_{ext}|^{2} if ℓ > 1.

Note that if (g^{∗})^{ℓ} = id as in Example 2.18 the matrix Ad_{g^{∗}} is semisimple so that Proposition 2.20 can be applied. Again a cautious note: in the case G = E(2) the meandering spiral wave can not start drifting unboundedly if its spatial symmetry group K contains a nontrivial rotation. In general by periodic forcing the isotropy group of the relative periodic orbit is not changed. So the group element g(θ, μ) satisfying Φ_{jT_{ext}(θ,μ)}(u(θ, μ)) = ρ_{g(θ,μ)}u(θ, μ) is in N(K) where K is the isotropy of u^{∗} for properly chosen u(θ, μ). Note that we chose G = N(K)/K in the whole section. Proof of Proposition 2.20. Let W (t, 0) = DΦ_{t}(u^{∗}) denote the solution of the variation equation along Φ_{t}(u^{∗}) and let W (t, s) := W (t, 0)(W (s, 0))^{−1}, that is, W (t, s) = DΦ_{t−s}(Φ_{s}(u_{∗})). We have

∂_{μ_{ext}}Φ_{T_{new}}(u^{∗}, 0) = ∫_{0}^{ℓT^{∗}} W (ℓT^{∗}, s)f̃(Φ_{s}(u^{∗})) cos(\frac{2πjs}{ℓT^{∗}})ds
= ∫_{0}^{T^{∗}}(…)ds + … + ∫_{(ℓ−1)T^{∗}}^{ℓT^{∗}}(…)ds

= Re (C ∫_{0}^{T^{∗}} W (T^{∗}, s)f̃(Φ_{s}(u^{∗}))e^{\frac{2πijs}{ℓT^{∗}}} ds)

where

C = ρ_{g^{∗}}^{ℓ} ∑_{i=0}^{ℓ−1}(ρ_{g^{∗}}^{−1}W (T^{∗}, 0))^{ℓ−i−1}e^{2πiji/ℓ}ρ_{g^{∗}}^{−1}.

Here we used that

W (t + iT^{∗}, s + iT^{∗}) = DΦ_{t−s}(Φ_{s+iT^{∗}}(u_{∗})) = DΦ_{t−s}(ρ_{g^{∗}}^{i}Φ_{s}(u^{∗}))
= ρ_{g^{∗}}^{i}W (t, s)ρ_{g^{∗}}^{−i}

and that

W (ℓT^{∗}, iT^{∗}) = DΦ_{(ℓ−i)T^{∗}}(Φ_{iT} (u^{∗})) = ρ_{g^{∗}}^{i}DΦ_{(ℓ−i)T^{∗}}(u^{∗})ρ_{g^{∗}}^{−i}
= ρ_{g^{∗}}^{ℓ}ρ_{g^{∗}}^{i−ℓ}DΦ_{(ℓ−i)T^{∗}}(u^{∗})ρ_{g^{∗}}^{−i}
= ρ_{g^{∗}}^{ℓ}(ρ_{g^{∗}}^{−1}W (T^{∗}, 0))^{ℓ−i}ρ_{g^{∗}}^{−i},

and that therefore for s ∈ [0, T^{∗})

W (ℓT^{∗}, s + iT^{∗})f̃(Φ_{iT^{∗}+s}(u^{∗})) ; \begin{matrix}= & W (ℓT^{∗}, (i + 1)T^{∗})W (iT^{∗} + T^{∗}, iT^{∗} + s)ρ_{g^{∗}}^{i} f̃(Φ_{s}(u^{∗})) \\ = & ρ_{g^{∗}}^{ℓ}(B^{∗})^{ℓ−i−1}ρ_{g^{∗}}^{−1}W (T^{∗}, s)f̃(Φ_{s}(u^{∗})), \end{matrix}

where B^{∗} := ρ_{g^{∗}}^{−1}W (T^{∗}, 0). Let P be the spectral projection of B^{∗} to the eigenvalue 1. We have

P ρ_{g^{∗}}^{−ℓ}∂_{μ_{ext}}Φ_{T_{new}}(u^{∗}, 0) = Re (cP ρ_{g^{∗}}^{−1} ∫_{0}^{T^{∗}} W (T^{∗}, s)f̃(Φ_{s}(u^{∗}))e^{\frac{2πijs}{ℓT^{∗}}} ds)

(2.14)

where c = ∑_{i=0}^{ℓ−1} e^{2πiji/ℓ}. So P ∂_{μ_{ext}}Φ_{T_{new}}(u^{∗}; 0) = 0 if ℓ > 1. Differentiating (2.13) in the solution (u, g, ω_{ext})(μ) with respect to μ_{ext} in μ = 0 yields with g(θ, μ_{ext})(g^{∗})^{−ℓ} = exp(∑_{i=1}^{m} jT_{ext}(θ, μ_{ext})ζ_{i}(θ, μ_{ext})ξ_{i})

0 = ((B^{∗})^{ℓ} − 1)∂_{μ_{ext}}u(θ, μ_{ext})|_{θ,μ_{ext}=0} − ℓT^{∗} ∑_{i=1}^{m} ∂_{μ_{ext}}ζ_{i}(0)ξ_{i}u^{∗}

−\frac{2πj∂_{μ_{ext}}ω_{ext}(0)}{ω_{ext}^{2}(0)}∂_{t}Φ_{t}(u^{∗})|_{t=0} + ρ_{g^{∗}}^{−ℓ}∂_{μ_{ext}}Φ_{T_{new},0}(u^{∗}, μ)|_{μ=0}.

Applying the projection P to the eigenvalue 1 of B^{∗} we see that \frac{∂ζ_{i}}{∂_{μ_{ext}}} (0) = 0

for all i with Ad_{g^{∗}} ξ_{i} = ξ_{i} and that \frac{∂ω_{ext}}{∂μ_{ext}} (0) = 0 provided that ℓ > 1.

3 Hopf bifurcation from relative equilibria

In this section we study transitions from relative equlibria to relative periodic orbits in autonomous systems caused by Hopf bifurcation. For experiments on Hopf bifurcation from rotating waves – the meandering transition – in the Belousov-Zhabotinsky reaction see [26], [32], [27]. First we state a general theorem for Hopf bifurcation from relative equilibria. The proof of the Hopf theorem can be found in Subsection 4.6. In Subsection 3.2 we explain the drift phenomena caused by resonance which were observed in experiments. In Subsection 3.3 we discuss equivariant Hopf bifurcation. In the whole section we assume that the nonlinearity f in (1.3) is autonomous. So we consider the differential equation

\frac{du}{dt} = −Au + f(u, μ).  (3.1)

In the applications we have in mind (3.1) is an autonomous reaction-diffusion system

\frac{∂u_{i}}{∂t} = δ_{i}Δu_{i} + f_{i}(u, μ), i = 1, …, M,  (3.2)

cf. (1.1).

3.1 The theorem on Hopf bifurcation

Let u^{∗} be a relative equilibrium of (3.1) for μ = 0 satisfying Hypothesis (S). We will show in Section 4 below that Hypothesis (S) implies that ρ_{g}u^{∗} is C^{k} in g. In this subsection we assume that the isotropy K of the relative equilibrium is trivial K = {id} or we exchange G by N(K), Y by Fix(K). We assume that ±i are eigenvalues of the linearization L^{∗} = −A − ξ^{∗} + Df(u^{∗}) in the comoving frame which are not only caused by symmetry, i.e., if Q is the spectral projection of L^{∗} to the i then there is some w ∈ QY with w ∉ alg(G)u^{∗}. Furthermore assume that

ni ∈ spec(L^{∗}), n ∈ ℤ ⟹ QY ⊂ span(w, w̄) ⊕ alg(G)u^{∗}.

Let u^{∗}(μ) be the C^{k}-smooth path of relative equilibria with

Φ_{t}(u^{∗}(μ)) = ρ_{exp(tξ^{∗}(μ))}u^{∗}(μ), l_{i}(u^{∗}(μ) − u^{∗}) = 0, i = 1, …, m, u(0) = u^{∗}.

Note that we can obtain the path of relative equilibria u^{∗}(μ) near u^{∗} by applying Theorem 2.3 with non-resonant period T_{ext}. As before the functionals l_{i}, i = 1, …, m, determine a section S_{l} = u^{∗} + Ŝ_{l} transversal to the group orbit of the relative equilibrium u^{∗}. We choose the functionals l_{i} such that l_{i}(w) = 0, i = 1, …, m (e.g. by using the spectral projection of L^{∗} to the symmetry eigenvalues to construct the functionals l_{i}.).

Lemma 3.1 Under the above assumptions there is a C^{k−1}-path β(μ) of eigenvalues of the linearization

L^{∗}(μ) = −A + Df(u^{∗}(μ)) − ξ^{∗}(μ)

such that β(0) = i.

This lemma will be proved in section 4.6 below. We write μ = (μ_{1}, μ_{2}) where μ_{1} ∈ ℝ and μ_{2} ∈ ℝ^{p−1}. If the transversality condition

Re \frac{∂β(0)}{∂μ_{1}} ≠ 0  (3.3)

holds then we can assume w.l.o.g. that μ_{1} = 0 parametrizes the relative equilibria u^{∗}(μ) which are Hopf points.

Theorem 3.2 Under the above assumptions there are relative periodic orbits u(s, μ_{2}), s ∈ ℝ_{0}^{+} small, of relative period T (s, μ_{2}) near u^{∗} to the parameter μ_{1}(s) satisfying

Φ_{T (s)}(u(s, μ_{2}), (μ_{1}(s), μ_{2})) = ρ_{g(s,μ_{2})}u(s, μ_{2})  (3.4)

and u(0) = u^{∗}, μ(0) = 0, g(0) = e^{2πξ^{∗}}, T (0) = 2π provided that the transversality condition (3.3) is satisfied. For each small s a circle u_{l}(s_{1}, s_{2}, μ_{2}), s_{1} = s cos τ, s_{2} = s sin τ, τ ∈ [0, 2π], of the relative periodic orbit to the parameter s lies in the section S_{l} with corresponding pull-back element g_{l}(s_{1}, s_{2}, μ_{2}), and we fix the phase by setting u(s, μ_{2}) = u_{l}(s, 0, μ_{2}), g(s, μ_{2}) = g_{l}(s, 0, μ_{2}), such that ∂_{s}u(0) = Re w. The functions u_{l}(s_{1}, s_{2}, μ_{2}), μ_{1}(s, μ_{2}), g_{l}(s_{1}, s_{2}, μ_{2}), T (s, μ_{2}) are C^{k−1} in s_{1}, s_{2} ∈ ℝ and μ_{2} ∈ ℝ^{p−1}, and μ_{1}(s, μ_{2}) and T (s, μ_{2}) only depend on s = ∥(s_{1}, s_{2})∥ and μ_{2}.

Theorem 3.2 is proved in section 4.6 below. The Hopf bifurcation from relative equilibria to relative periodic orbits is called relative Hopf bifurcation because it is a Hopf bifurcation in the space of group orbits. Formally we can define a semiflow Ψ_{t}(·) on Ŝ_{l} in a comoving frame by

Ψ_{t}(u; μ) = ρ_{g(Φ_{t}(u,μ))}^{−1}Φ_{t}(u + u^{∗}(μ); μ) − u^{∗}(μ)  (3.5)

where g(u) is such that l_{i}(ρ_{g(u)}^{−1}u − u^{∗}) = 0, i = 1, …, m, ie. ρ_{g(u)}^{−1}u ∈ S_{l}. Under the above assumptions Ψ_{t}(·) undergoes a usual Hopf bifurcation with two simple Hopf eigenvalues ±i and without any resonances. To see this note that the linearization e^{L̃t} of Ψ_{t}(u) in the Hopf point u = 0 is given by L̃ = P_{l}L^{∗}P_{l} where P_{l} is the projection onto the space l_{i}(u) = 0, i = 1, …, m such that P_{l} alg(G)u^{∗} = 0. Choosing l_{i}, i = 1, …, m, such that l_{i}(ρ_{g}y) is C^{1} in g for y ∈ Y (which is possible as we will see in Lemma 4.3 below) we see that the semiflow Ψ_{t}(u) is strongly continuous on Y. But it is only smooth in u if the group action is smooth on Φ_{t}(u), t > 0, u ∈ Y, which is not the case in applications as we saw in the introduction, cf. subsection 1.5. Often we need not use the full symmetry G of (3.1) to prove the Hopf theorem. The situation is analogous to the case of periodic forcing of relative equilibria, see section 2.1: If L^{∗} does not have eigenvalues ij, j ∈ ℤ, forced by symmetry then ξ^{∗} = 0 and we have an ordinary Hopf bifurcation from an equilibrium. If [ξ^{∗}, ·] has eigenvalues in iℤ, then the corresponding (generalized) eigenvectors form a Lie subalgebra of alg(G). We call the group generated by this Lie subalgebra the minimal symmetry group for the Hopf bifurcation.

Example 3.3 Consider again the reaction-diffusion system (3.2) on the domain Ω = ℝ^{2}. Then the symmetry group is G = E(2). Let u^{∗} be a rigidly rotating spiral wave Φ_{t}(u^{∗}) = ρ_{(ω_{rot}^{∗}t,0)}u^{∗} of the reaction-diffusion system (3.2). The meandering transition mentioned in the introduction corresponds to a relative Hopf bifurcation from the rotating wave u^{∗}.

3.2 Resonance drift and scaling of drift velocity

In this section we deal with resonant Hopf bifurcation. Again we assume that the isotropy K of the relative equilibrium u^{∗} is trivial, K = {id} or we choose Y as Fix(K), G as N(K)/K. In the next subsection we will deal with equivariant Hopf bifurcation where K ≠ {id}. Let the assumptions of Theorem 3.2 hold, let again u^{∗} be a Hopf point with Hopf eigenvalues ±i, let again μ ∈ ℝ^{p} and let u^{∗}(μ) be relative equilibria satisfying l_{i}(u^{∗}(μ) − u^{∗}) = 0, i = 1, …, m, and

Φ_{t}(u^{∗}(μ)) = ρ_{exp(ξ^{∗}(μ)t)}u^{∗}(μ), ξ^{∗}(μ) = ∑_{i=1}^{m} ζ_{i}^{∗}(μ)ξ_{i},

with u^{∗}(0) = u^{∗}, ξ^{∗}(0) = ξ^{∗}. Here we again denote by {ξ_{i}; i = 1, …, m} a basis of the Lie algebra alg(G) of G. We have

L^{∗}ξu^{∗} = [ξ, ξ^{∗}]u^{∗}, e^{2πL^{∗}}ξu^{∗} = Ad_{exp(−2πξ^{∗})} ξu^{∗} = (e^{2π[·,ξ^{∗}]}ξ)u^{∗}, ξ ∈ alg(G).

If exp(·) is not locally surjective near 2πξ^{∗} then there may be relative periodic orbits bifurcating from the relative equilibrium with all average drift velocities completely different from the drift velocity ξ^{∗} of the relative equilibrium at the Hopf bifurcation. We talk of resonance drift as introduced in subsection 2.2. For resonance drift to occur it is necessary that the Hopf bifurcation is resonant which means that the linearization L^{∗} of the relative equilibrium in the comoving frame has a symmetry eigenvalue in iℤ \ {0}. In group-theoretical terms, the linear map [·, ξ^{∗}] has eigenvalues in iℤ\{0}. Otherwise exp(·) would be surjective near 2πξ^{∗} and the relative periodic orbits u(s) generated by Hopf bifurcation would drift with velocity ξ(s) ≈ ξ^{∗}, cf. subsection 2.2. Let g = g̃(χ) be an n-dimensional hyper-surface in G, χ ∈ ℝ^{n}, |χ| ≤ 1 such that g̃(0) = g^{∗} = e^{ξ^{∗}2π}. Write g̃(χ) = exp(ζ̃(χ))g^{∗} where ζ̃ = ∑_{i=1}^{dim(G)} ζ̃_{i}(χ)ξ_{i}, ζ_{i}(0) = 0, i = 1, …, dim(G), and assume that (2.9) and (2.10) hold. As in section 2.2 the hyper-surface g = g̃(χ) may consist of elements with average drift velocity far away from the drift velocity ξ^{∗} of the relative equilibrium. Again let μ = (μ_{1}, μ_{2}) with μ_{1} ∈ ℝ, μ_{2} ∈ ℝ^{p−1}.

Proposition 3.4 Let the assumptions of Theorem 3.2 and the above assumptions hold and let K = \{id\}. If \frac{∂}{∂μ_{1}} Re β(0) ≠ 0 and if the matrix

∂_{μ_{2}}(\frac{Im β(μ)}{ζ_{i}^{∗}(μ)})|_{μ=0}\}_{i=n+1,...,m}  (3.6)

has full rank then there are relative periodic orbits with average drift inside the hypersurface g = g̃(χ), more precisely: there are C^{k−1}-smooth functions u(s, ν), T (s, ν), μ(s, ν), χ(s, ν) such that

Φ_{T (s,ν)}(u(s, ν)) = ρ_{g̃(χ(s,ν))}u(s, ν).

Here ν ∈ ℝ^{d}, d = p − 1 − (dim(G) − n), χ(0) = 0, u(0) = u^{∗}.

Proof. By Theorem 3.2 there are relative periodic orbits u(s, μ_{2}), g(s, μ_{2}), T (s, μ_{2}) bifurcating from u^{∗}(μ)|_{μ_{1}=0}. We want to solve the equation g̃(χ)^{−1}g(s, μ_{2}) = id by the implicit function theorem. Since T (0, μ_{2}) = \frac{2π}{Im β(0,μ_{2})} we have

∂_{μ_{2}}g(0, μ_{2})|_{μ_{2}=0} = ∂_{μ_{2}} exp(∑_{i=1}^{m} \frac{2πζ_{i}^{∗}(0, μ_{2})}{Im β(0, μ_{2})}ξ_{i})|_{μ_{2}=0}

= 2π(∑_{i=1}^{m} ∂_{μ_{2}}ζ_{i}^{∗}(0)ξ_{i} − Im ∂_{μ_{2}}β(0)ξ^{∗})g^{∗}.

We need that ∂_{(χ,μ_{2})}g̃(χ)^{−1}g(s, μ_{2})_{(s,χ,μ_{2})=(0,0,0)} has full rank. Therefore the matrix

{∂_{μ_{2}}ζ_{i}^{∗}(0) − ζ_{i}^{∗}(0)∂_{μ_{2}} Im β(0)}_{i=n+1,...,dim(G)}

has to be invertible, that is, we need that

\{∂_{μ_{2}}(\frac{Im β(μ)}{ζ_{i}^{∗}(μ)})|_{μ=0}\}_{i=n+1,...,dim(G)}

has full rank. Now we study the scaling behaviour of the drift velocities. Let μ ∈ ℝ and write the pull-back elements g(s) of the bifurcating relative periodic orbits u(s) as

g(s) = exp(T (s)ζ(s))g^{∗}, ζ(s) = ∑_{i=1}^{dim(G)} ζ_{i}(s)ξ_{i}.  (3.7)

Remark 3.5 Let [ξ_{i}, ξ^{∗}] = 0. Then \frac{d}{ds}ζ_{i}(0) = 0. In a j : 1-resonance

[ξ_{1} + iξ_{2}, ξ^{∗}] = ij(ξ_{1} + iξ_{2}), j ∈ ℕ,

we have \frac{d^{ℓ}}{ds^{ℓ}} ζ_{i}(0) = 0, i = 1, 2, ℓ = 1, …, min(j, k) − 1.

Proof. Differentiating

ρ_{g(s)}^{−1}Φ_{T (s)}(u(s); μ(s)) − u(s) = 0

w.r.t. s in s = 0 gives

−2π ∑_{i=1}^{m} ζ′_{i}(0)ξ_{i}u^{∗} + (e^{2πL^{∗}} − 1) Re w = 0.

Applying the spectral projection P_{0} of L^{∗} to the eigenvalue 0 gives

P_{0} ∑_{i=1}^{m} ζ′_{i}(0)ξ_{i}u^{∗} = 0.

If [ξ_{i}, ξ^{∗}] = 0 then P_{0}ξ_{i}u^{∗} = ξ_{i}u^{∗}, so ζ′_{i}(0) = 0. We have

Φ_{T (s)}(u_{l}(s_{1}, s_{2}), μ(s)) = ρ_{g_{l}(s_{1},s_{2})}u_{l}(s_{1}, s_{2})  (3.8)

where s_{1} = s cos τ, s_{2} = s sin τ, τ ∈ [0, 2π]. By Theorem 3.2 u_{l}(s_{1}, s_{2}) and g_{l}(s_{1}, s_{2}) are C^{k−1}-smooth in s_{1}, s_{2}. We write g_{l}(s_{1}, s_{2}) as in (3.7):

g_{l}(s_{1}, s_{2}) = exp(T (s)ζ_{l}(s_{1}, s_{2}))g^{∗}, ζ_{l}(s_{1}, s_{2}) = ∑_{i=1}^{dim(G)} ζ_{l,i}(s_{1}, s_{2})ξ_{i}.

Since u_{l}(s_{1}, s_{2}) ∈ GΦ_{τT (s)/2π}(u(s), μ(s)) there are C^{k−1}-functions ĝ(τ, s) ∈ G, ζ̂(τ, s) ∈ alg(G) such that ĝ(τ, 0) = id, ζ̂(τ, 0) = 0,

ĝ(τ, s) = exp(ζ̂(τ, s)), ζ̂(τ, s) = ∑_{i=1}^{m} ζ̂_{i}(τ, s)ξ_{i}

and

u_{l}(s_{1}, s_{2}) = ρ_{ĝ(τ,s) exp(−ξ^{∗}τT (s)/2π)}Φ_{τT (s)/2π}(u(s), μ(s)).  (3.9)

From (3.4), (3.8), (3.9) we conclude that

g_{l}(s_{1}, s_{2}) = ĝ(τ, s) exp(−\frac{τT (s)}{2π}ξ^{∗})g(s) exp(\frac{τT (s)}{2π}ξ^{∗})ĝ(τ, s)^{−1}.

Hence

e^{T (s)ζ(s_{1},s_{2})} = e^{ζ̂(τ,s)} exp(T (s) Ad_{exp(− \frac{τT (s)}{2π} ξ^{∗})} ζ(s))e^{− Ad_{g^{∗}} ζ̂(τ,s)}.

We can choose G minimal such that Ad_{g^{∗}} = id on alg(G). Therefore we conclude that for each i

ζ_{l,i}(s_{1}, s_{2})ξ_{i} = Ad_{exp(ζ̂(τ,s)) exp( \frac{τT (s)}{2π} ξ^{∗})} ζ_{i}(s)ξ_{i}.

Since exp(ξ^{∗}τ)(ξ_{1} + iξ_{2}) = exp(ijτ)(ξ_{1} + iξ_{2}) we see that

ζ_{l,1}(s_{1}, s_{2})ξ_{1} + iζ_{l,2}(s_{1}, s_{2})ξ_{2} = (1 + sM(s)) exp(ijτ)(ζ_{1}(s)ξ_{1} + iζ_{2}(s)ξ_{2})

where M(s) ∈ Mat(2) is a C^{k−2} smooth function. Therefore since ζ_{l,i}(s_{1}, s_{2}), i = 1, 2, is C^{k−1}-smooth in s_{1}, s_{2} we conclude that \frac{d^{ℓ}}{ds^{ℓ}} ζ_{i}(0) = 0, ℓ = 0, …, min(j, k) − 1, i = 1, 2.

Example 3.6 Again let G = E(2) and let u^{∗} be a rotating wave Φ_{t}(u^{∗}) = ρ_{(ω_{rot}^{∗}t,0)}u^{∗} of (3.1), e.g. a rigidly rotating spiral wave of (3.2). Assume that the parameter space is two-dimensional, μ ∈ ℝ^{2}, as in Fig. 3, and that parameters are chosen such that the rotating waves u^{∗}(μ) which are Hopf points lie on the line μ_{1} = 0 in parameter space. Note that ±iω_{rot}^{∗} are eigenvalues of [·, ξ^{∗}] with eigenvectors ξ_{2} ± iξ_{3}, cf. Example 2.1. Choosing the hypersurface g = g̃(χ) in Proposition 3.4 as the subgroup of translations we now understand Fig. 3: If the rotation frequency ω_{rot}^{∗} is resonant to the Hopf frequency ω_{Hopf}^{∗} = 1, ω_{rot}^{∗} = jω_{Hopf}^{∗} ∈ ℤ and the resonance is crossed with nonzero speed ∂_{μ_{2}}(\frac{Im β(μ)}{ω_{rot}^{∗}(μ)})|_{μ=0} ≠ 0 (which is generically satisfied) then there is a path μ(s) in parameter space ℝ^{2} of modulated travelling waves (drifting spiral waves)

Φ_{T (s)}(u(s); μ(s)) = ρ_{(0,a(s))}u(s).

From Remark 3.5 we see that the drift velocity v(s) = |a(s)|/T (s) generically scales like |μ|^{j/2}, see [4], [8].

3.3 Equivariant relative Hopf bifurcation

In this subsection we study relative Hopf bifurcation in the case of a compact isotropy K ≠ {id} of the relative equilibrium u^{∗}. We consider the case when the spatial isotropy K of the relative equilibrium is broken. If the bifurcating solutions are relative periodic solutions and not relative equilibria we talk of equivariant or symmetry-breaking relative Hopf bifurcation. Assume that the linearization L^{∗} at the relative equilibrium u^{∗} has an eigenvalue i with a generalized eigenvector w ∉ alg(G)u^{∗}, i.e., the eigenvalue i of L^{∗} is not (only) caused by symmetry. The generalized eigenspace to the Hopf eigenvalues ±i is K-invariant and may be forced by K-equivariance of L^{∗} to have higher dimension than two even if ±i are not eigenvalues of [·, ξ^{∗}]. see [11]. Let again S_{l} = u^{∗} +Ŝ_{l} denote a section transversal to the group orbit Gu^{∗} at u^{∗} defined by functionals l_{i}, i = 1, …, m_{K} where m_{K} = dim(G/K) and denote by P_{l} the projection from Y to the subspace Ŝ_{l} = {y; l_{i}(y) = 0, i = 1, …, m_{K}}. Since K is compact we can choose P_{l} to be K-equivariant and P_{l}Y = Ŝ_{l} to be K-invariant: for example choose P = P_{s} + Q where P_{s} is the projection onto the stable eigenspace of L^{∗} and Q is an orthogonal projection from the finite-dimensional center-unstable eigenspace E_{cu} to (alg(G)u^{∗})^{⊥}. Since ξ^{∗} commutes with the elements of K the operator L^{∗} = −A + Df(u^{∗}) − ξ^{∗} is Kequivariant and therefore E_{cu} is invariant and P_{s} is K-equivariant. If we choose the scalar-product on E_{cu} to be K-invariant then also Q is K-equivariant. Define L̃ = P_{l}LP_{l}. Denote the eigenspace of L̃ to the eigenvalues ±i by V. In the generic case when i is a simple eigenvalue of L̃ the matrices e^{L̃τ}, τ ∈ [0, 2π], define an S^{1}-action on V. We consider the subgroups H of K × S^{1} with two-dimensional fixed point spaces. They are called axial subgroups [11]. Let π : K × S^{1} → K be the projection of K ×S^{1} onto its first component. For each axial subgroup H there is a homomorphism Θ : K → S^{1} = ℝ/ℤ such that H = {(h, Θ(h)) | h ∈ π(H)}, see [11], [7]. There are two cases, Θ(K) = S^{1} or Θ(K) = Z_{ℓ}. Let K_{bif} denote the kernel of Θ. Then the following lemma holds:

Lemma 3.7 Let the assumptions of Theorem 3.2 and the above assumptions hold. If Θ(K) = S^{1} then there is a symmetry breaking transition from relative equilibria to relative equilibria.

If Θ(K) = Z_{ℓ} then a symmetry breaking relative Hopf bifurcation takes place: Let h^{∗} = Θ^{−1}(1/ℓ) ∈ K. There is a path of relative periodic solutions u(s) which emanates from the relative equilibrium u^{∗} by equivariant relative Hopf bifurcation and satisfies

Φ_{T (s)/ℓ}(u(s)) = ρ_{g(s)h^{∗}}u(s), T (0) = 2π, u(0) = u^{∗}, g(0) = e^{2πξ^{∗}/ℓ}.

The isotropy of the bifurcating solutions is K_{bif} = ker(Θ) in both cases.

The proof is a small modification of the proof of Theorem 3.2 and can be found in subsection 4.6, see also [7]. Again the pull-back element g(s)h^{∗} of the relative periodic orbit u(s) has to lie in N(K_{bif}). In the following discussion assume that G = N(K_{bif})/K_{bif}. In the case of symmetry breaking Hopf bifurcation the average velocity of the bifurcating relative periodic orbits is often far away from the drift velocity of the relative equilibrium, as we see from the following example.

Example 3.8 (See also [7], [10]) Again let G = E(2) and let u^{∗} be a rotating wave Φ_{t}(u^{∗}) = ρ_{(ω_{rot}^{∗}t,0)}u^{∗} with isotropy K = Z_{ℓ}, for example a rigidly rotating spiral wave of (3.2) with ℓ identical arms. Consider a representation of K on the critical eigenspace V = span_{ℂ}(w, w̄) which is faithful, i.e., Θ^{−1}(1/ℓ) = 2πn/ℓ, gcd(ℓ, n) = 1. If the rotating wave is a Hopf point then under the usual transversality condition and in the non-resonant case a Hopf bifurcation to modulated rotating waves takes place. The average rotation frequency ω_{rot}(s) of the bifurcating modulated rotating waves is given as ω_{rot}(s) = (h^{∗} + ϕ(s))/(T (s)/ℓ). Note that h^{∗} = 2πn/ℓ and that g(s) = (ϕ(s), a(s)) satisfies g(0) = (ω_{rot}^{∗}2π/ℓ, 0). Hence we get

ω_{rot}(s = 0) = (2πn/ℓ + ω_{rot}^{∗}2π/ℓ)/(2π/ℓ) = n + ω_{rot}^{∗}.

But in physical space the bifurcating modulated rotating waves in Example 3.8 still seem to drift in a similar direction as the rotating wave u^{∗}. So what is a useful definition of resonance drift in the case of symmetry-breaking Hopf bifurcation? We first continue our example:

Example 3.9 (Example 3.8 continued) We recall the condition for noncompact drift of relative periodic orbits nearby the Hopf point in Example 3.8, see also [7], [10]. Since g(s) ∈ N(K_{bif}) we can only get noncompact drift if K_{bif} ⊆ K is trivial. So we consider again, as above, a faithful representation of K on the critical eigenspace V = span_{ℂ}(w, w̄), where Θ^{−1}(1/ℓ) = 2πn/ℓ, gcd(ℓ, n) = 1. Resonance drift occurs if ω_{rot}^{∗} = jℓ − n, j ∈ ℤ, since for noncompact drift ϕ(0) = 2πω_{rot}^{∗}/ℓ + 2πn/ℓ = 0 mod 2π has to be satisfied. Since iω_{rot}^{∗} is in the spectrum of [·, ξ^{∗}] with eigenvectors ξ_{1} + iξ_{2} we see from Remark 3.5 that the drift velocity v(s) = |a(s)|/T (s) generically grows as |μ|^{|jℓ−n|/2}.

In the case of noncompact drift in the above example we clearly want to speak of resonance drift. Since we do not want to care about the (small) effects of the broken spatial Z_{ℓ} symmetry of the bifurcating relative periodic orbits in the comoving system (3.5) we only talk of resonance whenever the drift (g(s)h^{∗})^{ℓ} of the relative periodic orbits after time T (s) is not of the form exp(2πξ) with ξ ≈ ξ^{∗}. Note that a necessary condition for resonance drift is that exp(·) is not locally surjective at ξ = 2πξ^{∗}, but since g(s) and h^{∗} need not commute (in contrast to h^{∗} and ξ^{∗}) this condition is not sufficient: in Example 3.6 where the isotropy is trivial the condition for unbounded drift is ω_{rot}^{∗} ∈ ℤ, in the case of Z_{ℓ}-isotropy the condition for noncompact drift is more restrictive, see Example 3.9.

4 Proof of the main theorems

This section is devoted to the proof of the theorems on periodic forcing and Hopf bifurcation which we presented in Sections 2 and 3. First, in subsections 4.1 – 4.4 we present a general method how to continue relative periodic orbits that satisfy the spectral hypothesis (S). In subsection 4.5 we prove Theorem 2.3 on periodic forcing. In subsection 4.6 below we use the developed methods to prove the Hopf theorem 3.2 by use of Lyapunov-Schmidt reduction.

4.1 The method of proof

Assume that we are given a relative periodic orbit u^{∗} = ρ_{g^{∗}}^{−1}Φ_{2π/ω_{ext}^{∗},0}(u^{∗}) of (2.1) that satisfies the spectral hypothesis (S). We want to continue this relative periodic orbit wrt. the parameters μ and ω_{ext}, i.e., we want to solve the equation F = 0 where F is given by

F (u, g, ω_{ext}, μ) = ( \begin{matrix}ρ_{g}^{−1}Φ_{T_{ext},0}(u; ω_{ext}, μ) − u \\ l_{i}(u − u^{∗}), i = 1, …, m \end{matrix} ) .  (4.1)

We consider (4.1) for u in the fixed point space Fix(K) where K is the isotropy of the relative periodic orbit. W.l.o.g. we assume that Y = Fix(K) and G = N(K) is the normalizer of K. The functionals l_{i}, i = 1, …, m, define a section transversal to the group orbit Gu^{∗} at u^{∗}. We will show in Lemma 4.5 below that hypothesis (S) implies that ρ_{g}u^{∗} is C^{1} in g so that it makes sense to talk about a transverse section to Gu^{∗}. We can not solve (4.1) by the ordinary implicit function theorem because in general F (u, g, ω_{ext}, μ) is only continuous in g. This comes from the fact that the G-action is only strongly continuous and the Lie algebra elements ξ ∈ alg(G) act in general as unbounded operators on Y. Furthermore, the time-evolution does not smoothen the group action, that is, ρ_{g}Φ_{T_{ext},0}(u) is not differentiable in g in general. This is due to the fact that the operators ξ ∈ alg(G) are not assumed to be bounded w.r.t. A (in the case of the reaction-diffusion system (1.1) the operator \frac{∂}{∂ϕ} is not bounded w.r.t. Δ, see Proposition 1.2). Therefore the operator \frac{∂F}{∂u} (u, g, ω_{ext}, μ) is in general not continuous in g with respect to the norm ∥ · ∥_{L(Y)}. We overcome these difficulties as follows:

We will solve the fixed point equation

y = Π(y, q, g, ω_{ext}, μ) = (1 − P̂)ρ_{g}^{−1}Φ_{T_{ext},0}(y + q; ω_{ext}, μ),  (4.2)

y ∈ (1 − P̂)Y, q ∈ P̂Y, by Banach’s contraction mapping theorem. Here P̂ is a projector which is near the projection P onto the center-unstable eigenspace E_{cu} of B^{∗} = ρ_{g^{∗}}^{−1}DΦ_{T_{ext},0}(u^{∗}) in the L(Y)-norm. Furthermore we will show that the solution y(q, g, ω_{ext}, μ) of this fixed point equation depends smoothly on the parameters (q, g, ω_{ext}, μ) and that the G-action on the solutions is smooth. Then we solve the reduced equation F_{red} = 0

F_{red}(q, g, ω_{ext}, μ) = ( \begin{matrix}P̂ρ_{g}^{−1}Φ_{T_{ext},0}(y(q, g, ω_{ext}, μ) + q; ω_{ext}, μ) − q \\ l_{i}(y(q, g, ω_{ext}, μ) + q − u^{∗}) = 0, i = 1, …, m \end{matrix} )

(4.3)

by the implicit function theorem. In this way we can solve (4.1).

4.2 The scale of Banach spaces {Y_{j}}_{j=0,...,k}

For j > 1, define inductively

Y_{j} := {u ∈ Y_{j−1}; ξu ∈ Y_{j−1} for any ξ ∈ alg(G)}, Y_{0} = Y,  (4.4)

equipped with the graph norm | · |_{Y_{j}} given by

|u|_{Y_{j}} = |u|_{Y_{j−1}} + sup_{ξ∈alg(G),|ξ|=1} |ξu|_{Y_{j−1}}.

Let Y^{⋆} be the dual space to Y and define

Z_{0}^{⋆} := {y^{⋆} ∈ Y^{⋆}; ρ_{g}^{⋆}y^{⋆} is C^{0} in g},

where ρ_{g}^{⋆} denotes the adjoint operator of ρ_{g} in Y^{⋆}. For j > 1, we define the spaces Z_{j}^{⋆} with norm | · |_{Z_{j}^{⋆}} for the adjoint group action as in (4.4) with Y_{0} replaced by Z_{0}^{⋆}. In the following we will often use that P ρ_{g} and ρ_{g}P are continuous in g with respect to the norm ∥ · ∥_{L(Y)}. For the second operator this is clear since ρ_{g} is strongly continuous in g and P Y is finite-dimensional. The operator P ρ_{g} is continuous in g with respect to the norm ∥ · ∥_{L(Y)} iff ρ_{g}^{⋆}P^{⋆} is continuous with respect to the norm ∥ · ∥_{L(Y^{⋆})} where P^{⋆} is the spectral projection in Y^{⋆} onto the left center-unstable eigenspace of L^{∗}.

Lemma 4.1 P^{⋆} maps Y^{⋆} into Z_{0}^{⋆}.

If the group G acts strongly continuously on the dual space, for example in the case G = E(2) acting on Y = L^{2}(ℝ^{2}, ℝ^{M}), then Lemma 4.1 is automatically satisfied. Therefore we will skip the proof which is elementary, but technical and can be found in [23, Lemmata 5.1,5.2].

Remark 4.2 If we replace the assumption of a C^{0}-action of G by the assumption that ρ_{g}u^{∗} is continuous in g and the group action is weakly continuous then the theorems in sections 2, 3 still hold.

This is due to the fact that P Y ⊂ Y_{0} is still satisfied, see [23, Lemmata 5.1,5.2] and we can therefore restrict the problem onto Y_{0}. Since ξΦ_{t,t_{0}}(u) = DΦ_{t,t_{0}}(u)ξu we see that Φ_{t,t_{0}} maps Y_{1} into Y_{1}. Inductively we see that the time-evolution Φ_{t,t_{0}} maps each Y_{j}, j ≤ k, into itself. Further Φ_{t,t_{0}} is C^{k−j}-smooth from Y_{j} into Y_{j}. Now we need the following lemma:

Lemma 4.3 Y_{1} is dense in Y_{0} and Z_{1}^{⋆} is dense in Z_{0}^{⋆}. Moreover, G acts as C^{0}-group on Y_{j}, Z_{j}^{⋆}.

The proof can be found in [23, Lemma 4.1]. If dim(G) = 1 this is usual semigroup theory. From this lemma we can deduce

Lemma 4.4 There is a projector P̂ near P such that ρ_{g}P̂ and P̂ρ_{g} are C^{k} in g.

This was shown in [23, Lemma 5.3]. The idea is the following: let e_{i}, i = 1, …, dim(P Y) be a basis of P Y, and e_{i}^{⋆}, i = 1, …, dim(P Y), be a basis for P^{⋆}Y. Then by the foregoing lemma we can find ê_{i} ∈ Y_{k}, ê_{i}^{⋆} ∈ Z_{k}^{⋆} which are near e_{i} rsp. e_{i}^{⋆} in the Y -norm rsp. Y^{⋆}-norm. From these vectors ê_{i}, ê_{i}^{⋆} we ”build” the projection P̂.

4.3 Regularity of the relative periodic orbit

Now we need the following main lemma which will inductively yield C^{k}- regularity of Gu^{∗} and ρ_{g}P, P ρ_{g}:

Lemma 4.5 If Hypothesis (S) is satisfied then u^{∗} ∈ Y_{1}.

Proof. For a proof involving exponential dichotomies see [23]. Here we will give a more elementary proof. In a first step we define a formal expression for ξu^{∗}, ξ ∈ alg G, and in a second step we will show that ξu^{∗} exists and indeed equals this expression. Let P̂ be a projector near P such that ρ_{g}P̂ and P̂ρ_{g} are C^{1} in g in the operator norm on Y and denote Φ = Φ_{T_{ext},0}. Since u^{∗} = ρ_{g^{∗}}^{−1}Φ(u^{∗}) and ξρ_{g}^{−1} = ρ_{g}^{−1} Ad_{g} ξ we have

ξu^{∗} = ρ_{g^{∗}}^{−1}(Ad_{g^{∗}}ξ)Φ(u^{∗}) = ρ_{g^{∗}}^{−1}DΦ(u^{∗})(Ad_{g^{∗}}ξ)u^{∗} = B^{∗}(Ad_{g^{∗}}ξ)u^{∗}

where B^{∗} = ρ_{g^{∗}}^{−1}DΦ(u^{∗}) and so we formally get

z(ξ) = B_{s}z(Ad_{g^{∗}}ξ) + η(ξ).  (4.5)

Here

z(ξ) := (1 − P̂)ξu^{∗}, B_{s} := (1 − P̂)B^{∗}, η(ξ) = B_{s}P̂(Ad_{g^{∗}}ξ)u^{∗}.

Note that z(ξ) and η(ξ) are linear in ξ. Since P̂ is near P and since the spectral radius of (1 − P)B^{∗} is smaller than one also the spectral radius of B_{s} is smaller than one. Let {ξ_{i}, i = 1, …, m} be a basis of alg(G). By our overall hypothesis the operator Ad_{g^{∗}} : alg(G) → alg(G) has spectrum on the unit circle. Let (Ad_{g^{∗}})_{ij} be the matrix associated to the operator Ad_{g^{∗}} with respect to the basis {ξ_{i}, i = 1, …, m} of alg(G). We can define Ad_{g^{∗}} as operator in Y^{m} = Y × … × Y by setting

Ad_{g^{∗}}(z_{1}, …, z_{m}) := (s_{1}, …, s_{m}), s_{i} = ∑_{j=1}^{m}(Ad_{g^{∗}})_{ij}z_{j}, z_{i} ∈ Y, i = 1, …, m.

Also the operator B_{s} can be extended to an operator on Y^{m} by defining

B_{s}(z_{1}, …, z_{m}) := (B_{s}z_{1}, …, B_{s}z_{m}), z_{i} ∈ Y, i = 1, …, m.

Hence the operator B_{s}Ad_{g^{∗}} = Ad_{g^{∗}}B_{s} on Y^{m} has also spectral radius smaller than one. Changing Φ = Φ_{T_{ext},0} to Φ_{ℓT_{ext},0} and accordingly B^{∗} to (B^{∗})^{ℓ} and g^{∗} to (g^{∗})^{ℓ} with ℓ large enough we can achieve that ∥B_{s}∥∥Ad_{g^{∗}}∥ < 1. W.l.o.g. we assume that ℓ = 1. We rewrite (4.5) as

(1 − B_{s}Ad_{g^{∗}})z = η, z = (z_{1}, …, z_{m}), η = (η_{1}, …, η_{m})  (4.6)

where η_{i} = η(ξ_{i}), i = 1, …, m, are well-defined since P̂ρ_{g} is C^{1} in g in the operator norm. The system of equations (4.6) can be solved uniquely for z_{i} = z(ξ_{i}), i = 1, …, m. So we have proved that ξ_{i}u^{∗} = z_{i}+P̂ξ_{i}u^{∗} formally exists for all ξ_{i}, i = 1, …, m, and hence by linear combination we get for each ξ ∈ alg(G) a formal expression z(ξ) + P̂ξu^{∗} which we know equals ξu^{∗} if u^{∗} ∈ Y_{1}. To show that the formal expression z(ξ) is indeed (1 − P̂)ξu^{∗} we argue as follows. Let z(ξ, t) = \frac{1}{t}(1 − P̂)(ρ_{exp(ξt)}u^{∗} − u^{∗}). We have

\begin{aligned}z(ξ, t) & = & \frac{1}{t}(1 − P̂)(ρ_{exp(ξt)}ρ_{g^{∗}}^{−1}Φ(u^{∗}) − ρ_{g^{∗}}^{−1}Φ(u^{∗})) \\ & = & B_{s}(t)z(Ad_{g^{∗}} ξ, t) + η(ξ, t) \end{aligned}  (4.7)

where

B_{s}(t) := (1 − P̂)ρ_{g^{∗}}^{−1}DΦ(u^{∗} + Θ(t)(ρ_{exp(Ad_{g^{∗}} ξt)}u^{∗} − u^{∗}))

with 0 ≤ Θ(t) ≤ 1 and

η(ξ, t) = \frac{1}{t}B_{s}(t)P̂(ρ_{exp(Ad_{g^{∗}} ξt)}u^{∗} − u^{∗}).

Here we applied the mean value theorem. Let δ_{z}(ξ, t) = z(ξ, t) − z(ξ). Then

δ_{z}(ξ, t) = B_{s}(t)δ_{z}(Ad_{g^{∗}}ξ, t) + δ_{η}(ξ, t)  (4.8)

where

δ_{η}(ξ, t) = (B_{s}(t) − B_{s})z(ξ) + η(ξ, t) − η(ξ)

converges to zero as t → 0. Let

ϵ_{z}(t) = sup_{∥ξ∥≤1,|τ|≤t} δ_{z}(ξ, τ), ϵ_{η}(t) = sup_{∥ξ∥≤1,|τ|≤t} δ_{η}(ξ, τ).

Here ∥ξ∥ = (∑_{i=1}^{m} ζ_{i}^{2})^{1/2} for ξ = ∑_{i=1}^{m} ζ_{i}ξ_{i} is a norm on alg G. We define B_{s}(t) like B_{s} as operator from Y^{m} into Y^{m}. Since B_{s}(t) is continuous in t in the L(Y)-norm and ∥B_{s}∥∥ Ad_{g^{∗}} ∥ < 1 we get ∥B_{s}(t)∥∥Ad_{g^{∗}}∥ = c < 1 for t small enough . From (4.8) we get

ϵ_{z}(t) ≤ cϵ_{z}(∥Ad_{g^{∗}}∥t) + ϵ_{η}(t)  (4.9)

with ϵ_{η}(t) → 0 as t → 0. Here we used that

z(Ad_{g^{∗}}ξ, t) = z(\frac{1}{∥Ad_{g^{∗}}∥}Ad_{g^{∗}}ξ, ∥Ad_{g^{∗}}∥t)∥Ad_{g^{∗}}∥,
z(Ad_{g^{∗}}ξ) = z(\frac{1}{∥Ad_{g^{∗}}∥}Ad_{g^{∗}}ξ)∥Ad_{g^{∗}}∥

and that therefore

δ_{z}(Ad_{g^{∗}}ξ, t) = δ_{z}(\frac{1}{∥Ad_{g^{∗}}∥}Ad_{g^{∗}}ξ, ∥Ad_{g^{∗}}∥t)∥Ad_{g^{∗}}∥

and consequently

sup_{∥ξ∥≤1} δ_{z}(Ad_{g^{∗}}ξ, t) ≤ ∥Ad_{g^{∗}}∥ sup_{∥ξ∥≤1} δ_{z}(ξ, ∥Ad_{g^{∗}}∥t).

From (4.9) we conclude that

ϵ_{z}(t) ≤ c^{ℓ}ϵ_{z}(∥ Ad_{g^{∗}} ∥^{ℓ}t) + ∑_{i=0}^{ℓ−1} c^{i}ϵ_{η}(∥ Ad_{g^{∗}} ∥^{i}t),

and hence that

ϵ_{z}(t/∥ Ad_{g^{∗}} ∥^{ℓ}) ≤ \frac{1 − c^{ℓ}}{1 − c} ϵ_{η}(t) + c^{ℓ}ϵ_{z}(t).

Choosing t small enough and ℓ large enough we see that ϵ_{z}(t) → 0 as t → 0.

4.4 Contractions on a scale of Banach spaces

We first show (Lemma 4.6) that Π^{ℓ} is a contraction in (1 − P̂)Y for some ℓ ∈ ℕ. Afterwards, in Lemma 4.7, we show that we can apply the contraction theorem on the Banach scale {(1−P̂)Y_{j}}_{j=0,...,k−1}. Finally Theorem 4.8 below guarantees that the solution we obtained depends smoothly on parameters.

Lemma 4.6 Let u^{∗} be a relative periodic orbit of (1.3) to the parameters (ω_{ext}^{∗}, μ^{∗}) fulfilling the spectral condition (S). Let P̂ be a projection which is L(Y)-near P. Let (g, ω_{ext}, μ) be near (g^{∗}, ω_{ext}^{∗}, μ^{∗}) and let (y + q) be near (y^{∗} + q^{∗}) in the Y -norm with y^{∗}, y ∈ (1 − P̂)Y, q, q^{∗} ∈ P̂Y, q^{∗} + y^{∗} = u^{∗}. Then Π satisfies

∥\frac{∂Π^{ℓ}}{∂y} (y, q, g, ω_{ext}, μ)∥ ≤ c < 1,

where ℓ ∈ ℕ is sufficiently large.

Proof of Lemma 4.6. Again let B^{∗} = ρ_{g^{∗}}^{−1}DΦ_{T_{ext},0}(u^{∗}). We have ∥(B^{∗}(1 − P))^{ℓ}∥ ≤ MC^{ℓ}, C < 1. Let ℓ ∈ ℕ be so large that for g in a neighborhood U_{G} of id in G

∥(1 − P)ρ_{g}(B^{∗})^{ℓ}(1 − P)∥ ≤ ∥(1 − P)∥M_{G}C^{ℓ}M < 1.

Here we used that for g ∈ U_{G} there is a uniform bound M_{G} of ∥ρ_{g}∥. Then (1 − P)ρ_{g}(B^{∗})^{ℓ}(1 − P) is a uniform contraction for g ∈ U_{G}. We have

D_{y}Π^{ℓ}(y) = ∏_{i=0}^{ℓ−1} DΠ(Π^{i}(y)) = ∏_{i=0}^{ℓ−1}(1 − P̂)ρ_{g}^{−1}D_{z}Φ_{T_{ext},0}(q + z)|_{z=Π^{i}(y)}.

Since y is near y^{∗}, q is near q^{∗} and g is near g^{∗} we know that Π^{i}(y) ≈ y^{∗} and that

ρ_{g}^{−1}D_{u}Φ_{T_{ext},0}(u)|_{u=q+Π^{i}(y)} ≈ ρ_{g}^{−1}ρ_{g^{∗}}B^{∗}

in the operator norm. Since P̂ is near P in the ∥ · ∥_{L(Y)}-norm we conclude that

D_{y}Π^{ℓ}(y) ≈ ∏_{i=0}^{ℓ−1}(1 − P)ρ_{g^{−1}g^{∗}}B^{∗}(1 − P)

in the norm on L(Y). Further we compute

(ρ_{g^{−1}g^{∗}}B^{∗})^{2} = ρ_{g^{−1}}D_{y}Φ_{T_{ext},0}(u^{∗})ρ_{g^{−1}g^{∗}}B^{∗} ≈ ρ_{g^{−1}}ρ_{g^{−1}g^{∗}}D_{y}Φ_{T_{ext},0}(u^{∗})B^{∗}
= ρ_{g^{−2}(g^{∗})^{2}}(B^{∗})^{2}.

Similarly we get

(ρ_{g^{−1}g^{∗}}B^{∗})^{ℓ} ≈ ρ_{g^{−ℓ}(g^{∗})^{ℓ}}(B^{∗})^{ℓ}.

Since ρ_{g}P and P ρ_{g} are continuous in g in the operator norm we conclude that D_{y}Π^{ℓ}(y) is near (1−P)ρ_{g^{−ℓ}(g^{∗})^{ℓ}}(B^{∗})^{ℓ}(1−P) for g near g^{∗}, y near y^{∗}, q near q^{∗} in the operator norm. Hence \frac{∂Π^{ℓ}}{∂y} (y, q, g, ω_{ext}, μ) is a contraction if we choose (y + q, g, ω_{ext}, μ) near (y^{∗} + q^{∗}, g^{∗}, ω_{ext}^{∗}, μ^{∗}) (here we measure y − y^{∗}, q − q^{∗} in the Y -norm). Now we show that Π^{ℓ} is a contraction on the scale of Banach spaces {(1 − P̂)Y_{j}}_{j=0,...,k−1} for some ℓ = ℓ(k) ∈ ℕ.

Lemma 4.7 Let u^{∗} be a relative periodic orbit of (1.3) to the parameters (ω_{ext}^{∗}, μ^{∗}) fulfilling Hypothesis (S). If f is C^{k}-smooth, k ∈ ℕ, then we have:

(i) u^{∗} ∈ Y_{k}.

(ii) B^{∗}Y_{j} ⊆ Y_{j}, (B^{∗})^{⋆}Z_{j}^{⋆} ⊆ Z_{j}^{⋆}, and spec(B_{j}^{∗}) ⊂ spec(B^{∗}), j = 1, …, k − 1, where B_{j}^{∗} is the operator B^{∗} considered as map from Y_{j} into itself. Further, P ∈ L(Y, Y_{k−1}), P^{⋆} ∈ L(Y^{⋆}, Z_{k−1}^{⋆}).

(iii) u^{∗} satisfies Hypothesis (S) on each Y_{j}, 0 ≤ j ≤ k − 1.

(iv) Let P̂ be L(Y, Y_{k−1})-near P. If ℓ = ℓ(k) ∈ ℕ is large enough then the function y → Π^{ℓ}(y, q, g, ω_{ext}, μ) from (4.2) is a uniform contraction on each Y_{j}, 0 ≤ j ≤ k − 1, for y + q Y_{j}-near u^{∗} and (g, ω_{ext}, μ) near (g^{∗}, ω_{ext}^{∗}, μ^{∗}).

(v) Let P̂ be as in (iv) and assume that ρ_{g}P̂ and P̂ρ_{g} are C^{k}-smooth in the L(Y)-norm. Then there is a locally unique solution y(q, g, ω_{ext}, μ) ∈ (1 − P̂)Y of (4.2) which is continuous in (q, g, ω_{ext}, μ) with respect to the norm ∥ · ∥_{Y_{k}}.

Part (i) of this lemma can also be found in [23]. Proof of Lemma 4.7. Suppose that u^{∗} ∈ Y_{j} for some j with j ≥ 1, j < k. Since Φ_{t,t_{0}} is a time-evolution on each Y_{j} and G acts as C^{0}-group on each Y_{j} w.r.t. the Y_{j}-norm by Lemma 4.3 we know that B^{∗} ∈ L(Y_{i}), i ≤ j. We have

ξ(B^{∗} − λ) = (B^{∗} − λ) Ad_{g^{∗}} ξ + V (ξ),

with

V (ξ) := ∂_{u}^{2}Φ_{T_{ext},0}(u^{∗})(Ad_{g^{∗}} ξ)u^{∗} ∈ L(Y_{j−1}).

Let λ ∈ ℂ \ spec(B^{∗}) lie in the resolvent set of B^{∗}. Then we get

Ad_{g^{∗}} ξ(B^{∗} − λ)^{−1} = (B^{∗} − λ)^{−1}ξ − (B^{∗} − λ)^{−1}V (ξ)(B^{∗} − λ)^{−1}. (4.10)

Let B_{j}^{∗} be the operator B^{∗} considered as element of L(Y_{j}). From (4.10) we deduce that spec(B_{j}^{∗}) ⊂ spec(B_{j−1}^{∗}) ⊂ … ⊂ spec(B_{0}^{∗}). Let σ be the spectral set of the center-unstable eigenvalues of B^{∗}. Then

P = \frac{1}{2πi} ∮_{around σ}(λ − B^{∗})^{−1}dλ.  (4.11)

From (4.11) we see that P maps Y_{j} into itself if u^{∗} ∈ Y_{j}. Since Y_{j} is dense in Y by iterative application of Lemma 4.3 we can find w_{i} ∈ Y_{j}, i = 1, … dim(P Y), such that P w_{i}, i = 1, … dim(P Y), span P Y. Hence P Y ⊆ Y_{j}. Since ξu^{∗} ∈ P Y, ξ ∈ alg(G), we infer u^{∗} ∈ Y_{j+1}. According to Lemma 4.5 we have u^{∗} ∈ Y_{1} if k ≥ 1. Hence by induction we obtain

u^{∗} ∈ Y_{k}, P Y ⊆ Y_{k−1}.

By computing the adjoints on both sides of equation (4.10) we see that B^{⋆}Z_{j}^{⋆} ⊂ Z_{j}^{⋆}, 0 ≤ j ≤ k − 1. Analogously as above we obtain P^{⋆}Y^{⋆} ⊂ Z_{k−1}^{⋆}. Using (i) and (ii) we conclude that u^{∗} satisfies condition (S) on each Y_{j}, j ≤ k − 1. To prove (iv) we apply Lemma 4.6 on each Y_{j}, j ≤ k − 1. Applying the contraction principle on each Y_{j}, j ≤ k − 1, we obtain solutions y_{j}(q, g, ω_{ext}, μ) of y = Π^{ℓ}(y) which are continuous in the parameters and locally unique in Y_{j} and therefore solutions of (4.2). Since Y_{j} ⊆ Y for all j the solutions are the same solution y(q, g, ω_{ext}, μ). Since with y = y(q, g, ω_{ext}, μ) also Π^{i}(y), i ∈ ℤ, are solutions of y = Π^{ℓ}(y) and the solution is locally uniqute we know that y(q, g, ω_{ext}, μ) is a solution of (4.2). In the same way as in Lemma 4.5 we can show that y = y(q, g, ω_{ext}, μ) ∈ Y_{k}: From y = Π(y) we formally get the identity

z(ξ) = B_{s}z(Ad_{g^{∗}} ξ) + η(ξ)

on Y_{k−1} where z(ξ) = (1 − P̂)ξy(q, g, ω_{ext}, μ), B_{s} = DΠ(q + y)(1 − P̂) and

η(ξ) = −(1 − P̂)ξP̂ρ_{g}^{−1}Φ_{T_{ext},0}(y + q)
+(1 − P̂)ρ_{g}^{−1}DΦ_{T_{ext},0}(y + q)(P̂ Ad_{g^{∗}} ξy + Ad_{g^{∗}} ξq).

The operator η(ξ) is well-defined for all y ∈ Y and maps into Y_{k−1} because ρ_{g}P̂ and P̂ρ_{g} are C^{k}-smooth in the L(Y)-norm. Since B_{s} has spectral radius smaller than one this equation can be solved uniquely for z(ξ_{i}), i = 1, …, m. In the same way as in the proof of Lemma 4.5 we can now show that the formal derivative z(ξ) + P̂ξy(q, g, ω_{ext}, μ) is indeed the derivative ξy(q, g, ω_{ext}, μ). We infer that y(q, g, ω_{ext}, μ) is continuous in its parameters in the norm of Y_{k}. In order to show that the solutions really depend C^{k}-smoothly on their parameters we will use a contraction mapping theorem on a scale of Banach spaces. This idea has frequently been used in the literature, for example it is used to prove the smoothness of center manifolds (Vanderbauwhede & Van Gils [30], Vanderbauwhede & Iooss [29]). Renardy [19] proved a generalized implicit function theorem on a scale of Banach spaces {Y_{j}}_{0≤j≤k}; he required that the derivative of the nonlinear equation to be solved evaluated at the starting solution depends continuously on the parameter with respect to the norm ∥·∥_{L(Y_{j})}. As in [30] we will assume that the derivative is a contraction. Hard implicit function theorems can be found in Nirenberg [17]. We will employ the following theorem which is stated in general form in [30] for k = 1.

Theorem 4.8 Let Y = Y_{0} ⊃ Y_{1} ⊃ … ⊃ Y_{k}, k ≥ 1, be a scale of Banach spaces with norms ∥ · ∥_{Y_{j}}, j ≤ k, and let Y_{j} be continuously embedded in Y_{j−1}. Let (u, ν) → Π(u, ν) be a nonlinear map from some open set U ⊂ Y × ℝ^{p} into Y. Assume the following:

(i) Π maps U_{j} := (Y_{j} × ℝ^{p}) ∩ U into Y_{j} and Π is C^{ℓ−j}-smooth from U_{ℓ} to Y_{j}, j, ℓ ∈ ℕ_{0}, k ≥ ℓ ≥ j ≥ 0.

(ii) (ν, u, w_{1}, …, w_{j}) → \frac{∂^{j+ℓ}}{∂u^{j}∂ν^{ℓ}} Π(u, ν)(w_{1}, …, w_{j}) is continuous as map from U_{i} × (Y_{i})^{j} into L^{ℓ}(ℝ^{p}, Y_{i−ℓ}), for i, j, ℓ ∈ ℕ_{0}, ℓ ≤ i ≤ k, j + ℓ ≤ k, where L^{0}(ℝ^{p}, Y_{i}) := Y_{i}.

(iii) Π(·, ν) is a uniform contraction as map from U_{j} into Y_{j}, 0 ≤ j ≤ k − 1, with contraction constant c < 1.

Then

a) there is a unique solution u(ν) ∈ Y_{k−1} to Π(u, ν) = u and u(ν) is a C^{k−1}-function of ν with respect to the norm ∥ · ∥_{Y}.

b) If we require in addition

(iv) u(ν) is continuous in the norm ∥ · ∥_{Y_{k}}

then u(ν) is a C^{k}-function of ν with respect to the norm ∥ · ∥_{Y}.

Proof. We can apply Banach’s fixed point theorem on each Y_{j}, 0 ≤ j ≤ k −1, and since Y_{j} ⊂ Y for 0 ≤ j ≤ k the solutions are all equal to u(ν). Under assumptions (i)–(iii) we can formally compute the first (k − 1) derivatives of u(ν) considered as lying in Y, if we assume hypotheses (i)–(iv) then we can even compute the formal k-th derivative of u(ν) considered as lying in Y. It remains to be shown that the formal derivatives are indeed the derivatives of u(ν). For k = 1 the proof can be found in [30]. The rest is induction over k. Since this theorem is the main technical tool of our results we present the whole proof of the theorem.

1. Step. We first show that the solution u(ν) is a C^{1}-function of ν with respect to the norm of Y. Assuming that u(ν) is C^{0} in ν in the Y_{1}-norm the formal derivative κ(ν) is given by the equation

κ(ν) − (∂_{u}Π)(u(ν), ν)κ(ν) = (∂_{ν}Π)(u, ν)|_{u=u(ν)}.

Since ∥(∂_{u}Π)(u(ν), ν)∥_{L(Y)} ≤ c < 1 this equation can be solved uniquely for κ(ν) ∈ Y. Furthermore, due to our assumption, κ(ν) ∈ Y depends continuously on ν. We consider a fixed ν. In order to prove that κ(ν) = ∂_{ν}u(ν) we have to show that

∥u(ν + ν̃) − u(ν) − κ(ν)ν̃∥_{Y} = o(ν̃).  (4.12)

Multiplying

u(ν + ν̃) − u(ν) − κ(ν)ν̃ = u(ν + ν̃) − u(ν)
−ν̃(1 − (∂_{u}Π)(u(ν), ν))^{−1}(∂_{ν}Π)(u, ν)|_{u=u(ν)}

by (1 − (∂_{u}Π)(u(ν), ν)) we see that (4.12) is equivalent to ∥θ(ũ, ν̃)∥_{Y} = o(ν̃) where ũ = u(ν + ν̃) − u(ν) and

θ(ũ, ν̃) = Π(u(ν + ν̃), ν + ν̃) − Π(u(ν), ν) − (∂_{u}Π)(u(ν), ν)ũ − (∂_{ν}Π)(u(ν), ν)ν̃.

We can estimate

∥θ(ũ, ν̃)∥_{Y} ≤ ∥Π(u(ν + ν̃), ν + ν̃) − Π(u(ν + ν̃), ν) − (∂_{ν}Π)(u(ν), ν)ν̃∥_{Y}
+∥Π(u(ν + ν̃), ν) − Π(u(ν), ν) − (∂_{u}Π)(u(ν), ν)ũ∥_{Y}
= ∥(∂_{ν}Π)(u(ν + ν̃), ν)ν̃ − (∂_{ν}Π)(u(ν), ν)ν̃∥_{Y}
+o(∥ũ∥_{Y}) + o(ν̃)
≤ o(ν̃) + o(∥ũ∥_{Y}).

(4.13)

Here we used that u(ν) is a C^{0}-function of ν with respect to the norm ∥ · ∥_{Y_{1}}. This follows from Banach’s contraction mapping theorem applied onto Y_{1} or, if k = 1, from the additional assumption (iv). It holds

θ(ũ, ν̃) = (1 − (∂_{u}Π)(u(ν), ν))ũ − (∂_{ν}Π)(u(ν), ν)ν̃.

Hence

∥ũ∥_{Y} ≤ \frac{1}{1 − c}(∥(∂_{ν}Π)(u(ν), ν)∥_{Y}|ν̃| + ∥θ(ũ, ν̃)∥_{Y}) ≤ \frac{1}{1 − c}(c̃|ν̃| + o(∥ũ∥_{Y})).

Thus, we obtain ∥ũ∥_{Y} ≤ ĉ|ν̃| for |ν̃| small. From (4.13) we conclude that ∥θ(ũ, ν̃)∥_{Y} = o(ν̃). Hence u(ν) is a C^{1}-function of ν with respect to the norm ∥ · ∥_{Y}.

2. Step. To show that u(ν) is a C^{i}-function of ν, i > 1, we proceed by induction. If the theorem holds for k = (j − 1), j ≥ 2, and Π satisfies assumptions (i) –(iii) of the theorem with k = j then by the contraction principle applied on Y_{j−1} the function u(ν) is continuous in ν with respect to the norm ∥ · ∥_{Y_{j−1}}. Hence by part b) of the theorem for k = (j − 1) we conclude that u(ν) is C^{j−1}- smooth in ν when considered as lying in Y. This proves part a) of the theorem for k = j. Now we come to part b). If Π satisfies assumptions (i)–(iv) of the theorem for k = j then u(ν) is a C^{j−1}-function of ν in the Y-norm and u(ν) is a C^{j−ℓ}-function of ν with respect to the norm ∥ · ∥_{Y_{ℓ}}, j ≥ ℓ ≥ 1. Therefore we can apply part b) of the theorem with k = (j − 1) onto the differentiated equation

(1 − (∂_{u}Π)(u(ν), ν))∂_{ν}u(ν) = (∂_{ν}Π)(u, ν)|_{u=u(ν)}.

and conclude that ∂_{ν}u(ν) is a C^{j−1}-function of ν and that u is a C^{j}-function of ν with respect to the norm ∥ · ∥_{Y}.

4.5 Proof of the theorems on periodic forcing

We prove Theorem 2.3 by applying Theorem 4.8 onto (4.2) with ν = (g, q, ω_{ext}, μ) and with the hierarchy Y_{j} of Banach spaces defined by

Y_{j} = (1 − P̂)Y_{j}, Y_{j} given by (4.4), 0 ≤ j ≤ k.

As before P̂ is a projection which is L(Y, Y_{k−1})-near the spectral projection P onto the center-unstable eigenspace and such that P̂ρ_{g} and ρ_{g}P̂ are C^{k}-smooth in g in the L(Y)-norm. We consider the fixed point equation y = Π^{ℓ}(y) with ℓ so large that DΠ^{ℓ} is a contraction on each Y_{j}, 0 ≤ j ≤ k − 1, k ∈ ℕ. Because of Lemma 4.7 all assumptions of Theorem 4.8 are satisfied. So there is a locally unique solution y(q, g, ω_{ext}, μ) ∈ Y_{k} of Π^{ℓ}(y) = y if (q, g, ω_{ext}, μ) is near (q^{∗}, g^{∗}, ω_{ext}^{∗}, μ^{∗}) satisfying y(q^{∗}, g^{∗}, ω_{ext}^{∗}, μ^{∗}) = y^{∗} and y(q, g, ω_{ext}, μ) is a C^{k−j}-function of (q, g, ω_{ext}, μ) with respect to the norm ∥ · ∥_{Y_{j}}, 0 ≤ j ≤ k. As in the proof of Lemma 4.7 we can argue that y(q, g, ω_{ext}, μ) is also a solution of (4.2) since with y = y(q, g, ω_{ext}, μ) also Π^{i}(y), i ∈ ℤ, are solutions of Π^{ℓ}(y) = y and since the solution of y = Π^{ℓ}(y) is locally unique. The reduced equation (4.3) is C^{k−j}-smooth in its variables if y(q, g, ω_{ext}, μ) is considered as lying in Y_{j}. Solving the reduced equation by the ordinary implicit function theorem we obtain relative periodic orbits Φ_{T_{ext},0}(u(ω_{ext}, μ)) = ρ_{g(ω_{ext},μ)}u(ω_{ext}, μ) of (1.3) to the parameters ω_{ext}, μ with (ω_{ext}, μ) near (ω_{ext}^{∗}, μ^{∗}). Here g(ω_{ext}, μ) is C^{k}-smooth in (ω_{ext}, μ) and u(ω_{ext}, μ) depends C^{k−j}-smoothly on (ω_{ext}, μ) when considered as lying in Y_{j}. Proposition 2.17 is proved along the same lines.

4.6 Proof of the results on Hopf bifurcation by use of Lyapunov-Schmidt-reduction

In this section we will prove Theorem 3.2 on Hopf bifurcation and Proposition 3.7 on equivariant Hopf bifurcation by Lyapunov-Schmidt-reduction. First we will prove Lemma 3.1 on the eigenvalue path β(μ).

4.6.1 Proof of Lemma 3.1

Let P be the projection onto the center-unstable eigenspace of u^{∗}. By Lemma 4.7 we have P ∈ L(Y, Y_{k−1}) and P^{⋆} ∈ L(Y^{⋆}, Z_{k−1}^{⋆}) with the hierarchy of Banach spaces {Y_{j}}_{0≤j≤k} defined by (4.4). Let u^{∗} satisfy Hypothesis (S) and let u^{∗}(μ) ∈ S_{l}, μ small, be the C^{k}-smooth manifold of relative equilibria of (1.3) such that u^{∗}(0) = u^{∗} (cf. Section 3.1). We will show:

Lemma 4.9 If the above assumptions hold then u^{∗}(μ) satisfies Hypothesis (S) and the center-unstable spectral projection P of L^{∗} can be continued to a spectral projection P (μ) of L^{∗}(μ) such that P (μ) is C^{k−1} in μ in the space L(Y).

Proof. Let B(μ) = ρ_{exp(−ξ^{∗}(μ)t)}DΦ_{t}(u^{∗}(μ); μ) be the linearization at the relative equilibrium u^{∗}(μ) in the comoving frame and choose t so large that ∥(1 − P)B(μ)∥ ≤ c < 1 for small μ. Let λ ∈ ℂ, |λ| > c. Due to Lemma 4.7 the equation

y = \frac{1}{λ}(1 − P)(B(μ)y − w + (B(μ) − λ)q), q ∈ P Y, w ∈ Y_{k−1}

can be solved to get a solution y(q, μ, λ) for μ small enough. By Theorem 4.8 the solution y(q, μ, λ) is C^{k−1−j} in μ in the norm ∥ · ∥_{Y_{j}}. Now we solve the equation

P (B(μ) − λ)(y(q, μ, λ) + q) = P w

by the implicit function theorem. We conclude that (B(μ) − λ)^{−1}w is C^{k−j−1} in μ in the space Y_{j}. Let σ denote the center-unstable spectral set of B(0) and denote

P (μ) = \frac{1}{2πi} ∮_{around σ}(λ − B(μ))^{−1}dλ.

Since P ∈ L(Y, Y_{k−1}) we know that ρ_{g}P (μ)P is C^{k−1}-smooth in (g, μ) in the space L(Y). Since P^{⋆} ∈ L(Y^{⋆}, Z_{k−1}^{⋆}) we can apply the same arguments on the dual space which yields that ρ_{g}^{⋆}P^{⋆}(μ)P^{⋆} is C^{k−1} in (g, μ) in the space L(Y). The operator P (μ) is a linear combination of the operators ⟨P^{⋆}(μ)e_{i}^{⋆}, ·⟩P (μ)e_{i} where {e_{i}}_{i=1,...,m} is a basis of P Y and {e_{i}^{⋆}}_{i=1,...,m} is a basis of P^{⋆}Y. Consequently ρ_{g}P (μ) and P (μ)ρ_{g} are C^{k−1} in (g, μ) in the space L(Y).

Let Ψ_{t}(·) be the semiflow on Ŝ_{l} in a comoving frame

Ψ_{t}(u; μ) = ρ_{g(Φ_{t}(u,μ))}^{−1}Φ_{t}(u + u^{∗}(μ); μ) − u^{∗}(μ)

where u ∈ Ŝ_{l} is Y_{1}-near u^{∗}, and g(u) is such that ρ_{g(u)}^{−1}u ∈ S_{l}, see also (3.5).

Let DΨ_{t}(0; 0) = e^{L̃t} and denote by P_{l} the projection onto the space Ŝ_{l} such that P_{l} alg(G)u^{∗} = 0. Then L̃ = P_{l}L^{∗}P_{l}. Similarly denote DΨ_{t}(0; μ) = e^{L̃(μ)t}. Then L̃(μ) = P_{l}(μ)L^{∗}(μ)P_{l}(μ) where

P_{l}(μ)y = y − ∑_{i=1}^{m} α_{i}(μ)(y)ξ_{i}u^{∗}(μ)

and α_{i}(μ) ∈ Y^{⋆} are such that l_{i}(y − ∑_{i=1}^{m} α_{i}(μ)(y)ξ_{i}u^{∗}(μ)) = 0, i = 1, …, m. By the above Lemma 4.9 P_{l}(μ) is C^{k−1} in μ in the space L(Y) and the operator L̃(μ)P (μ) is C^{k−1} in μ in the space L(Y). So the simple eigenvalue β(0) = i of L̃ can be continued to a C^{k−1} smooth path of eigenvalues β(μ) of L̃(μ) with C^{k−1} smooth path of eigenvectors w(μ). Note that β(μ) is an eigenvalue of L(μ) as well.

4.6.2 Proof of Theorem 3.2

We will study the solutions of the equation

0 = F (u, g, T, μ) := ( \begin{matrix}ρ_{g}^{−1}Φ_{T} (u, μ) − u \\ l_{i}(u − u^{∗}(μ)), i = 1, …, m \end{matrix} ) ,  (4.14)

where l_{i} ∈ Y^{⋆} and the conditions l_{i}(u − u^{∗}) = 0, i = 1, …, m, define a section transversely to the G-orbit of u^{∗}. Later on, we will need an additional condition to take care of the time-shift symmetry of the relative periodic orbits which we want to find. The map F is smooth in u, μ, T for T > 0, but only continuous in g. Further ∂_{u}F (u, g, T, μ) is not continuous in g with respect to the norm ∥ · ∥_{L(Y)}. So we can not use the usual Lyapunov-Schmidt-reduction to solve equation (4.14), but (4.14) fits into the setting which we treated in the preceding subsections, and we will use the techniques developed in these subsections to solve (4.14). We can find a projector P̂ which is near P in the norm of L(Y, Y_{k−1}) and such that ρ_{g}P̂ and P̂ρ_{g} are C^{k}-smooth in g in the norm of L(Y). Consider the fixed point equation

y = Π(y, q, g, T, μ) := (1 − P̂)ρ_{g}^{−1}Φ_{T} (y + q, μ),

with y ∈ (1 − P̂)Y, q ∈ P̂Y, on the scale of Banach spaces Y_{j}, 0 ≤ j ≤ k. This fixed point equation equals (4.2) from Section 4 with T_{ext} replaced by T and Φ_{t}(·) autonomous. So we get a solution y(q, g, T, μ) of the fixed point equation which is C^{k−j}-smooth in its parameters in the Y_{j}-norm. Now we are ready to solve the reduced equation F_{red}(q, g, T, μ) = 0 with F_{red} given by

F_{red}(q, g, T, μ) = ( \begin{matrix}P (ρ_{g}^{−1}Φ_{T} (q + y(q, g, T, μ), μ) − q − y(q, g, T, μ)) \\ l_{i}(u − u^{∗}), i = 1, …, m \end{matrix} ) .

(4.15)

The map F_{red} is C^{k−j}-smooth in its variables when considered as map from Y_{j} into Y_{j}. The rest of the proof is standard, see [6]: Let w be the eigenvector of L̃ to the eigenvalue i. Let ⟨w^{⋆}, ·⟩ belong to the left eigenspace of L^{∗} to the eigenvalue i such that

⟨Re w^{⋆}, Re w⟩ = ⟨Im w^{⋆}, Im w⟩ = 1,
⟨Re w^{⋆}, Im w⟩ = ⟨Im w^{⋆}, Re w⟩ = 0

(4.16)

is satisfied and ⟨w^{⋆}, alg Gu^{∗}⟩ = 0, i = 1 … m. Let

s(q, g, T, μ) := \frac{1}{2π} ⟨Re w^{⋆}, ∫_{0}^{2π} e^{L̃(2π−t)}Ψ_{\frac{T t}{2π}} (y(q, g, T, μ) + q − u^{∗}(μ); μ)dt⟩

where Ψ_{t} is the semiflow in a comoving frame as defined in (3.5). We first compute q = q(s, T, μ) and g = g(s, T, μ) as functions of s, T and μ by solving

F_{red} − ⟨Re w^{⋆}, F_{red}⟩ Re w − ⟨Im w^{⋆}, F_{red}⟩ Im w = 0.

and

⟨Im w^{⋆}, ∫_{0}^{2π} e^{L̃(2π−t)}Ψ_{\frac{T t}{2π}} (y(q, g, T, μ) + q − u^{∗}(μ); μ)dt⟩ = 0.

The last condition fixes the time-shift. Now we still have to solve the C^{k}- function F̂ : ℝ^{3} → ℝ^{2}, (s, T, μ) → F̂(s, T, μ) given by

F̂(s, T, μ) = \binom{⟨Im w^{⋆}, F_{red}(q(s, T, μ), g(s, T, μ), T, μ)⟩}{⟨Re w^{⋆}, F_{red}(q(s, T, μ), g(s, T, μ), T, μ)⟩} = 0.

Obviously F̂(0, T, μ) = 0, ∂_{(s,T,μ)}F̂(0, 2π, 0) = 0. We define

F_{LS}(s, T, μ) := \frac{1}{s}F̂(s, T, μ).

Since

lim_{s→0} \frac{1}{s}F̂(s, T, μ) = ∂_{s}F_{LS}(0, T, μ)

and ∂_{s}F̂(0, 2π, 0) = 0 we have F_{LS}(0, 2π, 0) = 0. Furthermore ∂_{T} F_{LS}(0, T, μ) = ∂_{T} ∂_{s}F̂(0, T, μ) and ∂_{μ}F_{LS}(0, T, μ) = ∂_{μ}∂_{s}F̂(0, T, μ), since ∂_{(T,μ)}F̂(0, T, μ) = 0.

Lemma 4.10 Under the assumptions of Theorem 3.2 the derivative

∂_{(T,μ)}F_{LS}(0, 2π, 0)

of F_{LS} in (s, T, μ) = (0, 2π, 0) has full rank.

Proof of Lemma 4.10. We have

∂_{(T,μ)}F_{LS}(0, 2π, 0) = \binom{⟨Im w^{⋆}, ∂_{(T,μ)}∂_{u}Ψ_{2π}(0; 0) Re w⟩}{⟨Re w^{⋆}, ∂_{(T,μ)}∂_{u}Ψ_{2π}(0; 0) Re w⟩}  (4.17)

We invoke the following lemma which is the adaption of a lemma in Crandall & Rabinowitz [6] to our setting.

Lemma 4.11 Let assumptions (i)–(iii) of Theorem 3.2 hold. Then

⟨Re w^{⋆}, ∂_{μ}∂_{u}Ψ_{2π}(0; 0) Re w⟩ = 2π Re \frac{∂β}{∂μ}(0),
⟨Im w^{⋆}, ∂_{μ}∂_{u}Ψ_{2π}(0; 0) Re w⟩ = −2π Im \frac{∂β}{∂μ}(0), (4.18)

where ⟨w^{⋆}, ·⟩ is the left eigenvector of L̃ to the eigenvalue i which satisfies (4.16).

We have

∂_{T} ∂_{u}Ψ_{2π}(0; 0) Re w = L̃e^{L̃2π} Re w = Im w.

Using Lemma 4.11 and condition (iv) we conclude that ∂_{(T,μ)}F_{LS}(0, 2π, 0) has full rank. Because of Lemma 4.10 we can apply the ordinary implicit function theorem to obtain solutions u(s, μ_{2}) := u^{∗}(μ(s, μ_{2}))+z(s, μ_{2}), g(s, μ_{2}), T (s, μ_{2}), μ(s, μ_{2}) = (μ_{1}(s, μ_{2}), μ_{2}) of (4.14) which are relative periodic orbits with l_{i}(z(s, μ_{2})) = 0, i = 1, …, m. Here μ(s, μ_{2}), g(s, μ_{2}), T (s, μ_{2}) are C^{k−1}-smooth in s, μ_{2}. Moreover, z(s, μ_{2}) is C^{k−1}-smooth in the ∥ · ∥_{Y} -norm and C^{k−j−1}-smooth in the ∥ · ∥_{Y_{j}}-norm, 1 ≤ j ≤ k − 1. We have z(−s, μ_{2}) = Ψ_{\frac{T (s,μ_{2})}{2}} (z(s, μ_{2}); μ(s, μ_{2})). Since μ(s, μ_{2}), T (s, μ_{2}) do not depend on the time-shift, they are even in s. The solutions u(s) = u^{∗}(μ(s)) + z(s), g(s), T (s), μ(s) of (4.14) which we obtained above (for convenience we ignore the μ_{2}-dependence of the solutions in the notation) depend C^{k−1}-smoothly on the chosen eigenvector of L̃ to the eigenvalue i. We can also consider z_{l} ∈ Y, g_{l}, T_{l}, μ_{l} as C^{k−1}-smooth functions of (s_{1} + is_{2})w where s_{1}, s_{2} ∈ ℝ and w is the originally chosen eigenvector of L̃ to i. As before, z_{l}(s_{1}, s_{2}) is C^{k−j−1}-smooth in the Y_{j}-norm, 1 ≤ j < k. We have z(s) = z_{l}(s_{1}, 0) and

e^{iτ} = ⟨Re w^{⋆} + i Im w^{⋆}, ∫_{0}^{2π} e^{L̃(2π−t)}Ψ_{\frac{(t+τ)T (s)}{2π}} (z(s); μ(s))dt⟩.

Obviously μ(s) = μ_{l}(s_{1}, s_{2}), T (s) = T_{l}(s_{1}, s_{2}) only depend on s = ∥(s_{1}, s_{2})∥,

z_{l}(s_{1}, s_{2}) := Ψ_{\frac{τT (s)}{2π}} (z(s); μ(s)), where s_{1} = s cos τ, s_{2} = s sin τ,

and u_{l}(s_{1}, s_{2}) = u^{∗}(μ(s)) + z_{l}(s_{1}, s_{2}).

4.6.3 Equivariant Hopf bifurcation

In this subsection we prove Lemma 3.7, see also section 3.3. If the isotropy K of the relative equilibrium u^{∗} is non-trivial it may happen that forced by symmetry the eigenspace of the K-equivariant matrix L̃ to the Hopf eigenvalue i has dimension higher than 2. Then the assumptions of Theorem 3.2 are not satisfied any more. We choose functionals l_{i}, i = 1, …, m_{K}, which define a section S_{l} = u^{∗} + Ŝ_{l} transversal to the group orbit Gu^{∗} in u^{∗} such that Ŝ_{l} is K-invariant and that P_{l} is K-equivariant, see subsection 3.3. Here m_{K} = dim(G/K). If Θ(K) = Z_{ℓ} then we solve the equation

F (u, g, T, μ) = ( \begin{matrix}ρ(gh^{∗})^{−1}Φ_{T} (u, μ) − u \\ l_{i}(u − u^{∗}), i = 1, … m_{K} \end{matrix} ) = 0  (4.19)

on Fix(K_{bif}) where T ≈ 2π/ℓ, g ≈ e^{\frac{2π}{ℓ} ξ^{∗}}, the group H is generated by h^{∗} ∈ K and K_{bif} = ker(Θ) is axial. By our assumptions DF_{u}(u^{∗}, e^{\frac{2π}{ℓ} ξ^{∗}}, 2π/ℓ, 0)|Ŝ_{l} has a two-dimensional kernel and therefore (4.19) can be solved by the methods of subsection 4.6.2. If Θ(K) = S^{1} we solve

F (u, g, T, μ) = ( \begin{matrix}ρ_{g exp(−χ^{∗}T)}^{−1}Φ_{T} (u, μ) − u \\ l_{i}(u − u^{∗}), i = 1, … m_{K} \end{matrix} ) = 0  (4.20)

on Fix(K_{bif}). In this case the axial group H is generated by χ^{∗} ∈ alg(K) and K_{bif} = ker(Θ). We choose T such that (ρ_{e^{−χ^{∗}T}} e^{L̃T} −1)|Ŝ_{l} has a two-dimensional kernel. This is possible because the number of center eigenvalues of L̃ is finite. Then we can proceed as before.

References

[1] P. Ashwin and I. Melbourne. Noncompact drift for relative equilibria and relative periodic orbits. Nonlinearity, 10:595 – 616, 1997.

[2] P. Ashwin, I. Melbourne and M. Nicol. Drift bifurcations of relative equilibria and transitions of spiral waves. Nonlinearity, 12:741–755, 1999.

[3] D. Barkley. Linear stability analysis of rotating spiral waves in excitable media. Phys. Rev. Lett., 68(13):2090–2093, 1992.

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

[5] M. Braune and H. Engel. Compound rotation of spiral waves in a lightsensitive Belousov-Zhabotinsky medium. Chem. Phys. Lett., 204(3,4):257– 264, 1993.

[6] M. Crandall and P. Rabinowitz. The Hopf bifurcation theorem in infinite dimensions. Arch. Rat. Mech. Anal., 67:53–72, 1977.

[7] B. Fiedler, B. Sandstede, A. Scheel, and C. Wulff. Bifurcation from relative equilibria to non-compact group actions: Skew products, meanders, and drifts. Doc. Math. J. DMV, 1:479 – 505, 1996.

[8] B. Fiedler, D. Turaev. Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions. Arch. Rat. Mech. Anal., 145:129 – 159, 1998.

[9] M.J. Field, Local structure for equivariant dynamics, Singularity Theory and its Applications, Part II (M. Roberts and I. Stewart, eds.), Lecture Notes in Math., vol. 1463, Springer, New York, 1991, pp. 142–166.

[10] M. Golubitsky, V. LeBlanc, and I. Melbourne. Meandering of the spiral tip – an alternative approach. J. Nonl. Sc., 7:557–586, 1997.

[11] M. Golubitsky, I. Stewart, and D. Schaeffer. Singularities and Groups in Bifurcation Theory, Vol. II, Applied Mathematical Sciences, vol. 69, Springer, Berlin, 1988.

[12] F. P. Greenleaf. Invariant Means on Topological Groups. Number 16 in Van Nostrand Mathematical Studies. American Book Company, U.S.A., 1969.

[13] Dan Henry. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics. Springer Verlag, Berlin, Heidelberg, New York, 1981.

[14] 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.

[15] R. M. Mantel and D. Barkley. Parametric forcing of scroll-wave patterns in three-dimensional excitable media. Preprint, IMA, 1998.

[16] 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.

[17] L. Nirenberg. Topics in Nonlinear Functional Analysis. Lecture Notes Courant Inst. Math. Sci. New York University, 1974.

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

[19] M. Renardy. Bifurcation from rotating waves. Arch. Rat. Mech. Anal., 79:49–84, 1982.

[20] M. Renardy and H. Haken. Bifurcations of solutions of the Laser equations. Physica D, 8:57–89, 1983.

[21] B. Sandstede, A. Scheel, and C. Wulff. Center-manifold reduction for spiral waves. C. R. Acad. Sc., Série I, Math., 324:153 – 158, 1997.

[22] B. Sandstede, A. Scheel, and C. Wulff. Dynamics of spiral waves on unbounded domains using center-manifold reductions. J. Diff. Eq., 141:122 –149, 1997.

[23] B. Sandstede, A. Scheel, and C. Wulff. Bifurcations and dynamics of spiral waves. J. Nonlin. Sci., 9(4):439–478, 1999.

[24] A. Scheel. Subcritical bifurcation to infinitely many rotating waves. J. Math. Anal. Appl., 215:252–261, 1997.

[25] A. Scheel. Bifurcation to spiral waves in reaction-diffusion systems. SIAM J. Math. Anal. , 29:1399–1418, 1998.

[26] G.S. Skinner and H.L. Swinney. Periodic to quasiperiodic transition of chemical spiral rotation. Physica D, 48:1–16, 1991.

[27] O. Steinbock, S.C. Müller and V.S. Zykov. Control of spiral wave dynamics in active media by periodic modulation of excitability. Nature, 366:322– 324, 1993.

[28] A. Vanderbauwhede. Centre manifolds, normal forms and elementary bifurcations. Dyn. Rep., 2:89–169, 1989.

[29] A. Vanderbauwhede and G. Iooss. Centre manifold theory in infinite dimensions. Dyn. Rep., 1:125–163, 1992.

[30] A. Vanderbauwhede, S. A. van Gils. Center manifolds and contractions on a scale of Banach spaces. J. Funct. Anal., 72:209 – 224, 1987.

[31] V.S. Varadarajan. Lie groups, Lie algebras and their representations. Prentice-Hall, 1974.

[32] A.T. Winfree. Varieties of spiral wave behaviour: an experimentalist’s approach to the theory of excitable media. Chaos, 1:303–334, 1991.

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

[34] C. Wulff, J.S.W. Lamb, and I. Melbourne. Bifurcations from relative periodic solutions. Erg. Th. Dyn. Syst., to appear.

[35] V. Zykov, O. Steinbock, and S.C. Müller. External forcing of spiral waves. Chaos, 4(3):509–518, 1994.

[36] V. S. Zykov and S. C. Müller. Spiral waves on circular and spherical domains of excitable medium. Physica D, 97:322 – 332, 1996.

Claudia Wulff

Institut für Mathematik I Freie Universität Berlin Arnimallee 2-6 14195 Berlin, Germany wulff@math.fu-berlin.de