Singularity theory and forced symmetry breaking in equations

A theory of bifurcation equivalence for forced symmetry breaking bifurcation problems is developed. We classify (O(2), 1) problems of corank 2 of low codimension and discuss examples of bifurcation problems leading to such symmetry breaking.


Introduction
The importance of symmetries in understanding and influencing bifurcation problems has been recognised basically since the beginning of bifurcation theory. Symmetries influence bifurcation problems via two broad classes of mechanisms: • spontaneous symmetry breaking, • forced (or induced) symmetry breaking.
Spontaneous symmetry breaking occurs when the symmetry of the equations is constant whereas solutions bifurcate and lose (or gain) internal symmetry as the parameters vary. From the many problems leading to bifurcation equations with spontaneous symmetry breaking, we mention for later reference, the analysis of the behaviour of an homogeneous, isotropic elastic cube under uniform traction (its symmetry group is the permutation group S 3 , see [20]) or the bifurcation of periodic orbits in autonomous systems (its symmetry group for the non-resonant degenerate Hopf bifurcation is the rotation group S 1 , see [19]). A singularity theory approach to the local study of the bifurcation diagrams, based on equivariant contact equivalence, has been developed by Golubitsky and Schaeffer ([19], [20]). The original set-up is as follows.

Singularity Theory for Spontaneous Symmetry Breaking.
First, because such analysis is local, it is sensible to consider germs at the origin to be able to state results that will persist on any neighbourhood of the origin. A germ of a function around a point x 0 is an equivalence class when two functions are identified if they coincide in a neighbourhood of that point. We use the notation f : (R n , x 0 ) → R m to denote the germ of f around x 0 . To make sense of equations like f = 0, when f : (R n , x 0 ) → R m , we define germs of sets, or germs of varieties, at x 0 using the same identification process as for germ of functions at x 0 . Let Γ be a compact group acting on R n and R m , we say that f : (R n , 0) → (R m , 0) is Γ-equivariant, resp. Γ-invariant, if f (γx) = γf (x), resp. f (γx) = f (x), ∀ x ∈ (R n , 0), ∀ γ ∈ Γ. A bifurcation germ with l parameters is a germ f : (R n+l , 0) → (R m , 0). Its bifurcation diagram is its zero-set f −1 (0). Let Γ be a compact group acting on R n and R m , trivially on parameters, two Γ-equivariant bifurcation germs (with one distinguished bifurcation parameter) f, g : (R n+1 , 0) → (R m , 0) are bifurcation, or K Γ λ , equivalent if there exist Γ-equivariant changes of coordinates (T, X, L) such that (1.1) f (x, λ) = T (x, λ) g(X(x, λ), L(λ)), where T (x, λ) is an invertible matrix and (X, L) a local diffeomorphism around the origin. Note the special role of λ which means that λ-slices of the bifurcation diagrams of f and g are diffeomorphic via (X, L). The use of singularity theory leads to a systematic understanding and systematic calculation of the relative roles of the different terms of a bifurcation germ, and, via the notion of miniversal unfolding, of its perturbations. The theory based on (1.1) established itself for the local study of bifurcation germs, giving efficient algebraic calculations to establish classifications of normal forms, special polynomial members of the orbits under bifurcation equivalence, and their miniversal deformations, perturbations of the normal forms with the minimal number of parameters necessary to represent all possible perturbations modulo changes of co-ordinates. Given a bifurcation germ f , the key algebraic ingredient of the theory is the extended tangent space of f , denoted by T e K Γ λ (f ). It has a so-called mixed module structure because it is the sum of a module generated over the ring of Γ-invariant germs in (x, λ) and of a module generated over germs in λ only. An important number associated with f is the codimension of T e K Γ λ (f ) as a vector subspace of the space of Γ-equivariant maps. It is called the K Γ λ -codimension of f , denoted by K Γ λ -cod(f ). When K Γ λ -cod(f ) is finite, f is K Γ λ -equivalent to a polynomial normal form and has a miniversal unfolding with K Γ λcod(f ) parameters, formed using a basis a complement of T e K Γ λ (f ) in the space of Γ-equivariant bifurcation maps, the normal space of f , denoted by N e K Γ λ (f ).

Forced Symmetry Breaking.
In the second mechanism, that of forced symmetry breaking, the symmetries of the equations change when some parameters are switched on. For example, in the elastic cube problem, the traction load distribution could change first to remain equal only on two opposite faces before losing any symmetry with a second additional parameter. The forced symmetry breaking is then S 3 → Z 2 → 1 (with one, then two additional parameters coming into play, see [13]).
There are many other buckling problems in elasticity exhibiting forced symmetry breaking because there is often an interaction between the internal symmetries of the material, the geometry of the object and of the externally applied forces. An example is the work of Pierce [31] on the bifurcation of straight circular rods subject to an axially symmetric compressive load and perturbations by additional loads breaking the axial symmetry. In previous works on prismatic rods, like in [3], the typical diagrams are to be found in the miniversal unfolding of the non degenerate double cusp because the symmetry group is finite. When the rod is circular the double cusp is degenerate because the acting symmetry group is now O (2) and so the problem has no miniversal unfolding within the classical theory.
Similarly, the forcing of an autonomous equationü + q(u, λ) = 0 by either a T -periodic function p, like in the following model problem (1.2)ü + q(u, λ) + µ p(t) = 0, or non linear boundary conditions of the type (g 1 , g 2 are (non local) non linear functions of u): u(0) − u(1) = µg 1 (u, λ, µ),u(0) −u(1) = µg 2 (u, λ, µ), gives rise to a bifurcation problem with the forced symmetry breaking O(2) → 1 near the values of λ where the linear mode of the autonomous part is of period-T also. We can extend the symmetry breaking bifurcation for any pair of compact subgroups ∆ ⊂ Γ ⊂ O(2), by considering a generalisation of the differential equation (1.2) with time periodic terms g (nonlinear) and p with rationally dependent periods (see [17] for a classical analysis).

Main Results.
A special version of bifurcation equivalence, stronger than (1.1), is called for to classify systematically the bifurcation germs and their unfoldings arising in forced symmetry breaking problems.
In Section 2, we present a general theory of unfoldings, finite determinacy and the recognition problem for forced symmetry breaking bifurcation germs of the type where f is Γ-equivariant when µ = 0 and ∆-equivariant when µ = 0, for ∆ a closed subgroup of Γ. The essential part of the theory was advanced in [18] but not much action was taken on it, the main theory drifting instead towards the very rich field of spontaneous symmetry breaking. But, fundamentally, Damon in [8] adapted his general framework to clear the way for the abstract theory to work successfully in that case. So, we may write (1.3) f (x, λ, µ) = f 1 (x, λ) + µf 2 (x, λ, µ) with f 1 Γ-equivariant and f 2 ∆-equivariant. A first approach to study such problems, is to view f as a perturbation of f 1 in the K ∆ (λ,µ) -theory, the bifurcation equivalence of ∆-equivariant maps with two bifurcation parameters (λ, µ). In general that approach fails. For instance, when Γ and ∆ have different dimensions, the K ∆ (λ,µ) -codimension of f 1 is not finite (even when Γ is finite, it may fail). So, the group of change of coordinates K Γ,∆ (λ,µ) we define in Section 2.3 will have the property that (T, X, L) is Γ-equivariant when µ = 0, but only ∆-equivariant when µ = 0. This (Γ, ∆)-equivariant structure of the group of contact equivalences will be transported to the tangent spaces (see Section 2.3.4). They are modules over Γ-invariant functions when µ = 0 and over ∆-invariant functions when µ = 0. This is unusual but, in his general framework, Damon did define in [8] the necessary extended concepts to deal with the new situation.
The first point is to make sure that a version of the Preparation Theorem applies. As a consequence, we can work with the algebraic structure of the tangent spaces. For the usual theory of K Γ λ -equivalence, the ring of invariant functions has a structure of differentiable DA-algebra (see Section 2.2), and so the Preparation Theorem holds true. When Γ is a continuous Lie group and ∆ is finite, our rings of invariant functions will not be DA-algebras, but Damon showed that the main properties of DA-algebras can be extended to this situation of so-called extended DA-algebras. In Corollary 2.2 we give an explicit criterion we use to test if the rings of invariants under consideration are actually DA-algebras.
In a second step, we look into the structure of our group of contact equivalences K Γ,∆ (λ,µ) and the tangent spaces. In Section 2.3 we show that K Γ,∆ (λ,µ) decomposes into three subgroups:K(Γ, ∆),M(Γ, ∆) andŜ(Γ, ∆). When Γ is continuous and ∆ is finite, for instance, each subgroup add a non trivial contribution to the extended tangent space of K Γ,∆ (λ,µ) . In Section 2.3.5 we show that K Γ,∆ (λ,µ) is a geometric subgroup of K (contact equivalences), hence it satisfies the abstract theorems of Damon [8] about miniversal unfoldings and finite determinacy. We show in Theorem 2.7 that the explicit description of K Γ,∆ (λ,µ) in [18] corresponds to the best possible situation, because its extended tangent space contains the extended tangent space of any other geometric subgroup of K fixing globally the Γ-equivariant maps when µ = 0.
Finally, we discuss topological K Γ,∆ (λ,µ) -equivalence where the changes of co-ordinates (T, X, L) are only continuous, not smooth, germs. In Section 2.5, we summarise the results of Damon [10], [11] we need. Topological equivalence is more efficient in our context, because it preserves the topological properties of the bifurcation germs and their (smooth) miniversal unfoldings. Our smooth normal forms and their smooth miniversal unfoldings have many moduli (parameters without topological significance). They appear because the geometry of the bifurcation diagrams is intricate (see Figure 4). As most two parameter situations, the K Γ,∆ (λ,µ) -equivalence preserves the respective position of the regions of the parameter plane with a different zero structure. In our case the only one dimensional slice to be preserved is f −1 1 (0), when µ = 0.
We believe that we have a complete list up to topological codimension 2, adding 4 more normal forms, but we cannot fully prove it at present, so we simply mention that list as a remark following Theorem 3.4. Interestingly, one normal form has f 2 (0, 0, 0) = 0. Note that the highest smooth codimension of those germs can reach is 12. As a consequence of our computations of the tangent spaces, the list of the (Z 2 , 1)-symmetry breaking germs f in one dimension, that is, f (y, λ) = f 1 (y 2 , λ) y + µf 2 (y, λ, µ), is basically the same as the (O(2), 1)-classification list when f 2 (0, 0, 0) = 0. Nevertheless, the stability properties of the solutions are different (see [12]). In the corank two case, the symmetry breaking term selects only one stable solution from the O(2)-orbit. In the one dimensional case, the obvious obstructions from the unstable solutions mean that the stability properties of the pairs of solutions remain unchanged.
In this section, we look at the bifurcation diagrams for the (O(2), 1)symmetry breaking bifurcation germs f we classified in Theorem 3.4, cases I 0 , II and III. In Figure 2 of Section 4.2.1, we give a simple illustration on a potential dynamics of what happen when orbits of steady states are destroyed by the perturbation of the miniversal unfolding of case I 0 . More generally, we describe the regions of parameter space (λ, µ) where the zero set structure of f is invariant, including the stability of the solution with respect to the sign of the eigenvalues of the linearisation of f . They are portrayed in Figures 1, 3 and 4, respectively. For bifurcation equivalent germs, those regions are diffeomorphic but the only one dimensional slice preserved is µ = 0.

Section 5: examples.
In the final section we describe examples leading to bifurcation equations satisfying our framework. Our results are readily applicable to the bifurcation of period-2π solutions of u + u + u q(u, λ) + µ p(t, u,u, λ, µ) = 0 where p is 2π-periodic in t and q(0, 0) = 0. This example is technically easy to manipulate. As it is well-known, when | π −π p(t, 0) e it dt| = 0, the symmetry breaking term selects a pair of solutions (that is, selects a phase shift) from the O(2)-orbit, keeping only one solution possibly stable. In that case the problem reduces to the corank one symmetry breaking (Z 2 , 1).
Another problem is concerned with the problem of contact resistance in homogeneous metal rings obtained from welding the end points of a piece of metal. Usually the joining is not perfect and give rise to some 'contact resistance' to heat conduction. This resistance is characterised by a continuous gradient of the temperature across the join (the material is the same across the join on the left and the right) and a loss of temperature at first approximation proportional to its gradient. More explicitly, if the piece of metal is [0, 1], the simplest boundary conditions are The boundary conditions are periodic if µ = 0 and have no symmetries when µ = 0. Combined with an autonomous non linear heat equation they give rise to a forced (O(2), 1)-symmetry breaking (see [34] and [29]).
Another type of problems are reaction-diffusion equations. Let u be the density of a population living in the unit disk. Suppose u satisfies a parametrised semilinear parabolic equation (1.5) u t = ∆u + g(u,ū, λ) + µ p(x, y, u,ū, λ, µ) subject to non-flux boundary conditions ∂u ∂n = 0 on the unit circle. The termū represents a non local contribution of the averaged densityū(t) =

1
|Ω| Ω u(s, t) ds. Under the hypotheses of Section 4, (1.5) has a constant steady state which looses stability with a 2-D kernel for the linearisation. For general p, the resulting bifurcation equation fits our framework. As an example of (Z 2 , 1)-symmetry breaking, take a generic thin rectangular plate under a uniform compression along its boundary. Apply a generic normal load distribution and use the von Kármán approximation for the buckling equations (see [36]). The unperturbed equations have a Z 2 -symmetry, destroyed by the perturbing load distribution. Bifurcation equations in elasticity are often the gradients of some parametrised functional (see [31]). Bifurcation equivalences do not preserve in general the gradient structure, but they still induce an equivalence relation on the set of gradient bifurcation germs and their unfoldings. A theory using a path formulation has been developed in [2], but it is not clear how to extend those ideas to the (O(2), 1)-symmetry breaking, but no more powerful theory is needed here. Our normal forms (with case I 2 for some values of its moduli) and their miniversal unfoldings are already gradients. If any other gradient bifurcation germ satisfies the recognition conditions, then it is bifurcation equivalent to the representative normal form, which is a gradient. Moreover, the same applies to the miniversal unfoldings that are also gradients. This means that the classification of gradient problems mirrors the general classification.

Symmetries on parameters.
Forced symmetry breaking arises also when some symmetries are retained by the parameters (see [15]). A typical example is when we have a non trivial 'diagonal' action of Γ on the space of (x, λ), γ → (γx, γλ).
In that case the theory in this paper is not necessary. The general structure of the problem remains 'classical' because the invariants form a DA-algebra (see [15]). But, if we combine both ideas: some symmetry is lost, some is retained by the parameters when they are switched on, we get a theory which is still contained into the abstract framework we describe here because the invariants still form a DA-algebra. When combined with the symmetry breaking terms, we have a structure of extended DA-algebras for the invariants. More explicitly, the problems represented by the functions f (x, λ, µ) = f 1 (x, λ) + f 2 (x, λ, µ), with f 1 (γx, γλ) = γf 1 (x, λ) and f 2 (δx, δλ, δµ) = δf 2 (x, λ, µ), fall into a similar framework.

Bifurcations from orbits of solutions under perturbations.
Some previous work have been concerned with the symmetry of bifurcating solutions of (1.3). In the Γ-equivariant situation one can sometimes show that all bifurcating solutions have at least isotropy ∆ for some subgroup ∆ ⊂ Γ (see [36] for an abstract analysis). An example of such analysis is the generalised Duffing's equation (see [22], [23]) (1.6)ü + u + uq(u, λ) + µ p(t) = 0 where q(0, 0) = 0 and p is an even 2π-periodic function. One can then show that all solutions of (1.6) are also even. Similar results hold when q is an odd function and p is odd in time.
Systematic approaches can be traced back to Vanderbauwhede [35] and Lauterbach and Roberts [26], giving rise to Hou and Golubitsky [25] (see Chapter 10 of [6] for a more recent discussion). The bifurcation from Γ-orbits of solutions is studied when a ∆-equivariant perturbation is applied to the original Γ-equivariant equations. For instance, a simple example in tune with previous example is the equationü + q(u, λ) + µh(t, u,u) = 0 when h is 2π-periodic in t andü + q(u) = 0 has nontrivial periodic orbits (see [36] and [4]). More generally, there has been more recently interest about the effect of forced symmetry breaking on the dynamics around some special solutions, invariant manifolds (Galante and Rodrigues [17], Chillingworth and Lauterbach [5], Comanici [7], Parker et al. [30] to mention but a few). In these papers, somewhat different techniques are used, linked with equivariant differential geometry. A complete unfolding of the underlying singularity near bifurcation is difficult. However, in this paper, we are able to provide just such analysis.

General Theory
In this section we present a general theory of unfoldings, finite determinacy and the recognition problem for the bifurcation problems with forced symmetry breaking. The main ideas and definitions were sketched in [18] and [8]. Here we give a full, careful and organised account of the theory with some extended results. This needs many ingredients with many definitions and results. The abstract theory of Damon [8] works with modules over systems of rings that are DA-algebras. In general, the ring of forced symmetry breaking invariant functions has only the structure of an extended DA-algebra. In Corollary 2.2 we give a criterion we use to test if an extended DA-algebra is actually a DA-algebra. In that case we do not need any extension of the abstract theory. On the contrary, like for the (O(2), 1)-symmetry breaking, an extended theory is required. In Section 2.3, we discuss the structure of the group of bifurcation equivalence K Γ,∆ (λ,µ) we are going to use. It requires the composition of any finite string of elements of three subgroups. One subgroup,K(Γ, ∆), is natural, the two others,M(Γ, ∆) andŜ(Γ, ∆), are more surprising. In Lemma 2.4 we show that the order of the elements is irrelevant: we can always recombine any string as 3 elements, one in each subgroup. The group K Γ,∆ (λ,µ) is a geometric subgroup in the sense of [8] over an adequately ordered extended system of DA-algebras. To characterise the tangent spaces we calculate the unusual contributions ofŜ(Γ, ∆) and M(Γ, ∆) in Lemmas 2.5 and 2.6, respectively. In Theorem 2.7 we show that the tangent space of any other group of equivalence respecting the forced symmetry breaking structure cannot be larger than the tangent space of K Γ,∆ (λ,µ) , indicating that the choices ofŜ(Γ, ∆) andM(Γ, ∆) are optimal. In Section 2.4 we state the main results we can deduce from the abstract theory of [8] about the unfolding (Theorem 2.8) and determinacy (Theorem 2.9) theories for K Γ,∆ (λ,µ) . Finally, most normal forms have moduli for the smooth equivalence and so we use the topological equivalence theory of [10], [11] to regroup orbits of the smooth classification into classes with equivalent topological behaviour. The main results are stated in Theorems 2.13 and 2.14.

Notation and Preliminary Definitions.
The state variable is x ∈ (R n , 0) and the distinguished bifurcation parameters are (λ, µ) ∈ (R 2 , 0). The derivatives are denoted by subscripts, f x for ∂f ∂x , . . . and the superscript o denotes the value of any function at the origin, Let E x denote the ring of smooth germs f : (R n , 0) → R and M x its maximal ideal. For y ∈ R m , let E x,y denote the E x -module of smooth germs g : (R n , 0) → R m , and M x,y the submodule of germs vanishing at the origin. When y is clear from the context, we denote E x,y by E x and M x,y by M x . When we would like to emphasise only the dimension of the source we denote E x , E y by E n , E m , etc. To represent invariant and equivariant germs in terms of invariant polynomials, it is convenient to use the following concept. Let X, Y , Z be sets, f : Let GL(n) be the group of all invertible n × n-real matrices and O(n) the n-dimensional orthogonal group. Let Γ be a compact Lie group acting on R n via an orthogonal representation ρ : Γ → O(n). We denote by Γ 0 the connected component of the identity in Γ, identify γ with ρ(γ), ∀ γ ∈ Γ, and denote by γ the action on R n induced by ρ. We denote by GL Γ (n) the group {M ∈ GL(n) : M γ = γM, ∀ γ ∈ Γ} of Γ-equivariant matrices in GL(n) and by L o Γ (n) its connected component of the identity.

More Algebraic Structures.
Before we discuss the contact equivalence we need, we turn our attention to some important algebraic properties of E Γ,∆ (x,λ,µ) . We need them to be able to use some basic definitions and results due to Damon [8] to establish the Unfolding and Finite Determinacy Theorems. A DA-algebra A (differentiable algebra) consists of an R-algebra A and a surjective algebra homomorphism ψ : E n → A (for some n). These algebras are local rings with maximal ideals M A . If φ : E m → B defines another DA-algebra, then a homomorphism of DA-algebras α : A → B is an algebra homomorphism which lifts toα : E n → E m withα = g * for some g ∈ E n,m and α • ψ = φ •α.
Example. From Schwarz's Theorem [33], E Γ (x,λ) and E ∆ (x,λ,µ) are DA-algebras. As an example, in the first case, let φ : The ring E Γ,∆ (x,λ,µ) is not necessarily a DA-algebra. Nevertheless, it behaves quite well. Damon has introduced the notion of extended DA-algebra to fit his general framework.

Extended DA-algebra.
Let A ⊂ B be two DA-algebras and I ⊂ B a B-finitely generated ideal. Then R = A + I is still a local R-algebra with maximal ideal A module over an extended DA-algebra is an R-module which satisfies the additional property that the B-module I induces a B-module structure over I · M . This is always the case if M is contained in a B-module. The next result is fundamental to ascertain if an extended DA-algebra is actually still a DA-algebra.
is a set of generators for I/I 2 as an A-module.
Conversely, we need only to show that I is a finitely generated R-ideal.
Then we see that the union of those two sets of generators generates any element of M R up to flat germs and the result follows from the theory of graded modules.
When M R is not finitely generated, M k R is not of finite codimension for k > 1. For later use in the discussion of finite determinacy, Damon defines in [8] the ideal which has now finite codimension, because R/M (k) x is a finite E Γ x -module. Proof: (a) Let R be a DA-algebra with surjective homomorphism φ : E y → R. As M y is finitely generated, M R is itself finitely R-generated and so I/I 2 is a finitely generated A-module.
Conversely, if I/I 2 is a finitely generated A-module, we know from Lemma 2.1 that M R is finitely generated, say by . As our groups do not act on the parameters, Examples.
(1) Let Γ = O(2) acting on C via the usual orthogonal representation on C generated by θ · z = e iθ z, θ ∈ S 1 , and κ · z =z. Take ∆ = 1, the trivial subgroup of Γ. The ring of invariants E -module because the local algebra E z / zz cannot be finite.
(2) When Γ is a finite group acting on R n , the origin is an isolated singularity for the quotient space in the complexification, and so E ∆ x is a finite E Γ x -module. When Γ is a faithful representation of a compact continuous Lie group and ∆ is finite the result is false.

Systems of algebras.
Let (D, ≤) be a finite partially ordered set of indices. A system of (extended) DA-algebras consists of a set of (extended) DA-algebras {R α } α∈D together with homomorphisms of (extended) DA-algebras φ αβ : R α → R β defined for α ≤ β so that φ γβ •φ αγ = φ αβ , for α ≤ β ≤ γ, and φ αα = id. We only allow extended DA-algebras for indices maximal in D. In our context, Then, {(R α , I α )} α∈D denotes a system of (extended) DA-algebras and ideals. We use for the unfolding theory the system of (extended) To specify in this last system the maximal ideals we define We say that {R (x,λ,µ,β) } is an adequately ordered system of (extended) DA-algebras over E β if each connecting homomorphism is an E β -algebra homomorphism and each ring has one predecessor. For such systems a version of the Preparation Theorem holds true.

Bifurcation Equivalence.
Next we introduce some equivalence relations we need to organise the theory.
by GL Γ,∆ (x,λ,µ) the corresponding subsets of matrix-valued functions with values in GL(n). We also define M Λ = M λ + µ E (λ,µ) . The group of equivalences we are going to work with must preserve the Γ-equivariance of the (µ = 0)-slice. A straightforward group of equivalences is obtained by combining K Γ (x,λ) and K ∆ (x,λ,µ) in the groupK(Γ, ∆), defined as the connected component of the identity in This key fact when Γ is a continuous group was already recognised in [18]. We identify two groups of such equivalences: In Lemmas 2.5 and 2.6 we give alternative descriptions.
Remark. We can always assume that the changes of coordinates in M(Γ, ∆) andŜ(Γ, ∆) are ∆-equivariant because nothing is gained by relaxing this condition (because we can always average non equivariant change of coordinates over ∆).
Finally, considering words of finite length on the elements of the three subgroupsM(Γ, ∆),Ŝ(Γ, ∆) andK(Γ, ∆), our changes of coordinates form a group, denoted by K Γ,∆ (λ,µ) , defined as the free product ( [24]) is defined in the following way: • for (T, X, L) ∈K(Γ, ∆), we have (T, X, The groups of contact equivalences K, or K Γ λ , are semi-direct products. What about K Γ,∆ (λ,µ) ? AlthoughM(Γ, ∆) andŜ(Γ, ∆) are not normal subgroups in K Γ,∆ (λ,µ) , we show next that we can use the changes of coordinates in K Γ,∆ (λ,µ) in any order we may wish, and so we can reduce the words of any length in K Γ,∆ (λ,µ) into words of length at most 3 with one element in each subgroup.
The elements ofΛ commute with the elements of the other groups, so we do not need to consider their influence. We deal with the relations betweenM(Γ, ∆), S(Γ, ∆),X(Γ, ∆) andT(Γ, ∆), showing how we can make their elements to interchange, to combine finally together the different terms of each of them. In the following we consider the different cases, in turn. The proofs are straightforward verifications from the definitions.
Example. To show why composing elements ofŜ(Γ, ∆) andT(Γ, ∆) does not form a direct product, consider the case we are going to look at in Section 3, Γ = O(2) and ∆ = 1.
-equivariant because it has cubic terms in x and y. In the other direction, takeT = 1 µ 0). The definitions of Section 2.3.1 clearly extend to their β-parametrised versions for unfoldings: and M (λ,µ,β) . Perturbations of any f ∈ E Γ,∆ (x,λ,µ) are described by q-parameter unfoldings of f . They are map-germs (λ,µ,β) the group of (Γ, ∆)-equivariant equivalences for q-parameter unfoldings. It is a natural extension of K Γ,∆ (λ,µ) in the following sense: and Φ is a diffeomorphism germ}. We begin by showing that the elements ofŜ(Γ, ∆) are matrices of Then its tangent spaceŝ(Γ, ∆) is formed of the matrices M ∆ n (R) annihilating all the generators of E Γ (x,λ) .
(2) ClearlyM(Γ, ∆) ⊂M and is connected and contains the identity. For the converse, we establish first that we can test M ∈M on its behaviour on the Γ-invariant generators and Γ-equivariant generators: Let t ∈ (R, 0) and M (x, λ, µ, t) be a family of elements inM, unfolding the identity. Differentiating the conditionsū i (M (x, λ, µ, t)x) =ū i (x), 1 ≤ i ≤ r, with respect to t (at the origin), the tangent space ofM consists of ∆-equivariant maps m such that ∇ xūi (x), m(x)x = 0, 1 ≤ i ≤ r. If we introduce the orbit-map for Γ, Φ : R n /Γ → R r , defined by Φ(Γ · x) = (ū 1 (x), . . . ,ū r (x)), then ∇ x Φ(x), m(x)x = 0 which means that m(x) ∈ ℓ Γ (n) so thatM andM(Γ, ∆) have the same tangent spaces. The local changes of coordinates form a local Lie group (see [28]). By uniqueness between the Lie algebras and the connected component of the identity of local Lie groups we can conclude that every element in the connected component of the identity inM takes its value in Γ 0 and so the conclusion.

Geometric subgroups: the 'best' situation.
We have now all the ingredient needed to check that our theory satisfies the abstract framework of Damon [8], so that the major theorems about unfolding and determinacy theories hold true. Efficient summaries of what is needed can be found in [9] or [1, pp. 170-178]. We refer to Damon's notation in [9], followed in [1]. The results and discussions from the previous subsections show that G = K Γ,∆ (λ,µ) is a geometric subgroup of K, that is, it satisfies the four conditions: naturality (with respect to pullback), algebraic structure of tangent spaces, exponential map and filtration property. Our group K Γ,∆ (λ,µ) acts on the linear subspace F = E Γ,∆ (x,λ,µ) . For any q ∈ N, let β ∈ (R q , 0), the linear subspace of q-parameter unfoldings are F un (q) = E Γ,∆ (x,λ,µ,β) . The subgroups acting on F un (q) are G un (q) = {(T, X, L, Φ) ∈ K Γ,∆ (λ,µ,β) : Φ = I β } and, finally, is an unfolding of the identity in K Γ,∆ (λ,µ) }.
The algebraic structure of the tangent spaces is achieved via the (extended) system of DA-algebras defined in (2.1), (2.2), that are nice (adequately ordered). Explicitly, In an ideal world, we would be able to show that if f, g ∈ E Γ,∆ (x,λ,µ) are bifurcation equivalent with (T, X, L) only in K ∆ (λ,µ) , then there exists a K Γ,∆ (λ,µ) -equivalence between f and g. We cannot prove it, but we can show that the definition of K Γ,∆ (λ,µ) corresponds to the largest extended tangent space in the class of geometric subgroups respecting the structure of E Γ,∆ (x,λ,µ) .
The unfolding F is called versal if any unfolding G of f maps into F . A versal unfolding with minimal number of parameter is called miniversal. As a direct consequence of the abstract results of Damon [8, Theorems 9.3 and 11.4, and Corollary 9.9], we get the usual results from unfolding theory.  (., ., 0), . . . , F α k (., ., 0) R .
(2) Two versal unfoldings of a germ in E Γ,∆ (x,λ,µ) are equivalent as unfoldings if and only if they have the same number of unfolding parameters.
be a basis for W . Then a miniversal unfolding of g is (4) If f and g ∈ E Γ,∆ (x,λ,µ) are two K Γ,∆ (λ,µ) -equivalent germs of finite codimension and F and G ∈ E Γ (x,λ,α) , with α = (α 1 , . . . , α k ), are two miniversal unfoldings of f and g, respectively, then F and G are K Γ un (k)-equivalent. For any germ of map f at the origin, we denote by j k (f ) the Taylor polynomial of order k (or k-jet ) of f at the origin. A germ f ∈ E Γ,∆ (λ,µ) -determined for some integer k. As usual, there is a close relationship between being finitely determined and being of finite codimension. The first theorem follows from the general theory (see [ The recognition problem seeks conditions under which a germ g ∈ E Γ,∆ (x,λ,µ) is K Γ,∆ (λ,µ) -equivalent to a given normal form. To solve a particular recognition problem means to characterise the K Γ,∆ (λ,µ) -equivalence class in terms of a finite number of polynomial equalities and inequalities to be satisfied by the Taylor coefficients of the elements of that class. Let Φ = (T, X, L) ∈ K Γ,∆ (λ,µ) and consider the mapping , denoted by Itr V , is the largest intrinsic subspace of E Γ,∆ (x,λ,µ) contained in V . Note that any power of the maximal ideal is intrinsic because the contact equivalence preserves the gradation of the jet space by degrees.

Topological Equivalence.
We have described a theory of smooth equivalences to classify the bifurcation diagrams in E Γ,∆ (x,λ,µ) and their deformations. It is a wellknown problem of smooth equivalences that they often lead to too fine a classification, containing parametrised families of normal forms. The continuous parameters of such families are called moduli. Although we are mainly interested in topological properties of the diagrams, like number and position of solutions (and some stability questions), because we are also interested in the deformations of those diagrams, we could not simply use homeomorphisms for change of coordinates. For instance, the homeomorphic diagrams of the type x 2 − λ = 0 and x 4 − λ = 0 are considered different because their perturbations are different. On the other hand, there are also higher order terms in the smooth normal forms (or their unfoldings) with no topological influence or values of the moduli which can be lumped together because the germs (and their miniversal unfolding) are topologically equivalent. The theory of Damon [10], [11] deals with two preoccupations: • when are two smooth normal forms topologically equivalent?
• when is an unfolding topologically versal?
In this paper we use the theory in an utilitarian way; to deal with our germs which can have a large smooth codimension (up to 12 when we cap the topological codimension at 2). We present here the main results we are going to need to carry out the classification and check the topological versality of the unfoldings (see Theorem 3.4). Our cases are very simple, basically the homogeneous terms of lower order with respect to a weight filtration of the variables and parameters determine the topological type of the normal form (and of its miniversal unfolding).
For the rest of the section, we have a set of weights (a, b, c) for (x, λ, µ), with a, b, c ∈ N. Let f ∈ E Γ,∆ (x,λ,µ) , the weight of f , wt(f ), is the minimum weight of the nonzero monomials in the Taylor series expansion of f . The initial part of f , denoted by f in , is the sum of those monomials of degree exactly wt(f ). We say that two germs f, g ∈ E Γ,∆ (x,λ,µ) are topologically K Γ,∆ (λ,µ) -equivalent if the changes of coordinates involved in K Γ,∆ (λ,µ) are only homeomorphisms instead of being diffeomorphisms. Similarly, we define the notion of topological mapping between unfoldings which induces the notion of topological K Γ,∆ (λ,µ) -versality. To discuss the topological versality of a K Γ,∆ (λ,µ) -miniversal unfolding F of f , we construct the following sub-unfolding of F . Suppose projecting onto a basis of N Γ,∆ e (f ). We reorganise that basis such that we can assume that the p i 's are weighted homogeneous. We define then is of finite dimension as a R-vector space. Suppose that F is versal in positive weight then the number j is called the topological codimension of f . Theorem 2.13 (Topological Determinacy, [10], [11]). Let f ∈ E Γ,∆ (x,λ,µ) with cod Γ ∆ (f in ) < ∞. Then, any g ∈ E Γ,∆ (x,λ,µ) such that g in = f in is topologically K Γ,∆ (λ,µ) -equivalent to f . Theorem 2.14 (Topological Versality, [10], [11]). Let f ∈ E Γ,∆ (x,λ,µ) with cod Γ ∆ (f in ) < ∞. Let F be an unfolding of f , versal in positive weight, then F is topologically K Γ,∆ (λ,µ) -versal. Moreover, F is topologically K Γ,∆ (λ,µ) -trivial along the directions of weight larger or equal to wt(f ).

(O(2), 1)-Symmetry Breaking Theory
Via the notation z = x + iy we identify C and R 2 . We would like to classify the (O(2), 1)-symmetry breaking problems on R 2 with the usual orthogonal representation of O(2) on C, generated by θ : z → e iθ z, θ ∈ S 1 , and κ : z →z. We first look at the O(2)-part of the problem.
Proof: We use the equivalent definition ofŜ(O(2), 1) given in Lemma 2.5. Let S ∈Ŝ(O(2), 1), then S can be written as Sw = aw + bw for a, b two complex valued functions of (z, λ, µ). As Sz = z, we see that (a − 1) z + bz ≡ 0. Writing a = a 1 + ia 2 and b = b 1 + ib 2 , we see that We can then conclude of the existence of g, h ∈ E (z,λ,µ) such that and that a 2 + b 2 = g y. Thus, in complex notation, we find that and the result follows passing to the real coordinates.
The first part of the tangent space represents the usual extended tangent space in E By definition the kernel of φ f1 1 , denoted by Φ 1 , has no action on f 1 , so its elements can make effective, independent contributions to the µ-dependent part of the tangent space when applied to f 2 ; let ζ ∈ Φ 1 , then ζ(f ) = ζ(f 1 + µf 2 ) = µζ(f 2 ). In general, Φ 1 is the sum of E λ and E (u,λ) -modules. In Section 3.5.3 we shall determine Φ 1 for our normal forms. The calculations for the extended tangent space T O(2),1 e (f ) of f = h z + µ(r, s) can be split into two parts. First λ) ) and the second in E (z,λ,µ) where Φ 1 (f ) = µΦ 1 (f 2 ) represents the result of the action of the elements of Φ 1 on f .
Remark. The generators of (3.1) also give non trivial contributions in µE (z,λ,µ) but they are not relevant for the extended normal space because we consider the quotient E (λ,µ) . Although its structure is similar to that of the extended tangent space, we are interested in its totality to be able to estimate its maximal intrinsic subspace (hence P(f )) where Φ 2 represents the kernel of the tangent map φ f1 (z,λ) . In our cases, Φ 2 can be larger than M (u,λ,µ) · Φ 1 (see Proposition 3.8). The contributions by T 2 , X 2 andŝ are the same as previously, but, for the last ones, we get M (z,λ,µ) ·m(f ) and for (3.2) we have
Proof: The proof is in Section 3.5.2.
Remark. The pre-normal form (3.3) is variational, that is, it is a gradient with respect to (x, y), as are the O(2)-miniversal unfoldings for cases II-VI: H 2 , . . . , H 6 .

Classification theorems.
The smooth classification involves many moduli and is rapidly complicated. Moreover, the main information of practical importance for the study of the bifurcation diagrams is the topological type of germs and unfoldings. So, our first result is about the topological classification of germs. case miniversal unfolding/normal form top-cod cod The Greek letters δ i and ǫ j represent normalised non-zero coefficients (±1). The coefficients a, b, c are topologically irrelevant. They can be put to 0, normalised or be genuine moduli depending on the situation (see Theorem 3.5 for more details in the cases I 0 , II and III).
Remarks. 1. We believe that we have also a complete list up to topological codimension 2, but we cannot prove that the following unfoldings are of topological codimension 2. As previously we list the miniversal unfoldings with (topologically relevant) unfolding parameters α and β. The normal forms are obtained by setting α = β = 0. The symbols δ i , ǫ j , ν 11 and ν 14 represent normalised non-zero coefficients (±1) as m is a modulus with topological implication. The coefficients a, b, c, d,e, n 12 , n 13 and the functions e i (λ) are topologically irrelevant. They can be put to 0, normalised or be genuine moduli depending on the situation. We discuss in more details those cases in [12].
2. Cases I 0 to VI correspond to the classification of the (Z 2 , 1)-symmetry breaking problems such that f o 2 = 0. The normal forms are obtained by setting x = 0 in the second component of the (O(2), 1)-normal forms and they have the same topological and smooth codimensions. When f o 2 = 0, for case I 2 , we have a richer structure than the (Z 2 , 1)-symmetry breaking problems in one dimension.
In the next theorem we give the smooth classification of the families of germs I, II and III of topological codimension 0 and 1.
We express the higher order terms of g as higher order terms of s; terms which can be removed from s for any representative of the K O(2),1 (λ,µ) -orbit of g. We denote them by P(s) and they satisfy P(s) ⊃ E (y,λ) · y 2 + M m−1 λ · y + M λ , for (3.4), and for k = 2n − 2, for (3.5), Proof: (1) The pre-normal form (3.4) is a direct consequence of Proposition 3.3. That s has no topological influence follows from Damon's theory of topological equivalence (see Section 2.5). For (3.4), we introduce the weights wt(z, λ, µ) = (m, 2, 3m). With respect to that grading, g in (z, λ, µ) = (ǫ 1 u + δ m λ m ) z + µ ( 0 1 ). To prove the result, we show that g in is of finite K O(2),1 (λ,µ) -codimension. Recall that the calculations for the various tangent spaces of f can be split into two parts. In E O(2) (z,λ) , the extended tangent space for h is given in (3.1). In E (z,λ,µ) , using the pre-normal form g(z, λ, µ) = h(u, λ) z + µ 0 s(y,λ) , a calculation shows that the generators for the extended tangent space in E (z,λ,µ) (given in Section 3.3) simplify to the following: Using the results of Proposition 3.7 in Section 3.5.3, the contributions of the vectors Φ 11 (0, s) and Φ 12 (0, s) do not bring any new information for our cases. Because h is of finite O(2)-codimension, we only need to check the part of the tangent space given by (3.6): for our g in . Therefore, the part of N O(2),1 e (g in ) coming from (3.6) is E (y,λ,µ) /(E (y,λ,µ) · ǫ 1 y 3 + δ m λ m y + µ, 3ǫ 1 y 2 + δ m λ m + E (λ,µ) · λ m−1 y, 1 ).
(3) That both (3.4), (3.5) have finite smooth codimension follows from our previous calculations because their initial parts have finite smooth codimension. The generators of the part of the unipotent tangent space of g inside E (z,λ,µ) simplify into the following (3.7) From (3.7) we can evaluate the intrinsic part of the unipotent tangent space of g which gives us an estimate of the terms we can ignore in s.

Proof of Theorem 3.4:
The first point to remark is that the O(2)-equivariant part of f , h, give a lower bound for the codimension. That is, As a consequence of Damon's theory of topological equivalence we are going to see that the two codimensions in (3.8) are equal for our cases.
We can cast f 1 into one of the normal forms mentioned in Section 3.1. Using Proposition 3.3 we can cast f into the pre-normal form (3.3) for some s and use Proposition 3.6 to conclude in all cases about the germs. Finally, to verify that the topological codimension of our three cases is at most one, we use Theorem 2.14.
Proof of Theorem 3.5: The calculations for the various tangent spaces can be carried out like in the proof of Proposition 3.6. Here the terms coming from Φ 11 (0, s) and Φ 12 (0, s) cannot be ignored. First we compute an estimate on the set of higher order terms P(f ) using the intrinsic part of T U O(2),1 (λ,µ) (f ). In our situation, because of the component mixing role ofM(O(2), 1), we take the two second components of intrinsic submodules to be equal. So we find that the following intrinsic submodules I + µ J J are in P(f ): for I 0 : I = M 2 (u,λ) and J = M (y,λ,µ) , for II 1,2 : I = M (u,λ) · u + λ 3 and J = M 2 (y,λ,µ) + λ, µ , for III 1,2,3,4 : I = M 3 (u,λ) + M (u,λ) · λ and J = M 4 (y,λ,µ) + λ, µ . We use those estimates to a priori cut off all irrelevant terms from the Taylor series expansion of f . For the cases III, Mather's Lemma [27] shows that the terms in y are irrelevant in s.
Let s = 1 + ay + by 2 + cy 3 + dλy 3 and define f a,b,c,d = h + µs. A calculation shows that the unipotent tangent space of f a,b,c,d is equal to the unipotent tangent space of f 0,b,c,d . Because Then scaling of the coordinates gives the normal form. The last calculation is for the miniversal unfolding when we need to compute the normal space of our normal forms.

Pre-normal forms.
Proof of Proposition 3.3: We are going to proceed in 3 steps.
Step 2. Recall that if M ∈M(O(2), 1) then M (z, λ, µ) ∈ O(2), and so we are going to bringr to 0 andŝ to somes(z, λ) using the rotation aspect of M . Using the Implicit Function Theorem we can find θ ∈ E (z,λ,µ) such that the first component of M · f pn1 is 0. Thus we find Step 3. Finally, let S ∈Ŝ(O(2), 1) be of the form 1 0 −yg xg+1 for some g ∈ E (z,λ) . To eliminate the variable x in (3.11), we consider .
Proposition 3.8. For our normal forms h i , i = 5, the part of Φ 2 , not inŝ, is generated over E (u,λ) by Φ 11 and over E λ by λΦ 12 .

General Properties of Bifurcation Diagrams.
The miniversal unfolding F of the normal forms of Theorem 3.4 are given by where H is an O(2)-invariant unfolding and s satisfies s o = 1. Note that F in (4.1) is the gradient of the potential (4.2) V λ,µ,α (z) = 1 2Ĥ (u, λ, α) + µS(y, λ) whereĤ u = H and S y = s. Hence, the solutions of F = 0 are the critical points of V λ,µ,α . The solutions of F = 0 belong to two families. When µ = 0, we get the origin x = y = 0 and the S 1 -orbits x 2 + y 2 = u determined by the solutions u of H(u, λ, α) = 0 of the original O(2)-equivariant diagrams. When µ = 0, the symmetry breaking solutions appear. For the normal form (4.1), they are in the hyperplane x = 0. In that hyperplane, we have the perturbed branches y H(y 2 , λ, α) + µs(y, λ) = 0. In general, this means that the symmetry breaking selects one hypersurface in the (x, y, λ, µ, α)-space where the symmetry breaking solutions live.
The stability of the solutions of F = 0 are determined from the Jacobian F z of (4.1) When µ = 0, the eigenvalues of the O(2)-equivariant problem reduce to Note that the sign of the first eigenvalue of (4.4) changes with the sign of y, which means that only one of the pair of solutions ±y of the O(2)-problem remains stable after the symmetry breaking. The second eigenvalue is only a small perturbation of 2uH u . We shall come back to that issue when analysing case I 0 .
Remark. Using the approach of Proposition 3.3, we can show that (4.1) is a general form for all miniversal unfoldings of normal forms with f o 2 = 0. Therefore, the results of this section will also apply to those cases.

Bifurcation Diagrams for Germs up to Topological Codimension 1.
In the following we describe explicitly the bifurcation diagrams associated with the cases of topological codimension at most 1, cases I 0 , II and III. Bifurcation equivalence preserves the µ = 0-slice and the distribution of regions in the (λ, µ)-plane with the same pattern of solutions. The boundaries of those regions form the bifurcation variety (at fixed values of α). It is constituted of the values of (λ, µ) where there is at least one fold in the bifurcation diagram, that is, the values of (λ, µ) such that there exist solutions of F = 0 with det(F z ) = 0. Bifurcation equivalence does not preserve one dimensional slices of the bifurcation diagrams apart from along the λ-axis (when µ = 0). We sketch the bifurcation varieties in the (λ, µ)-plane and the set of critical points of V λ,µ,α in the connected component of the complement of the bifurcation variety.

Case I 0 .
For case I 0 , H(u, λ) = ǫ 1 u + δ 1 λ and s = 1, and so the potential (4.2) is V I0 λ,µ (x, y) = 1 4 ǫ 1 (x 2 + y 2 ) 2 + 1 2 δ 1 λ(x 2 + y 2 ) + µy. The part of the zero-set of case I 0 when µ = 0 is ǫ 1 y 3 + δ 1 λy + µ = 0 and so the bifurcation variety for case I 0 is (27µ 2 + 4ǫ 1 δ 1 λ 3 ) µ = 0. When ǫ 1 = −δ 1 = 1, the bifurcation variety is sketched in Figure 1. We have also indicated the number of critical points of V I0 λ,µ with their stability, stable if it is a minimum, unstable otherwise.  The corank one (Z 2 , 1)-symmetry breaking problems when f o 2 = 0 have the same normal forms ( [12]), but stability is preserved under perturbation when µ = 0. This is not the case here, only one solution of the O(2)-orbit remains stable. To understand why, we represent in Figure 2 the potential V I0 µ,λ . When µ = 0, V I0 0,λ has a unique minimum for λ ≤ 0, but has a 'cardinal hat'-like shape for λ > 0 (see Figure 2 on the right). The influence of the term µy is to add a planar contribution to V I0 µ,λ , so V I0 µ,λ will look tilted into the y-direction. Obviously this has only a marginal effect when λ ≤ 0, so we still have a unique minimum there. But, when λ is positive, one can see that the minimum rim of the hat is now tilted, only one minimum is left from the original circle orbit (see Figure 2 on the right). This is a typical planar phenomena. In corank one, the potential of the normal form corresponding to case I 0 is W µ,λ (y) = V I0 λ,µ (0, y) = 1 4 ǫ 1 y 4 + 1 2 δ 1 λy 2 + µy, but, when slightly tilted when µ = 0, the local minima retain their nature.

Case II.
For case II, H(u, λ, α) = ǫ 1 u + δ 2 λ 2 + α and s = 1 + ay, and so the potential (4.2) is The part of the zero-set of case II when µ = 0 is ǫ 1 y 3 + (δ 2 λ 2 + α + aµ)y + µ = 0 and so the bifurcation variety for case II is The parameter a is topologically irrelevant. It breaks the Z 2 -symmetry in µ of the bifurcation variety. The Taylor series expansion in term of a of the extremum value of µ on the bifurcation variety when λ = 0 is µ ± (a) = ± −4ǫ 1 α 3 /27 − (2ǫ 1 α 2 /9) a + · · · , when sign(ǫ 1 α) = −1. For ǫ 1 = δ 2 = 1, the bifurcation variety is sketched in Figure 3 for α < 0. When α > 0, the bifurcation variety is µ = 0. We get a unique minimum for V II λ,µ , perturbation of the trivial branch when µ = 0. The main difference between cases I 0 and II is in the behaviour of the parameters. Indeed, the potential V II λ,µ behaves like in Figure 2, but for a more complicated dependence in the parameters, as can be seen in Figure 3. Note that the bifurcation variety for case II is formed from the gluing of two bifurcation varieties for case I 0 in opposite directions. When ǫ 1 δ 2 = 1, the bifurcation variety is bounded when µ = 0 ('elliptic' case). On the contrary, when ǫ 1 δ 2 = −1, the bifurcation variety is unbounded for all values of µ ('hyperbolic' case).

Examples
In this final section we consider a few examples leading to bifurcation equations of the type we analysed in Section 4.

Periodic Solutions of Autonomous Equations.
We want to study the existence of period-2π solutions of (5.1)ü + u + u g(u, λ, α) + µ p(t, u,u, λ, µ, α) = 0 where p is 2π-periodic in t and g(0, 0) = 0. As usual we assume that we have made a change in coordinates in u so that the origin is the trivial solution for all λ.
The autonomous part of equation (5.1) is O(2)-equivariant with the shift action t → t + θ, θ ∈ S 1 , and time-reversal action t → −t for the symmetry generator. It has been normalised so that the bifurcation point is at λ = 0, with the kernel of the linearisationü + u = 0 generated by cos t and sin t. If p has period 2π in t, the equation (5.1) has generically no additional symmetries (see [14] for a broader discussion of subharmonic symmetry breaking). When π −π p(t, 0) e it dt = 0, we are in one of our cases I 0 , II or III provided that the bifurcation equation of the autonomous equation is of topological codimension not more than 1. The coefficients we need to monitor for the cases classified in Theorem 3.4 are more or less well-known and are given by: Depending on those values, we find the diagrams we have illustrated in Section 4. When ǫ 1 = −δ 1 = 1, we have case I 0 illustrated in Figure 1. When ǫ 1 · δ 1 = 0, we get problems of topological codimension at least 1. We assume that ǫ 1 or δ 1 = 0 due to the variation of g on another parameter we denote by α. Then we get Figure 3 when δ 1 = 0 but ǫ 1 = δ 2 = 1 and g o λα < 0, and we get Figure 4 when ǫ 1 = 0, but ǫ 2 = −δ 1 = 1 and 9g o vvv − 20g o vv 2 α > 0. In each case the symmetry breaking term selects a pair of solutions from the O(2)-orbits, keeping only one solution possibly stable.

Reaction-Diffusion Equations.
Consider a population of density u living in a bounded regular domain Ω ⊂ R n . Suppose that u satisfies the following parametrised semilinear parabolic evolution equation (5.4) u t = ∆u + g(u,ū, λ) subject to non-flux (Neumann) boundary conditions ∂u ∂n = 0. The termū represents a nonlocal contribution of the averaged densityū = 1 |Ω| Ω u(s) ds, for instance if there is some sharing of resources on a time scale faster than the reproductive cycle of u. The diffusion coefficient has been normalised to 1 by rescaling time.
We assume that there is a constant positive steady state u 0 (λ). To study the bifurcation from u 0 of nontrivial steady states, consider the linearisation of the steady state equation of (5.4), subject to Neumann boundary conditions: where a(λ) = g u (u 0 (λ),ū 0 (λ), λ) and b(λ) = gū(u 0 (λ),ū 0 (λ), λ). The values of λ for which L has a non-trivial kernel are candidates for bifurcation points. Moreover, the stability of the steady state solutions of (5.4) obeys the Principle of Linearised Stability. We are looking for values of λ when u 0 looses stability to nontrivial steady-states.
Take Ω to be the unit disk in the plane. Then (5.4) is O(2)-equivariant under the action (γu)(x) = u(γx), where γ is the usual orthogonal action of O(2) in the plane, and so the bifurcation equations are O(2)-equivariant. When the kernel of L is two dimensional we can apply our results to see how the perturbation of the symmetry (both in the domain or in the equation) affects the solution set. Because we have Neumann boundary conditions, the study of the spectrum of the nonlocal operator L can be explicitly carried out. LetLv = ∆v+a 2 v for a ∈ R. Then, using polar coordinates and separation of variables, we can find the eigenvalues and eigenfunctions ofL in terms of Fourier series in the angle variable θ and Bessel functions J n of the radius r. Let v(r, θ) = R(r)Θ(θ) with Θ 2π-periodic and R bounded on [0, 1] such that R ′ (1) = 0. The eigenvalue problemLv = µv becomes And so, for k = 0, . . . , Θ satisfies Θ ′′ + k 2 Θ = 0 and R the following Bessel equation (5.5) r 2 R ′′ + rR ′ + ((a 2 − µ) r 2 − k 2 ) R = 0.