Symmetric spiral patterns on spheres

Symmetric spiral patterns on spheres Rachel Sigrist and Paul Matthews December 13, 2010

Abstract Spiral patterns on the surface of a sphere have been seen in laboratory experiments and in numerical simulations of reaction–diffusion equations and convection. We classify the possible symmetries of spirals on spheres, which are quite different from the planar case since spirals typically have tips at opposite points on the sphere. We concentrate on the case where the system has an additional sign-change symmetry, in which case the resulting spiral patterns do not rotate. Spiral patterns arise through a mode interaction between spherical harmonics degree ℓ and ℓ+1. Using the methods of equivariant bifurcation theory, possible symmetry types are determined for each ℓ. For small values of ℓ, the centre manifold equations are constructed and spiral solutions are found explicitly. Bifurcation diagrams are obtained showing how spiral states can appear at secondary bifurcations from primary solutions, or tertiary bifurcations. The results are consistent with numerical simulations of a model pattern-forming system.

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Introduction

Spiral patterns or spiral waves arise in various chemical and biological systems as well as in numerical simulations of reaction–diffusion systems. For example, spiral waves have been observed in the Belousov–Zhabotinsky chemical reaction [39], Rayleigh–Bénard convection and in the oxidation of carbon monoxide on the surface of a platinum catalyst [28]; an excitable system which can be described using the FitzHugh–Nagumo model [15, 27]. This model has been shown to exhibit a wide range of spiral behaviours in the plane depending on parameter values [4, 5]. It is thought that spiral waves and their threedimensional analogues, scroll waves, appear in heart muscle during cardiac arrhythmias (see for example [6, 19, 29, 35]). In addition, there is speculation that spiral waves may be involved in epileptic seizures where a spiral wave manifests as the local synchronization of large groups of neurons [26], and in migraines [12]. Spiral waves in planar domains have been widely studied and observed in numerical simulations and experiments (see [2, 3, 4, 5, 13] for example). In planar domains, spiral waves rotate rigidly about a centre where the front of the wave has a tip. Far from the rotation centre the spiral wave is well approximated by an Archimedean spiral (see for example [17]). This rigid rotation is a relative equilibrium since in a frame rotating at the same speed as the spiral the tip position is fixed. In addition, spiral waves can meander (the tip traces out a flower pattern with either inward or outward petals depending on parameter values) or drift (the tip drifts off along a line forming loops as it goes). These motions are 1

SYMMETRIC SPIRAL PATTERNS ON SPHERES

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two-frequency quasiperiodic and as such are examples of relative periodic orbits. Barkley [4] realized that these spiral wave dynamics could be explained by the Euclidean symmetry SE(2) of the plane. His ideas were extended by Sandstede et al. [30, 31] and Wulff [37]. In particular, the Euclidean symmetry can be used to study the transition (via Hopf bifurcation) from rigidly rotating to meandering planar spirals [31]. Spiral waves can also occur in spherical geometry. In contrast to the large volume of work on planar spirals, there has been relatively little research concerning spiral patterns on spheres. A spiral wave on the surface of a sphere must have two tips (see [14] for a more precise statement in terms of phase singularities) and so the dynamics of such patterns are expected to be qualitatively different from the planar case. In this paper we will investigate a related difference between planar and spherical spirals; while one-armed planar spirals have trivial isotropy (i.e. no symmetries) one-armed spherical spirals typically have a rotation symmetry. Spiral wave patterns on a sphere have been observed experimentally in the Belousov– Zhabotinsky reaction [23]. They have also been found many times in numerical simulations of reaction–diffusion systems on a sphere [1, 18, 25, 38, 41], and have recently appeared on the cover of SIAM Review [7]. A further motivation for the study of spirals on spheres is the potential applications in neuroscience [12, 26], where a spherical model is more relevant than a planar one. The transition from rotating spiral waves to meandering spiral waves on a sphere has been studied using the group of rotations of a sphere, SO(3), by Wulff [36], Chan [8] and Comanici [10]. They independently studied the Hopf bifurcation of a rotating spiral relative equilibrium, which leads to the meandering of the spiral wave. In addition to rotating spirals, stationary spiral patterns have also been observed in spherical geometries. For example, numerical simulations of Rayleigh–Bénard convection in a thin spherical shell have been found to give a stable stationary spiral roll covering the whole surface of the sphere [21, 40]. A similar stable stationary spiral pattern has also been found in numerical simulations of a variation of the Swift–Hohenberg equation [24]; depending on parameter values, single-armed spirals or double spirals were found. In this paper we first describe the results of further numerical simulations of variations of the Swift–Hohenberg model. We demonstrate a range of spherical spiral patterns with different symmetries. The remainder of the paper investigates analytically the existence of spiral patterns with specific symmetries which can occur as a result of a stationary bifurcation from spherical symmetry. These spiral patterns and their symmetries are discussed in section 3. In section 4 we discuss how we can apply results from equivariant bifurcation theory (which can be found in [9, 16, 17], for example) to the study of spiral patterns on spheres and in sections 5–9 we carry out an analysis of the generic bifurcations from spherical symmetry which can result in solution branches with the symmetries of spiral patterns. We also use the results of the general analysis to study the specific case of the Swift–Hohenberg equation analytically. This enables us to find spiral patterns which are unstable in addition to those stable spiral patterns which were found numerically.


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3

Spirals in numerical simulations

The work presented in this paper was motivated by the ease with which spiral patterns with symmetry can be found in numerical simulations of variations of a pattern-forming partial differential equation on the sphere [1, 7, 18, 21, 24, 40]. In this section we present some numerical results obtained from the Swift–Hohenberg model ∂w = µw − (1 + ∇2 )2 w + sw2 − w3 . ∂t

(1)

This equation is widely used as a model for convection and other pattern-forming systems [34]. We demonstrate that it is not unusual for (1) to possess a stable stationary spiral pattern solution. The PDE given by (1) has a basic state solution w = 0 which is potentially unstable when the parameter µ becomes positive. Since we are considering the solutions of (1) which exist in a spherical domain, w = 0 is a solution with spherical symmetry. When this solution becomes unstable we expect simulations to yield a solution which is stable near a bifurcation from spherical symmetry. Note that (1) can be written in variational form, so the large-time behaviour is a stationary state. If s = 0 in (1) then the equation has w → −w symmetry in addition to the spherical symmetry imposed by the geometry. This means that if w is a solution of (1) then −w is also a solution. At a bifurcation from spherical symmetry the eigenfunctions are the spherical harmonics Yℓm (θ, ϕ) of integer degree ℓ for −ℓ ≤ m ≤ ℓ, where (θ, ϕ) is a point on the surface of the sphere in spherical polar coordinates for 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π. The spherical harmonics of degree ℓ are the eigenfunctions of the angular part of the spherical Laplacian operator with eigenvalue −ℓ(ℓ + 1)/R2 where R is the radius of the sphere and is constant. Thus the spherical harmonics satisfy ∇2 Yℓm (θ, ϕ) = −

ℓ(ℓ + 1) m Yℓ (θ, ϕ). R2

They also satisfy Yℓ−m (θ, ϕ) = (−1)m Yℓm (θ, ϕ), where the overbar denotes the complex conjugate [9]. The functions Yℓm (θ, ϕ) form an orthonormal basis of the subspace of L2 of square integrable functions on the sphere of constant radius. Hence the variable w(θ, ϕ, t) can be written as ℓ ∑ ∑ w(θ, ϕ, t) = xℓ,m (t)Yℓm (θ, ϕ), (2) ℓ≥0 m=−ℓ

where xℓ,0 is real for all values of ℓ, and xℓ,m is complex for m ̸= 0. Also xℓ,−m = (−1)m xℓ,m since w is real. This expansion holds in the sense that 2 ∫ 2π ∫ π N ∑ ℓ ∑ m xℓ,m (t)Yℓ (θ, ϕ) sin θ dθ dϕ = 0. lim w(θ, ϕ, t) − N →∞ 0 0 ℓ=0 m=−ℓ

By linearizing (1) using the expansion (2) we see that for a particular value of ℓ all the modes xℓ,m have the same growth rate ( )2 λ = µ − 1 − ℓ(ℓ + 1)/R2 , (3)


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where R is the radius of the sphere. Thus there is instability when µ > µc where µc = )2 ( 1 − ℓ(ℓ + 1)/R2 . A plot of µc as a function of R for different values of ℓ is given in Figure 1. The function µc (R) takes its minimum value of zero when R2 = ℓ(ℓ + 1). When R = ℓ + 1 the modes of degrees ℓ and ℓ + 1 both have the same growth rate and µc = 1/(ℓ + 1)2 . This is the point in Figure 1 where the ℓ and ℓ + 1 plots cross.

Figure 1: A plot of µc as a function of R for ℓ = 2, 3, 4 and 5. The dot at the intersection of the plots for ℓ and ℓ+1 indicates the location of the ℓ, ℓ+1 mode interaction point.

The model equation (1) has been solved numerically using a pseudo-spectral method. The expansion (2) is truncated at some value ℓ = L, so that a total of (L + 1)2 modes are used. To ensure adequate resolution of the pattern, we choose L ≥ 5ℓc , where ℓc is the value of ℓ that minimises µc for the chosen value of R. For the linear terms in the equation, each mode is multiplied by a factor eλ∆t where ∆t is the timestep; thus the linear terms are integrated in time exactly. This avoids the small-timestep restriction that would otherwise arise from the four spatial derivatives. The nonlinear terms are computed by evaluating w(θ, ϕ) at (L+1)2 mesh points on the sphere. The choice of these points is important since the transformation from spherical harmonics to mesh points must be well conditioned and the mesh approximation must be accurate. We use the extremal points provided by Sloan and Womersley [33] that have excellent interpolation properties. The nonlinear terms are advanced in time using the exponential time differencing method [11]; an advantage of this method is that fixed points of the truncated system of ordinary differential equations for the xℓ,m (t) are obtained exactly by the time-stepping method. Many simulations were carried out, for different values of the parameters R and µ. The parameter s was set to zero, so that the system has the additional w → −w symmetry. Each simulation was started from a small-amplitude random perturbation of the equilibrium, and approached a steady state at large time, as required by the variational nature of (1). In general, many different solutions were found, and it is often the case that several different steady states are stable for the same parameter values. Only the spiral states are discussed here. Several different types of stable spiral patterns were found, and some of these are shown in Figure 2. In each case the viewpoint is from above the spiral tips, and the appearance of the pattern from the other side of the sphere is the same; more precisely, there is a π rotation symmetry about an axis lying in the plane of the diagram. For R = 6.2, µ = 0.2, a single spiral pattern was found (Figure 2 (a)). This pattern has a symmetry between the red and blue spiral tip, corresponding to a π rotation combined with the w → −w symmetry. For the same parameters a double spiral pattern was also found (Figure 2


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Figure 2: Steady state spiral solutions of (1) for (a), (b) R = 6.2, µ = 0.2, (c) R = 5.0, µ = 0.1.

(b)); this state has no symmetries involving w → −w. A triple spiral pattern is shown in Figure 2 (c), for R = 5.0, µ = 0.1, with a symmetry of a rotation through π/3 combined with w → −w. In summary, single and multi-armed spiral patterns with symmetries such as those in Figure 2 can be found easily. The single-armed spiral patterns in particular become more common the larger the value of R (i.e. the larger the degrees ℓ of the unstable modes). We have found stable single-armed spirals right down to values of R near 4 where the unstable modes are of degrees 3 and 4. All of the simulations of (1) result in stable stationary solutions since (1) is variational. In order to find rotating spirals (relative equilibria) we must use a non-variational PDE such as the nonvariational version of the Swift–Hohenberg model given by ∂w = µw − (1 + ∇2 )2 w + qw∇2 w − w3 ∂t

(4)

for q ̸= 0 [20]. The nonvariational term qw∇2 w ensures that (4) does not have a Lyapunov functional so it has the potential to support periodic solutions such as rotating spiral patterns on the sphere. Indeed, we have found that numerical simulations of this equation can give single-armed spirals which rotate such as that depicted in Figure 13. These rotating spirals will be discussed in more detail in section 9. Numerical simulations can only provide information about solutions which are stable for the chosen parameter values. By studying the problem analytically we can find unstable solutions and determine the bifurcation structure. The remainder of this paper is devoted to the analytical study of bifurcations from spherical symmetry, both generically and in the Swift–Hohenberg equation, to determine which spiral patterns can exist and how they bifurcate from other solutions.

3

Symmetries of spiral patterns on spheres

Throughout the remainder of this paper we will discuss the existence and stability properties of spiral patterns on spheres which can be found analytically. An essential first step is to introduce a classification and notation for the different possible symmetry types of spirals. We consider here patterns which are functions on the sphere where the areas on which w > 0 and w < 0 form intertwined spirals such as those in Figure 4. The contours along which w = 0 are Archimedean spherical spirals which originate at a single point on the


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surface of the sphere and terminate at the antipodal point. Figure 3 shows an example of such an Archimedean spiral. We say that the spiral is m-armed if near the tips, or point of origin, there are m areas where the function is positive. This means that for an m-armed spiral pattern there are 2m zero contour Archimedean spirals.

Figure 3: An example of an Archimedean spiral on a sphere. The spiral has a rotationthrough-π symmetry about the axis in the plane of the equator indicated by the dot–dashed line.

Figure 4: A view of (a) one-armed, (b) two-armed and (c) three-armed spherical spirals looking directly at the point of origin. The red areas show where the functions are positive and blue areas show where they are negative.

The patterns which are studied in this paper fall into two categories; those with and those without symmetries involving w → −w. A pattern has a symmetry involving w → −w when the −w solution is simply a rotation or reflection of the original pattern w. Notice that the patterns in Figure 4 all have such symmetries whilst Figure 2(b) does not. For patterns without symmetries involving w → −w, the symmetries of the spiral pattern can be described purely in terms of rotations and reflections of the sphere. These are elements of the group O(3) and thus the symmetry group or isotropy type of each spiral without w → −w symmetry is a subgroup of O(3). For details of the subgroups of O(3) see [16, p.103 and p.120]. Every spiral pattern which we consider in this paper has a rotation-through-π symmetry, Rπe , with an axis lying in the plane of the equator when the tips are considered to be lying at the poles. In addition, the m-armed spiral has rotationt through-2π/m symmetry, R2π/m , in the axis through the spiral tips. Thus the symmetry group of the one-armed spiral is Z2 and an m-armed spiral for m ≥ 2 has symmetry group Dm , generated by the combination of the two rotation symmetries. Notice that the group of symmetries of a one-armed spiral is contained in that of an m-armed spiral for any value of m. Patterns with symmetries involving w → −w are the most symmetric spiral patterns on


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the sphere. In addition to the rotational symmetries contained in O(3) described above, an m-armed spiral can have the symmetry ) ( t Rπ/m , −1 ∈ O(3) × Z2 , (5) t where Rπ/m is a rotation through π/m in the axis through the spiral tips and −1 is the non-identity element in Z2 which acts as multiplication by −1 sending w → −w. Spiral patterns with this additional symmetry have a symmetry group (or isotropy type) which is a subgroup of O(3) × Z2 . The symmetry group of the most symmetric one-armed spiral is then ⟨ ( )⟩ f2 = (Re , 1) , Rt , −1 D (6) π π

and the symmetry group of the most symmetric m-armed spiral for m ≥ 2 is ⟨ ( )⟩ e t ] D = (R , 1) , R , −1 . 2m π π/m

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The tildes indicate that these are twisted subgroups of O(3) × Z2 . That is, some of the elements in the subgroup have the non-identity element of Z2 as their second component f2 ⊂ D ] (see [16, 17] for a precise definition of twisted subgroups). Notice that D 2m for all ] values of m; when m is odd this is obvious but for m even observe that(D2m with the ) ( ) f2 with generators Rt , 1 and Re Rt , −1 generators above contains the subgroup D π π π/m t where Rπe Rπ/m is a rotation through π in a different axis in the plane of the equator. This gives the symmetry group of a one-armed spiral with its tips on the equator. The twisted subgroups of O(3) × Z2 are discussed in more detail in section 5.1.

Henceforth in this paper, when we refer to a spiral pattern we mean a spiral pattern with one of the symmetry groups described above.

3.1

Description of spirals in terms of spherical harmonics

Algebraically O(3) = SO(3) × Zc2 where SO(3) is the group of all rotations of the sphere and Zc2 = {I, −I} where I is the identity element and −I is inversion in the centre of the sphere i.e. the element which takes (θ, ϕ) → (π−θ, π+ϕ) for any point (θ, ϕ) on the surface of the sphere. The group SO(3) has precisely one irreducible representation in each odd dimension 2ℓ + 1 for ℓ ≥ 0, denoted by Vℓ , where Vℓ is the space of spherical harmonics of degree ℓ. For each irreducible representation of SO(3) there are two irreducible representations of O(3), where the element −I either acts on the spherical harmonics of degree ℓ as multiplication by ±1, giving rise to the plus and minus representations of O(3). In the natural representation of O(3) on Vℓ , −I acts as multiplication by (−1)ℓ . This is the plus representation if ℓ is even and the minus representation if ℓ is odd. We are interested in the spiral patterns which occur in natural representations of O(3) since these are the representations which occur for scalar PDEs such as (1). In this paper we will be considering spirals, such as those in Figures 2 and 4, for which a rotation–through–π symmetry takes one spiral tip to the other. This rotation symmetry can be seen clearly in the diagram of the Archimedean spiral given in Figure 3. From this diagram and the images of spirals in Figures 2 and 4, we can see that inversion in the origin, −I ∈ O(3) does not act as the identity or minus the identity on any spiral


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pattern we consider in this paper since the symmetry which maps tips to tips is a rotation in every case. If a pattern, w(θ, ϕ, t), can be made with a linear combination of spherical harmonics of even degree then since −I acts as the identity on all spherical harmonics of even degree in the natural representation, −I must act as the identity on w(θ, ϕ, t). Similarly if w(θ, ϕ, t) can be made with a linear combination of spherical harmonics of odd degree then in the natural representation of O(3), −I must act as minus the identity on w(θ, ϕ, t). Since −I acts as neither plus nor minus the identity on the spiral patterns we are studying we conclude that these patterns can only be made through linear combinations of spherical harmonics of both odd and even degrees. Indeed, we find that spiral patterns such as those in Figures 2 and 4 can be made with linear combinations of spherical harmonics of degrees ℓ and ℓ + 1 and hence are patterns of the form ℓ ℓ+1 ∑ ∑ n w(θ, ϕ, t) = xm (t)Yℓm (θ, ϕ) + yn (t)Yℓ+1 (θ, ϕ) (8) m=−ℓ

(−1)m xm

n=−ℓ−1

(−1)n y

where x−m = and y−n = n since w(θ, ϕ, t) must be real. Hence the spiral patterns we are studying can only exist when there is a mode interaction between the modes of degrees ℓ and ℓ + 1 since we require that modes of an odd and even degree are simultaneously unstable to get patterns of the form (8).

4

Application of equivariant bifurcation theory to spiral patterns

In this paper we use equivariant bifurcation theory to show that spiral patterns with symmetries as described in section 3 can be created through a stationary bifurcation from O(3) symmetry and subsequent secondary bifurcations. For a full account of the definitions and results of equivariant bifurcation theory see [16] for example. Here we define only the terms we require in this paper. We also concentrate on one particular type of representation of the group O(3) rather than stating more general results. We have seen in section 3.1 that for it to be possible for spiral patterns to exist as a result of a stationary bifurcation with O(3) symmetry, the representation of O(3) must be a reducible representation on the 4(ℓ + 1) dimensional space Vℓ ⊕ Vℓ+1 . Consider the system of ordinary differential equations dz = f (z, λ, ρ) dt where λ, ρ ∈ R are bifurcation parameters, ( ) z = (x ; y) = x−ℓ , x−(ℓ−1) , . . . , xℓ ; y−(ℓ+1) , y−ℓ , . . . y(ℓ+1)

(9)

(10)

is the vector of amplitudes of the spherical harmonics of degrees ℓ and ℓ + 1 and f is a smooth mapping which commutes with the action of O(3) on Vℓ ⊕ Vℓ+1 . By this we mean that γ · f (z, λ, ρ) = f (γ · z, λ, ρ) ∀ z ∈ Vℓ ⊕ Vℓ+1 , ∀γ ∈ O(3) (11) where · denotes the action of O(3) on Vℓ ⊕ Vℓ+1 . This action of γ ∈ O(3) on z ∈ Vℓ ⊕ Vℓ+1 is executed by multiplication on the left by the 4(ℓ + 1) × 4(ℓ + 1) matrix Mγ . The form


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of these matrices for a set of generators of O(3) is given in Appendix A. We say that f is a O(3) equivariant vector field. Suppose that z0 is an equilibrium of (9) for some values of λ and ρ. Since f satisfies (11), γz0 is also an equilibrium. Equilibria of (9) exist in group orbits, (O(3))z0 = {γz0 : γ ∈ O(3)}.

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The symmetry of an equilibrium z0 ∈ Vℓ ⊕ Vℓ+1 is the set of all γ ∈ O(3) that leaves z0 invariant. This set is a subgroup of O(3) called the isotropy subgroup of z0 and is denoted Σz0 = {γ ∈ O(3) : γz0 = z0 }.

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The group orbit (O(3))z0 is a smooth manifold of dimension 3 − dim(Σz0 ). This means that it is possible for a group orbit to be flow invariant rather than just consisting of equilibria. In this case the group orbit is called a relative equilibrium. Relative equilibria are quasiperiodic motions with k-frequencies where generically k = rank(N (Σz0 )/Σz0 ),

(14)

(see, for example, [17, Theorem 6.4]) where the rank of a group is the maximal dimension of any torus subgroups contained in that group. Here N (Σz0 ) = {γ ∈ O(3) : γΣz0 γ −1 = Σz0 } is the normalizer of Σz0 in O(3). A subgroup Σ ⊂ O(3) is an isotropy subgroup if it fixes some vector z ∈ Vℓ ⊕ Vℓ+1 and contains all the group elements that fix z. Given a value of ℓ it is possible to compute all conjugacy classes of isotropy subgroups of O(3) in the representation on Vℓ ⊕ Vℓ+1 using a slight modification of the ‘massive chain criterion’ of Linehan and Stedman [22] which will be explained in section 5. The fixed-point subspace of subgroup Σ ⊂ O(3) for the action of O(3) on Vℓ ⊕ Vℓ+1 is the subspace FixVℓ ⊕Vℓ+1 (Σ) = {z ∈ Vℓ ⊕ Vℓ+1 : σz = z

∀σ ∈ Σ}.

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We say that a subgroup Σ ⊂ O(3) is an axial isotropy subgroup if it is an isotropy subgroup which has a one-dimensional fixed-point subspace. The subgroups of O(3) fall into three classes I The subgroups of SO(3): O(2), SO(2), I, O, T, Dn and Zn for (n ≥ 1). II Subgroups of the form K × Zc2 where K is a subgroup of SO(3). III Subgroups which are not contained in SO(3) and do not contain the inversion element −I ∈ O(3). These subgroups Σ are denoted by O(2)− , O− , Dd2m , Dzm and Dd2m . Let Π : O(3) → SO(3) be the homomorphism whose kernel is Zc2 . Each class III subgroup Σ ⊂ O(3) is isomorphic (but not conjugate) to the subgroup Π(Σ) ⊂ SO(3) whose symbol is the same as Σ except without the superscript, for example Π(O(2)− ) = O(2). The notation used in this paper for these subgroups is the standard notation for the subgroups of O(3) which is used in particular in [16] where precise definitions of these subgroups can be found.


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For each subgroup Σ ⊂ O(3) the dimension of the subspace FixVℓ ⊕Vℓ+1 (Σ) can be computed using the formulae for dim FixVℓ (Σ) given by Theorems 8.1 and 9.5 of [16, Chapter XIII]. If Σ is a class I subgroup of O(3) then dim FixVℓ ⊕Vℓ+1 (Σ) = dim FixVℓ (Σ) + dim FixVℓ+1 (Σ). If Σ = K × Zc2 where K is a subgroup of SO(3) then { dim FixVℓ (K) for ℓ even dim FixVℓ ⊕Vℓ+1 (Σ) = dim FixVℓ+1 (K) for ℓ odd. Finally if Σ is a class III subgroup of O(3) then { dim FixVℓ (Π(Σ)) + dim FixVℓ+1 (Σ) for ℓ even dim FixVℓ ⊕Vℓ+1 (Σ) = dim FixVℓ (Σ) + dim FixVℓ+1 (Π(Σ)) for ℓ odd.

(16)

(17)

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Since FixVℓ ⊕Vℓ+1 (O(3)) = {0}, (9) has a trivial equilibrium z = 0 with O(3) symmetry for all values of λ and ρ. We assume that at λ = 0 the modes of degree ℓ (those in Vℓ ) become unstable and the equilibrium z = 0 undergoes a stationary bifurcation. At this stationary bifurcation branches of equilibria bifurcate. Branches with certain symmetries are guaranteed to exist by the equivariant branching lemma: Theorem 1 (Equivariant branching lemma). Let Γ be a Lie group acting on a vector space V . Assume 1. Fix(Γ) = {0}, 2. Σ ⊂ Γ is an axial isotropy subgroup 3. f : V × R → V commutes with the action of Γ on V and satisfies f (0, 0) = 0, (df )(0,0) = 0 and (dfλ )(0,0) v ̸= 0 (19) for some nonzero v ∈ Fix(Σ). Then there exists a unique branch of solutions to f (z, λ) = 0 emanating from (0, 0) where the symmetry of the solution is Σ. Here (dfλ ) is defined by (dfλ )ij =

∂(df )ij . ∂λ

Proof. See [16, Chapter XIII, Theorem 3.5] By restricting our problem where Γ = O(3) to Vℓ we see that by the equivariant branching lemma, equilibria with the symmetries of isotropy subgroups Σ ⊂ O(3) which fix a onedimensional subspace of Vℓ bifurcate from the trivial solution when λ = 0. We also assume that at λ = −ρ the modes of degree ℓ + 1 (those in Vℓ+1 ) become unstable and the equilibrium z = 0 undergoes a further stationary bifurcation. By the equivariant branching lemma, equilibria with the symmetries of isotropy subgroups Σ ⊂ O(3) which fix a one-dimensional subspace of Vℓ+1 emanate from this bifurcation.


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The equilibria which are guaranteed to exist by the equivariant branching lemma are not the only solutions of (9). Depending on the values of λ, ρ and coefficients in the Taylor expansion of the vector field f it may be possible for solutions with the symmetries of other isotropy subgroups to exist. If Σ is an isotropy subgroup of O(3) in the representation on Vℓ ⊕ Vℓ+1 which fixes a subspace of dimension larger than one then it may be possible for a solution to (9) with this symmetry to exist, however we must find this solution directly from (9). In other words, to establish whether a solution with Σ symmetry can exist we must compute the form of the equivariant mapping f directly and look for this solution in the restriction to the invariant subspace FixVℓ ⊕Vℓ+1 (Σ). If such a solution exists then we call it a submaximal solution. The results outlined in this section can also be applied to the stationary bifurcation with O(3)×Z2 symmetry where the isotropy subgroups are then twisted subgroups of O(3)×Z2 . Bifurcations with this symmetry are considered in section 5.

4.1

Aims of our analysis

Our aim is to show that (9) can have one-armed spiral solutions with Z2 symmetry or m-armed spiral solutions for m ≥ 2 with Dm symmetry. The solutions of (9) which are guaranteed to exist by the equivariant branching lemma have the symmetries of the axial isotropy subgroups of O(3) in the representations on Vℓ (which bifurcate at λ = 0) and Vℓ+1 (which bifurcate at λ = −ρ). Since by (16) dim FixVℓ ⊕Vℓ+1 (Z2 ) = dim FixVℓ (Z2 ) + dim FixVℓ+1 (Z2 ) = 2ℓ + 2,

(20)

even if Z2 is an isotropy subgroup of O(3) for some representation on Vℓ ⊕Vℓ+1 , it never has a one dimensional fixed point subspace and hence it is never an axial isotropy subgroup. Also by (16) dim FixVℓ ⊕Vℓ+1 (Dm ) = dim FixVℓ (Dm ) + dim FixVℓ+1 (Dm ) = [(ℓ + 1)/m] + [ℓ/m] + 1,

(21)

where [x] is the greatest integer less than or equal to x. Hence dim FixVℓ ⊕Vℓ+1 (Dm ) = 1 when ℓ < m − 1. However Dm ⊂ O(2) × Zc2 for all values of m and dim FixVℓ ⊕Vℓ+1 (O(2) × Zc2 ) = 1 for all values of ℓ so Dm is never an axial isotropy subgroup of O(3). The existence of submaximal solutions of (9) with the symmetries of the spiral patterns is not guaranteed by the equivariant branching lemma but it may be possible for such solutions to exist if Z2 or Dm are isotropy subgroups which fix a subspace of dimension larger than one. Suppose that this is the case for some values of m in the representation on Vℓ ⊕ Vℓ+1 for some value of ℓ. Then to determine whether solutions with these symmetries can exist we must look for them directly in the O(3) equivariant vector field f . In the representation of O(3) on Vℓ ⊕Vℓ+1 this vector field is 4(ℓ+1) dimensional. Since Z2 ⊂ Dm for all values of m, if any spiral solutions with Z2 or Dm symmetry exist then they can be found in the restriction of the vector field f to the 2(ℓ + 1) dimensional vector space FixVℓ ⊕Vℓ+1 (Z2 ). Even for low values of ℓ this vector space is large so to find one-armed spirals with Z2 symmetry in this space is not a simple task. Remark 2. Since N (Z2 ) = O(2) × Zc2 but N (Dm ) = D2m × Zc2 , rank(N (Z2 )/Z2 ) = 1

and

rank(N (Dm )/Dm ) = 0,

(22)


12

by (14), generically solutions with Dm symmetry (if they exist) are equilibria (stationary m-armed spirals for m ≥ 2) and one-armed spirals with Z2 symmetry are generically relative equilibria with one period i.e. they are rotating waves. These rotating spiral waves have a twisted subgroup Σrot of O(3)×S 1 as their group of spatiotemporal symmetries. By (20) a linear combination of 2ℓ+2 spherical harmonics has Z2 symmetry so a rotating wave with Σrot symmetry is a time dependent linear combination of 2ℓ + 2 spherical harmonics and hence the equivariant Hopf theorem (see for example [16]) cannot be used to guarantee the existence of such solutions. The rotating waves with Σrot symmetry must be found analytically in a space of 2ℓ + 2 complex dimensions. This again increases the difficulty of identifying such solutions. We can make the following simplification which will be the basis for most of the work in this paper. Rather than look for spiral patterns without w → −w symmetries directly we first consider the most symmetric spiral patterns on spheres. Recall from section 3 that these patterns have the additional symmetry (5) and hence the most symmetric m-armed ] spiral has symmetry group D 2m where the tilde indicates that this is a twisted subgroup of O(3) × Z2 . Remark 3. By (14), since ] ] rank(N (D 2m )/D2m ) = 0

(23)

for all values of m, all spiral solutions with the additional symmetry (5) (if they exist) are equilibria. We will see in section 5 that by considering the spiral patterns with the additional symmetry (5) rather than those without, we halve the number of equations we must solve in order to identify such solutions. This, and the fact that all such solutions are equilibria by Remark 3, greatly simplifies the analysis of the spirals which can exist. We identify all such spiral patterns and how they bifurcate from other solutions for two particular representations in sections 6 and 7. In section 8 we consider spiral patterns which can exist in the general ℓ, ℓ + 1 mode interaction. Finally, in section 9, we consider the spiral patterns which can persist when the additional symmetry (5) is slightly broken.

5

Stationary bifurcation with O(3) × Z2 symmetry

In this section we consider the types of solutions which can be created at a general stationary bifurcation with O(3) × Z2 symmetry by computing the isotropy subgroups of O(3) × Z2 in various representations. This work is motivated by the fact that we wish to ] determine if the subgroup D 2m of symmetries of the most symmetric m-armed spiral on a sphere can be an isotropy subgroup for any values of m in a reducible representation on Vℓ ⊕ Vℓ+1 . This is a necessary (but not sufficient) condition for solutions with the symmetries of these spiral patterns to exist through the stationary bifurcation with O(3) × Z2 symmetry and subsequent secondary bifurcations. Although we are motivated by our search for solutions with spiral pattern symmetries such as those in Figure 4, the stationary bifurcation with O(3) × Z2 symmetry is an interesting problem in its own right which has, up until now, not been investigated. Many dynamical


13

systems, including pattern forming systems such as Boussinesq Rayleigh–Bénard convection, are invariant under a change in sign of the physical variable w. For example, the Swift–Hohenberg equation (1) with no quadratic term is invariant under the transformation w → −w. Hence if w is a solution then −w is also a solution. When we study the dynamics of such a system on a sphere the overall symmetry is O(3) × Z2 . When such a system is reduced to the centre manifold of a stationary bifurcation of a trivial solution with O(3) × Z2 symmetry we have dz = fe(z, λ) dt

(24)

where z ∈ V (here V = Vℓ for irreducible representations and V = Vℓ ⊕ Vℓ+1 for reducible representations), λ ∈ R is a bifurcation parameter. Here the smooth mapping fe not only commutes with the action of O(3) on V (as in (11)) but is also equivariant with respect to the action of −1 ∈ Z2 . Hence it satisfies fe(−z, λ) = −fe(z, λ)

(25)

i.e. the vector field fe is odd in z so a Taylor expansion of this vector field will not contain any terms of even order. An O(3) × Z2 equivariant vector field is just an O(3) equivariant vector field with all terms of even order removed. We now compute the isotropy subgroups of O(3) × Z2 . These are the symmetry groups of the possible solutions to (24) which may exist at a stationary bifurcation. The equivariant branching lemma guarantees the existence of branches of solutions with the symmetries of the axial isotropy subgroups.

5.1

Isotropy subgroups of O(3) × Z2

For any representation of O(3) × Z2 the isotropy subgroups are twisted subgroups H θ where H is a subgroup of O(3) and θ : H → Z2 is a group homomorphism. These twisted subgroups are a subset of the twisted subgroups of O(3) × S 1 containing only the subgroups, H θ , with twist types Z2 or 1. The conjugacy classes of twisted subgroups of O(3) × S 1 are listed in [32]. The twisted subgroups of O(3) × Z2 are uniquely determined by pairs of subgroups (H, K) where H is a subgroup of O(3) and K is a (normal) subgroup of H such that H/K ∼ = Z2 or 1. The group homomorphism θ : H → H/K is always given by { 1 if h ∈ K θ(h) = (26) −1 if h ∈ H − K. A complete list of the twisted subgroups of O(3) × Z2 is given in Table 1 We are interested in the twisted subgroups which can be isotropy subgroups when the representation of O(3) × Z2 is the reducible natural representation on Vℓ ⊕ Vℓ+1 . To compute which of the twisted subgroups are isotropy subgroups we use the ‘massive chain criterion’ of Linehan and Stedman [22]. This says that a twisted subgroup H θ ⊂ O(3)×Z2 is an isotropy subgroup in the representation on Vℓ ⊕ Vℓ+1 if and only if for each strictly larger and adjacent group ∆ (so that H θ ⊂ ∆ ⊂ · · · ⊂ O(3) × Z2 ) dim FixVℓ ⊕Vℓ+1 (∆) − q(∆) < dim FixVℓ ⊕Vℓ+1 (H θ ) − q(H θ )

(27)

H class II K class II

H class I

O(3) O(2) × Zc2 O(2) × Zc2 SO(2) × Zc2 Dn × Zc2 Dn × Zc2 D2n × Zc2 Zn × Zc2 Z2n × Zc2 T × Zc2 O × Zc2 O × Zc2 I × Zc2

O(3) O(2) × Zc2 SO(2) × Zc2 SO(2) × Zc2 Dn × Zc2 Zn × Zc2 Dn × Zc2 Zn × Zc2 Zn × Zc2 T × Zc2 O × Zc2 T × Zc2 I × Zc2

1

Z2

1 1

Z2

1

Z2 Z2

1 1

Z2

1 1

0 1 0 1 [(ℓ + 1)/n] + 1 [(ℓ + 1)/n] [(ℓ + 1 + n)/2n] 2 [(ℓ + 1)/n] + 1 2 [(ℓ + 1 + n)/2n] 2 [(ℓ + 1)/3] + [(ℓ + 1)/2] − ℓ [(ℓ + 1)/4] + [(ℓ + 1)/3] + [(ℓ + 1)/2] − ℓ [(ℓ + 1)/4] − [(ℓ + 1)/3] [(ℓ + 1)/5] + [(ℓ + 1)/3] + [(ℓ + 1)/2] − ℓ

Continued on next page

0 1 0 1 [ℓ/n] + 1 [ℓ/n] [(ℓ + n)/2n] 2 [ℓ/n] + 1 2 [(ℓ + n)/2n] 2 [ℓ/3] + [ℓ/2] − ℓ + 1 [ℓ/4] + [ℓ/3] + [ℓ/2] − ℓ + 1 [ℓ/4] − [ℓ/3] [ℓ/5] + [ℓ/3] + [ℓ/2] − ℓ + 1

Table 1 Twisted subgroups H θ of O(3) × Z2 and formulae for dim Fix(H θ ) for the reducible representations on Vℓ ⊕ Vℓ+1 . H K H/K dim Fix(H θ ) for ℓ odd dim Fix(H θ ) for ℓ even SO(3) SO(3) 1 0 0 O(2) O(2) 1 1 1 O(2) SO(2) Z2 1 1 SO(2) SO(2) 1 2 2 Dn Dn 1 [ℓ/n] + [(ℓ + 1)/n] + 1 [ℓ/n] + [(ℓ + 1)/n] + 1 D2n Dn Z2 [(ℓ + n)/2n] + [(ℓ + 1 + n)/2n] [(ℓ + n)/2n] + [(ℓ + 1 + n)/2n] Dn Zn Z2 [ℓ/n] + [(ℓ + 1)/n] + 1 [ℓ/n] + [(ℓ + 1)/n] + 1 Zn Zn 1 2 [ℓ/n] + 2 [(ℓ + 1)/n] + 2 2 [ℓ/n] + 2 [(ℓ + 1)/n] + 2 Z2n Zn Z2 2 [(ℓ + n)/2n] + 2 [(ℓ + 1 + n)/2n] 2 [(ℓ + n)/2n] + 2 [(ℓ + 1 + n)/2n] T T 1 2 [ℓ/3] + 2 [(ℓ + 1)/3] − ℓ + 1 2 [ℓ/3] + 2 [(ℓ + 1)/3] − ℓ + 1 O O 1 [ℓ/4] + [(ℓ + 1)/4] + [ℓ/3] + [(ℓ + 1)/3] − ℓ + 1 [ℓ/4] + [(ℓ + 1)/4] + [ℓ/3] + [(ℓ + 1)/3] − ℓ + 1 O T Z2 [ℓ/3] + [(ℓ + 1)/3] − [(ℓ + 1)/4] − [ℓ/4] [ℓ/3] + [(ℓ + 1)/3] − [(ℓ + 1)/4] − [ℓ/4] I I 1 [ℓ/5] + [(ℓ + 1)/5] + [ℓ/3] + [(ℓ + 1)/3] − ℓ + 1 [ℓ/5] + [(ℓ + 1)/5] + [ℓ/3] + [(ℓ + 1)/3] − ℓ + 1

SYMMETRIC SPIRAL PATTERNS ON SPHERES 14

H class III

I or III

Z2 Z2 Z2 Z2

Z− 2n Dzn Zn Dzn Z− 2

1

[ℓ/2n] + [(ℓ + 1 + n)/2n]

Z2

3ℓ 2

−

1 2

− [ℓ/2]

[(ℓ + n)/2n] + [(ℓ + 1)/2n] [ℓ/n] + [(ℓ + 1)/n] + 2 [ℓ/n] + [(ℓ + 1)/n] [(ℓ + n)/2n] + [(ℓ + 1 + n)/2n]

[ℓ/2n] + [(ℓ + 1 + n)/2n] + 1

Z2

Dd2n Dd2n Dd2n Dzn Dzn Dz2n Dz2

Z2

Dn

Dd2n

Dd2n

1

Z2

1

2 0 [ℓ/3] − [ℓ/4] + [(ℓ + 1)/4] + [ℓ + 1/3] + [(ℓ + 1)/2] − ℓ [ℓ/4] + [ℓ/3] − [(ℓ + 1)/4] + [(ℓ + 1)/3] − [(ℓ + 1)/2] + 1 2 [(ℓ + n)/2n] + 2 [(ℓ + 1)/2n] + 1 2 [ℓ/2n] + 2 [(ℓ + 1 + n)/2n] + 1 [(ℓ + n)/2n] + [(ℓ + 1)/2n] + 1

T Z− 2n Zn

O− Z− 2n Z− 2n

1

Z2

[(ℓ + n)/2n] 2 [ℓ/n] + 1 2 [(ℓ + n)/2n] 2 [ℓ/3] + [ℓ/2] − ℓ + 1 [ℓ/4] + [ℓ/3] + [ℓ/2] − ℓ + 1 [ℓ/4] − [ℓ/3] [ℓ/5] + [ℓ/3] + [ℓ/2] − ℓ + 1

dim Fix(H θ ) for ℓ odd 0 0 1 1 [ℓ/n] [ℓ/n] + 1

1

O(2)− SO(2) O−

O(2)− O(2)− O−

Z2 Z2 Z2 Z2 Z2 Z2 Z2

H/K Z2 Z2 Z2 Z2 Z2 Z2

Dzn

Dd2n Zn Z− 2n T O O− I

D2n × Zc2 Zn × Zc2 Z2n × Zc2 T × Zc2 O × Zc2 O × Zc2 I × Zc2

Table 1 – continued from previous page H K O(3) SO(3) O(2) × Zc2 O(2) O(2) × Zc2 O(2)− SO(2) × Zc2 SO(2) H class II Dn × Zc2 Dn c K class Dn × Z2 Dzn

3ℓ 2

+ 1 − [(ℓ + 1)/2]

[(ℓ + n + 1)/2n] + [ℓ/2n] [ℓ/n] + [(ℓ + 1)/n] + 2 [ℓ/n] + [(ℓ + 1)/n] [(ℓ + n)/2n] + [(ℓ + 1 + n)/2n]

[(ℓ + n)/2n] + [(ℓ + 1)/2n]

[(ℓ + n)/2n] + [(ℓ + 1)/2n] + 1

[(ℓ + 1 + n)/2n] + [ℓ/2n] + 1

2 0 [(ℓ + 1)/3] − [(ℓ + 1)/4] + [ℓ/4] + [ℓ/3] + [ℓ/2] − ℓ + 1 [(ℓ + 1)/4] + [(ℓ + 1)/3] − [ℓ/4] + [ℓ/3] − [ℓ/2] 2 [(ℓ + 1 + n)/2n] + 2 [ℓ/2n] + 1 2 [(ℓ + n)/2n] + 2 [(ℓ + 1)/2n] + 1

[(ℓ + n + 1)/2n] 2 [(ℓ + 1)/n] + 1 2 [(ℓ + 1 + n)/2n] 2 [(ℓ + 1)/3] + [(ℓ + 1)/2] − ℓ [(ℓ + 1)/4] + [(ℓ + 1)/3] + [(ℓ + 1)/2] − ℓ [(ℓ + 1)/4] − [(ℓ + 1)/3] [(ℓ + 1)/5] + [(ℓ + 1)/3] + [(ℓ + 1)/2] − ℓ

dim Fix(H θ ) for ℓ even 0 0 1 1 [(ℓ + 1)/n] [(ℓ + 1)/n] + 1



16

where q(H θ ) = dim NO(3)×Z2 (H θ ) − dim(H θ ) and NO(3)×Z2 (H θ ) is the normalizer of H θ in O(3) × Z2 . Note that we have adapted the ‘massive chain criterion’ of [22] for isotropy subgroups of O(3) × Z2 in reducible representations. All of the twisted subgroups H θ in Table 1 have q(H θ ) = 0 with the following exceptions. The twisted subgroups given by pairs (Z2n , Zn ),

(Z2n × Zc2 , Zn × Zc2 ),

(Z2n × Zc2 , Z− 2n ),

− (Z− 2n , Z2n )

and

(Z− 2n , Zn )

have q(H θ ) = 1 for all values of n and the twisted subgroups given by pairs (Zn , Zn ),

(Zn × Zc2 , Zn × Zc2 )

and

(Zn × Zc2 , Zn )

(28)

also have q(H θ ) = 1 except when n = 1 in which case q(H θ ) = 3. To use the ‘massive chain criterion’ we need the values dim FixVℓ ⊕Vℓ+1 (H θ ) for each twisted subgroup in each representation on Vℓ ⊕ Vℓ+1 . Formulae for these values are given for both even and odd values of ℓ in Table 1. These formulae are computed using Theorems 8.1 and 9.5 of [16, Chapter XIII], the formulae given by (16)–(18) and the fact that when θ(H) = 1, dim FixVℓ ⊕Vℓ+1 (H θ ) = dim FixVℓ ⊕Vℓ+1 (H) and whenθ(H) = Z2 the twisted subgroup H θ is given by the pair (H, K) and dim FixVℓ ⊕Vℓ+1 (H θ ) = dim FixVℓ ⊕Vℓ+1 (K) − dim FixVℓ ⊕Vℓ+1 (H). See Theorem 8.3 of [16, Chapter XVI] or [32] for a proof of this fact. Recall that the equivariant branching lemma guarantees the existence of branches of solutions with the symmetries of the axial isotropy subgroups. With the reducible natural action of O(3) × Z2 on Vℓ ⊕ Vℓ+1 the axial isotropy subgroups for the representation on Vℓ ⊕ Vℓ+1 are precisely the axial isotropy subgroups in the representations on Vℓ and Vℓ+1 since all one-dimensional invariant subspaces of Vℓ ⊕ Vℓ+1 are either one-dimensional subspaces of Vℓ or Vℓ+1 . All isotropy subgroups H θ of O(3) × Z2 in the natural representation on Vℓ when ℓ is even are given by pairs (H, K) where both H and K are class II subgroups of O(3) since H θ must contain the element (−I, 1) ∈ O(3) × Z2 . Similarly in the natural representation on Vℓ when ℓ is odd, every isotropy subgroup must contain the element (−I, −1) ∈ O(3) × Z2 and so the isotropy subgroups are given by pairs (H, K) where H is a class II subgroup and K is either a class I or III subgroup of O(3). Using this information and the ‘massive chain criterion’ we find that the axial isotropy subgroups of O(3) × Z2 in the natural representation on Vℓ are as given in Table 2. Given a particular value of ℓ it is possible to compute all of the isotropy subgroups of O(3) × Z2 in the reducible representation on Vℓ ⊕ Vℓ+1 using Table 1 and the ‘massive chain criterion’. See [32] for the results in the case where ℓ = 2. f2 , We are particularly interested in whether it is possible for the twisted subgroups D ] given by the pair (H, K) = (D2 , Z2 ), and D2m , given by pairs (H, K) = (D2m , Dm ), to be isotropy subgroups for any representation Vℓ ⊕ Vℓ+1 . These are the symmetry groups of most symmetric spiral patterns on spheres. Note that these subgroups are never axial isotropy subgroups. f2 is an isotropy Using Table 1 and the ‘massive chain criterion’ it can be seen that D ] subgroup which fixes a subspace of dimension ℓ + 1 for all values of ℓ ≥ 1. Also D 2m is an isotropy subgroup when ℓ ≥ m. A consequence of this is that in the representation on Vℓ ⊕ Vℓ+1 it may be possible for spiral patterns with symmetry groups (6) and (7) with


J

K

θ(H)

O(2) O(2) I

O(2) × Zc2 O(2)− I × Zc2

1 Z2 1

I

I

Z2

O O O O D2m

O × Zc2 O T × Zc2 O− Dm × Zc2

1 Z2 Z2 Z2 Z2

D2m

Dd2m

Z2

D4 D4

D2 × Zc2 Dd4

Z2 Z2

ℓ even (plus representation)

17

ℓ odd (minus representation)

all even ℓ All odd ℓ 6, 10, 12, 16, 18, 20, 22, 24, 26, 28, 32, 34, 38, 44 15, 21, 25, 27, 31, 33, 35, 37, 39, 41, 43, 47, 49, 53, 59 4, 6, 8, 10, 14 9, 13, 15, 17, 19, 23 6, 10, 12, 14, 16, 20 3, 7, 9, 11, 13, 17 even ℓ satisfying m ≤ ℓ < 3m, (m ≥ 3) odd ℓ satisfying m ≤ ℓ < 3m, (m ≥ 3) 2, 4 5

Table 2: The axial isotropy subgroups of O(3) × Z2 for the natural representation on Vℓ . The last two columns give the values of ℓ for which the subgroups are isotropy subgroups. Here H = J × Zc2 .

m ≤ ℓ to exist depending on values of the coefficients in the O(3) × Z2 equivariant vector field fe. These solutions must be found directly using the general form of the vector field. The general form of a O(3)×Z2 equivariant vector field for the representation on Vℓ ⊕Vℓ+1 for a particular value of ℓ can be computed to any order using the fact that it must satisfy (11) and (25). The computation becomes increasingly messy as the value of ℓ increases. ] Our aim is to find solutions with D 2m symmetry for m ≥ 1 within the ℓ + 1 dimensional f space dim FixVℓ ⊕Vℓ+1 (D2 ). This too becomes increasingly difficult for large values of ℓ.

6

Spiral patterns in the representation of O(3)×Z2 on V2 ⊕V3

In this section we demonstrate that solutions with the symmetries of one- and two-armed spirals can exist within the O(3) × Z2 equivariant vector field for the representation on f2 and D f4 , the V2 ⊕ V3 . From the results of section 5 we can that in this representation D symmetry groups of the most symmetric one- and two-armed spirals on a sphere respectively, are isotropy subgroups in this representation. To show that patterns with these symmetries can exist generically we must find them directly in the O(3) × Z2 equivariant vector field. Recall from Remark 3 that any such solutions will be stationary solutions. The general form of the cubic order truncation of this vector field is given in Appendix B. f2 ⊂ D f4 there are copies of the fixed-point subspaces of these subgroups which Since D f f2 ), i.e. for every choice of generators of D f2 , Fix(D f2 ) contains a satisfy Fix(D4 ) ⊂ Fix(D f4 ). For example we can choose generators which give copy of Fix(D f2 ) = {(ia, 0, 0, 0, −ia ; 0, b, 0, c, 0, b, 0)} Fix(D f4 ) = {(ia, 0, 0, 0, −ia ; 0, b, 0, 0, 0, b, 0)} Fix(D

(29) (30)


18

f4 symmetry) has its tips at where a, b, c ∈ R. In this case, the two-armed spiral (with D f2 symmetry) the North and South poles of the sphere and the one-armed spiral (with D has its tips on the equator. By restricting the general form of the O(3) × Z2 equivariant f2 ) given by (29) we have the set of ODEs vector field in Appendix B to Fix(D a˙ = µx a + 2α1 a3 + (2β1 + 8γ1 − 6γ2 ) ab2 + (β1 + γ2 ) ac2 b˙ = µy b + (2β2 − 50δ1 ) b3 + (2α2 − 30δ2 + 10δ3 ) ba2 + (β2 + 15δ1 ) bc2

(31)

c˙ = µy c + (β2 − 9δ1 ) c3 + (2α2 − 30δ2 + 18δ3 ) ca2 + (2β2 + 30δ1 ) cb2

(33)

(32)

where µx , µy , α1 , β1 , γ1 , γ2 , α2 , β2 , δ1 , δ2 and δ3 are real functions of λ and are the coefficients in the full vector field (52)–(53). The trivial solution z = 0 of (24) undergoes stationary bifurcations when µx = 0 and µy = 0. We assume that µx = λ and µy = λ + ρ where |ρ| ≪ 1. Then the trivial solution is stable when λ < min(0, −ρ). At λ = 0 the ℓ = 2 modes become unstable and the equivariant branching lemma guarantees that the unrestricted system (52)–(53) has solution branches with the symmetries of axial isotropy subgroups of O(3) × Z2 in the representation on V2 which bifurcate at λ = 0. Similarly at λ = −ρ the ℓ = 3 modes become unstable and solution branches with the symmetries of axial isotropy subgroups of O(3) × Z2 in the representation on V3 bifurcate. f2 . The Any stationary solution of (31)–(33) has an isotropy subgroup which contains D isotropy subgroups of O(3) × Z2 in the representation on V2 ⊕ V3 can be determined from f2 Table 1 using the ‘massive chain criterion’. Those isotropy subgroups which contain D are given in Table 3 along with the form of their fixed-point subspace which lies inside f2 ) given by (29). The subsection of the lattice of isotropy subgroups including only Fix(D f2 is as in Figure 5. those subgroups which contain D Isotropy subgroup c D^ 4 × Z2 ^ O(2) × Zc2 c D^ 6 × Z2

H

K

D4 × Zc2

D2 × Zc2

O(2) × Zc2

{(ia, 0, 0, 0, −ia ; 0, 0, 0, 0, 0, 0, 0)}

−

{(

{(0, 0, 0, 0, 0 ; 0, 0, 0, c, 0, 0, 0)} √ √ )} 3 3 c, 0, c, 0, 10 c, 0 0, 0, 0, 0, 0 ; 0, 10

Zc2

Dd6

O^ × Zc2 f4 D

O × Zc2

O−

{(0, 0, 0, 0, 0 ; 0, b, 0, 0, 0, b, 0)}

D4

D2

{(ia, 0, 0, 0, −ia ; 0, b, 0, 0, 0, b, 0)}

fd D 4

Dd4

Dz2

{(ia, 0, 0, 0, −ia ; 0, 0, 0, c, 0, 0, 0)}

Dz2

{(0, 0, 0, 0, 0 ; 0, b, 0, c, 0, b, 0)}

Z2

{(ia, 0, 0, 0, −ia ; 0, b, 0, c, 0, b, 0)}

c D^ 2 × Z2

f2 D

D6 ×

O(2)

Fixed-point subspace

D2 × D2

Zc2

Table 3: Isotropy subgroups for the representation of O(3) × Z2 on V2 ⊕ V3 which conf2 . Also shown is the form of the fixed-point subspace which lies inside tain D f2 ). Fix(D

Branches of solutions with the symmetries of the five axial isotropy subgroups of O(3)×Z2 f2 are guaranteed to exist in (31)–(33). in the representation on V2 ⊕ V3 which contain D Analysis of the equations reveals that for certain values of the coefficients λ, ρ, α1 , β1 , γ1 , γ2 , α2 , β2 , δ1 , δ2 and δ3 it is possible for stationary solutions with each of the symmetry groups given in Table 3 which fix a subspace of dimension greater than one to exist with c the exception of the group D^ 2 × Z2 - there are no stationary solutions of (31)–(33) with


19

Figure 5: Subsection of lattice of isotropy subgroups of O(3) × Z2 in the representation f2 . on V2 ⊕ V3 including only those isotropy subgroups which contain D c isotropy D^ 2 × Z2 . See [32] for proof of these facts. Images of solutions to (31)–(33) with each of the possible symmetries are given in Figure 6. f2 and D f4 (one- and two-armed To demonstrate that solutions with symmetry groups D spirals respectively) can exist for some values of the coefficients λ, ρ, α1 , β1 , γ1 , γ2 , α2 , β2 , δ1 , δ2 and δ3 we consider some examples.

6.1

f2 symmetry Example 1: Coefficient values where a solution with D exists and is stable

Suppose that for some pattern forming system the values of the coefficients of the cubic order terms in the O(3) × Z2 equivariant vector field are α1 = −1, γ1 =

1 2,

α2 = −1,

γ2 =

1 2,

δ1 =

β1 = − 13 , 1 60 ,

δ2 =

β2 = −1, 1 2,

δ3 =

(34)

− 15 .

A linear stability analysis of the equilibria of (31)–(33) results in the gyratory bifurcation diagram in Figure 7. We can see that as a path is traversed around the codimension 2 point λ = ρ = 0 there is a range of values of λ and ρ (between B and F) where a solution f2 symmetry (a one-armed spiral) exists and is stable within the subspace Fix(D f2 ). with D f2 symmetry bifurcates from the axial solution with The one-armed spiral pattern with D c f D^ 6 × Z2 symmetry at B and the branch of solutions with D4 symmetry (a two-armed f4 symmetryare is stable in spiral) at F. We can also see that the two-armed spirals with D f2 ) for values of λ and ρ between F and H. Fix(D

6.2

Example 2: Coefficient values for the Swift–Hohenberg equation

In Appendix C the method for computing the values of the coefficients in the general O(3) × Z2 equivariant vector field for the Swift–Hohenberg model is outlined. Using this method we can compute using the vector field (52)–(53) that in the representation on


20

c D^ 4 × Z2

^ O(2) × Zc2

O^ × Zc2

c D^ 6 × Z2

fd (i) D 4

fd (ii) D 4

f4 (i) D

f4 (ii) D

f2 (i) D

f2 (ii) D

Figure 6: Images of solutions to (31)–(33). These solutions all have symmetry groups f2 . In some cases two views of the solutions are given to show the containing D symmetries more clearly.

V2 ⊕ V3 where the radius of the sphere is R = 3 + ϵ2 R2 the coefficient values are µx = µ2 − γ2 =

1 44π ,

α2 =

8 27 R2 , 25 − 22π ,

µy = µ2 + β2 =

16 27 R2 ,

175 − 286π ,

15 6 α1 = − 28π , β1 = − 11π , γ1 =

δ1 =

7 − 2860π ,

δ2 =

3 − 44π

where µ = 19 + ϵ2 µ2 . Let µx = λ then µy = λ + ρ where ρ = values into equations (31)–(33) we have

8 9 R2 .

15 2 23 2 15 3 a − ab − ac 14π 22π 44π 315 3 15 2 371 2 b˙ = (λ + ρ) b − b − ba − bc 286π 22π 572π 23 2 371 2 1687 3 c − ca − cb . c˙ = (λ + ρ) c − 2860π 22π 286π

a˙ = λa −

and

3 44π , 1 δ3 = − 22π

(35)

Substituting these

(36) (37) (38)

A linear stability analysis of the equilibria of these equations results in the gyratory bifurcation diagram in Figure 8. See [32] for a full exposition of this analysis. We can see that f4 symmetry (two-armed for values of λ and ρ between C and I solution branches with D f2 ). It can be computed using (52)–(53) that, up to spirals) exist and are stable in Fix(D f4 symmetry is in fact stable in degeneracy in one eigenvalue, the solution branch with D the whole space when it exists (see [32]). Quintic order terms in the equivariant vector field are required to establish the sign of the real part of the remaining eigenvalue. f2 symmetry (one-armed spirals) exists between D and F but A branch of solutions with D this solution branch is never stable. It bifurcates from the branch of axial solutions with


Figure 7: Bifurcation diagram for the solution branches in the representation on V2 ⊕ V3 for the coefficient values in example 1. The top diagram is an unfolding diagram showing the lines on which bifurcations of the solution branches occur as the circle around the codimension 2 point, λ = ρ = 0, is traversed. The gyratory bifurcation diagram at the bottom of this figure shows which branch f2 ). each bifurcation lies on and the stability of the solution branches in Fix(D All bifurcations are pitchfork bifurcations.

21


22

fd c D^ 6 × Z2 symmetry at D and from the branch of submaximal solutions with D4 symmetry at F. The fact that this solution is unstable explains why in the numerical simulations detailed in section 2 no one-armed spiral patterns are found on spheres of radius R near 3 when s = 0. Sufficiently close to the codimension 2 point λ = ρ = 0 numerical simulations f2 ) agree with Figure 8 of the Swift–Hohenberg model (1) with s = 0 in this subspace Fix(D as to the stable solution branches and the locations of the secondary pitchfork bifurcations on these stable branches.

7

Spiral patterns in the representation of O(3)×Z2 on V3 ⊕V4

We now consider the solutions with the symmetries of one-, two- and three-armed spirals which can exist within the O(3) × Z2 equivariant vector field for the representation on f2 , D f4 and V3 ⊕ V4 . From the results of section 5 we can see that in this representation D f D6 (the symmetry groups of the most symmetric one- two- and three-armed spirals on a sphere respectively) are isotropy subgroups in this representation. An analysis identical to that of the case for the representation on V2 ⊕ V3 in section 6 reveals that the isotropy f2 are those listed in Table 4. The section of subgroups of O(3) × Z2 which contain D the lattice of isotropy subgroups of O(3) × Z2 in this representation including only those f2 is as in Figure 9. For each isotropy subgroup, the isotropy subgroups with contain D final column of Table 4 gives the form of the fixed-point subspace which lies inside f2 ) = {(0, a, 0, b, 0, a, 0 ; ic, 0, id, 0, 0, 0, −id, 0, −ic)}. Fix(D

(39)

The equivariant branching lemma guarantees the existence of solution branches with the symmetries of the axial isotropy subgroups in Table 4. Other solutions must be found f2 of the general O(3) × Z2 equivariant vector field. See directly using the restriction to D [32] for the form of the general cubic order mapping which commutes with the action of f2 consists O(3) × Z2 on V3 ⊕ V4 . In this representation this restriction of this mapping to D of four ODEs for which we would like to find all equilibria. Recall from Remark 3 that f2 , D f4 or D f6 will be stationary solutions. We find any spiral solutions with symmetries D that it is not possible to find formulae for every equilibrium of these ODEs in the general case so we focus on the specific case where the values of the coefficients are those which can be computed, using the method of Appendix C, for the Swift–Hohenberg model (1) with s = 0. We then have the four ODEs √ 371 2 315 2 225 2 3 210 315 3 a − ab − ac − ad + bcd (40) a˙ = λa − 286π 572π 286π 286π 286π √ 1687 3 371 2 21 2 219 2 3 210 b˙ = λb − b − ba − bc − bd + acd (41) 2860π 286π 22π 286π 143π √ 6615 3 315 2 21 2 5535 2 3 210 c˙ = (λ + ρ) c − c − ca − cb − cd + abd (42) 4862π 286π 44π 4862π 286π √ 39825 3 225 2 219 2 5535 2 3 210 d˙ = (λ + ρ) d − d − da − db − dc + abc.(43) 34034π 286π 572π 4862π 286π By studying these equations analytically we can find branches of solutions with the symc metries of each of the isotropy subgroups in Table 4 with the exceptions of (D^ 2 × Z2 )Dz2 ,


Figure 8: Bifurcation diagram for the solution branches in the representation on V2 ⊕ V3 for the coefficient values in example 2. The top diagram is an unfolding diagram showing the lines on which bifurcations of the solution branches occur for the Swift–Hohenberg equation as the circle around the codimension 2 point, λ = ρ = 0, is traversed. The gyratory bifurcation diagram at the bottom of this figure shows which branch each bifurcation lies on and the f2 ). All bifurcations are pitchfork stability of the solution branches in Fix(D bifurcations.

23


24

c f (D^ 2 × Z2 )Z2 ×Zc2 and D2 . It can be shown analytically in the general case that solution c ^c branches with symmetry (D^ 2 × Z2 )Dz2 or (D2 × Z2 )Z2 ×Zc2 never exist [32]. To find branches f2 symmetry we then compute (analytically) the stability of each of the of solutions with D solution branches we have found. Points at which the real part of one of the eigenvalues of these solution branches changes sign can be located and this information, along with the numerical branch continuation package AUTO, can then be used to establish the existence f2 symmetry [32]. Images of each of the solution types which of solution branches with D exist are given in Figure 10.

The linear stability analysis and numerical evidence from AUTO leads to the unfolding and gyratory bifurcation diagrams in Figures 11 and 12. We can see that for values of λ and ρ f4 symmetry (two-armed spirals) are stable in between K and Q a solution branch with D f2 ). Furthermore, between C and M a branch of solutions with D f6 symmetry (threeFix(D armed spirals) are stable. At the transcritical bifurcation at M this solution loses stability f2 symmetry (one-armed spirals) which is stable until the to a branch of solutions with D pitchfork bifurcation at N. These stable solutions can be found in numerical simulations of the Swift–Hohenberg model (1) on spheres of radius R near 4 when s = 0. Sufficiently close to the codimension 2 point λ = ρ = 0 these numerical simulations in the subspace f2 ) agree with Figure 12 as to the stable solution branches and the locations of the Fix(D secondary pitchfork bifurcations on these stable branches. Isotropy subgroup O^ × Zc2 ^ O(2) × Zc2

K

O × Zc2

O−

O(2) × Zc2

c (D^ 6 × Z2 )Dd c D^ 4 × Z2

H

6

c (D^ 6 × Z2 )D3 ×Zc 2 c D^ 8 × Z2 c (D^ 2 × Z2 )Dz 2

D6 ×

Zc2

Fixed-point subspace {(0, a, 0, 0, 0, a, 0 ; 0, 0, 0, 0, 0, 0, 0, 0, 0)}

O(2)

−

Dd6

{(0, 0, 0, b, 0, 0, 0 ; 0, 0, 0, 0, 0, 0, 0, 0, 0)} √ )} {( √ 3 3 b, 0, b, 0, 10 b, 0 ; 0, 0, 0, 0, 0, 0, 0, 0, 0 0, 10

D4 × Zc2

D2 × Zc2

D6 ×

Zc2

D3 ×

Zc2

{(0, 0, 0, 0, 0, 0, 0 ; 0, 0, id, 0, 0, 0, −id, 0, 0)} √ √ {( )} 0, 0, 0, 0, 0, 0, 0 ; ic, 0, 7ic, 0, 0, 0, − 7ic, 0, −ic

D8 ×

Zc2

D4 × Zc2

D2 ×

Zc2

Dz2

{(0, a, 0, b, 0, a, 0 ; 0, 0, 0, 0, 0, 0, 0, 0, 0)}

Z2 × Zc2

{(0, 0, 0, 0, 0, 0, 0 ; ic, 0, 0, 0, 0, 0, 0, 0, −ic)}

c (D^ 2 × Z2 )Z2 ×Zc 2

D2 × Zc2

f6 D f4 D

D6

D3

{(0, 0, 0, 0, 0, 0, 0 ; ic, 0, id, 0, 0, 0, −id, 0, −ic)} √ )} {( √ √ √ 3 3 b, 0, b, 0, 10 b, 0 ; ic, 0, 7ic, 0, 0, 0, − 7ic, 0, −ic 0, 10

D4

D2

{(0, a, 0, 0, 0, a, 0 ; 0, 0, id, 0, 0, 0, id, 0, 0)}

fd D 8 fd ) (D

Dd8

Dz4

{(0, 0, 0, b, 0, 0, 0 ; ic, 0, 0, 0, 0, 0, 0, 0, −ic)}

Dd4

Dz2

{(0, 0, 0, b, 0, 0, 0 ; 0, 0, id, 0, 0, 0, −id, 0, 0)}

fd ) − (D 4 Z 4 f2 D

Dd4

Z− 4

{(0, a, 0, 0, 0, a, 0 ; ic, 0, 0, 0, 0, 0, 0, 0, −ic)}

D2

Z2

{(0, a, 0, b, 0, a, 0 ; ic, 0, id, 0, 0, 0, −id, 0, −ic)}

z 4 D2

Table 4: Isotropy subgroups of O(3) × Z2 in the representation on V3 ⊕ V4 which contain f2 . D


f2 for the Figure 9: The lattice of isotropy subgroups of O(3) × Z2 which contain D representation on V3 ⊕ V4 .

25


26

O^ × Zc2

^ O(2) × Zc2

c (D^ 6 × Z2 )Dd

c D^ 4 × Z2

c (D^ 6 × Z2 )D3 ×Zc 2

c D^ 8 × Z2

f6 (i) D

f6 (ii) D

f4 (i) D

f4 (ii) D

fd (i) D 8

fd (ii) D 8

fd ) z (i) (D 4 D2

fd ) z (ii) (D 4 D2

fd ) − (i) (D 4 Z

fd ) − (ii) (D 4 Z

6

4

4

f2 D

Figure 10: Images of solutions to (40)–(43). These solutions all have symmetry groups f2 In some cases two images of the solutions are given to show containing D the symmetries more clearly.


f2 ) for the Figure 11: Unfolding diagram for the Swift–Hohenberg equation in Fix(D representation on V3 ⊕V4 . This diagram shows the lines on which bifurcations of the solution branches occur as the circle around the codimension 2 point, λ = ρ = 0, is traversed.

27

Figure 12: Gyratory bifurcation diagram which shows the stability of each of the branches of solutions and the locations of the bifurcation points as the circle in Figure 11 is traversed. All bifurcations are pitchfork bifurcations with the exception of the bifurcation at M which is transcritical.



8

29

Spiral patterns in the ℓ, ℓ + 1 mode interaction

In the general case of the codimension-two mode interaction between the ℓ and ℓ + 1 f2 ) is of dimension ℓ + 1, so analytical progress in this representations, the subspace Fix(D ] subspace is not possible. However, the subspace Fix(D 2m ), where (l + 1)/3 < m ≤ l, is two-dimensional. In this subspace, a rotation through 2π/m acts as the identity and a rotation through π/m acts as −1. An explicit realisation of this subspace is to choose xm = a,

x−m = (−1)m a,

ym = ib,

y−m = (−1)m+1 ib,

(44)

in (8), with a, b ∈ R and xj = yj = 0 for j ̸= ±m. This two-dimensional subspace includes one-dimensional subspaces a = 0 and b = 0 corresponding to axial isotropy subgroups (D2m × Zc2 , Dm × Zc2 ) and (D2m × Zc2 , Dd2m ) for m ≥ 3, or (D4 × Zc2 , D2 × Zc2 ) and (O × Zc2 , O− ) when m = 2. The general form of the equivariant vector field in this subspace can easily be deduced up to cubic order. The rotation symmetry acting as −1 means that no quadratic terms occur. Also, the point inversion symmetry acts as +1 on the even-ℓ mode and −1 on the odd-ℓ mode, so the equations must be equivariant with respect to a sign change of either mode. Hence the cubic truncation must be of the form a˙ = µx a + αa3 + βab2 b˙ = µy b + γb3 + δba2 .

(45) (46)

In addition to the axial solution branches with a = 0 or b = 0 there is a mixed-mode solution βµy − γµx δµx − αµy a2 = , b2 = , (47) αγ − βδ αγ − βδ which exists for certain values of µx and µy (for example if αγ −βδ > 0, this solution exists in the region of the (µx , µy ) plane where βµy − γµx > 0, δµx − αµy > 0). This mixed-mode ] solution with D 2m symmetry bifurcates directly from one of the axial branches. ] Hence, an m-armed spiral solution with isotropy D 2m , for (l + 1)/3 < m ≤ l, exists generically in some region of the bifurcation diagram near the ℓ, ℓ + 1 mode interaction point.

9

Weak symmetry breaking from O(3) × Z2 to O(3)

In sections 6–8 of this paper we have demonstrated that for certain values of m ≥ 1, ] m-armed stationary spiral patterns on spheres with symmetry groups D 2m ⊂ O(3) × Z2 can exist in representations on Vℓ ⊕ Vℓ+1 . The next question to address is what happens to these stationary solutions when the symmetry is weakly broken from O(3) × Z2 to O(3) by introducing small terms which are only equivariant with respect to O(3). This can be done by adding small even order terms to the general O(3) × Z2 equivariant vector field. This is equivalent to having s or q small in (1) or (4) respectively which breaks the w → −w symmetry which is present when s = 0 or q = 0. Recall that solutions which can exist in an O(3)×Z2 equivariant vector field have isotropy subgroup H θ which is uniquely defined by the pair (H, K) where H/K = Z2 or 1. Suppose that, for some values of the coefficients of the terms in the cubic truncation of the


30

equivariant vector field, a solution with H θ symmetry exists. Suppose then that small quadratic terms are added which break the symmetry to O(3). The solution with H θ (if it persists) would have only K symmetry. This means that one-armed spirals would have only Z2 symmetry and m armed-spirals for m ≥ 2 would have Dm symmetry provided such solutions persist. Recall from Remark 2 that the one-arm spiral solution (if it persists) is a periodic solution so the spiral wave generically rotates. We now demonstrate that it is possible for multi-armed spiral patterns to persist under this symmetry breaking. g Proposition 4. Stationary solutions z0 with symmetry D 2m (m ≥ 2) which exist within O(3)×Z2 equivariant vector fields persist as stationary solutions with Dm symmetry when the O(3) × Z2 symmetry is broken to O(3) by adding small even order terms to the vector field. Proof. In order to use the Implicit Function Theorem to demonstrate the persistence of the ] solution we must show that the solution z0 with D 2m symmetry has no zero eigenvalues within Fix(Dm ). ] The solution with D 2m symmetry generically has ] dim(O(3) × Z2 ) − dim(D 2m ) = 3 zero eigenvalues in Vℓ ⊕ Vℓ+1 . In the restriction of the O(3) equivariant vector field to Fix(Dm ) the equations are equivariant with respect to NO(3) (Dm )/Dm . Since dim(NO(3) (Dm )/Dm ) = 0

∀m ≥ 2

the group orbit of the solution is zero dimensional and hence generically the solution has no zero eigenvectors in Fix(Dm ). The three eigenvalues which are forced to be zero by symmetry must lie in the complement of Fix(Dm ). ] Furthermore, in a small enough neighbourhood of the solution with D 2m symmetry, solutions with Dm symmetry will have the same stability properties. Remark 5. Since dim(NO(3) (Z2 )/Z2 ) = 1 every solution has a zero eigenvector in Fix(Z2 ) and hence the Implicit Function Theorem cannot be used to show the persistence of onearmed spiral solutions as periodic solutions with Z2 symmetry. The persistence of one-armed spirals can be considered on a case by case basis. We find f2 symmetry which we found in the O(3) × Z2 that the unstable single-armed spiral with D equivariant vector field on V2 ⊕ V3 with the values of the coefficients given by the Swift– Hohenberg model can persist as 1. a family of stationary spiral solutions in the variational equation (1) with s of order ϵ or 2. a periodic solution in the nonvariational equation (4) with q of order ϵ.


31

In either case we compute the terms of quadratic order which commute with the action O(3) on V2 ⊕ V3 . The coefficients of these equivariant mappings depend linearly on the value of s or q. By expanding in powers of s or q (where s and q are small) one can find solutions with Z2 symmetry as above in the vector field. See [32] for details. The periodic solution in the nonvariational equation (4) is found to rotate at a speed proportional to q. Numerical simulations of (4) on a sphere of radius R ≈ 4 starting from the stable singlearmed spiral pattern found in section 7 with q small positive result in a single-armed spiral pattern with Z2 symmetry which rotates in the direction indicated in Figure 13. Notice that unlike the planar case, the spiral does not rotate about its tips but about a point midway between the two spiral tips. The speed of rotation is again found to be proportional to q.

Figure 13: The rotating single-armed spiral which results from numerical simulations of (4) for small positive value of q on a sphere of radius near 4. The inif2 symmetry. The tial condition is a stationary single-armed spiral with D arrow indicates the direction and axis of rotation. The speed of rotation is proportional to q.

10

Conclusions

In this paper we have investigated how spiral patterns can appear in a pattern-forming system near a stationary bifurcation in spherical geometry. Equivariant bifurcation theory for the symmetry group O(3) is complicated, because of the large number of subgroups and the large dimension of the irreducible representations [16, 24]. The case of spiral patterns is considerably more involved, since spirals cannot occur in the irreducible representations; it is necessary to consider reducible representations involving a mode interaction of spherical harmonics of degree ℓ and ℓ + 1. The symmetries of spirals on a sphere are greater than those of spirals on a plane, since spherical spirals generally have two tips at opposite points on the sphere. We have concentrated on the case where an additional Z2 symmetry is present in the system. This introduces some further complications in the classification of the isotropy subgroups, but also has three significant simplifying features. The first is that all spiral patterns have sufficient symmetry that they are prevented from rotating, so they appear at stationary bifurcations. Secondly, the extra symmetry reduces the size of the invariant subspaces by


32

approximately a factor of two. Thirdly, this symmetry rules out quadratic terms from the equations on the centre manifold, simplifying the study of these equations in their cubic truncation. If this extra symmetry is weakly broken, the one-armed stationary spiral solutions persist in the form of slowly rotating waves (although the rotation is not about the spiral tip), but the multi-armed spirals remain as stationary states. A large part of the work is the computation of the dimension of the fixed-point subspaces of the twisted subgroups of O(3) × Z2 ; this enables us to determine which subgroups are isotropy subgroups for any given value of ℓ. We have shown that an isotropy subgroup corresponding to the symmetry of m-armed spirals exists for 1 ≤ m ≤ l. However, since this subgroup is never maximal, the existence of equilibria with this symmetry needs to be investigated on a case-by-case basis. For l = 2 and l = 3 we have constructed detailed bifurcation diagrams showing how some symmetric spirals can bifurcate directly from the primary axial solutions. These spiral patterns can themselves undergo bifurcations to less symmetric spirals. Hence, spiral patterns can exist and can be stable for certain values of the control parameter. In the general ℓ, ℓ + 1 mode interaction, a similar existence result was found for the most symmetric spirals. A possible extension of this work would be to consider the case where the primary bifurcation is oscillatory. This would be more directly relevant in the context of spiral waves in excitable media [2, 13, 38], but would add considerable complexity to the analysis.

Acknowledgements The authors would like to thank Stephen Cox for many helpful discussions. We are also grateful to Rob Womersley for making available his tables of points on the sphere via his web page. RS would also like to thank the EPSRC for providing financial support for this work through a doctoral training account.

A

Matrices for the action of O(3) on Vℓ ⊕ Vℓ+1

The natural action of the element γ ∈ O(3) on the vector z ∈ Vℓ ⊕ Vℓ+1 is given by multiplication on the left by the 4(ℓ + 1) × 4(ℓ + 1) matrix Mγ . A set of generators of O(3) is 1. An infinitesimal rotation ϕ′ in the ϕ direction. 2. An infinitesimal rotation θ′ in the θ direction. 3. The inversion element −I. The matrices Mγ for each of these generators are given by ] [ Mγℓ 0ℓ,ℓ+1 Mγ = 0ℓ+1,ℓ Mγℓ+1

(48)

where Mγℓ is the matrix for the action of γ on Vℓ and 0j,k is the (2j + 1) × (2k + 1) zero matrix.

SYMMETRIC SPIRAL PATTERNS ON SPHERES Here Mϕℓ ′ = diag

( ) ′ ′ ′ ′ e−iℓϕ , ei(−ℓ+1)ϕ , . . . , ei(ℓ−1)ϕ , eiℓϕ ℓ M−I = (−1)ℓ I2ℓ+1

where I2ℓ+1 is the (2ℓ + 1) × (2ℓ + 1) identity matrix and [ T ] T Mθℓ′ = v−ℓ | v−ℓ+1 | · · · | vℓT ,

33

(49) (50)

(51)

T where i.e. the matrix with columns vm ( ) 1√ 1√ ′ ′ vm = 0, . . . , 0, − (ℓ + m)(ℓ − m + 1)θ , 1, (ℓ − m)(ℓ + m + 1) θ , 0, . . . , 0 2 2

for m = −ℓ, . . . , ℓ where the entry 1 lies in the mth position.

Cubic truncation of the O(3) × Z2 equivariant vector field for the representation on V2 ⊕ V3

B

Using the fact that a O(3) × Z2 equivariant vector field must satisfy (11) and (25) we can compute that to cubic order the equivariant vector field on V2 ⊕ V3 has general form f (z, λ) = (g(z, λ); h(z, λ)) where g(z, λ) = µx x + α1 x|x|2 + β1 x|y|2 + γ1 P(x, y) + γ2 Q(x, y) 2

2

h(z, λ) = µy y + α2 y|x| + β2 y|y| + δ1 R(y) + δ2 S(x, y) + δ3 T(x, y).

(52) (53)

Here µ1 , µ2 , α1 , α2 , β1 , β2 , γ1 , γ2 , δ1 , δ2 and δ3 ∈ R are smooth functions of λ, |x|2 = |x−2 |2 + |x−1 |2 + |x0 |2 + |x1 |2 + |x2 |2 = x20 − 2x1 x−1 + 2x2 x−2 |y|2 = |y−3 |2 + |y−2 |2 + |y−1 |2 + |y0 |2 + |y1 |2 + |y2 |2 + |y3 |2 = y02 − 2y1 y−1 + 2y2 y−2 − 2y3 y−3 and • P(x, y) = (P−2 , P−1 , P0 , P1 , P2 ) where P−k = (−1)k P k and √ √ 2 P−2 (x, y) = 18y−3 y3 x−2 − 2y−2 y2 x−2 − 10y−2 x2 + 4 15y−3 y−1 x2 − 5 6y−3 y2 x−1 √ √ + 10 (y−2 y1 x−1 + 4y−3 y1 x0 + 2y−2 y−1 x1 ) − 2 5y0 (y−2 x0 + 3y−3 x1 ) 2 P−1 (x, y) = 10y−1 x1 + 5y−3 y2 x0 − 3y−3 y3 x−1 − 18y−2 y2 x−1 + 5y−1 y1 x−1 √ √ √ √ −2 30y−2 y0 x1 − 5 2y−1 y0 x0 + 6 5y−3 y0 x2 + 5 6y3 y−2 x−2 √ √ − 10y−1 (y2 x−2 + 2y−2 x2 ) + 15y1 (3y−2 x0 − 2y−3 x1 ) √ √ P0 (x, y) = 4 10 (y−1 y3 x−2 + y−3 y1 x2 ) − 3 15 (y−2 y1 x1 + y−1 y2 x−1 ) √ √ −2 5y0 (y−2 x2 + y2 x−2 ) + 5 2y0 (y1 x−1 + y−1 x1 ) ( ) −5 (y−3 y2 x1 + y−2 y3 x−1 ) + 2x0 y−3 y3 + 4y−2 y2 + 5y−1 y1 − 5y02


34

• Q(x, y) = (Q−2 , Q−1 , Q0 , Q1 , Q2 ) where Q−k = (−1)k Qk and √ √ ( ) Q−2 (x, y) = x−2 y02 − 6y−2 y2 + 16y−3 y3 + 3 10y−2 y1 x−1 + 2 10y−3 y1 x0 √ √ √ 2 √ −4 5y−2 y0 x0 − 2 3y−1 y0 x−1 + 2 6y−1 x0 − 5 6y−3 y2 x−1 ( ) 2 Q−1 (x, y) = x−1 y−3 y3 − 6y−2 y2 + 9y−1 y1 − 5y02 + 6y−1 − 5y−3 y2 x0 √ √ √ √ −3 10y−1 y2 x−2 + 2 3y1 y0 x−2 − 2 30y−2 y0 x1 − 2y0 y−1 x0 √ √ + 5 6y3 y−2 x−2 + 15y1 (y−2 x0 + 2y−3 x1 ) ( ) Q0 (x, y) = x0 −7y02 + 12y−1 y1 − 6y−2 y2 − 4y−3 y3 + 5y−3 y2 x1 + 5y−2 y3 x−1 √ √ − 15 (y1 y−2 x1 + y−1 y2 x−1 ) + 2 10 (y−3 y1 x2 + y−1y3 x−2 ) √ √ − 4 5y0 (y−2 x2 + y2 x−2 ) + 2y0 (y1 x−1 + y−1 x1 ) √ ( ) 2 + 2 6 y12 x−2 + y−1 x2 • R(y) = (R−3 , R−2 , R−1 , R0 , R1 , R2 , R3 ) where R−k = (−1)k Rk and √ √ √ ( ) 2 3 R−3 (y) = 15y−3 4y−1 y1 − 3y02 − 10 15y−2 y1 − 4 15y−1 + 30 2y−2 y−1 y0 √ √ ( ) R−2 (y) = 5y−2 3y02 + 4y−1 y1 − 10y−2 y2 − 20 15y−3 y−1 y2 − 30 2y−3 y0 y1 √ 2 −2 30y−1 y0 √ ( ) R−1 (y) = y−1 60y−3 y3 − 20y−2 y2 + 8y−1 y1 − 9y02 + 30 2y−3 y0 y2 √ √ √ 2 −12 15y12 y−3 − 10 15y−2 y3 + 4 30y−2 y0 y1 √ ( ( ) ) 2 R0 (y) = 3y0 30y−3 y3 + 10y−2 y2 + 6y−1 y1 − 3y02 − 2 30 y−2 y12 + y−1 y2 √ −30 2 (y−3 y1 y2 + y3 y−2 y−1 ) • S(x, y) = (S−3 , S−2 , S−1 , S0 , S1 , S2 , S3 ) where S−k = (−1)k Sk and √ √ √ ) ( S−3 (x, y) = 15y−3 2x−1 x1 − x20 − 10 6x−2 x1 y−2 + 2 15x2−2 y1 − 6 5x−2 x−1 y0 √ + 6 10x0 x−2 y−1 √ ) ( S−2 (x, y) = −5y−2 3x20 − 2x−1 x1 + 4x−2 x2 − 2 10x−2 (2x1 y−1 + x−1 y1 ) √ √ √ + 4 15x−1 x0 y−1 − 2 30x2−1 y0 + 10x2−2 y2 + 10 6x−1 x2 y−3 √ + 6 5x−2 x0 y0 √ ( ) S−1 (x, y) = −y−1 3x20 − 14x−1 x1 + 28x−2 x2 + 2 10x−1 (2x2 y−2 + x−2 y2 ) √ √ √ √ + 6 10x0 x2 y−3 + 2 15x2−2 y3 − 4 15x0 x1 y−2 − 2 3x−2 x1 y0 √ √ + 4 6x−2 x0 y1 − 16x2−1 y1 + 6 2x0 x−1 y0 √ ( ) S0 (x, y) = 3y0 x20 + 6x−1 x1 − 10x−2 x2 + 2 3 (x−2 x1 y1 + x2 x−1 y−1 ) √ √ ( ) −6 2x0 (x1 y−1 + x−1 y1 ) − 2 30 x2−1 y2 + x21 y−2 √ √ + 6 5x0 (x2 y−2 + x−2 y2 ) + 6 5 (x1 x2 y−3 + x−1 x−2 y3 )


35

• T(x, y) = (T−3 , T−2 , T−1 , T0 , T1 , T2 , T3 ) where T−k = (−1)k Tk and √ √ ( ) T−3 (x, y) = 5y−3 2x20 − 3x−1 x1 + 5 6x−2 x1 y−2 − 5x−1 x0 y−2 − 2 10x0 x−2 y−1 √ + 15x2−1 y−1 √ ( ) T−2 (x, y) = 5y−2 x20 − 2x−1 x1 + 2x−2 x2 − 5 6x−1 x2 y−3 + 5x0 x1 y−3 √ √ √ √ − 15x1 x−2 y−1 + 30x2−1 y0 − 4 5x−2 x0 y0 + 3 10x−2 x1 y−1 √ √ ( ) T−1 (x, y) = y−1 2x20 − 7x−1 x1 + 16x−2 x2 − 2 10x0 x2 y−3 + 15x21 y−3 √ √ √ + 6x2−1 y1 + 2 3x−2 x1 y0 − 4 6x−2 x0 y1 + 15x0 x1 y−2 √ √ − 2x0 x−1 y0 − 3 10x−1 x2 y−2 √ ( ) T0 (x, y) = y0 x20 − 6x1 x−1 + 18x2 x−2 − 4 5x0 (x2 y−2 + x−2 y2 ) √ ( √ ) + 30 x21 y−2 + x2−1 y2 − 2 3 (x−2 x1 y1 + x−1 x2 y−1 ) √ + 2x0 (x1 y−1 + x−1 y1 )

C

Computing values of coefficients in general vector fields for the Swift–Hohenberg model

In this section we show how to compute the values of the coefficients in the general O(3) × Z2 equivariant vector field for the Swift–Hohenberg model. These values were used in Example 2 of section 6. Recall from section 2 that the ℓ and ℓ+1 modes are simultaneously unstable when Rc = ℓ+1 and µc = 1/(ℓ + 1)2 . By letting µ = µc + ϵ2 µ2 ,

R = Rc + ϵ 2 R 2 ,

T = ϵ2 t,

w = ϵw1 + ϵ2 w2 + ϵ3 w3

(54)

in (1) with s = 0 we find that to cubic order in ϵ ϵ3

∂w1 = ϵL0 w1 + ϵ2 L0 w2 + ϵ3 (L0 w3 + L2 w1 − w13 ). ∂T

(55)

where the operators L0 and L2 act on the spherical harmonics of degree ℓ as follows. ) ℓ(ℓ + 1) 2 = µc − 1 − Rc2 ( )( ) ℓ(ℓ + 1) 2ℓ(ℓ + 1) = µ2 − 2 1 − R2 Rc2 Rc3 (

L0 L2

(56) (57)

If we let w1 =

ℓ ∑ m=−ℓ

xm (T )Yℓm (θ, ϕ)

+

ℓ+1 ∑

n yn (T )Yℓ+1 (θ, ϕ),

and

w2 = 0

(58)

n=−ℓ−1

then (55) is satisfied at orders ϵ and ϵ2 . At order ϵ3 we have ∂w1 = L0 w3 + L2 w1 − w13 ∂T

(59)


36

p

If we multiply (59) by Y ℓ and integrate over the sphere we find ( ) ∫ 2π ∫ π 4ℓ p x˙ p = µ2 − R w13 Y ℓ sin θ dθdϕ. x − 2 p (ℓ + 1)3 0 0

(60)

since the other terms from the right-hand-side of (59) become zero due to orthogonality p of the spherical harmonics. Similarly if we multiply (59) by Y ℓ+1 and integrate over the sphere we find ) ( ∫ 2π ∫ π 4(ℓ + 2) p y˙ p = µ2 + R2 y p − w13 Y ℓ+1 sin θ dθdϕ. (61) (ℓ + 1)3 0 0 Comparing these equations with the general O(3) × Z2 equivariant vector field for some value of ℓ determines the values of the coefficients of the terms in that vector field which correspond to the Swift–Hohenberg model.

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