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. 1 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 2 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 SE2) 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, SO3), 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.

SYMMETRIC SPIRAL PATTERNS ON SPHERES 3 2 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 t = µw 1 + 2 ) 2 w + sw 2 w 3. 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 Yl m 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 ll+1)/r 2 where R is the radius of the sphere and is constant. Thus the spherical harmonics satisfy 2 Y m l θ, ϕ) = ll + 1) R 2 Yl m θ, ϕ). They also satisfy Y m l θ, ϕ) = 1) m Yl m θ, ϕ), where the overbar denotes the complex conjugate [9]. The functions Yl m θ, ϕ) form an orthonormal basis of the subspace of L 2 of square integrable functions on the sphere of constant radius. Hence the variable wθ, ϕ, t) can be written as wθ, ϕ, t) = l x l,m t)yl m θ, ϕ), 2) l 0 m= l where x l,0 is real for all values of l, and x l,m is complex for m 0. Also x l, m = 1) m x l,m since w is real. This expansion holds in the sense that 2π π N 2 lim N 0 0 wθ, ϕ, t) l x l,m t)yl m θ, ϕ) sin θ dθ dϕ = 0. l=0 m= l By linearizing 1) using the expansion 2) we see that for a particular value of l all the modes x l,m have the same growth rate λ = µ 1 ll + 1)/R 2) 2, 3)

SYMMETRIC SPIRAL PATTERNS ON SPHERES 4 where R is the radius of the sphere. Thus there is instability when µ > µ c where µ c = 1 ll + 1)/R 2 ) 2. A plot of µc as a function of R for different values of l is given in Figure 1. The function µ c R) takes its minimum value of zero when R 2 = ll + 1). When R = l + 1 the modes of degrees l and l + 1 both have the same growth rate and µ c = 1/l + 1) 2. This is the point in Figure 1 where the l and l + 1 plots cross. Figure 1: A plot of µ c 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 5l c, where l c is the value of l that minimises µ c for the chosen value of R. For the linear terms in the equation, each mode is multiplied by a factor e λ 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 wθ, 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 x l,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

SYMMETRIC SPIRAL PATTERNS ON SPHERES 5 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 t = µw 1 + 2 ) 2 w + qw 2 w w 3 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

SYMMETRIC SPIRAL PATTERNS ON SPHERES 6 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 2b) 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 O3) and thus the symmetry group or isotropy type of each spiral without w w symmetry is a subgroup of O3). For details of the subgroups of O3) see [16, p.103 and p.120]. Every spiral pattern which we consider in this paper has a rotation-through-π symmetry, Rπ, e 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 rotationthrough-2π/m symmetry, R2π/m t, in the axis through the spiral tips. Thus the symmetry group of the one-armed spiral is Z 2 and an m-armed spiral for m 2 has symmetry group D m, 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

SYMMETRIC SPIRAL PATTERNS ON SPHERES 7 the sphere. In addition to the rotational symmetries contained in O3) described above, an m-armed spiral can have the symmetry R t π/m, 1 ) O3) Z 2, 5) where Rπ/m t is a rotation through π/m in the axis through the spiral tips and 1 is the non-identity element in Z 2 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 O3) Z 2. The symmetry group of the most symmetric one-armed spiral is then D 2 = Rπ, e 1), Rπ, t 1 ) 6) and the symmetry group of the most symmetric m-armed spiral for m 2 is ) D 2m = Rπ, e 1), Rπ/m t, 1. 7) The tildes indicate that these are twisted subgroups of O3) Z 2. That is, some of the elements in the subgroup have the non-identity element of Z 2 as their second component see [16, 17] for a precise definition of twisted subgroups). Notice that D 2 D 2m for all values of m; when m is odd this is obvious but for m even observe that D 2m with the generators above contains the subgroup D 2 with generators Rπ, t 1 ) ) and RπR e π/m t, 1 where RπR e π/m t 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 O3) Z 2 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 O3) = SO3) Z c 2 where SO3) is the group of all rotations of the sphere and Z c 2 = {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 SO3) has precisely one irreducible representation in each odd dimension 2l + 1 for l 0, denoted by V l, where V l is the space of spherical harmonics of degree l. For each irreducible representation of SO3) there are two irreducible representations of O3), 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 O3). In the natural representation of O3) on V l, 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 O3) 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 O3) does not act as the identity or minus the identity on any spiral

SYMMETRIC SPIRAL PATTERNS ON SPHERES 8 pattern we consider in this paper since the symmetry which maps tips to tips is a rotation in every case. If a pattern, wθ, 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 wθ, ϕ, t). Similarly if wθ, ϕ, t) can be made with a linear combination of spherical harmonics of odd degree then in the natural representation of O3), I must act as minus the identity on wθ, 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 wθ, ϕ, t) = l m= l x m t)y m l θ, ϕ) + l+1 n= l 1 y n t)yl+1 n θ, ϕ) 8) where x m = 1) m x m and y n = 1) n y n since wθ, 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 O3) 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 O3) 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 O3) symmetry, the representation of O3) must be a reducible representation on the 4l + 1) dimensional space V l V l+1. Consider the system of ordinary differential equations dz dt where λ, ρ R are bifurcation parameters, = fz, λ, ρ) 9) z = x ; y) = x l, x l 1),..., x l ; y l+1), y l,... y l+1) ) 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 O3) on V l V l+1. By this we mean that γ fz, λ, ρ) = fγ z, λ, ρ) z V l V l+1, γ O3) 11) where denotes the action of O3) on V l V l+1. This action of γ O3) on z V l V l+1 is executed by multiplication on the left by the 4l + 1) 4l + 1) matrix M γ. The form 10)

SYMMETRIC SPIRAL PATTERNS ON SPHERES 9 of these matrices for a set of generators of O3) is given in Appendix A. We say that f is a O3) equivariant vector field. Suppose that z 0 is an equilibrium of 9) for some values of λ and ρ. Since f satisfies 11), γz 0 is also an equilibrium. Equilibria of 9) exist in group orbits, O3))z 0 = {γz 0 : γ O3)}. 12) The symmetry of an equilibrium z 0 V l V l+1 is the set of all γ O3) that leaves z 0 invariant. This set is a subgroup of O3) called the isotropy subgroup of z 0 and is denoted Σ z0 = {γ O3) : γz 0 = z 0 }. 13) The group orbit O3))z 0 is a smooth manifold of dimension 3 dimσ z0 ). 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 = ranknσ z0 )/Σ z0 ), 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 NΣ z0 ) = {γ O3) : γσ z0 γ 1 = Σ z0 } is the normalizer of Σ z0 in O3). A subgroup Σ O3) is an isotropy subgroup if it fixes some vector z V l V l+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 O3) in the representation on V l V l+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 O3) for the action of O3) on V l V l+1 is the subspace Fix Vl V l+1 Σ) = {z V l V l+1 : σz = z σ Σ}. 15) We say that a subgroup Σ O3) is an axial isotropy subgroup if it is an isotropy subgroup which has a one-dimensional fixed-point subspace. The subgroups of O3) fall into three classes I The subgroups of SO3): O2), SO2), I, O, T, D n and Z n for n 1). II Subgroups of the form K Z c 2 where K is a subgroup of SO3). III Subgroups which are not contained in SO3) and do not contain the inversion element I O3). These subgroups Σ are denoted by O2), O, D d 2m, Dz m and D d 2m. Let Π : O3) SO3) be the homomorphism whose kernel is Zc 2. Each class III subgroup Σ O3) is isomorphic but not conjugate) to the subgroup ΠΣ) SO3) whose symbol is the same as Σ except without the superscript, for example ΠO2) ) = O2). The notation used in this paper for these subgroups is the standard notation for the subgroups of O3) which is used in particular in [16] where precise definitions of these subgroups can be found.

SYMMETRIC SPIRAL PATTERNS ON SPHERES 10 For each subgroup Σ O3) the dimension of the subspace Fix Vl V l+1 Σ) can be computed using the formulae for dim Fix Vl Σ) given by Theorems 8.1 and 9.5 of [16, Chapter XIII]. If Σ is a class I subgroup of O3) then dim Fix Vl V l+1 Σ) = dim Fix Vl Σ) + dim Fix Vl+1 Σ). 16) If Σ = K Z c 2 where K is a subgroup of SO3) then { dim FixVl K) for l even dim Fix Vl V l+1 Σ) = dim Fix Vl+1 K) for l odd. 17) Finally if Σ is a class III subgroup of O3) then dim Fix Vl V l+1 Σ) = { dim FixVl ΠΣ)) + dim Fix Vl+1 Σ) for l even dim Fix Vl Σ) + dim Fix Vl+1 ΠΣ)) for l odd. 18) Since Fix Vl V l+1 O3)) = {0}, 9) has a trivial equilibrium z = 0 with O3) symmetry for all values of λ and ρ. We assume that at λ = 0 the modes of degree l those in V l ) 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. FixΓ) = {0}, 2. Σ Γ is an axial isotropy subgroup 3. f : V R V commutes with the action of Γ on V and satisfies f0, 0) = 0, df) 0,0) = 0 and df λ ) 0,0) v 0 19) for some nonzero v FixΣ). Then there exists a unique branch of solutions to fz, λ) = 0 emanating from 0, 0) where the symmetry of the solution is Σ. Here df λ ) is defined by df λ ) ij = df) ij λ. Proof. See [16, Chapter XIII, Theorem 3.5] By restricting our problem where Γ = O3) to V l we see that by the equivariant branching lemma, equilibria with the symmetries of isotropy subgroups Σ O3) which fix a onedimensional subspace of V l bifurcate from the trivial solution when λ = 0. We also assume that at λ = ρ the modes of degree l + 1 those in V l+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 O3) which fix a one-dimensional subspace of V l+1 emanate from this bifurcation.

SYMMETRIC SPIRAL PATTERNS ON SPHERES 11 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 O3) in the representation on V l V l+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 Fix Vl V l+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 O3) Z 2 symmetry where the isotropy subgroups are then twisted subgroups of O3) Z 2. 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 Z 2 symmetry or m-armed spiral solutions for m 2 with D m symmetry. The solutions of 9) which are guaranteed to exist by the equivariant branching lemma have the symmetries of the axial isotropy subgroups of O3) in the representations on V l which bifurcate at λ = 0) and V l+1 which bifurcate at λ = ρ). Since by 16) dim Fix Vl V l+1 Z 2 ) = dim Fix Vl Z 2 ) + dim Fix Vl+1 Z 2 ) = 2l + 2, 20) even if Z 2 is an isotropy subgroup of O3) for some representation on V l V l+1, it never has a one dimensional fixed point subspace and hence it is never an axial isotropy subgroup. Also by 16) dim Fix Vl V l+1 D m ) = dim Fix Vl D m ) + dim Fix Vl+1 D m ) = [l + 1)/m] + [l/m] + 1, 21) where [x] is the greatest integer less than or equal to x. Hence dim Fix Vl V l+1 D m ) = 1 when l < m 1. However D m O2) Z c 2 for all values of m and dim Fix V l V l+1 O2) Z c 2 ) = 1 for all values of l so D m is never an axial isotropy subgroup of O3). 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 Z 2 or D m 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 V l V l+1 for some value of l. Then to determine whether solutions with these symmetries can exist we must look for them directly in the O3) equivariant vector field f. In the representation of O3) on V l V l+1 this vector field is 4l+1) dimensional. Since Z 2 D m for all values of m, if any spiral solutions with Z 2 or D m symmetry exist then they can be found in the restriction of the vector field f to the 2l + 1) dimensional vector space Fix Vl V l+1 Z 2 ). Even for low values of l this vector space is large so to find one-armed spirals with Z 2 symmetry in this space is not a simple task. Remark 2. Since NZ 2 ) = O2) Z c 2 but ND m) = D 2m Z c 2, ranknz 2 )/Z 2 ) = 1 and ranknd m )/D m ) = 0, 22)

SYMMETRIC SPIRAL PATTERNS ON SPHERES 12 by 14), generically solutions with D m symmetry if they exist) are equilibria stationary m-armed spirals for m 2) and one-armed spirals with Z 2 symmetry are generically relative equilibria with one period i.e. they are rotating waves. These rotating spiral waves have a twisted subgroup Σ rot of O3) S 1 as their group of spatiotemporal symmetries. By 20) a linear combination of 2l+2 spherical harmonics has Z 2 symmetry so a rotating wave with Σ rot 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 Σ rot 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 O3) Z 2. Remark 3. By 14), since rankn D 2m )/ D 2m ) = 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 O3) Z 2 symmetry In this section we consider the types of solutions which can be created at a general stationary bifurcation with O3) Z 2 symmetry by computing the isotropy subgroups of O3) Z 2 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 V l V l+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 O3) Z 2 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 O3) Z 2 symmetry is an interesting problem in its own right which has, up until now, not been investigated. Many dynamical

SYMMETRIC SPIRAL PATTERNS ON SPHERES 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 O3) Z 2. When such a system is reduced to the centre manifold of a stationary bifurcation of a trivial solution with O3) Z 2 symmetry we have dz dt = fz, λ) 24) where z V here V = V l for irreducible representations and V = V l V l+1 for reducible representations), λ R is a bifurcation parameter. Here the smooth mapping f not only commutes with the action of O3) on V as in 11)) but is also equivariant with respect to the action of 1 Z 2. Hence it satisfies f z, λ) = fz, λ) 25) i.e. the vector field f is odd in z so a Taylor expansion of this vector field will not contain any terms of even order. An O3) Z 2 equivariant vector field is just an O3) equivariant vector field with all terms of even order removed. We now compute the isotropy subgroups of O3) Z 2. 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 O3) Z 2 For any representation of O3) Z 2 the isotropy subgroups are twisted subgroups H θ where H is a subgroup of O3) and θ : H Z 2 is a group homomorphism. These twisted subgroups are a subset of the twisted subgroups of O3) S 1 containing only the subgroups, H θ, with twist types Z 2 or 1. The conjugacy classes of twisted subgroups of O3) S 1 are listed in [32]. The twisted subgroups of O3) Z 2 are uniquely determined by pairs of subgroups H, K) where H is a subgroup of O3) and K is a normal) subgroup of H such that H/K = Z 2 or 1. The group homomorphism θ : H H/K is always given by { 1 if h K θh) = 26) 1 if h H K. A complete list of the twisted subgroups of O3) Z 2 is given in Table 1 We are interested in the twisted subgroups which can be isotropy subgroups when the representation of O3) Z 2 is the reducible natural representation on V l V l+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 O3) Z 2 is an isotropy subgroup in the representation on V l V l+1 if and only if for each strictly larger and adjacent group so that H θ O3) Z 2 ) dim Fix Vl V l+1 ) q ) < dim Fix Vl V l+1 H θ ) qh θ ) 27)

SYMMETRIC SPIRAL PATTERNS ON SPHERES 14 Table 1 Twisted subgroups H θ of O3) Z2 and formulae for dim FixH θ ) for the reducible representations on Vl Vl+1. H K H/K dim FixH θ ) for l odd dim FixH θ ) for l even SO3) SO3) 1 0 0 O2) O2) 1 1 1 O2) SO2) Z2 1 1 SO2) SO2) 1 2 2 H class I 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 O3) O3) 1 0 0 O2) Z c 2 O2) Z c 2 1 1 1 O2) Z c 2 SO2) Z c 2 Z2 0 0 SO2) Z c 2 SO2) Z c 2 1 1 1 Dn Z c 2 Dn Z c 2 1 [l + 1)/n] + 1 [l/n] + 1 Dn Z c 2 Zn Z c 2 Z2 [l + 1)/n] [l/n] H class II D2n Z c 2 Dn Z c 2 Z2 [l + 1 + n)/2n] [l + n)/2n] K class II Zn Z c 2 Zn Z c 2 1 2 [l + 1)/n] + 1 2 [l/n] + 1 Z2n Z c 2 Zn Z c 2 Z2 2 [l + 1 + n)/2n] 2 [l + n)/2n] T Z c 2 T Z c 2 1 2 [l + 1)/3] + [l + 1)/2] l 2 [l/3] + [l/2] l + 1 O Z c 2 O Z c 2 1 [l + 1)/4] + [l + 1)/3] + [l + 1)/2] l [l/4] + [l/3] + [l/2] l + 1 O Z c 2 T Z c 2 Z2 [l + 1)/4] [l + 1)/3] [l/4] [l/3] I Z c 2 I Z c 2 1 [l + 1)/5] + [l + 1)/3] + [l + 1)/2] l [l/5] + [l/3] + [l/2] l + 1 Continued on next page

SYMMETRIC SPIRAL PATTERNS ON SPHERES 15 Table 1 continued from previous page H K H/K dim FixH θ ) for l odd dim FixH θ ) for l even O3) SO3) Z2 0 0 O2) Z c 2 O2) Z2 0 0 O2) Z c 2 O2) Z2 1 1 SO2) Z c 2 SO2) Z2 1 1 H class II Dn Z c 2 Dn Z2 [l/n] [l + 1)/n] K class Dn Z c 2 D z n Z2 [l/n] + 1 [l + 1)/n] + 1 I or III D2n Z c 2 D d 2n Z2 [l + n)/2n] [l + n + 1)/2n] Zn Z c 2 Zn Z2 2 [l/n] + 1 2 [l + 1)/n] + 1 Z2n Z c 2 Z 2n Z2 2 [l + n)/2n] 2 [l + 1 + n)/2n] T Z c 2 T Z2 2 [l/3] + [l/2] l + 1 2 [l + 1)/3] + [l + 1)/2] l O Z c 2 O Z2 [l/4] + [l/3] + [l/2] l + 1 [l + 1)/4] + [l + 1)/3] + [l + 1)/2] l O Z c 2 O Z2 [l/4] [l/3] [l + 1)/4] [l + 1)/3] I Z c 2 I Z2 [l/5] + [l/3] + [l/2] l + 1 [l + 1)/5] + [l + 1)/3] + [l + 1)/2] l O2) O2) 1 2 2 O2) SO2) Z2 0 0 O O 1 [l/3] [l/4] + [l + 1)/4] [l + 1)/3] [l + 1)/4] + [l/4] + [l + 1/3] + [l + 1)/2] l + [l/3] + [l/2] l + 1 O T Z2 [l/4] + [l/3] [l + 1)/4] + [l + 1)/3] [l + 1)/2] + 1 [l + 1)/4] + [l + 1)/3] [l/4] + [l/3] [l/2] Z 2n Z 2n 1 2 [l + n)/2n] + 2 [l + 1)/2n] + 1 2 [l + 1 + n)/2n] + 2 [l/2n] + 1 Z 2n Zn Z2 2 [l/2n] + 2 [l + 1 + n)/2n] + 1 2 [l + n)/2n] + 2 [l + 1)/2n] + 1 H class III D d 2n D d 2n 1 [l + n)/2n] + [l + 1)/2n] + 1 [l + 1 + n)/2n] + [l/2n] + 1 D d 2n D z n Z2 [l/2n] + [l + 1 + n)/2n] + 1 [l + n)/2n] + [l + 1)/2n] + 1 D d 2n Dn Z2 [l/2n] + [l + 1 + n)/2n] [l + n)/2n] + [l + 1)/2n] D d 2n Z 2n Z2 [l + n)/2n] + [l + 1)/2n] [l + n + 1)/2n] + [l/2n] D z n D z n 1 [l/n] + [l + 1)/n] + 2 [l/n] + [l + 1)/n] + 2 D z n Zn Z2 [l/n] + [l + 1)/n] [l/n] + [l + 1)/n] D z 2n D z n Z2 [l + n)/2n] + [l + 1 + n)/2n] [l + n)/2n] + [l + 1 + n)/2n] D z 2 Z 2 Z2 3l 3l 2 2 2 + 1 [l + 1)/2]

SYMMETRIC SPIRAL PATTERNS ON SPHERES 16 where qh θ ) = dim N O3) Z2 H θ ) dimh θ ) and N O3) Z2 H θ ) is the normalizer of H θ in O3) Z 2. Note that we have adapted the massive chain criterion of [22] for isotropy subgroups of O3) Z 2 in reducible representations. All of the twisted subgroups H θ in Table 1 have qh θ ) = 0 with the following exceptions. The twisted subgroups given by pairs Z 2n, Z n ), Z 2n Z c 2, Z n Z c 2), Z 2n Z c 2, Z 2n ), Z 2n, Z 2n ) and Z 2n, Z n) have qh θ ) = 1 for all values of n and the twisted subgroups given by pairs Z n, Z n ), Z n Z c 2, Z n Z c 2) and Z n Z c 2, Z n ) 28) also have qh θ ) = 1 except when n = 1 in which case qh θ ) = 3. To use the massive chain criterion we need the values dim Fix Vl V l+1 H θ ) for each twisted subgroup in each representation on V l V l+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 θh) =, dim Fix Vl V l+1 H θ ) = dim Fix Vl V l+1 H) and whenθh) = Z 2 the twisted subgroup H θ is given by the pair H, K) and dim Fix Vl V l+1 H θ ) = dim Fix Vl V l+1 K) dim Fix Vl V l+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 O3) Z 2 on V l V l+1 the axial isotropy subgroups for the representation on V l V l+1 are precisely the axial isotropy subgroups in the representations on V l and V l+1 since all one-dimensional invariant subspaces of V l V l+1 are either one-dimensional subspaces of V l or V l+1. All isotropy subgroups H θ of O3) Z 2 in the natural representation on V l when l is even are given by pairs H, K) where both H and K are class II subgroups of O3) since H θ must contain the element I, 1) O3) Z 2. Similarly in the natural representation on V l when l is odd, every isotropy subgroup must contain the element I, 1) O3) Z 2 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 O3). Using this information and the massive chain criterion we find that the axial isotropy subgroups of O3) Z 2 in the natural representation on V l are as given in Table 2. Given a particular value of l it is possible to compute all of the isotropy subgroups of O3) Z 2 in the reducible representation on V l V l+1 using Table 1 and the massive chain criterion. See [32] for the results in the case where l = 2. We are particularly interested in whether it is possible for the twisted subgroups D 2, given by the pair H, K) = D 2, Z 2 ), and D 2m, given by pairs H, K) = D 2m, D m ), to be isotropy subgroups for any representation V l V l+1. These are the symmetry groups of most symmetric spiral patterns on spheres. Note that these subgroups are never axial isotropy subgroups. Using Table 1 and the massive chain criterion it can be seen that D 2 is an isotropy 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 V l V l+1 it may be possible for spiral patterns with symmetry groups 6) and 7) with

SYMMETRIC SPIRAL PATTERNS ON SPHERES 17 J K θh) l even l odd plus representation) minus representation) O2) O2) Z c 2 all even l O2) O2) Z 2 All odd l I I Z c 2 6, 10, 12, 16, 18, 20, 22, 24, 26, 28, 32, 34, 38, 44 I I Z 2 15, 21, 25, 27, 31, 33, 35, 37, 39, 41, 43, 47, 49, 53, 59 O O Z c 2 4, 6, 8, 10, 14 O O Z 2 9, 13, 15, 17, 19, 23 O T Z c 2 Z 2 6, 10, 12, 14, 16, 20 O O Z 2 3, 7, 9, 11, 13, 17 D 2m D m Z c 2 Z 2 even l satisfying m l < 3m, m 3) D 2m D d 2m Z 2 odd l satisfying m l < 3m, m 3) D 4 D 2 Z c 2 Z 2 2, 4 D 4 D d 4 Z 2 5 Table 2: The axial isotropy subgroups of O3) Z 2 for the natural representation on V l. The last two columns give the values of l for which the subgroups are isotropy subgroups. Here H = J Z c 2. m l to exist depending on values of the coefficients in the O3) Z 2 equivariant vector field f. These solutions must be found directly using the general form of the vector field. The general form of a O3) Z 2 equivariant vector field for the representation on V l V l+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 space dim Fix Vl V l+1 D 2 ). This too becomes increasingly difficult for large values of l. 6 Spiral patterns in the representation of O3) Z 2 on V 2 V 3 In this section we demonstrate that solutions with the symmetries of one- and two-armed spirals can exist within the O3) Z 2 equivariant vector field for the representation on V 2 V 3. From the results of section 5 we can that in this representation D 2 and D 4, the 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 O3) Z 2 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. Since D 2 D 4 there are copies of the fixed-point subspaces of these subgroups which satisfy Fix D 4 ) Fix D 2 ), i.e. for every choice of generators of D 2, Fix D 2 ) contains a copy of Fix D 4 ). For example we can choose generators which give Fix D 2 ) = {ia, 0, 0, 0, ia ; 0, b, 0, c, 0, b, 0)} 29) Fix D 4 ) = {ia, 0, 0, 0, ia ; 0, b, 0, 0, 0, b, 0)} 30)

SYMMETRIC SPIRAL PATTERNS ON SPHERES 18 where a, b, c R. In this case, the two-armed spiral with D 4 symmetry) has its tips at the North and South poles of the sphere and the one-armed spiral with D 2 symmetry) has its tips on the equator. By restricting the general form of the O3) Z 2 equivariant vector field in Appendix B to Fix D 2 ) given by 29) we have the set of ODEs ȧ = µ x a + 2α 1 a 3 + 2β 1 + 8γ 1 6γ 2 ) ab 2 + β 1 + γ 2 ) ac 2 31) ḃ = µ y b + 2β 2 50δ 1 ) b 3 + 2α 2 30δ 2 + 10δ 3 ) ba 2 + β 2 + 15δ 1 ) bc 2 32) ċ = µ y c + β 2 9δ 1 ) c 3 + 2α 2 30δ 2 + 18δ 3 ) ca 2 + 2β 2 + 30δ 1 ) cb 2 33) where µ x, µ y, α 1, β 1, γ 1, γ 2, α 2, β 2, δ 1, δ 2 and δ 3 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 µ x = 0 and µ y = 0. We assume that µ x = λ and µ y = λ + ρ where ρ 1. Then the trivial solution is stable when λ < min0, 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 O3) Z 2 in the representation on V 2 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 O3) Z 2 in the representation on V 3 bifurcate. Any stationary solution of 31) 33) has an isotropy subgroup which contains D 2. The isotropy subgroups of O3) Z 2 in the representation on V 2 V 3 can be determined from Table 1 using the massive chain criterion. Those isotropy subgroups which contain D 2 are given in Table 3 along with the form of their fixed-point subspace which lies inside Fix D 2 ) given by 29). The subsection of the lattice of isotropy subgroups including only those subgroups which contain D 2 is as in Figure 5. Isotropy H K Fixed-point subspace subgroup D 4 Z c 2 D 4 Z c 2 D 2 Z c 2 {ia, 0, 0, 0, ia ; 0, 0, 0, 0, 0, 0, 0)} O2) Z c 2 O2) Z c 2 O2) {0, 0, 0, 0, 0 ; 0, 0, 0, c, 0, 0, 0)} { )} D 6 Z c 2 D 6 Z c 2 D d 3 6 0, 0, 0, 0, 0 ; 0, 3 10 10 O Z c 2 O Z c 2 O {0, 0, 0, 0, 0 ; 0, b, 0, 0, 0, b, 0)} D 4 D 4 D 2 {ia, 0, 0, 0, ia ; 0, b, 0, 0, 0, b, 0)} D d 4 D d 4 D z 2 {ia, 0, 0, 0, ia ; 0, 0, 0, c, 0, 0, 0)} D 2 Z c 2 D 2 Z c 2 D z 2 {0, 0, 0, 0, 0 ; 0, b, 0, c, 0, b, 0)} D 2 D 2 Z 2 {ia, 0, 0, 0, ia ; 0, b, 0, c, 0, b, 0)} Table 3: Isotropy subgroups for the representation of O3) Z 2 on V 2 V 3 which contain D 2. Also shown is the form of the fixed-point subspace which lies inside Fix D 2 ). Branches of solutions with the symmetries of the five axial isotropy subgroups of O3) Z 2 in the representation on V 2 V 3 which contain D 2 are guaranteed to exist in 31) 33). Analysis of the equations reveals that for certain values of the coefficients λ, ρ, α 1, β 1, γ 1, γ 2, α 2, β 2, δ 1, δ 2 and δ 3 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 the exception of the group D 2 Z c 2 - there are no stationary solutions of 31) 33) with

SYMMETRIC SPIRAL PATTERNS ON SPHERES 19 Figure 5: Subsection of lattice of isotropy subgroups of O3) Z 2 in the representation on V 2 V 3 including only those isotropy subgroups which contain D 2. isotropy D 2 Z c 2. See [32] for proof of these facts. Images of solutions to 31) 33) with each of the possible symmetries are given in Figure 6. To demonstrate that solutions with symmetry groups D 2 and D 4 one- and two-armed spirals respectively) can exist for some values of the coefficients λ, ρ, α 1, β 1, γ 1, γ 2, α 2, β 2, δ 1, δ 2 and δ 3 we consider some examples. 6.1 Example 1: Coefficient values where a solution with D 2 symmetry exists and is stable Suppose that for some pattern forming system the values of the coefficients of the cubic order terms in the O3) Z 2 equivariant vector field are α 1 = 1, α 2 = 1, β 1 = 1 3, β 2 = 1, 34) γ 1 = 1 2, γ 2 = 1 2, δ 1 = 1 60, δ 2 = 1 2, δ 3 = 1 5. 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 with D 2 symmetry a one-armed spiral) exists and is stable within the subspace Fix D 2 ). The one-armed spiral pattern with D 2 symmetry bifurcates from the axial solution with D 6 Z c 2 symmetry at B and the branch of solutions with D 4 symmetry a two-armed spiral) at F. We can also see that the two-armed spirals with D 4 symmetryare is stable in Fix D 2 ) for values of λ and ρ between F and H. 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 O3) Z 2 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

SYMMETRIC SPIRAL PATTERNS ON SPHERES 20 D 4 Z c 2 O2) Z c 2 O Z c 2 D 6 Z c 2 D d 4 i) Dd 4 ii) D4 i) D4 ii) D 2 i) D2 ii) Figure 6: Images of solutions to 31) 33). These solutions all have symmetry groups containing D 2. In some cases two views of the solutions are given to show the symmetries more clearly. V 2 V 3 where the radius of the sphere is R = 3 + ϵ 2 R 2 the coefficient values are µ x = µ 2 8 27 R 2, µ y = µ 2 + 16 27 R 2, α 1 = 15 28π, β 1 = 6 11π, γ 1 = 3 44π, 35) γ 2 = 1 44π, α 2 = 25 22π, β 2 = 175 286π, δ 1 = 7 2860π, δ 2 = 3 44π and δ 3 = 1 22π where µ = 1 9 + ϵ2 µ 2. Let µ x = λ then µ y = λ + ρ where ρ = 8 9 R 2. Substituting these values into equations 31) 33) we have ȧ = λa 15 14π a3 15 22π ab2 23 44π ac2 36) ḃ = λ + ρ) b 315 286π b3 15 22π ba2 371 572π bc2 37) ċ = λ + ρ) c 1687 2860π c3 23 22π ca2 371 286π cb2. 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 for values of λ and ρ between C and I solution branches with D 4 symmetry two-armed spirals) exist and are stable in Fix D 2 ). It can be computed using 52) 53) that, up to degeneracy in one eigenvalue, the solution branch with D 4 symmetry is in fact stable in 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. A branch of solutions with D 2 symmetry one-armed spirals) exists between D and F but this solution branch is never stable. It bifurcates from the branch of axial solutions with

SYMMETRIC SPIRAL PATTERNS ON SPHERES 21 Figure 7: Bifurcation diagram for the solution branches in the representation on V 2 V 3 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 each bifurcation lies on and the stability of the solution branches in Fix D 2 ). All bifurcations are pitchfork bifurcations.

SYMMETRIC SPIRAL PATTERNS ON SPHERES 22 D 6 Z c 2 symmetry at D and from the branch of submaximal solutions with D d 4 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 of the Swift Hohenberg model 1) with s = 0 in this subspace Fix D 2 ) agree with Figure 8 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 O3) Z 2 on V 3 V 4 We now consider the solutions with the symmetries of one-, two- and three-armed spirals which can exist within the O3) Z 2 equivariant vector field for the representation on V 3 V 4. From the results of section 5 we can see that in this representation D 2, D 4 and D 6 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 V 2 V 3 in section 6 reveals that the isotropy subgroups of O3) Z 2 which contain D 2 are those listed in Table 4. The section of the lattice of isotropy subgroups of O3) Z 2 in this representation including only those isotropy subgroups with contain D 2 is as in Figure 9. For each isotropy subgroup, the final column of Table 4 gives the form of the fixed-point subspace which lies inside Fix D 2 ) = {0, a, 0, b, 0, a, 0 ; ic, 0, id, 0, 0, 0, id, 0, ic)}. 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 directly using the restriction to D 2 of the general O3) Z 2 equivariant vector field. See [32] for the form of the general cubic order mapping which commutes with the action of O3) Z 2 on V 3 V 4. In this representation this restriction of this mapping to D 2 consists of four ODEs for which we would like to find all equilibria. Recall from Remark 3 that any spiral solutions with symmetries D 2, D 4 or D 6 will be stationary solutions. We find 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 ȧ = λa 315 286π a3 371 572π ab2 315 286π ac2 225 ḃ = λb 1687 2860π b3 371 286π ba2 21 22π bc2 219 286π ad2 + 3 210 286π 286π bd2 + 3 210 143π ċ = λ + ρ) c 6615 4862π c3 315 286π ca2 21 44π cb2 5535 d = λ + ρ) d 39825 34034π d3 225 286π da2 219 572π db2 5535 bcd 40) acd 41) 4862π cd2 + 3 210 abd 42) 286π 4862π dc2 + 3 210 286π abc.43) By studying these equations analytically we can find branches of solutions with the symmetries of each of the isotropy subgroups in Table 4 with the exceptions of D 2 Z c 2 ) D z 2,

SYMMETRIC SPIRAL PATTERNS ON SPHERES 23 Figure 8: Bifurcation diagram for the solution branches in the representation on V 2 V 3 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 stability of the solution branches in Fix D 2 ). All bifurcations are pitchfork bifurcations.

SYMMETRIC SPIRAL PATTERNS ON SPHERES 24 D 2 Z c 2 ) Z 2 Z c and D 2 2. It can be shown analytically in the general case that solution branches with symmetry D 2 Z c 2 ) D z or D 2 2 Z c 2 ) Z 2 Z c never exist [32]. To find branches 2 of solutions with D 2 symmetry we then compute analytically) the stability of each of the 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 of solution branches with D 2 symmetry [32]. Images of each of the solution types which 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 between K and Q a solution branch with D 4 symmetry two-armed spirals) are stable in Fix D 2 ). Furthermore, between C and M a branch of solutions with D 6 symmetry threearmed spirals) are stable. At the transcritical bifurcation at M this solution loses stability to a branch of solutions with D 2 symmetry one-armed spirals) which is stable until the 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 Fix D 2 ) agree with Figure 12 as to the stable solution branches and the locations of the secondary pitchfork bifurcations on these stable branches. Isotropy H K Fixed-point subspace subgroup O Z c 2 O Z c 2 O {0, a, 0, 0, 0, a, 0 ; 0, 0, 0, 0, 0, 0, 0, 0, 0)} O2) Z c 2 O2) Z c 2 O2) {0, 0, 0, b, 0, 0, 0 ; 0, 0, 0, 0, 0, 0, 0, 0, 0)} { )} D 6 Z c 2 ) D d D 6 Z c 2 D d 3 6 0, b, 0, b, 0, 3 b, 0 ; 0, 0, 0, 0, 0, 0, 0, 0, 0 6 10 10 D 4 Z c 2 D 4 Z c 2 D 2 Z c 2 {0, 0, 0, 0, 0, 0, 0 ; 0, 0, id, 0, 0, 0, id, 0, 0)} D 6 Z c 2 ) { )} D 3 Z c D 2 6 Z c 2 D 3 Z c 2 0, 0, 0, 0, 0, 0, 0 ; ic, 0, 7ic, 0, 0, 0, 7ic, 0, ic D 8 Z c 2 D 8 Z c 2 D 4 Z c 2 {0, 0, 0, 0, 0, 0, 0 ; ic, 0, 0, 0, 0, 0, 0, 0, ic)} D 2 Z c 2 ) D z D 2 2 Z c 2 D z 2 {0, a, 0, b, 0, a, 0 ; 0, 0, 0, 0, 0, 0, 0, 0, 0)} D 2 Z c 2 ) Z 2 Z c D 2 2 Z c 2 Z 2 Z c 2 {0, 0, 0, 0, 0, 0, 0 ; ic, 0, id, 0, 0, 0, id, 0, ic)} 3 D 6 D 6 D 3 {0, b, 0, b, 0, 3 b, 0 ; ic, 0, 7ic, 0, 0, 0, )} 7ic, 0, ic 10 10 D 4 D 4 D 2 {0, a, 0, 0, 0, a, 0 ; 0, 0, id, 0, 0, 0, id, 0, 0)} D d 8 D d 8 D z 4 {0, 0, 0, b, 0, 0, 0 ; ic, 0, 0, 0, 0, 0, 0, 0, ic)} D d 4 ) D z 2 D d 4 D z 2 {0, 0, 0, b, 0, 0, 0 ; 0, 0, id, 0, 0, 0, id, 0, 0)} D d 4 ) Z D d 4 Z 4 {0, a, 0, 0, 0, a, 0 ; ic, 0, 0, 0, 0, 0, 0, 0, ic)} 4 D 2 D 2 Z 2 {0, a, 0, b, 0, a, 0 ; ic, 0, id, 0, 0, 0, id, 0, ic)} Table 4: Isotropy subgroups of O3) Z 2 in the representation on V 3 V 4 which contain D 2.

SYMMETRIC SPIRAL PATTERNS ON SPHERES 25 Figure 9: The lattice of isotropy subgroups of O3) Z 2 which contain D 2 for the representation on V 3 V 4.

SYMMETRIC SPIRAL PATTERNS ON SPHERES 26 O^ Zc2 ^ O2) 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.