Bifurcation and Branching of Equilibria in Reversible Equivariant Vector Fields

Bifurcation and branching of equilibria in reversible equivariant vector ﬁelds

Pietro-Luciano Buono Faculty of Science, University of Ontario Institute of Technology 2000 Simcoe St. North Oshawa, ON L1H 7K4, Canada Jeroen S.W. Lamb Department of Mathematics, Imperial College London London, SW7 2AZ, UK Mark Roberts Department of Mathematics & Statistics, University of Surrey Guildford, GU2 7XH, UK

November 30, 2004

Abstract We study steady-state bifurcation in reversible equivariant vector fields. That is, we assume the action on the phase space of a compact Lie group G with a normal subgroup H of index two, and vector fields that are H-equivariant and have all elements of the complement G \ H as time-reversal symmetries. We focus on separable bifurcation problems that can be reduced to equivariant steady-state bifurcation problems, possibly with parameter symmetry. We describe, both, bifurcations of equilibria that arise when external parameters are varied, and branching of families of equilibria that may arise in the phase space at a fixed value of external parameters. We also show how our results apply to bifurcation problems for reversible relative equilibria and reversible (relative) periodic orbits. The theory is illustrated by many examples. Contents

1 Introduction and summary 2

2 Nondegenerate local zero set 4

3 Separable bifurcation and branching 8 3.1 Bifurcation in the absence of a forced kernel ...... 9 3.2 Separability and organizing centres in the presence of a forced kernel ...... 10 3.3 Bifurcation with trivial forced kernel symmetry ...... 12 3.4 Bifurcation with nontrivial forced kernel symmetry ...... 16 3.5 Nonseparable bifurcation ...... 22

4 Branching and bifurcation of relative equilibria and (relative) periodic solutions 23 4.1 Relative equilibria ...... 23 4.2 Bifurcation and branching of (relative) periodic solutions ...... 26

5 Linear reversible equivariant systems 28

6 Equivariant Transversality Theory 30 6.1 Local zero sets and G-transversality ...... 30 6.2 Stratumwise transversality and dimension estimates ...... 32

7 Organizing centres 33 7.1 Existence of organizing centres ...... 34 7.2 Proof of Main Theorem ...... 38

1 2

1 Introduction and summary

Symmetry properties arise naturally and frequently in dynamical systems. In recent years, a lot of attention has been devoted to understand and use the interplay between dynamics and symmetry properties. The type of symmetries that have been studied most thoroughly, are symmetries that do not involve a transformation of the independent variable (i.e. the ‘time’-variable). Dynamical systems with such type of symmetries are usually called equivariant dynamical systems. Alterna- tively, there are also symmetry transformations that involve a transformation of the time-variable. In particular, time-reversibility (transformation of spatial variables and simultaneously an inver- sion t → −t of time) is a symmetry arising in many dynamical systems of interest. Examples are found in mechanical systems and in dynamical systems obtained from reductions of PDEs with symmetries involving a transformation of the independent variables. For a more extended discussion, see for instance the survey by Lamb and Roberts [25]. In this paper we address the problem of steady-state bifurcation in dynamical systems that are equivariant and reversible. The corresponding problem of steady-state bifurcation in equivariant (non-reversible) systems has been studied extensively, see for instance Field [10] or Golubitsky et al. [19]. The problem for reversible equivariant systems, however, has not been addressed sys- tematically before. Some examples of steady-state bifurcations in reversible (equivariant) vector fields, though, were discussed by Politi et al. [32], Lamb and Capel [22], Teixeira [35], da Rocha Medrado and Teixeira [6], and Lim and McComb [28, 29]. The work of the Lim and McComb highlighted the existence of two types of steady-state bifurcations which they named null-separable and null-nonseparable bifurcations. One of the problems in developing a bifurcation theory for reversible equivariant systems, was the lack of a systematic approach. The classification of linear reversible equivariant systems by Lamb and Roberts [26] opened the way for a systematic program of studies of reversible equivariant dynamical systems, extending the - by now - well established program for equivariant dynamical systems. In this paper we address the problem of steady-state bifurcation in reversible equivariant systems. For a treatment of Hopf bifurcation, see Vanderbauwhede [38] and Buzzi and Lamb [4]. Our main result shows that often (in the so called separable bifurcation cases) the reversible- equivariant steady-state bifurcation problem may be reduced to the setting of equivariant (non- reversible) steady-state bifurcation, possibly with parameter symmetry. More precisely, we find that in these cases a subset of the bifurcating branches are obtained from the analysis of such a steady-state bifurcation problem. In many cases of interest this subset provides the key to a complete description of the branching pattern. The remaining (nonseparable bifurcation) cases do not admit such a reduction, and may be studied using general singularity theoretical tools. As many applications of reversible equivariant dynamical systems can be found in Hamiltonian mechanics, one may view the theory developed here as an important step towards developing the corresponding theory in the reversible equivariant Hamiltonian setting. However, we want to emphasize that there are also examples of nonHamiltonian reversible equivariant dynamical systems to which our theory applies, e.g. the Stokeslet fluid model [17, 30] and the rolling disk [21] (an example of nonHamiltonian mechanical systems with nonholonomic constraints). In addition, dynamical systems obtained from reductions of PDEs with symmetries involving a transformation 3 of the independent variables need not be Hamiltonian. We now briefly introduce the setting of reversible equivariant dynamical systems. We consider a vector field on a manifold M and a symmetry group G acting on it. Let x ∈ M, the set

Gx = {g ∈ G | g.x = x} is a subgroup of G called the isotropy subgroup of x. Assuming that G is a compact Lie group and G is the isotropy subgroup of some point x ∈ M, by application of Bochner’s theorem [2], without loss of generality a neighborhood of x in M can be modelled by a vector space V on which G acts linearly and orthogonally. Let V be a vector space and let f : V × R` → V be a smooth vector ﬁeld. We consider the `-parameter family of dynamical systems d x = f(x, λ), (1) dt where λ ∈ R` is an `-dimensional parameter vector. Consider the following linear representations of G:

ρ : G → O(V ) σ : G → Z2 = {+1, −1} (2) ρσ : G → O(V ); ρσ(g) = σ(g)ρ(g).

Definition 1.1 The vector field (1) is G-reversible equivariant or (G, σ)-equivariant if for all g ∈ G f(ρ(g)x, λ) = ρσ(g)f(x, λ). (3) Note that our definition implies that x(t) is a solution of (1) if and only if ρ(g)x(σ(g)t) is for all g ∈ G. In this paper, we work towards a steady-state bifurcation theory for reversible-equivariant dynamical systems, i.e. we aim to describe the (bifurcation) set

{x ∈ V, λ ∈ R` | f(x, λ) = 0}. near an equilibrium solution (0, 0). Next to presenting a number of results regarding the structure of the bifurcation set, we also work through several examples illustrating our observations that have motivated our approach. Furthermore, we point out some remaining open problems. The proofs of our main results rely on tools from equivariant transversality theory, in particular the concept of G-transversality or equivariant general position and stratumwise transversality. The original definitions of G-transversality are found in Bierstone [1] and Field [7] in the context of smooth G-equivariant maps between arbitrary smooth G-manifolds. The equivalence between the definitions from the papers above is given in Field [8]. Bierstone developed the theory in order to extend Mather’s singularity theory to equivariant maps while Field’s interest was in equivariant dynamics. In the context of equivariant bifurcation theory, Field [9, 11, 10] and Field and Richardson [12] have been successful in using G-transversality theory for the development of a 4 general theory of equivariant bifurcation and to demonstrate that the Maximal Isotropy Subgroup Conjecture (MISC0) is incorrect for finite reflection groups [12]. We use G-transversality theory as part of our reduction procedure to G-equivariant (non-reversible) bifurcation problems by enabling us to obtain stable zero sets (which we call ”organizing centres”) of certain G-maps between non-isomorphic G-representations. This paper is divided into two main parts. The first part outlines step-by-step our approach to the bifurcation problem, stating our main results and illustrating them with examples. We begin in Section 2 with a discussion of local sets of equilibria near a nondegenerate equilibrium. We then divide the study of reversible-equivariant bifurcation problems into two classes depending on the absence or presence of a forced kernel for the associated linear reversible-equivariant vector field. In the absence of a forced kernel we show in Section 3.1 that reversible-equivariant bifurcation problems are equivalent to equivariant ones. In Section 3.2 we discuss bifurcations in the presence of a forced kernel, dividing them into two types that we call separable and non-separable (after Lim and McComb [28, 29]). We focus on separable bifurcations. Theorem 3.12 in Section 3.4 describes our main result about separable steady-state bifurcations, showing that part of the solution set in these cases is identical (up to homeomorphism) to the solution set of an associated equivariant bifurcation problem, possibly with parameter symmetry. Working towards this result, in Section 3.4, we first state in Theorem 3.6 the corresponding result in the special (important) case of trivial forced kernel symmetry. In Section 4 we discuss the application of our results to branching and bifurcation of relative equilibria and (relative) periodic obits. The second part of the paper is more technical and contains the proofs of our main results. In Section 5 we recall some properties of linear reversible equivariant vector fields, and discuss the correspondence between the occurrence of a forced kernel for such vector fields, and the existence of a linear invertible reversible equivariant map. In Section 6, we prove the results dealing with the structure of the nondegenerate local zero set in the neighbourhood of an equilibrium with given symmetry properties. In Section 7 proofs of our results on separable bifurcations are given with all the technical details.

2 Nondegenerate local zero set

Let Σ be an isotropy subgroup of G. It turns out that equilibria with isotropy subgroup Σ may arise in families. In order to get a feel for this phenomenon, we may view f as a G-equivariant map from the vector space V , with G acting as the representation ρ, to the vector space Vσ which is similar to V , though has G acting as the representation ρσ:

` f : V × R → Vσ, with fρ(g) = ρσ(g)f ∀g ∈ G. (4)

A G-reversible equivariant map may thus be viewed as a G-equivariant map with diﬀerent representations of G on domain and target. Correspondingly, as any equivariant map, f maps FixV (Σ) 5

to FixVσ Σ, where as usual

FixV (Σ) = {v ∈ V | ρ(g)v = v, ∀g ∈ Σ}, FixVσ (Σ) = {v ∈ Vσ | ρσ(g)v = v, ∀g ∈ Σ}.

Note that the set of points with isotropy subgroup Σ in FixV (Σ) is open and dense. However, as ρ may be nonisomorphic to ρσ, it may happen that dim FixV (Σ) 6= dim FixVσ (Σ), which is important when determining f −1(0). To keep track of this dimensional diﬀerence, we introduce the notion of the σ-index of an isotropy subgroup Σ ≤ G:

Deﬁnition 2.1 (index) Let Σ be a subgroup of G. Then, the σ-index of Σ on V is deﬁned as

sV (Σ) = dim FixV (Σ) − dim FixVσ (Σ). When it is clear from the context that we are dealing with index on V we also write s(Σ) instead of sV (Σ). We further deﬁne nΣ as N(Σ) n = dim , (5) Σ Σ where N(Σ) := {g ∈ G | gΣ = Σg} is the normalizer of Σ in G. Finally, if Σ is the isotropy subgroup of a point x, we let (Σ) be the conjugacy class of Σ in G, then we say that (Σ) is the isotropy type of x (with respect to a representation ρ of G on V ). The following result gives the dimension of families of equilibria with a given isotropy subgroup in terms of the index.

Proposition 2.2 Let f : V ×R` → V be an `-parameter family of G-reversible equivariant vector ﬁelds. Let dΣ := ` + s(Σ).

(1) If dΣ ≥ nΣ and (x, 0) is an equilibrium with isotropy subgroup Σ. Then generically, locally near (x, 0), equilibria with isotropy subgroup Σ form a dΣ-dimensional manifold

(2) if dΣ < nΣ, then generically f has no equilibrium with isotropy subgroup Σ.

Proof: (i) Let (x, 0) ∈ FixV (Σ) be a point with isotropy subgroup Σ such that f(x, 0) = 0. Consider the restriction of f to Fix(Σ)

` f : FixV (Σ) × R → FixVσ (Σ).

Generically (df)(x,0) has maximal rank, equal to dim FixV (Σ) − nΣ, where nΣ := dim N(Σ)/Σ. In order to be surjective - and hit 0 transversally in the target space FixVσ (Σ) - we need

dim FixV (Σ) + ` − nΣ ≥ dim FixVσ (Σ) ⇔ dΣ ≥ nΣ. Then, generically - if f meets 0 transversally -, locally in a neighbourhood of (x, 0), f −1(0) ∩ ` (FixV (Σ) × R ) is a dΣ-dimensional manifold. This proves part (i).

(ii) Since dΣ < nΣ then (df)(x,0) does not map FixV (Σ) surjectively into FixVσ (Σ) Hence an intersection with 0 cannot be transversal, and generically, we do not have equilibria with isotropy subgroup Σ near (x, 0). 6

Proposition 2.2 only gives information about the dimension of the set of equilibria with constant isotropy subgroup Σ. Of particular interest is the generic set of equilibria to be found in a full open neighbourhood of a generic equilibrium point with isotropy subgroup Σ of a G-reversible equivariant vector fields. We refer to this set as the nondegenerate local zero set of such an equilibrium. G-transversality theory shows that the nondegenerate local zero set has the structure of a “stratified set”. A stratification of a set is a locally finite partition of the set into smooth submanifolds called strata. Moreover, each strata has constant isotropy type. See [1, 7, 12, 10, 16] for more on G-transversality and stratifications. Obtaining the nondegenerate local zero set is important in order to obtain a complete picture of the bifurcation diagrams for G-reversible equivariant bifurcation problems. In this paper, we do not provide a general method to obtain nondegenerate local zero sets. Instead we compute the set for specific examples when needed. The problem of computing zero sets for G-equivariant maps between two arbitrary G-spaces depends on the actual computation of minimal sets of homogeneous generators for smooth G-equivariant maps, see for instance [12, 14] for examples with finite reflection groups or [40] for general computational and algorithmic aspects. Some progress has been made to obtain partial results on the zero set of G-equivariant maps between arbitrary G-spaces by using only the index of an isotropy subgroup s(Σ), see [3]. We illustrate the stratified structure of local zero sets below in some reversible equivariant examples.

3 3 3 Example 2.3 (Z2-reversible in R with positive index) Let V = R and G = Z2 acts on R by R.(x, y, z) = (x, y, −z). We consider a G-reversible vector ﬁeld in V . There are two isotropy subgroups Z2 and 1 with indices s(Z2) = 1 and s(1) = 0. Hence, if x is an equilibrium in Fix(Z2), generically it is part of a one-dimensional family of equilibria with the same isotropy subgroup. As 1 does not contain a reversing symmetry, equilibria with isotropy subgroup 1 appear isolated. Therefore, equilibria with isotropy subgroup Z2 and 1 are generically bounded away from each other.

3 Example 2.4 (Z2-reversible in R × R with negative index) Consider G = Z2 acting on V = R3 by R.(x, y, z) = (x, −y, −z). We consider a G-reversible vector ﬁeld in V . There are two isotropy subgroups G and 1. Their indices are dG = s(G) = −1 and d1 = s(1) = 0. Hence, when there is no external variable, equilibria with isotropy subgroup G generically do not arise because of the negative index of G. Equilibria with isotropy subgroup 1 generically appear isolated. In a one-parameter family, we have ` = 1 so that dG = s(G) + 1 = 0 and d1 = s(1) + 1 = 1. Hence, in one-parameter families, equilibria with isotropy subgroup 1 typically arise in one-parameter families with the same isotropy subgroup. Moreover, in R3 × R, it turns out that equilibria with isotropy subgroup G generically arise as a single point embedded in a one- dimensional family of equilibria with trivial isotropy subgroup. A general Z2-reversible equivariant system f has the form

2 2 2 2 2 2 2 2 x˙ = p1(x, y , z , λ)y + p2(x, y , z , λ)z, y˙ = q(x, y , z , λ), z˙ = r(x, y , z , λ).

Suppose that f(0, 0, 0, 0) = 0. Generically, we can solve these equations locally using the implicit function theorem by (x, y, λ) = φ(z2) with φ(0) = (0, 0, 0). Hence, the equilibrium with isotropy subgroup Z2 is embedded in a one dimensional family of equilibria with trivial isotropy subgroup. 7

3 3 Example 2.5 (SO(2) × Z2 in R × R) Consider G = SO(2)×Z2(R) acting on V = R ' C×R by R.(z, x) = (z, −x) and θ.(z) = (eiθz, x), θ ∈ SO(2). We consider a G-reversible equivariant vector ﬁeld in V , with σ-representation of G:

σ(R) = −1 and σ(θ) = 1.

The isotropy subgroups of G are G, Z (R) and 1. Here n = 0 and n = n = 1. Furthermore, 2 G Z2 1 d = s(G) = −1, d = s(Z (R)) = 1 and d = s(1) = 0. Hence, since d ≥ n , G Z2(R) 2 1 Z2(R) Z2(R) equilibria with isotropy subgroup Z2(R) form a one-dimensional SO(2) group orbit of equilibria. As dG < nG and d1 < n1, generically, equilibria with isotropy subgroup G and 1 do not appear. In a one-parameter family, we have ` = 1 so that d = 0, d = 2 and d = 1. The G Z2(R) 1 SO(2) × Z2(R)-reversible equivariant vector ﬁeld f is z˙ = p(x2, |z|2, λ)xz, x˙ = q(x2, |z|2, λ).

3 Suppose that f(0, 0, 0) = 0, then in R × R, generically (if qλ(0, 0, 0) 6= 0) an equilibrium with isotropy subgroup G is embedded in a 1-dimensional family of SO(2) group orbits of equilibria with isotropy subgroup Z2(R). In a one-parameter family, equilibria with isotropy subgroup 1 appear persistently as isolated SO(2)-group orbits of dimension one, as dΣ = nΣ = 1.

Example 2.6 (O(2) in R2) Consider G = O(2) acting on V = R2 ' C by

R.(z) =z ¯ and θ.(z) = eiθz, θ ∈ SO(2).

We consider a G-reversible equivariant vector ﬁeld in V , with σ-representation of G:

σ(R) = −1 and σ(θ) = 1.

The isotropy subgroups of G are G and Z (R). Here n = 0, n = 0, d = s(G) = 0, and 2 G Z2 G d = s(Z (R)) = 0. Since d = n , equilibria with isotropy subgroup Z (R) typically Z2(R) 2 Z2(R) Z2(R) 2 arise isolated. However, note that such equilibria still arise as an SO(2) group orbit of equilibria. In fact, the SO(2)-group orbit of equilibria is typically embedded in a one-parameter family of R-reversible relative equilibria passing transversally through Fix Z2. As dG = nG = 0, equilibria with full isotropy subgroup G generically arise as isolated points. (In this particular example it is actually the limit point of a one-parameter family of R-reversible relative equilibria).

3 Example 2.7 (Td-reversible equivariant in R ) Let G = Td be the (full) tetrahedral group 3 acting on V = R by its standard representation (symmetries of the tetrahedron). G = Td = T o Z2(R) where T is the orientation preserving tetrahedral group generated by γ and κ, and R is a reversing symmetry. The generators have the following representations in V = R3:       0 0 1 1 0 0 1 0 0 ρ(γ) =  1 0 0  , ρ(κ) =  0 −1 0  , and ρ(R) =  0 0 1  . 0 1 0 0 0 −1 0 1 0 8

Td [0] q8 Mf M qqq MMM qqq MMM qqq MM D2(κ, R) [1] D3(γ, R) [1] O Vj VV O VVVV VVVV VVVV VVVV Z (κR) [1] Z (R) [1] 2 f 82 MMM qq MM qqq MMM qq MM qqq 1 [0]

3 Figure 1: Isotropy lattice for the Td action on R in Example 2.7. The indices of the isotropy subgroups are indicated in square brackets.

The isotropy lattice of Td is given in Figure 1, in this ﬁgure for each isotropy subgroup Σ also the index s(Σ) is indicated. Note that since G is ﬁnite, we always have nΣ = 0. Now dim FixVσ (Σ) = {0} for Σ = D2 and Σ = D3. Hence, the equilibrium with isotropy subgroup Td is embedded in one-dimensional families of equilibria with isotropy subgroups D2 and D3. Equilibria with isotropy subgroups D2, D3, Z2(R) and Z2(κR) are typically locally embedded in one-dimensional manifolds of equilibria with the same isotropy subgroup. See Figure 2 for an illustration of the nondegenerate local zero set in R3.

3 Separable bifurcation and branching

If we are at a degenerate point of the set of equilibria, one of the two following situations may be at hand:

(i) We are at a local bifurcation point, i.e. the local zero set changes as we vary the external parameters.

(ii) We are at a branching point of the set of equilibria, i.e. the local zero set changes as we vary our reference point on the family of equilibria in phase space, without varying external parameters.

In fact, bifurcation points are also branching points if we view V × R` as the phase space. Our discussion naturally deals with branching and bifurcation at the same time. However, to emphasize that we may have branching families of equilibria in phase space without the need to vary parameters, we tend to reserve the term “bifurcation” exclusively to the cases where branching is induced by varying external parameters. Our aim is thus to describe bifurcation and branching of equilibria in parameter families of G-reversible equivariant vector fields. It is natural to address the question whether the problem of steady-state bifurcation for G-reversible equivariant vector fields is related to the problem of steady-state bifurcation in G- equivariant vector fields. It turns out that in certain circumstances, in the absence of a forced 9

Figure 2: Sketch of the nondegenerate local zero set near 0 for the Td-reversible equivariant vector ﬁeld in R3 discussed in Example 2.7. kernel or in the case of trivial forced kernel symmetry, these problems are indeed often closely related. However, our approach towards separable bifurcation leads more generally to steady state bifurcation problems for G-equivariant vector ﬁelds with parameter symmetry.

3.1 Bifurcation in the absence of a forced kernel The basis of our approach towards steady-state bifurcation and branching lies in the following observation:

Lemma 3.1 Let f : V → V be G-reversible equivariant, and T : V → V be a linear invertible G-reversible-equivariant map. Then,

(i) f˜ = T f is G-equivariant.

(ii) f˜(u, λ) = 0 ⇔ f(u, λ) = 0.

Proof: (i) f˜(gx) = T σ(g)gf(x) = σ(g)T gf(x) = σ(g)2gT f(x) = gf˜(x). (ii) Follows from invertibility of T .

Hence, reversible equivariant and equivariant steady-state bifurcation problems are equivalent whenever there exists a linear reversible equivariant invertible map T . From the work of Lamb and Roberts [26] we can readily answer the question when such a linear map exists. Namely, it turns out that such a linear map exists if and only if the representation of G is such that a G-reversible equivariant map L has no forced kernel, i.e. that it is possible for a linear reversible equivariant map L : V → V to have its rank equal to the dimension of V . Sometimes, the representation of G 10 prevents this. For more details see Section 5. This observation, in combination with Lemma 3.1, leads to our ﬁrst theorem:

Theorem 3.2 (Bifurcation in absence of forced kernel) Let f : V × R` → V be an `- parameter family of G-reversible equivariant vector fields, with f(0, 0) = 0. Then, whenever the representation of G is such that dV f(0, 0) has no forced kernel, steady-state bifurcation from (0, 0) reduces to the problem of equivariant steady-state bifurcation in an `-parameter family of G-equivariant vector fields. In particular, the set of equilibria in V ×R` near a G-reversible equivariant steady-state bifurcation is locally diffeomorphic to the set of equilibria in V × R` near a corresponding G-equivariant steady-state bifurcation. We illustrate the theorem with an elementary example.

Example 3.3 (Z2-reversible steady-state bifurcation: no forced kernel) Let f be a Z2- 2m m m m m reversible vector ﬁeld in R , where Z2 is generated by R : R × R → R × R , with R(x, y) = (x, −y). There exist a Z2-reversible linear invertible map T (x, y) = (y, x). Hence, generically, steady-state bifurcations of F are in one-to-one correspondence with steady-state bifurcations of 2m a Z2-equivariant vector ﬁeld g in R . In one-parameter families, such bifurcations are generically symmetry preserving folds (saddle-node or saddle-centre), or symmetry breaking pitchforks.

3.2 Separability and organizing centres in the presence of a forced kernel The existence of forced kernels is closely related with the fact that in reversible equivariant vector ﬁelds, equilibria may arise in families. Theorem 3.2 reduced the study to standard equivariant steady-state bifurcation theory, in case we have no forced kernel. It remains thus to consider the problem when there exists a forced kernel. We assume that (0, 0) is an equilibrium for f. To ﬁx notation, we decompose V into two parts

V = W1 ⊕ W2,

where W2 is the forced kernel, so that dW2 f(0, 0, 0) = 0. In case the forced kernel varies smoothly with the parameters, we can assume without loss of generality that this precise decomposition holds for all λ ∈ R` in a neighbourhood of 0, i.e. we may assume that W2 is the forced kernel for all λ. (For technical conditions see Proposition 5.4). If f at a branching point satisﬁes this condition, we call this branching point, and the corresponding branching or bifurcation, separable (following the terminology introduced by Lim and McComb [28, 29]). The exact deﬁnition is the following.

Deﬁnition 3.4 We say that (0, 0) is a separable bifurcation point if ker (df)(0,0) ∩ W1 6= ∅, and ker (df)(0,0) ∩W1 has no irreducible representations G-isomorphic to irreducible representations in W2. With this terminology, we see that bifurcation in the absence of a forced kernel - as described above - is the most trivial example of separable bifurcation. 11

We decompose the equation f(w1, w2, λ) = 0 into two parts ½ f (w , w , λ) = 0 1 1 2 (6) f2(w1, w2, λ) = 0

` ` where f1 : W1 ⊕ W2 × R → W1 and f2 : W1 ⊕ W2 × R → W2. Importantly, at separable bifurcation points, the separation V = W1 ⊕ W2 is persistent: W1 and W2 vary continuously and smoothly with the parameter and hence, by choosing appropriate coordinates, may be chosen to be constant.

We now note that since dW1 f1 has no forced kernel, we may replace (6) by the equivalent set of equations ½ f˜ (w , w , λ) = 0 1 1 2 (7) f2(w1, w2, λ) = 0 where f˜1(w1, w2, λ) := T f1(w1, w2, λ) = 0, (8) which, with the appropriate choice for the linear invertible reversible equivariant map T , is G- equivariant in the following way:

f˜1(ρ1(g)w1, ρ2(g)w2, λ) = ρ1(g)f˜1(w1, w2, λ), ∀g ∈ G (9)

with ρi := ρ|Wi . At the same time recall that for all g ∈ G,

f2(ρ1(g)w1, ρ2(g)w2, λ) = (ρσ)2(g)f2(w1, w2, λ). (10)

−1 Now we consider a point (w1, w2, λ) in f2 (0) with isotropy subgroup G. Locally, such points typically arise in a (dG + dim FixW1 G)-dimensional manifold. Note that we here and later quietly assume that we have sufficient parameters (ie ` sufficiently large) to make sure that we avoid drift −1 (ie dΣ ≥ nΣ, cf Proposition 2.2). The local structure of f2 (0) may be dependent on the point −1 chosen in this set. Our aim is to describe a (mostly smooth) G-invariant subset of f2 (0) which after substitution in the first (f˜1) equation, reduces the separable bifurcation problem f = 0 to that of a G-equivariant bifurcation problem on W1, possibly with parameter symmetry. In this way, we obtain a relationship between a subset of the solution set of f = 0 with the full solution set of this G-equivariant bifurcation problem.

Deﬁnition 3.5 (Organizing centre) A point (x, 0) ∈ V × R` with isotropy subgroup G, is an organizing centre for a G-reversible-equivariant steady-state bifurcation problem f = 0 if near −1 (x, 0), f2 (0) contains a family of G-invariant dim V -dimensional subsets Kν given by

φν(w1, w2) = µ,

k where φν : V → R (for some k) is continuous in V , and smooth within substrata of V with constant isotropy, and depends smoothly on ν ∈ R`−k. 12

It may be tempting to assume that the subsets Kν can be chosen as manifolds. In various examples this turns out to be the case. However, it is not a general fact, see Example 7.4 for a counterexample. Importantly, it turns out that the organizing centers we describe above arise typically in a persistent manner, see Section 7 for more details. The proof of the existence and persistence of organizing centres is obtained via G-transversality theory. Our approach towards separable bifurcations is to study unfoldings of organizing centres.

3.3 Bifurcation with trivial forced kernel symmetry

We consider ﬁrst the situation where ρ2 is the trivial representation, in which case we speak of trivial forced kernel symmetry. Under this assumption, f˜1 is equivariant in the following sense:

f˜1(ρ1(g)w1, w2, λ) := ρ1(g)f˜1(w1, w2, λ), ∀g ∈ G. (11)

That is, the steady-state bifurcation problem for f˜1 resembles that of a G-equivariant steady- state bifurcation problem on W1 with parameters (w2, λ). The bifurcation diagram is obtained ˜−1 −1 by intersecting the zero sets f1 (0) and f2 (0). Since the action of G on W2 is assumed to be trivial, we may begin by ﬁnding the bifurcating branches from the G-equivariant bifurcation problem f˜1(w1, w2, λ) = 0 with parameters (w2, λ). Because of the parameters of the system, these bifurcating branches come in dim W2 + ` dimensional families. Since the bifurcation point is an organizing centre, for each isotropy subgroup of the ρ1 representation, there is a submanifold of zeroes of the f2 = 0 equation which we intersect with the bifurcating family from the f˜1 equation to obtain a bifurcating branch for the full G-reversible equivariant bifurcation problem f = 0. −1 In the case of trivial forced kernel symmetry, the dimension of the submanifolds in f2 (0) with a given isotropy are easy to determine. In particular, let Σ be an isotropy subgroup of G then as ρ2(g) acts as σ(g)I on W2,σ, from (10) it follows that if Σ contains a reversing symmetry (and ` is suﬃciently large to avoid drift),

−1 dim(f2 (0) ∩ Fix(Σ)) = dim FixW1 Σ + dim W2 + `.

As the action of G on W2 is trivial, we may in fact consider the w2 variables as parameters for the f2-equation, and the organizing centre condition simpliﬁes to the existence of an `-parameter −1 family of dim W1-dimensional, G-invariant subsets of f2 (0) given by continuous maps φλ : W1 → W2 smooth on subsets of W1 with ﬁxed isotropy subgroup Σ of G (see Corollary 7.3). Hence, −1 at an organizing centre, if Σ does not contain reversing symmetries, dim(f2 (0) ∩ Fix(Σ)) = −1 dim FixW1 (Σ)+`. However, if not at an organizing centre, then typically dim(f2 (0)∩Fix(Σ)) = 0. As a result, the (G-transversal) intersection comprises a uniform dim W2-dimensional fattening ˜−1 of the branches of f1 (0) whose isotropy subgroups contain a reversing symmetry.

Theorem 3.6 (Separable bifurcation with trivial forced kernel symmetry) Let f : V ×R` → V be an `-parameter family of G-reversible equivariant vector ﬁelds, with f(0, 0) = 0 and (0, 0) has isotropy subgroup G. Suppose that f has a separable bifurcation at (0, 0), and 13

that the representation of G on the forced kernel of dV f(0, 0) is trivial. Let B denote the set of equilibria in a small neighbourhood of (0, 0). We write B = B0 ∪ Br, where Br contains all equilibria whose isotropy subgroup contains a reversing symmetry and B0 is the remainder. Then, the following situation arises persistently near an organizing centre: V × R` contains a subset B0 containing the origin, with the following properties: B0 is G-homeomorphic to a persistent bifurcation set of an `-parameter family of G-equivariant vector ﬁelds on W1. Let 0 0 0 0 0 dim W2 0 0 0 0 B0 = B ∩ B0 ⊂ B0, and (locally) Br ' (B \ B0) × R . Then B ⊇ B0 ∪ Br and B = B0 ∪ Br −1 if the W1-dimensional subset of f2 (0) found at an organizing centre is a smooth manifold. We recall that two sets are G-homeomorphic if there is a G-equivariant homeomorphism mapping one to the other. The proof of Theorem 3.6 is a consequence of the proof of Theorem 3.12 and is found in Section 7.2.

Remark 3.7 It should be noted that the local set of equilibria B can sometimes - but not always - be completely deduced from that of B0. As is mentioned in the theorem, this is guaranteed if the −1 W1-dimensional subset of f2 (0) found at an organizing centre is a smooth manifold, see Section 7 for the details. If this is not the case, the theorem can only guarantee that a subset of the full branching pattern has been obtained (yielding a type of result reminiscent of the Equivariant Branching Lemma, cf. [19]). If we are not at an organizing centre, but near it, one observes only the branches whose isotropy subgroups contain a reversing symmetry. In fact, branches can be found in phase space, without the need to vary parameters.

Theorem 3.8 (Branching of equilibria in phase space) Consider the context of Theorem 3.6, with ` = 0. Let k be positive such that k ≤ dim W2. Then, the following situation arises persistently: a G-invariant equilibrium of f is locally contained in a (dim W2)-dimensional set of branching equilibria that is G-homeomorphic to the set

0 dim W2−k B ' Br × R ,

0 where Br is G-homeomorphic to the subset of a persistent bifurcation set of a k-parameter family of G-equivariant vector ﬁelds on W1, composed of the union of branches whose isotropy subgroups contain a reversing symmetry. We refer to this phenomenon as branching of equilibria in phase space. To illustrate the result of Theorem 3.6 and Theorem 3.8 we discuss some elementary examples.

3 Example 3.9 (Z2-symmetry breaking pitchfork in R × R) Consider a Z2-reversible vector 3 ﬁeld in R , where Z2 is generated by R.(x, y, z) = (x, y, −z). Here we can choose W1 = {(x, y, z) | ⊥ x = 0} and W2 = W1 . Thus, s(Z2) = 1 and Z2-symmetric equilibria typically arise in one-dimensional families. The forced kernel W2 is one-dimensional and Z2 acts trivially on the forced kernel. Now we consider a one-parameter family of such vector ﬁelds, and we assume a separable bifurcation at (x, y, z, λ) = (0, 0, 0, 0). Then, following the above described procedure, we may assume that B0 is a generic Z2- symmetry breaking pitchfork bifurcation. Correspondingly, the bifurcation set B in R3×R consists 14

3 2 Figure 3: Projection of Z2-symmetry breaking pitchfork bifurcation in R × R to R × R. The equivariant Z2-symmetry breaking pitchfork bifurcation diagram forms the basis of this set. The Z2-invariant branch has dimension two and the branch with trivial isotropy subgroup has dimension one. of a set homeomorphic to the pitchfork bifurcation diagram, with additional one-dimensional “fattening” of the branch with isotropy subgroup Z2. The set is illustrated in Figure 3. We conclude by noting that in Example 7.5 we prove that the set Kν at the organizing centre is a manifold. Threfore, by Remark 3.7 all the bifurcating branches are accounted for by the analysis above and are G-diﬀeomorphic to the branches found using Theorem 3.6.

3 Example 3.10 (D2-symmetry breaking pitchfork branching in R ) Consider a D2 reversible 3 equivariant vector field in R , generated by the reversing symmetries R1.(x, y, z) = (x, y, −z) and R2.(x, y, z) = (x, −y, z) (with σ(R1) = σ(R2) = −1). A simple computation shows that a generic linear map commuting with the action of D2 on domain and range has a one-dimensional forced ⊥ kernel. We can choose W1 = {(x, y, z) | x = 0} and W2 = W1 . We have s(D2) = s(Z2(R1)) = s(Z2(R2)) = 1. Hence, generically equilibria with isotropy subgroup D2, Z2(R1) and Z2(R2) arise in one-dimensional families. Now, we consider a one-parameter family of such vector fields, and separable bifurcation from an equilibrium point with isotropy subgroup D2. The reduced D2-equivariant generic steady-state bifurcation has a pitchfork bifurcation diagram with a bifurcating branch of solutions with isotropy 3 subgroup Z2(R1) or Z2(R2). In R ×R, the bifurcation set of equilibria is a homogeneously fattened version of the pitchfork bifurcation diagram, as the isotropy subgroups of all branches contain a reversing symmetry, yielding two-dimensional branches with a one-dimensional intersection. We illustrate Theorem 3.8 about branching in phase space by fixing a value of the parameter. Then, we may observe a persistent pitchfork branching of equilibria, breaking from isotropy subgroup D2 to Z2(R1) or Z2(R2). In Theorem 3.8, we have k = 1 since the pitchfork is codimension 0 0 0 1 and the set B ' Br × R where Br is G-homeomorphic to the pitchfork bifurcation. These branching sets of equilibria are illustrated in Figure 4.

3 Example 3.11 (D4 branching and bifurcation in R × R) Consider a D4-reversible equiv- 15

F1 F1

PSfrag replacements PSfrag replacements Fix(R1) F2 Fix(R1) F2 Fix(R(a)2) Fix(R(b)2)

3 Figure 4: Generic D2-symmetry breaking pitchfork branching in R , as discussed in Example 3.10. The ﬁgures illustrate generic pitchfork branching from D2 to (a) Z2(R1), and (b) Z2(R2).

(a) (b)

Figure 5: Steady-state bifurcation and branching for a D4-reversible equivariant vector field in R3 × R, as discussed in Example 3.11. (a) Bifurcation set for the reduced bifurcation problem in R2 × R (b) Branching of equilibria in the phase space R3 at a fixed value of the parameter. 16 ariant vector field in R3, generated by the reversing symmetries

κ1.(x, y, z) = (x, −y, z) and κ2.(x, y, z) = (y, x, z) with γ = κ1κ2 and σ(κ1) = −σ(κ2) = −1.

We set W1 = {(x, y, z) | z = 0} since this subrepresentation is self-dual, therefore W2 = {(x, y, z) | x = y = 0} is the forced kernel. The lattice of isotropy subgroups along with indices are given in Figure 6. Hence, generi-

D4[1] oo7 Og OO ooo OOO ooo OOO ooo OO 2 2 D2(κ1, γ )[1] D2(κ2, γ )[0] O Og OO oo7 O OOO ooo OOO ooo OO ooo Z (κ )[1] Z (γ2)[0] Z (κ )[0] 2 1 Og 2 O 7 2 2 OOO ooo OOO ooo OOO ooo OO ooo 1[0]

3 Figure 6: Isotropy lattice for the D4 action on R with isotropy subgroups indices

2 cally equilibria with isotropy subgroups D4, D2(κ1, γ ), Z2(κ1) arise in one-dimensional families, other equilibria typically arise isolated. In a one-parameter family of such vector ﬁelds, we consider a separable bifurcation from an equilibrium point with isotropy subgroup D4. The reduced 2 D4-equivariant generic steady-state bifurcation in R × R has a bifurcation set with bifurcating branches of solutions with isotropy subgroup Z2(κ1) and Z2(κ2), see Figure 5(a). For the full equation, in R3 × R, the bifurcation set of equilibria is a fattened version of the bifurcation set for the reduced problem, where all branches whose isotropy subgroup contains a reversing symmetry are two-dimensional (in R3 × R), whereas the remaining branches are one-dimensional. Conse- quently (following Theorem 3.8), at a ﬁxed value of the parameter we may observe a persistent branching point, at which in the phase space R3 a one-dimensional branch of equilibria with isotropy subgroup D4 intersects a one-dimensional family of equilibria with isotropy subgroup 2 Z2(κ1) (and a one-dimensional family with conjugate isotropy subgroup Z2(κ1γ )), as illustrated in Figure 5(b).

3.4 Bifurcation with nontrivial forced kernel symmetry

We now discuss the (general) case in which G acts nontrivially on the kernel W2. There are two main diﬀerences with the case of trivial forced kernel symmetry:

• f˜1(x) = 0 is a G-equivariant bifurcation problem with parameter symmetry, viewing - as ` before - f˜1 as a family of endomorphisms of W1 with parameters in W2 × R :

f˜1(ρ1(g)w1, ρ2(g)w2, λ) = ρ1(g)f˜1(w1, w2, λ), ∀g ∈ G. (12) 17

That is, in general, ρ2(g) 6= 1. • Contrary to the trivial forced kernel case, the existence of an organizing centre may depend on the number of additional bifurcation parameters ` added to the system. We show later that it suﬃces to take ` = dim W2 +max{dim G, dim W2} parameters to enable the existence of an organizing centre. A general approach towards separable bifurcation problems is to solve the two equations

f˜1(x, λ) = 0 and f2(x, λ) = 0 separately, and then take the intersection of the two solution sets. Because we assume separable bifurcation, the intersection of the two zero sets won’t have any tangencies since the zero sets of f˜1 and f2 are tangent to subspaces which do not have common irreducible representations. The dimension of the set of intersections of solutions is then obtained by elementary calculations depending on isotropy type. By this procedure we need to solve the equivariant bifurcation problem with parameter symmetry. For the f2(x, λ) = 0 equation we need to calculate persistent codimension zero local conﬁgurations of equilibria. In analogy to the approach in the case of trivial forced kernel symmetry, we focus on organizing centres. To obtain the bifurcating branches of equilibria at the separable bifurcation point, we proceed in a similar way as before. The main diﬀerence with the trivial forced kernel symmetry case is that now we possibly need to solve an equivariant bifurcation problem with parameter symmetry.

Theorem 3.12 (Separable bifurcation) Let f : V × R` → V be an `-parameter family of G- reversible equivariant vector ﬁelds, such that f(0, 0) = 0, (0, 0) has isotropy subgroup G and is an organizing centre. Suppose f has a separable bifurcation at (0, 0). Let B denote the set of equilibria of f in a small neighbourhood of (0, 0). Then, the following situation arises persistently near an organizing centre: B ⊂ V × R` has a subset B0 containing the origin with the following properties: B0 is G-homeomorphic to a persistent bifurcation set of a smooth G-equivariant bifurcation problem with parameter symmetry

h1(ρ1(g)w1, ρ2(g)w2, ν) = ρ1(g)h1(w1, w2, ν), ∀g ∈ G. `−k on W1, with parameters (w2, ν) ∈ W2 × R with k = dim W2.

Remark 3.13 In the nontrivial forced kernel symmetry case, our approach does not detect local solutions whose isotropy subgroup (of G on V ) is not an isotropy subgroup of G on W1. If such isotropy subgroups arise, an explicit computation of the nondegenerate local zero set is necessary.

Remark 3.14 Theorem 3.12 assumes the existence of an organizing centre. In the unfolding of this organizing centre, related bifurcation diagrams can be identiﬁed. In this way all types of local bifurcations can be found. If the number of parameters ` is too low to enable the existence of an organizing centre, we may consider the system as arising in the unfolding of the organizing centre. In such a situation, information on the local bifurcation set can be obtained by taking the appropriate intersections with the bifurcation set. 18

We now illustrate these results with a few examples.

3 2 Example 3.15 (Z2-reversible saddle-node in R × R ) Consider a Z2-reversible vector field 3 in R , where Z2 is generated by R.(x, y, z) = (x, −y, −z). Consider a ` = 2 parameter family of 2 3 Z2-reversible equivariant vector fields f : W1 ⊕ W2 × R → R with f(0, 0, 0, 0, 0) = 0. Each coordinate subspace is an irreducible representation of G: the y-axis and z-axis are isomorphic irreducible representations, both σ-dual to the x-axis. Consequently, W2 is a one- dimensional subspace of x = 0, and without loss of generality we choose W2 = {(x, y, z)|x = y = 0} and so W1 = {(x, y, z)|z = 0}. Now, s(Z2) = −1 and Z2 acts as −1 (i.e. nontrivially) on W2. By definition of W1, f1 has a separable bifurcation at the origin if ker (df1)(0) = {(x, y)|y = 0}. By Theorem 3.12, the set of equilibria B0 is G-homeomorphic to the zero set of a bifurcation problem with parameter symmetry h˜1(x, y, z, µ) = 0 on W1. We study a generic Z2-equivariant 1 2 bifurcation problem with parameter symmetry h˜1 : W1 ⊕ W2 × R → R since k = dim W2 = 1. Bifurcation problem with parameter symmetry: Using the Lyapunov-Schmidt reduction we solve y in terms of (x, z, ν) to obtain the bifurcation equation in normal form p(x, z, ν) = ν + x2 + bz2 with b = ±1. As ν moves through 0, there are two scenarios depending on the coefficient b: an elliptic and a hyperbolic one. In the elliptic scenario (b = +1), the family of zeroes with isotropy subgroup 1 near the bifurcation point forms an ellipse when ν < 0 disappearing when ν ≥ 0. In the hyperbolic scenario (b = −1), the zero set with trivial isotropy subgroup forms a family of two hyperbola if ν 6= 0 and a transversal intersection of two lines when ν = 0. This situation is illustrated in Figure 7 in a projection to a two-dimensional subspace. In Example 7.6, it is shown that the set Kν at the organizing centre is indeed a manifold. Therefore, from Remark 3.7 B0 contains all bifurcating solutions and these are in fact G-diffeomorphic to the branching set of the bifurcation problem with parameter symmetry.

3 2 3 Example 3.16 (D2 symmetry in R × R ) Let D2(R1,R2) act on R by

R1.(x, y, z) = (x, y, −z) and R2.(x, y, z) = (−x, −y, z) and consider a two-parameter family of D2-reversible equivariant vector ﬁelds f with a separable bifurcation at (0, 0, 0, 0, 0). The group D2 can arise as a reversing symmetry group in two ways:

(a) σ(R2) = σ(R1) = −1 and (b) σ(R2) = −σ(R1) = 1.

The isotropy subgroup and index lattice for each type of reversibility is given in Figure 8. Case (b) naturally falls within the context of nonseparable bifurcations as any commuting linear map L : R3 → R3 is identically zero since there are no isomorphic representations in domain and range. Thus W1 = {0} and case (b) cannot have separable bifurcations by deﬁnition. We consider case (a) for which the kernel of a generic commuting linear map is one-dimensional. Each coordinate axis is an irreducible representation with the x and y axes isomorphic and nonisomorphic to the z-axis. We choose W2 = {(x, y, z)|y = z = 0}. Therefore, f has a separable bifurcation if ker(df)(0,0,0,0,0) ∩ W1 = {(y, z)|y = 0}. 0 By Theorem 3.12, B is G-homeomorphic to the zero set of a D2-equivariant bifurcation problem with parameter symmetry h˜1(x, y, z, ν) = 0 which we now study. 19

(a)

(b)

Figure 7: Generic codimension-one Z2 symmetry preserving turning point bifurcation in W1⊕W2× R2 (see Example 3.15). Depicted are projections to a two-dimensional subspace. (a) elliptic case (b) hyperbolic case. Also bifurcation diagrams corresponding to Example 3.16 and Example 3.18.

Bifurcation problem with parameter symmetry: Since W2 = x and the bifurcation direction is z, we use the Lyapunov-Schmidt reduction to solve y as a function of (x, z, ν) and obtain a bifurcation equation q(x2, z2, ν)z = 0. For z = 0 we obtain a two-dimensional manifold of solutions with isotropy subgroup Z2(R1). Since q(0, 0, 0) = 0 at a separable bifurcation, and if qν(0, 0, 0) 6= 0 is satisﬁed, then by the implicit function theorem we have a two-dimensional manifold of solutions which contains an open and dense set of solutions with isotropy subgroup 1, a one-dimensional submanifold of solutions with isotropy subgroup Z2(R2) and a one-dimensional submanifold of solutions with isotropy subgroup Z2(R1) (the intersection with the z = 0 manifold). Bifurcation problem with parameter symmetry: ` = 1 When one or more parameters are removed some branches of solution may disappear as this case shows. The bifurcation equation q = 0 has generic normal form ²(z2 + δx2)z = 0 where ² = ±1 and δ = ±1, see Golubitsky and

(a) D2[0] (b) D2[−1] s9 fLL r9 fLL ss LL rr LL ss LL rr LL ss LL rr LL ss LL rrr LL Z2(R1)[1] Z2(R2)[−1] Z2(R1)[1] Z2(R2)[0] Ke 9 fLL 9 KKK rrr LL rrr KK rrr LLL rrr KKK rr LL rr K rrr LL rrr 1[0] 1[0]

Figure 8: Isotropy subgroup and index lattice for the D2 action: (a) σ(R2) = σ(R1) = −1. (b) σ(R2) = −σ(R1) = 1. 20

(a) D4[0] (b) D4[0] t9 Le L t9 eJJ tt LL tt JJ tt LL tt JJ tt LL tt JJ tt LL tt J Z (κ )[1] Z (κ )[−1] Z (κ )[1] Z (κ )[0] 2 1 eJ 29 2 2 1 eJ 29 2 JJ rr JJ tt JJ rr JJ tt JJ rr JJ tt JJ rr JJ tt J rr J tt 1[0] 1[0]

Figure 9: Isotropy subgroup and index lattice for (a) σ(κ1) = σ(κ2) = 1 (b) σ(κ1) = −σ(κ2) = −1

Schaeﬀer [18]. If δ = +1, then no new solutions exist. If δ = −1, we obtain a zero set of two lines, z = ±x, crossing at the origin as seen in Figure 7. The origin is the only equilibrium with isotropy subgroup D2 and there are 1-dimensional families of solutions with trivial isotropy. In particular, there are no solutions with isotropy subgroup Z2(R2). Reversible-equivariant with ` = 2. The bifurcating branch with isotropy subgroup Z2(R1) in fact is fattened by one dimension from two to three since f2 ≡ 0 on Fix(Z2(R1)). The branches with isotropy subgroups Z2(R2) and 1 keep the same dimension in the full bifurcation problem. These dimensions correspond to dΣ. To determine if all bifurcating solutions are obtained from the analysis above, one would need to ﬁnd out if the set Kν at the organizing centre is a manifold. A similar analysis applies for the ` = 1 case.

3 2 3 Example 3.17 (D4 symmetry in R × R ) Consider the action of D4 on R generated by the reflections κ1.(z, x) = (z, x) and κ2.(z, x) = (iz, −x). where z ∈ C and x ∈ R. We consider a two-parameter family of D4-reversible equivariant vector fields f. Here, W1 = C is a self-dual representation, W2 = R and the choice of 2 dim W2 parameters (since dim G = 0) suffices to solve the f2 = 0 equation for an organizing centre. We look at two cases of reversing symmetries:

σ(κ1) = σ(κ2) = −1 and σ(κ1) = −σ(κ2) = −1.

The lattices of isotropy subgroups and indices are given in Figure 9. The bifurcation equation splits into the system

f˜1(z, x, λ, µ) = T f1(z, x, λ, µ) = 0 f2(z, x, λ, µ) = 0.

2 2 where f˜1 : C × R × R → C and f2 : C × R × R → R. 0 −1 By Theorem 3.12, there is a subset B of f (0), G-homeomorphic to the zero set of a D4- equivariant steady-state bifurcation problem with parameter symmetry h1(z, x, λ) = 0. Furter et al. [15] study this bifurcation problem. In particular, we consider their solution to the generic case; see Theorem 3.2.1 with ²5 = 1 and m > 1 (the case m < 1 is similar). For complete- ness, we describe this bifurcation diagram in details before returning to the original D4-reversible equivariant bifurcation problems. 21

PSfrag replacements C PSfrag replacements

Fix(κ1) F1 Fix(κ1) Fix(κ2κ1κ2) Fix(κ2κ1κ2)

C F2 C (a)x (b)x

Figure 10: Bifurcation diagram of Z2(κ1) (and conjugate) symmetric solutions in a generic D4 bifurcation problem with parameter symmetry in R3 for (a) λ < 0 and (b) λ = 0.

PSfrag replacements PSfrag replacements F2 Fix(κ1) Fix(κ1) Fix(κ(a)2) Fix(κ(b)2)

Figure 11: Bifurcating solutions for λ > 0. (a) Solutions with isotropy subgroup Z2(κ1) (b) Detail of solutions with isotropy subgroup Z2(κ2) and 1.

Bifurcation problem with parameter symmetry We consider the parameter λ2 of Furter et al. [15] as a phase space variable since it corresponds to our variable x. We draw the bifurcation diagram of [15] with a three-dimensional phase space and one parameter instead of the two- parameter bifurcation problem with two-dimensional phase space of [15]. See Figure 10 and Figure 11. Solutions with isotropy subgroups Z2(κ1) are given in Figure 10 for λ ≤ 0 and in Figure 11(a) for λ > 0. For λ ≤ 0 there are no solutions with isotropy subgroups Z2(κ2) and 1. For λ > 0, Figure 11(b) shows details of solutions with isotropy subgroups Z2(κ1), Z2(κ2) and 1 in one quadrant of R3. The solutions in the other quadrants are obtained by applying the order four 3 rotation symmetry κ1κ2. Thus in R × R, solutions with isotropy subgroup Z2(κ1) form a two- dimensional manifold, solutions with isotropy subgroup Z2(κ2) form a one-dimensional manifold and solutions with trivial isotropy subgroup form a two-dimensional manifold. 0 Reversible-equivariant case σ(κ1) = σ(κ2) = −1. The set B contains the bifurcating branches for isotropy subgroups Z2(κ1), Z2(κ2) and 1 described above. For isotropy subgroups Z2(κ2) and 1, the dimension of the bifurcating branches is given by dΣ = s(Σ) + 2 in accor- dance with Proposition 2.2, but not for isotropy subgroup Z2(κ1). Now, note that f2 ≡ 0 on 22

FixV (Z2(κ1)), therefore, the dimension of the bifurcating branch with isotropy subgroup Z2(κ1) is 3 which corresponds to dΣ = s(Σ) + 2. With a little extra work, see Example 3.17, we show that at the organizing centre, the dim V 0 subsets Kν are in fact manifolds. Therefore, following Remark 3.7, B contains all bifurcating branches given, up to G-diffeomorphism, by the bifurcation diagram of the D4-equivariant steady- state bifurcation problem with parameter symmetry studied by [15]. 0 Reversible-equivariant case σ(κ1) = −σ(κ2) = −1. Again, B contains the bifurcating branches found above in the bifurcation problem with parameter symmetry. However, the dimension of the branches may differ as we explain now. Note that f2 ≡ 0 on Fix(Z2(κi)) for i = 1, 2 since σ(κ1) and σ(κ2) act on W2,σ = R by −1. Hence, the bifurcating branches of h1 = 0 for isotropy subgroups Z2(κ1) and Z2(κ2) are also branches for the full bifurcation problem with an additionnal dimension for each since the second parameter µ extends the zero sets trivially. The branch with trivial isotropy subgroup remains unchanged as a two-dimensional branch and the dimensions of the three branches are given by dΣ = s(Σ) + 2. Alternate case: σ(κ1) = −σ(κ2) = −1 Suppose that f only depends on one parameter instead of two. Using Theorem 3.12 we would only find the branch with isotropy subgroup Z2(κ1). Again, with some extra work, see the second part of Example 7.7, we show that an organizing centre can be obtained as x = φ(z, λ) where φ is a smooth map. Substituting we obtain the equation f˜1(z, φ(z, λ), λ) = 0 which is a one parameter D4-equivariant bifurcation problem (with no parameter symmetry). Typically, we expect one dimensional branches of equilibria with isotropy subgroup Z2(κ1) and Z2(κ2). Note that the branch with isotropy subgroup Z2(κ1) has an extra dimension since s(Z2(κ1)) = 1. The alternate case of the last example shows that the minimal number of parameters to obtain all the bifurcating branches may be lower than the number necessary for our approach based on organizing centres and Theorem 3.12 to work. Moreover, the equivariant bifurcation problem may not even involve parameter symmetry. However, obtaining a sharp bound on the number of parameters and determining if there is parameter symmetry or not might depend on performing explicit calculations on the f2 = 0 equation. Rather than aiming to be efficient with parameters and bifurcation equations, in this paper we have given preference to the presentation of a method that is generally applicable and relies only on calculations of G-equivariant bifurcation problems with parameter symmetry.

3.5 Nonseparable bifurcation

Nonseparability of f at a branching point is induced by the fact that ker dV f contains more copies of an irreducible representation of G than the forced kernel. This gives f the opportunity to have its forced kernel changing discontinuously with external parameters. It is not our aim to discuss these bifurcations in detail here. We provide just one example as an illustration.

3 3 Example 3.18 (nonseparable bifurcation for Z2 in R ) Let G = Z2(R) and V = R be as in Example 2.3. Suppose that f is a one-parameter family of Z2(R)-reversible equivariant vector ﬁelds with f(0, 0, 0, 0) = 0, so that ker (df)(0,0,0,0) = {(x, y, z) | z = 0} and ker (df)(x,y,z,λ) has minimal dimension for (x, y, z, λ) in a small neighbourhood of (0, 0, 0, 0). In contrast to the sep- 23

arable bifurcation cases discussed before, the forced kernel W2(λ) now may display discontinuity at λ = 0. By Lyapunov-Schmidt reduction to ker (df)(0, 0, 0, 0) = FixV (Z2(R)), we are led to look for equilibria with isotropy subgroup Z2(R) that are zeroes of

fR : FixV (Z2(R)) × R → FixVσ (Z2(R)), where FixVσ (Z2(R)) = {(x, y, z) | x = y = 0} is one-dimensional. In particular, 2 2 fR(x, y, λ) = p(x , y , λ) with p(0, 0, 0) = 0, px(0, 0, 0) = 0 and py(0, 0, 0) = 0. Generically, zeroes of this function are determined by the normal form p(x2, y2, λ) = ax2 + by2 + cλ with a, b 6= 0. At λ = 0, the zero set of fR is locally a point if a, b have the same sign (elliptic case) and the intersection of two one-dimensional curves if a, b have diﬀerent signs (hyperbolic case). The unfoldings of these singularities at λ = 0 are illustrated by the sketches of the equilibria in R3 of Figure 7 where the plane corresponds to FixV (Z2(R)).

4 Branching and bifurcation of relative equilibria and (relative) periodic solutions

In the earlier discussion, we have focussed on symmetric equilibria that are ﬁxed by G. In particular, the main hypothesis that we have relied on is that the reference equilibrium does not lie on a continuous group orbit of equilibria. In that case the relevant group action in the neighbourhood of the equilibrium is the action of the isotropy subgroup of the equilibrium. An equilibrium that lies on a continuous group orbit of equilibria is an example of a relative equilibrium. In the Section 4.1 we discuss how branching and bifurcation of such equilibria and more generally relative equilibria can be understood from our results on bifurcation and branching of equilibria. Our results furthermore apply (via reduction) to a range of other local bifurcation problems, including nonHopf bifurcation and branching of periodic and relative periodic solutions. We will brieﬂy discuss this in Section 4.2.

4.1 Relative equilibria We consider a smooth G-reversible equivariant vector field f on a manifold M, with G acting properly. A solution x(t) ∈ W , with x(0) = x, is called a relative equilibrium if for all t, x(t) ∈ Gx, where the group orbit Gx is defined as Gx := {y ∈ M | y = g.x, ∀ g ∈ G}. Hence, such a solution is an equilibrium for the induced vector field on the quotient space M/H (recall that H = ker ρ). Relative equilibria, and their bifurcations, have received much attention in the literature. Of particular relevance to the discussion here is Lamb and Wulff’s treatment of relative equilibria in reversible equivariant systems [27]. We recall some of their results. 24

Let x ∈ W be a point on a relative equilibrium with isotropy subgroup Σ ≤ G, ie Σx = x, implying that ρ(σ)x(t) = x(σ(σ)t) for all t. Then we may consider a Σ-invariant slice V through x, that is transversal to Gx. V locally has the structure of a linear vector space, and evidently dim G + dim V = dim W . The slice may be used to define a tubular neighbourhood U of Gx, by taking the direct product of V and G, and quotienting out the Σ-action on V : V × G U ' . Σ One may lift U Σ-equivariantly to V × G, so that the vector field in U can now be described in V ×G coordinates. Now, let (v, g) ∈ V ×G, then one obtains the following form for the differential equation on V × G: ½ d dt v = fV (v), d (13) dt g = gfG(v).

Here, fV is a Σ-reversible equivariant vector ﬁeld on V : for all σ ∈ Σ

fV (ρ(σ)v) = ρσ(σ)fV (v), ∀v ∈ V where ρ is the representation of Σ on V , and ρσ its dual. fG is also Σ-reversible equivariant: for all σ ∈ Σ fG(ρ(σ)v) = σ(σ)AdσfG(v), ∀(v, g) ∈ V × G, where Adσ is the adjoint action of Σ on LG, the Lie algebra of G. It is important to note that the equations (13) form a skew product: the vector field fV on the slice V does not depend on the group-coordinate g ∈ G, but meanwhile drives the motion on the group orbit, with fG depending on v ∈ V . A group orbit of equilibria corresponds to a relative equilibrium G(v, 0) with fV (v) = fG(v) = 0. In this case, one also says that the relative equilibrium has no drift. A general relative equilibrium G(v, 0) satisfies fV (v) = 0. If fG(v) 6= 0 one says that the relative equilibrium drifts along the G-group orbit. From the skew-product equations, it is evident that our results for equilibria in Σ-reversible equivariant vector fields on V , also apply to relative equilibria with isotropy subgroup Σ and slice V . In particular, results about the nondegenerate local zero set of Section 2 and bifurcation theorems such as Theorem 3.6 and Theorem 3.12 apply also to relative equilibria with isotropy subgroup Σ and slice V , reading “relative equilibria” wherever statements concern “equilibria”. The main thrust of this paper has been to describe the set of equilibria in parameter families of reversible equivariant vector fields. This leads us to digress somewhat on the drift of relative equilibria. In order to know whether a relative equilibrium consists of group orbits of equilibria we need to determine whether there is a flow (drift) along group orbits, or not. We now recall a result on drift of reversible equivariant relative equilibria by Lamb and Wulff [27]. But first, we introduce some notation. Let LGσ denote the Lie algebra LG of G with an action of Σ, defined by

σ γ.ξ = Adγ (ξ) = σ(γ)Adγ(ξ), ξ ∈ LGσ, γ ∈ Σ. 25

This action is called the σ-adjoint action of Σ. Correspondingly,

FixLGσ Σ := {ξ ∈ LGσ | γ.ξ = ξ, ∀γ ∈ Σ}.

Theorem 4.1 (Drift of reversible equivariant relative equilibria [27]) Let Gx be a relative equilibrium for a G-reversible equivariant vector ﬁeld in W , and let Σ denote the isotropy subgroup of x. Then, generically, the relative equilibrium drifts along a maximal reversible torus of dimension σ nΣ = dim FixLGσ Σ − dim FixLΣσ Σ. σ The number nΣ is directly linked to the index sW (Σ), and nΣ, the generic drift dimension for relative equilibria with isotropy subgroup Σ in G-equivariant vector ﬁelds:

σ Theorem 4.2 (Generic drift dimension) Let nΣ be as deﬁned in Theorem 4.1, and let

sG(Σ) := sW (Σ) − sV (Σ), then σ nΣ = nΣ − sG(Σ). Before we prove this theorem, we ﬁrst provide some example of local steady-state bifurcations in the slice of relative equilibria. Taking into account drift, bifurcation from group orbits of equilibria (and relative equilibria in general) can be tedious. We here do not aim to provide general results on such bifurcations, but rather illustrate the use of the bundle equations.

Example 4.3 (Z2-reversible pitchfork bifurcation of relative equilibria) Consider now a 3 SO(2) × Z2-reversible equivariant vector field in R × R as discussed in Example 2.5. Suppose we have (x0, 0, 0), with x0 6= 0, lying on a one-dimensional SO(2)-group orbit of equilibria. Then V is a two-dimensional slice transversal to the SO(2)-group orbit, which may be chosen as a subset of the plane x = x0 and R acts on V as R.(y, z) = (y, −z). dfV has no forced kernel and by Theorem 3.2 the local “slice”-steady-state bifurcation problem of the relative equilibrium in V reduces to that of a steady-state bifurcation of a Z2-equivariant vector field. In a one-parameter family, on V we may hence observe a generic Z2-symmetry breaking pitchfork bifurcation of equilibria. For the full flow this implies that there is a Z2-symmetry breaking pitchfork bifurcation of relative equilibria. If we consider the flow on the group orbit, the Z2-invariant relative equilibrium is a one-dimensional family of equilibria (d = n = 1), but the bifurcating relative equilibria Z2 Z2 without symmetry do generically not consist of equilibria but are flow-invariant SO(2)-group orbits with a nonzero drift (d1 − n1 = 0 − 1 = −1 < 0).

We now consider the bundle equations more closely, and prove Theorem 4.2. Let fˆ = (fV , fG) be the vector ﬁeld on the Σ-equivariant lift V × G. Then fˆ is a Σ-equivariant map.

fˆ : V × G → Vσ × LGσ

Let −1 FixGΣ := {g ∈ G | σgσ = g, ∀σ ∈ Σ}. 26

σ Proof of Theorem 4.2 Let nΣ denote the generic drift dimension. Consider a reversible relative equilibrium with slice isotropy subgroup Σ. Let sW (Σ) and sV (Σ) denote the indices of Σ in W and V , respectively. Let ` be such that sV (Σ)+` ≥ 0 and sW (Σ)+` ≥ 0. Then, generically, relative equilibria with slice-isotropy Σ come in (sV (Σ) + `)-dimensional families. The generic drift dimension is precisely the codimension of the set of nondrifting relative equilibria (which are group orbits of equilibria), in this set of relative equilibria. We know that generically Σ- invariant equilibria arise in (sW (Σ) + `)-dimensional families. From the slice equations we obtain σ a sV (Σ) + ` − nΣ dimensional family of equilibria in the slice, leading to a set of Σ-invariant σ equilibria in W with dimension sV (Σ) + ` − nΣ + nΣ. Hence,

σ σ sV (Σ) + ` − nΣ + nΣ = sW (Σ) + ` ⇔ nΣ = nΣ − sG(Σ). Alternatively, one may verify the result by noticing that Σ-invariant equilibria will generically exist if and only if sW (Σ) + ` − nΣ ≥ 0. From the slice equations perspective the corresponding condition to see a persistent Σ-invariant equilibrium (relative equilibrium that does not drift) σ is: sV (Σ) + ` − nΣ ≥ 0. Equating the left hand sides of both inequalities conﬁrms the same σ relationship between nΣ, nΣ and sG(Σ). The expression for the reversible drift dimension in Theorem 4.1 can be derived from Theo- rem 4.2, by identifying an expression for sG directly from the deﬁnition of sW and sV . Lemma 4.4

sG(Σ) = dim FixG(Σ) − dim FixΣ(Σ) − (dim FixLGσ (Σ) − dim FixLΣσ (Σ)).

Proof: Recall that W = (V × G)/Σ.

sG(Σ) := sW (Σ) − sV (Σ) = dim FixW (Σ) − dim FixWσ (Σ) − dim FixV (Σ) − dim FixVσ (Σ),

= dim FixG\Σ(Σ) − dim FixLGσ\LGσΣ(Σ),

= dim FixG(Σ) − dim FixΣ(Σ) − (dim FixLGσ (Σ) − dim FixLΣσ (Σ)).

From Lemma 4.4, Theorem 4.2, and the fact that

dim FixG(Σ) − dim FixΣ(Σ) = nΣ, one recovers the expression for the generic reversible drift dimension in Theorem 4.1.

4.2 Bifurcation and branching of (relative) periodic solutions We can use the results of this paper to understand so-called nonHopf branching and bifurcation from periodic solutions in reversible equivariant systems. NonHopf bifurcations are bifurcations arising when Floquet multipliers go through ﬁnite (resonant) roots of unity [23, 24]. The main observation [24] is that the bifurcation and branching equations of nonHopf type always formally reduce to reversible equivariant steady-state bifurcation equations. The fact that one may conclude the true existence of solutions from the formal (normal form) analysis is 27 guaranteed by a general result on ﬁnite determinacy for such bifurcation problems by Field [11]. Here we merely discuss an elementary application and will not digress on the reduction, but for more details see [24].

Example 4.5 (Subharmonic branching of periodic solutions) Consider a vector ﬁeld f in 4 W = R that is Z2-reversible, where the time-reversal symmetry acts as R.(x, y, u, v) = (x, y, −u, −v). As s(Z2) = 0, generically equilibria with isotropy subgroup Z2 are isolated. Now, suppose we have a Z2-symmetric periodic solution, ie a solution x(t) = x(t + T ) for all t (T > 0), satisfying for all t

x(t) = R.x(−t).

We can construct a Poincarésection V for this periodic solution, so that the dynamics near the periodic solution may be represented by a Poincarémap F : V → V . We study the map using a formal normal form, which is closely associated to a Z2-reversible normal form vector field (formal approximation of F modulo formal equivariance, by a vector field on V ). We denote this vector field fV . Now, dfV has a one-dimensional forced kernel, as sV (Z2) = 1, even though sW (Z2) = 0. Hence, away from a branching point, we observe a one-dimensional family of Z2-symmetric equilibria in the normal form vector field. In turn, this branch of equilibria corresponds to a one- dimensional family of fixed points for the Poincaréreturn map F . In V , this family corresponds to a one-dimensional family of periodic solutions with slowly varying period. Somewhere at the one-dimensional branch of periodic solutions we may encounter a periodic solution with Floquet multipliers {exp(2πip/q), exp(−2πip/q), 1}. These Floquet multipliers lead to a Dq = Zq o Z2-reversible equivariant normal from vector field, the equivariance originating as a formal symmetry induced by the linear part dF of the Poincaréreturn map F . The Dq- reversible equivariant normal form vector field displays steady-state branching as discussed in Example 3.16 for the case q = 2, and in fact for general q as the bifurcation set of a codimension one steady-state Dq-equivariant bifuration problem (as discussed for instance in [19]), with 2q Z2-invariant branches of equilibria bifurcating from the main Dq-invariant branch. For the return map, the resulting branches correspond to families of symmetric period q orbits coming from the one-dimensional family of fixed points with Z2-symmetry. In V one observes branches of q-tupled symmetric periodic solutions from a one-dimensional family of symmetric periodic solutions. The above approach provides a proof of the generic occurrence of reversible subharmonic branching, alternative to those of Vanderbauwhede [37, 39] and Ciocci [5]. Using the same approach one can also study subharmonic branching in reversible equivariant vector fields. The pattern of subharmonic branching along a one-parameter family of periodic solutions and subsequent subharmonic branching from the branches yields an intricate structure, that is sometimes referred to as the subharmonic branching tree [39, 33]. Roberts and Lamb [33] studied the period-doubling part of such a branching tree numerically in various examples of reversible maps in R3, and observed that such a period-doubling branching tree has self-similar properties, with scaling factors similar to those familiar in period-doubling cascades of one-parameter families of area-preserving maps in R2. Our results can also be applied to describe local nonHopf bifurcations of relative periodic 28 solutions as these also essentially reduce to reversible equivariant steady-state bifurcation problems [41, 24].

5 Linear reversible equivariant systems

In this section we discuss some results on linear reversible equivariant maps, based on the treatment in [26]. These results form a basis for the further detailed treatments of nondegenerate local zero sets and bifurcations at organizing centres in Sections 6 and 7. Recall the deﬁnitions of the representations ρ and its σ-dual ρσ of G, from (2). We say that ρ and ρσ are (σ-)dual representations. A representation of G on V has an isotypic decomposition

V = V1 ⊕ · · · ⊕ Vm. where Vj is a direct sum of G-isomorphic irreducible representations. It is well known (by Schur’s lemma) that G-equivariant maps respect this isotypic decomposition. G-reversible equivariant maps, however, map an isotypic component Vj to the isotypic component of its σ-dual, which we here will denote Vj,σ (It is easy to verify that if ρ is irreducible, then so is ρσ.) Hence, from the isotypic components Vj, one may construct σ-isotypic components Vbj = Vj + Vj,σ, which are left invariant by G-reversible equivariant maps. Hence, Vbj = Vj if Vj is a direct sum of self-dual irreducible representations and Vbj = Vj ⊕ Vj,σ otherwise. Accordingly, we obtain a σ-isotypic decomposition V = Vc1 ⊕ · · · ⊕ Vcmˆ , which is respected by G-reversible equivariant maps. Lamb and Roberts [26] classify the diﬀerent types of reversible equivariant linear maps on σ- isotypic components. In particular, they show that there are three types of nonself-dual irreducible representation and seven types of self-dual irreducible representation. They also found that an irreducible representation V is self-dual if and only if there exists a linear G-reversible equivariant map T : V → V , satisfying T 2 = ±1. The question is whether the existence of nonself-dual irreducible representations prevent the existence of an invertible reversible equivariant map. It turns out that the answer depends on whether the dimensions of the isotypic components Vj and Vj,σ inside the σ-isotypic component Vbj are the same or not.

Lemma 5.1 Let Vbj = Vj ⊕ Vj,σ be a σ-isotypic component for a nonself-dual representation of G. Then there exists an invertible G-reversible equivariant map T : Vbj → Vbj if and only if dim Vj = dim Vj,σ.

Proof: If dim Vj = dim Vj,σ, the map T (v1, v2) = (v2, v1) is invertible and G-reversible equivariant. The only if part follows from noticing that it is suﬃcient to show that there does not exist a linear invertible map T . Note that T : Vj → Vj,σ and T : Vj,σ → Vj. Suppose a linear invertible G-reversible map T exists. Then, T is a bijection between Vj and Vj,σ, and hence they must have the same dimension. Hence, whenever dim Vj 6= dim Vj,σ, an invertible linear G-reversible equivariant map T does not exist. 29

The nonexistence of invertible linear reversible equivariant maps is intimately related to the existence of a forced kernel for such maps.

Lemma 5.2 Let Vbj = Vj ⊕ Vj,σ be a σ-isotypic component for a nonself-dual representation of G. Let T : Vbj → Vbj be a linear G-reversible equivariant map. Then, dim ker T ≥ | dim Vj − dim Vj,σ|. Moreover, there exists a T attaining the lower bound for the kernel.

Proof: Suppose dim Vj ≥ dim Vj,σ. Recall that T : Vj → Vj,σ and T : Vj,σ → Vj. Hence, the restriction T |Vj has a kernel of dimension great or equal than dim Vj − dim Vj,σ. The restriction

T |Vj,σ does not give rise to a forced kernel. To construct a T with minimal kernel, decompose Vj = Uj ⊕ Kj, where dim Uj = dim Vj,σ. Take coordinates (u, k, v) ∈ Uj ⊕ Kj ⊕ Vj,σ. Then T (u, k, v) = (v, k, u) is invertible, G-reversible equivariant and has dim ker T = dim Vj − dim Vj,σ. The proof is analogous when dim Vj ≤ dim Vj,σ. In this paper, our strategy towards steady-state bifurcation is to optimally use the existence of a linear G-reversible equivariant map T on a subspace of V with maximal dimension. The minimal kernel of a reversible linear map is called the forced kernel. By the arguments provided above, the forced kernel has a ﬁxed decomposition in terms of nonself-dual irreducible representations of G. In particular,

Corollary 5.3 Let Vbj = Vj ⊕ Vj,σ be a σ-isotypic component for a nonself-dual representation of G. Then there exists an invertible linear G-reversible equivariant map T : Vbj → Vbj if and only if G-reversible equivariant maps on Vbj do not have a forced kernel. When studying separable bifurcations, we rely on the following elementary property:

Proposition 5.4 Suppose that L(λ) is a parameter family of linear reversible equivariant maps. Suppose that ker L(0) ⊇ W2, where W2 is a forced kernel, and that for all irreducible representations Vj ∈ W2, ker L(0) \ W2 contains no irreducible representation G-isomorphic to Vj. Then there exists a smooth linear change of coordinates such that W2 ⊆ L(λ) for suﬃciently small values of λ. Proof: Recall that the irreducible representations that enter the forced kernel are determined up to isomorphism by the representation of the group. Since ker L(0) \ W2 has no irreducible representations isomorphic to irreducible representations in the forced kernel W2, then for suﬃciently small λ, ker L(λ) contains a forced kernel W2(λ), G-isomorphic to W2, so that ker L(λ) \ W2(λ) contains no irreducible representations of G that are isomorphic to irreducible representations in W2(λ). We follow in detail the argument in case W2 contains copies of only one irreducible representation. Let Vˆj = Vj ⊕ Vj,σ be the corresponding σ-isotypic component of V . On Vˆ , L(λ) has the form [26] · ¸ 0 A (λ) L(λ) = n×m Bm×n(λ) 0 where An×m(λ) and Bm×n(λ) are n ×m and m× n matrices whose entries depend smoothly on λ. Without loss of generality, we may assume that m > n. Then, the assumption in the proposition 30

implies that W2(λ) = ker Bm×n(λ). Hence, Bm×n(0) is surjective, and by the implicit function theorem W2(λ) depends smoothly on λ, as the basis vectors of W2(λ) do. Consequently, there exists a G-equivariant linear map P (λ) depending smoothly on λ mapping W2,λ to W2. If W2 contains irreducible representations of more than one σ-isotypic component of V , the same argument holds for each σ-isotypic component, and can be applied sequentially, as L(λ) respects the σ-isotypic decomposition of V .

6 Equivariant Transversality Theory

In this section we introduce the notation and concepts necessary to discuss G-transversality results. We work in the following general context. Let V and W be G-representations and f : V → W a smooth G-equivariant map.

6.1 Local zero sets and G-transversality If G acts trivially on V and W , the problem of finding generic solutions to an equation f = 0 can be expressed in terms of transversality conditions. Thus, if f : V → W is smooth and f(0) = 0, then the zero set of f is regular if f is transverse to 0 in W ; that is, df(0) is of maximal rank. This transversality condition, together with the implicit function theorem, implies that f −1(0) is a smooth submanifold of V near 0 ∈ V . See [20] for details. Suppose that G acts nontrivially on V and W and f(0) = 0. Following our previous example, we say that f is transverse to 0 in W at 0 in V if (df)0 is of maximal rank. However, unless V (resp W ) is isomorphic to a subrepresentation of W (resp V ), there will be no linear equivariant maps of maximal rank between V and W . In particular, we cannot generally use simple transversality conditions to determine the generic solutions of equivariant equations. To overcome this problem, Bierstone [1] and Field [7] developed a theory of G-transversality or equivariant general position. Conditions for G-transversality are framed in terms of transversality conditions to an algebraic variety. We follow the notation and approach of [1] to the definition of G-transversality. The definition given in [7] is equivalent as is shown in [8]. ∞ ∞ Denote by CG (V,W ) the space of G-equivariant C maps from V to W . It follows from Schwarz’s theorem [34, 19] on smooth invariants (see also Poénaru[31]) that there exists a ∞ minimal set of homogeneous polynomial generators {F1,...,Fk} for CG (V,W ). That is, any ∞ f ∈ CG (V,W ) can be written as Xk f(x) = hi(x)Fi(x) i=1 where hi : V → R are G-invariant smooth functions. k Pk −1 Consider the map F : V ×R → W defined by F (x, t) = i=1 tiFi(x) and the set E = F (0). k Define graphf : V → V × R by graphf (x) = (x, h1(x), . . . , hk(x)). The following result is straightforward.

∞ Lemma 6.1 Let f ∈ CG (V,W ). Then,

(1) f = F ◦ graphf 31

−1 −1 (2) f (0) = graphf (E) Therefore, the generic zero set of any function f is obtained by looking at the transversal intersection of graphf with the algebraic variety E. The set E is called the universal zero set and F is the universal map. Let Σ be an isotropy subgroup. Recall that the conjugacy class of Σ is called the isotropy type and is denoted by (Σ). If x ∈ V , we denote by ι(x) the isotropy type of x. Let V(Σ) = {x ∈ V |ι(x) = (Σ)}. Consider the sets

EΣ = {(x, t) ∈ E|x ∈ V(Σ)} over all isotropy types (Σ). The set E is the disjoint union of the sets EΣ and E admits a Whitney regular stratification (see below for a definition of Whitney regular stratification).

Definition 6.2 (G-transversality) The G-equivariant map f is G-transversal to 0 ∈ W at k 0 ∈ V if graphf : V → V × R is transverse to the (canonical) semi-algebraic Whitney regular stratification of E. Generic solutions of equivariant equations are represented locally as the zero sets of polynomial maps and typically have singularities. The characteristic transversality openness, density and isotopy theorems continue to hold for G-transversality though the transversality isotopy theorem only holds up to homeomorphism (we refer to [1] and [7] for details and precise statements of results). We now review some basic facts about stratifications of sets. See Field [12, 10] for clear and concise introductions to stratifications and Gibson et al [16] for a more detailed and general description.

Stratifications A stratification S of a subset X of Rn is a locally finite partition of X into smooth and connected submanifolds of Rn called strata. In particular, if X is semi-algebraic, then S is a semi-algebraic stratification if each stratum is semi-algebraic. A stratification is said to be Whitney regular if it satisfies a Whitney regularity condition (which we won’t mention here, see [10] or [16]). Of particular interest to us is the frontier condition of Whitney regular stratifications: if A, B are strata satisfying A ∩ B 6= ∅ then B ⊂ A or also B ⊂ ∂A. An important consequence of Whitney regularity in our context is the following. If A ∩ B 6= ∅, then any map transverse to B is also transverse to A near B. Denote the union of i-dimensional strata of S by Si, i ≥ 0. Then the following property holds.

Property 6.3 If S is a Whitney regular stratiﬁcation, then Si does not meet Sj unless i ≤ j.

Let Xi be the union of all strata of dimension less or equal than i. A Whitney regular stratiﬁcation S of a set X is said to be canonical if for each i, Si is the largest smooth submanifold of Xi for which Sj is Whitney regular over Si, for all j > i. The following result ties the Whitney regular stratiﬁcation of E to the submanifolds EΣ for all isotropy types (Σ). 32

Theorem 6.4 (Field [12]) The canonical semi-algebraic Whitney regular stratiﬁcation S of E induces a semi-algebraic Whitney stratiﬁcation of each EΣ. That is, each EΣ is a union of strata of S.

6.2 Stratumwise transversality and dimension estimates We remark that G-transversality implies stratumwise transversality – where strata are determined by isotropy type, see [1, 7]. In the context of equivariant equations this means that if f : V → W is G-transversal to 0 and Σ is an isotropy subgroup for the action of G on V, then f|V(Σ) : V(Σ) → FixW (Σ) is transverse to zero. We use this result below to get estimates on the dimension of zero sets with given isotropy types. Note also that stratumwise transversality does not imply G-transversality (for examples, see [1, 7]). We extend the deﬁnition of index to general representations V and W by letting s(Σ; V,W ) = dim FixV (Σ) − dim FixW (Σ). In this paper, we reserve the notation s(Σ) to the context of G- reversible equivariant maps. The next result gives the dimension of zero sets by isotropy types, it is an immediate consequence of stratumwise transversality.

Lemma 6.5 Suppose that graphf has nontrivial transversal intersection with EΣ. Then

−1 dim graphf (EΣ) = s(Σ; V,W ) + dim G − dim NG(Σ). In particular, −1 dim(graphf (EΣ) ∩ FixV (Σ)) = s(Σ; V,W ). Lemma 6.5 provides an alternate proof of Proposition 2.2. The following result is an extension of Proposition 3.6 of [12] needed for the proof of Lemma 6.5.

k Proposition 6.6 Let N(Σ) be the normalizer of Σ in G. Then, EΣ is a submanifold of V × R of dimension s(Σ; V,W ) + dim G − dim N(Σ) + k,

k Proof: Consider V × R as a k-dimensional bundle over E. Let (x, t) ∈ EΣ where x ∈ V(Σ) and d(x) = dim span{F1(x),...,Fl(x)}. Without loss of generality we can assume that {F1(x),...,Fd(x)(x)} is a set of linearly independent vectors, then

Xk Xd(x) 0 tiFi(x) = tiFi(x) = 0. i=1 i=1

0 The solution space of ti = 0 for i = 1, . . . , d(x) is k − d(x) = k − dim FixW (Σ) dimensional. Finally, sweeping over the whole of V(Σ) we obtain

dim E(Σ) = dim V(Σ) + k − dim FixW (Σ). (14)

Claim: dim V(Σ) = dim FixV (Σ) + dim G − dim N(Σ) 33

Let x ∈ FixV (Σ). If g ∈ N(Σ) then gx ∈ FixV (Σ). Consider the set Bx = {gx|g ∈ G \ N(Σ)}. Now dim(G/N(Σ)) = dim G − dim N(Σ). Therefore, dim Bx = dim G − dim N(Σ). Summing up over FixV (Σ) the claim is proved. k Proof of Lemma 6.5 The codimension of EΣ in V × R is dim V − s(Σ; V,W ) − dim G/Σ + nΣ −1 −1 and is equal to the codimension of graphf (EΣ) in V . Thus, dim graphf (EΣ) = s(Σ; V,W ) + dim G/Σ − nΣ. The intersection with FixV (Σ) leaves a s(Σ; V,W ) dimensional manifold.

7 Organizing centres

In this section we present the proofs of the separable bifurcation results reported in Section 3.3 and 3.4. For the study of separable bifurcations we assume that (0, 0) ∈ V × R` is an equilibrium with isotropy subgroup G. We use the splitting V = W1 ⊕ W2 where W2 is the forced kernel. At a separable bifurcation point, Proposition 5.4 guarantees that throughout the bifurcation, the forced kernel can be taken to be equal to W2. −1 It is our aim to ﬁnd organizing centres, where some part of the persistent solution set f2 (0) k can be locally parametrized by an (` − k)-parameter family of G-invariant functions φν : V → R . We will refer to this part of the solution set as K. The strategy is to substitute for µ ∈ Rk into the f˜1 equation, which subsequently can be interpreted as an equivariant bifurcation problem on W1 with parameter (i.e. W2) symmetry. −1 Our starting point is to aim for constructing K as the closure of the subset S1 of f2 (0) whose −1 points have trivial isotropy: K = S1. We here note that although the subsets SΣ of f2 (0) with constant isotropy Σ are manifolds, the closure of such a set in general need not be. We will deal with this issue later on. We also want K to be such that it contains points with all possible isotropies. We thus ask that K hits SΣ transversally for all isotropy subgroups Σ of G. A straightforward dimensional calculation yields that the dimension of such an intersection typically is given by

dim(SΣ ∩ K) = ` + s(Σ) + dim FixW1 Σ − dim W2. so that in order to have persistent nontrivial intersection, one would need

` > dim W2 − min{s(Σ) + dim FixW Σ}. Σ 1

In turn, it follows from the fact that s(Σ) + dim FixW1 Σ ≥ − dim W2, that a safe choice for ` would be ` = 2 dim W2. Secondly, in order to be able to write K as a parameter family of functions from V to Rk, we would like to ensure that the dimension of the stratum of K with isotropy Σ is indeed greater or equal than the dimension of the corresponding stratum in V , ie

` + s(Σ) + dim FixW1 Σ − dim W2 ≥ dim FixV Σ, implying that

` ≥ dim W2 + dim FixW2,σ Σ, 34

and again we find that the choice ` = 2 dim W2 is sufficient to satisfy this criterion. Incidentally, the latter criterion precisely guarantees that typically a neighbourhood of K near ` 0 projects to a neighbourhood of 0 in V , using the natural projection ΠV : V × R → V . We now construct φν such that it lifts a neighbourhood of 0 in V to a dim V -dimensional subset of K. As SΣ is a manifold, we can take φν smooth on subsets of V with fixed isotropy. However, when crossing isotropy strata we cannot always guarantee smoothness, but only continuity. In order to `−k fill K with the parameter family of surfaces φν(V ), we thus need ν ∈ R with dim V + ` − k = dim K = dim W1 + `, so that

k = dim V − dim W1 = dim W2, and correspondingly with ` = 2 dim W2 we have ` − k = dim W2. Finally, there is one remaining issue. We want to make sure to have enough parameters so that if we add yet even more parameters, say n, then the solution set will typically only extend in a trivial fashion (by some product with Rn. In the case of continuous groups, however, due to drift that may arise along group orbits, we must make sure to have a minimal number of parameters to observe possible groups orbits of equilibria with certain isotropies. To avoid drift of equilibria, we need to satisfy ` ≥ max nΣ − s(Σ), Σ see Proposition 2.2. Using the estimate s(Σ) ≥ − dim W2 and nΣ ≤ dim G, we obtain the suﬃcient condition ` ≥ dim G + dim W2, guaranteeing typical persistence of the structure of the local solution set with respect to adding parameters. Combining the above suﬃcient criteria into one, we will assume (conservatively) that

` = dim W2 + max{dim W2, dim G},

` k `−k and correspondingly write the parameter λ ∈ R as λ = (µ, ν) ∈ R × R with k = dim W2 and ` − k = max{dim W2, dim G}. Of structural importance to our results is the fact that the organizing centres we describe above, do in fact arise, and in a persistent fashion. This is discussed in detail in the following Subsection 7.1. Thereafter we prove Theorem 3.6 and Theorem 3.12.

7.1 Existence of organizing centres ∞ Let {F1,...,Fk} be a minimal set of homogeneous generators for CG (V,W2,σ). Then, Xk f2(x, λ) = pi(x, λ)Fi(x) i=1

` k where pi : V × R → R are G-invariant functions. Let F2 : V × R → W2,σ be the universal map and E be the universal zero set of the map F2. From Theorem 6.4, for all isotropy types Σ, we know that EΣ is made up of strata from the canonical Whitney regular stratiﬁcation of E. Consider the stratum S0 = {(x, t) ∈ E | x = t1 = ... = tk = 0} of EG, then S0 ⊂ ∂EΣ for all isotropy subgroups Σ of G. Let SG ⊂ EG be the largest stratum of EG such that SG ⊂ ∂EΣ for all isotropy subgroups Σ of G. 35

Theorem 7.1 Suppose that ` ≥ dim W2 + max{dim G, dim W2}. Then, (0, 0) is an organizing centre persistent to small perturbations if and only if graphf2 is transverse to SG at (0, 0) (or a smaller stratum with the same properties).

Proof: ⇐) Suppose that graphf2 is transverse to SG at (0, 0). Then, graphf2 meets EΣ transversally for each isotropy type Σ. Thus, graph−1(E ) is a manifold with f2 Σ

dim graph−1(E ) = dim Fix (Σ) + s(Σ) + ` − n + dim G/Σ. f2 Σ W1 Σ

Since ` ≥ dim W2 + max{dim G, dim W2}, then

dim graph−1(E ) ≥ dim Fix (Σ) + dim W − n + dim G/Σ. f2 Σ V 2 Σ

For a ﬁxed isotropy subgroup, the dimension of the manifold graph−1(E )∩Fix (Σ) is dim Fix (Σ)+ f2 Σ V V dim W . We can take K as in the construction at the beginning of the Section: K = graph−1(E ). 2 f2 1 Since the manifolds graph−1(E ) ∩ Fix (Σ) for all isotropy subgroups Σ have the correct di- f2 Σ V mensions, this makes K an organizing centre. The persistence is guaranteed by the transversal intersection of graphf2 (0, 0) with SG.

⇒) Suppose that graphf2 (0, 0) does not meet SG transversally (or a smaller stratum with the same properties as SG). Then, a ﬁrst case is that graphf2 is transversal to a stratum S at (0, 0) which does not belong to the boundary of EΣ for some isotropy subgroup Σ, therefore graph−1(E) does not contain points with every isotropy subgroup Σ of G near (0, 0). Then, (0, 0) f2 cannot be an organizing centre. The other case is the following. Suppose graphf2 (0, 0) meets SG ∞ ` but not transversally. Then, there exists ²-small perturbations of f2 in C (V × R ,W2,σ) such that graphf2,² (0, 0) does not meet SG at all. Thus, the zero set K described in the deﬁnition of organizing centre does not persist and (0, 0) is not a persistent organizing centre.

Remark 7.2 If graphf2 hits SG and ` is suﬃciently large, then generically, the intersection is transversal. See the ﬁrst case in Example 7.7 for a situation where the addition of a parameter is necessary for transversality. In the trivial forced kernel symmetry situation, we have the following results

Corollary 7.3 Suppose that W2 ⊂ FixV (G) is nonzero and graphf2 (0, 0) hits SG transversally. −1 Then K can be chosen as a dim W1 + ` subset of f2 (0) parametrised by a map φλ : W1 → W2 with φλ continuous and smooth on subsets of W1 with constant isotropy subgroup Σ.

Proof: Since graphf2 (0, 0) ∈ SG, then for isotropy subgroups Σ with no reversing symmetries, the manifold graph−1(Σ) ∩ Fix (Σ) has dimension dim Fix (Σ) + `. Because W ⊂ Fix (Σ), f2 V W1 2 V we treat the w2-variables as parameters. Let K be the union of the manifolds of zeroes over all isotropy subgroups Σ with no reversing symmetries (that is, with dimension dim FixW1 (Σ) + `). Then, we can write K as a family of maps w2 = φλ(w1) where φλ is continuous and smooth on subsets of W1 with constant isotropy subgroup Σ. Therefore, (0, 0) is an organizing centre for the bifurcation problem. 36

˜ Thus, in the trivial forced kernel symmetry case, using the maps φλ, we can replace w2 in f1 ˜ to obtain a bifurcation equation f1(w1, φλ(w1), λ) = 0. We now present some examples of organizing centres for simple cases of G-reversible equivariant maps beginning with an example which shows that all we can hope for in general is that −1 K ⊂ f2 (0) is a G-invariant surface of dimension dim W1 which is not necessarily a manifold.

4 2 Example 7.4 Let V = R × R with coordinates (x, y) and G = Z2(R) where R.(x, y) = (x, −y).

We split V = W1 ⊕ W2 where dim W1 = 4, dim W2 = 2 and R|W2 = +I. Then, f2 : V → W2,σ is

2 2 f2(x, y) = [pij(x, y1, y2)]y

2 2 t where [pij(x, y1, y2)] is a 2 × 2 matrix of Z2-invariant functions and y = (y1, y2) . The universal map is then F2(x, y, t) = [tij]y where [tij] is a 2 × 2 real matrix. So, E = EG ∪ E1 where

EG = {(x, y, t) | y = 0}, and E1 = {(x, y, t) | y 6= 0}.

The zero sets EΣ are stratiﬁed by rank of the matrix [tij]:

G G S0 = {(x, y, t) | rank [tij] = 0, y = 0},S1 = {(x, y, t) | rank [tij] = 1, y = 0} G S2 = {(x, y, t) | rank [tij] = 2, y = 0} and 1 1 S0 = {(x, y, t) | rank [tij] = 0, y 6= 0},S1 = {(x, y, t) | rank [tij] = 1, y 6= 0}. G G For (0, 0) to be an organizing centre, graphf2 (0, 0) must hit S0 transversally since S0 ⊂ ∂E1. The bifurcation problem has trivial forced kernel symmetry, we look for a dim W1 = 4 dimensional G set K that can be given by a map w2 = φ(w1). We define the set S := S0 ∪ E1. This set is not a manifold because it is not differentiable at the origin t11 = t12 = t21 = t22 = y = 0. Since codim E = 2, then the set K = graph−1(S) has dimension 4, but is not a manifold. The only 1 f2 G non-differentiable point being the inverse image of the stratum S0 . The examples that follow, some connected to previously worked out examples, have a sufficiently regular structure to allow K to be a G-invariant manifold of dimension dim W1. Sufficient conditions for the existence of organizing centres are easily computed in these examples.

Example 7.5 (Example 3.9) Recall that R.(x, y, z) = (x, y, −z) and σ(R) = −1. We choose 2 W2 = {(x, y, z)|y = z = 0}. Then f2(x, y, z) = p(x, y, z )z and F2(x, y, z, t) = tz. So,

E = {(x, y, z, t) | z = 0}, E = {(x, y, z, t) | t = 0}. Z2 1 Thus, S = {(x, y, z, t)|t = z = 0}. Now, graph (0, 0, 0) = (0, 0, 0, p(0, 0, 0)) ∈ S if and only Z2 f2 Z2 if p(0, 0, 0) = 0. To apply Theorem 7.1 we must check that graph t S at (0, 0, 0); that is, the f2 Z2 following equation must be satisﬁed:

d(graph ) (R3) + T S = R3 × R. f2 (0,0,0) graph (0,0,0) Z2 f2 37

In coordinates, we obtain     1 0 0   u x 1  0 1 0   u     y  +  2  = R3 × R  0 0 1   0  z px(0, 0, 0) py(0, 0, 0) 0 0 if the non-degeneracy condition px(0, 0, 0)x + py(0, 0, 0)y 6= 0 holds, in particular, we have generically that px(0, 0, 0) 6= 0. Note that p(0, 0, 0) = 0 and px(0, 0, 0) 6= 0 are suﬃcient conditions for using the implicit function theorem on p(x, y, z2) = 0 and the set K is given by x = φ(y, z2) where φ is smooth.

Example 7.6 (Example 3.15) In this case, R.(x, y, z) = (x, −y, −z), σ(R) = −1 and W2 = 2 2 {(x, y, z)|x = y = 0}. We have f2(x, y, z) = r(x, y , z ) and F2(x, y, z, t) = t. So, E = {(x, y, z, t) | y = z = t = 0}, E = {(x, y, z, t) | t = 0}. Z2 1 Thus, S = E and graph (0, 0, 0) ∈ S if and only if r(0, 0, 0) = 0. Now, the transversality Z2 Z2 f2 Z2 equation for graphf2 ,     1 0 0   u x 1  0 1 0   0     y  +   = R3 × R,  0 0 1   0  z rx(0, 0, 0) 0 0 0 is satisﬁed if the non-degeneracy condition rx(0, 0, 0) 6= 0 holds. Thus, Theorem 7.1 applies. Again, r(0, 0, 0) = 0 and rx(0, 0, 0) 6= 0 are suﬃcient conditions to apply the implicit function theorem on r = 0 to obtain x = φ(y2, z2).

Example 7.7 (Example 3.17) The group D4 acts on C×R as κ1.(z, x) = (z, x) and κ2.(z, x) = (iz, −x). Consider the case σ(κ1) = σ(κ2) = −1. We have

2 2 4 4 2 2 4 4 2 2 2 2 f2(z, x, µ) = (p(x , |z| , z + z , µ) + q(x , |z| , z + z , µ)(z + z )x)(z − z ).

2 2 2 2 2 2 So, F2(z, x, t1, t2) = t1(z − z ) + t2x(z − z )(z + z ) and E = {(z, x, t , t ) | z = x = 0}, D4 1 2 E = {(z, x, t , t ) | z = z}, Z2(κ1) 1 2 E = {(z, x, t , t ) | z = iz, x = t = 0}, Z2(κ2) 1 2 1 2 2 E1 = {(z, x, t1, t2) | t1 + t2x(z + z ) = 0}

Thus, SG = {(z, x, t1, t2) | z = x = t1 = 0} and graphf2 (0, 0, 0) = (0, 0, p(0, 0, 0, 0), q(0, 0, 0, 0)) ∈ SG if and only if p(0, 0, 0, 0) = 0. The transversality equation     1 0 0   0 z  0 1 0   0     x  +   = C × R × R2  0 0 p (0, 0, 0, 0)   0  µ µ 0 0 qµ(0, 0, 0, 0) u4 38

is satisﬁed if the non-degeneracy condition pµ(0, 0, 0, 0) 6= 0 holds. Note that if f2 is not embedded in a one-parameter family, then the transversal intersection cannot occur. So Theorem 7.1 applies. The conditions p(0, 0, 0, 0) = 0 and pµ(0, 0, 0, 0) 6= 0 are suﬃcient to use the implicit function theorem. In the case, σ(κ1) = −σ(κ2) = −1, we have

2 2 4 4 2 2 4 4 2 2 2 2 f2(z, x) = (p(x , |z| , z + z )x + q(x , |z| , z + z )(z + z ))(z − z ),

2 2 2 2 2 2 F2(z, x, t1, t2) = t1x(z − z ) + t2(z + z )(z − z ) and

E = {(z, x, t , t ) | z = x = 0}, D4 1 2 E = {(z, x, t , t ) | z = z}, Z2(κ1) 1 2 E = {(z, x, t , t ) | z = iz, x = 0}, Z2(κ2) 1 2 2 2 E1 = {(z, x, t1, t2) | t1x + t2(z + z )}.

Thus, SG = {(z, x, t1, t2) | z = x = 0} = EG and so graphf2 (0, 0) ∈ SG automatically. Moreover, the transversality equation is also satisﬁed     1 0 µ ¶ 0  0 1  z  0    +   = C × R × R2  0 p(0, 0, 0)  x  u3  0 q(0, 0, 0) u4 and Theorem 7.1 applies. If p(0, 0, 0) 6= 0 then the parametrization of K is given by the implicit function theorem.

7.2 Proof of Main Theorem Proof of Theorem 3.12 For (0, 0) an organizing centre of the bifurcation problem f = 0, we −1 ˜ consider the solution set K in f2 (0) parametrized by µ = φν(w1, w2). Substituting into the f1 equation we obtain g1(w1, w2, ν) = f˜1(w1, w2, φν(w1, w2), ν) = 0, with g1(ρ1(g)w1, ρ2(g)w2, ν) = ρ1(g)g1(w1, w2, ν).

Since φν is not necessarily smooth in V , we cannot guarantee smoothness of g1. We now show that we can perturb K so that it becomes smooth and construct a family of smooth G-equivariant ² mappings g1. Suppose that K is not smooth and let K∗ ⊂ K be the subset of points where K is not smooth, ∗ but only continuous. K is a G-invariant set by G-invariance of φν. The nonsmoothness of the set appears on subsets where smooth submanifolds of K with diﬀerent isotropy subgroups meet. In particular, points in K∗ have nontrivial isotropy. Since the action of G on V × R` is trivial on R` we can consider the nonsmoothness at the ∗ ∗ level of V . Let Vν ⊂ V be the subset of points for which φν(v) is nonsmooth for all v ∈ Vν . Since ∗ ∗ the points in K have nontrivial isotropy, Vν has the same property. 39

∗ Let Uδ,ν be a family of G-invariant δ neighborhood of Vν in Vν, that is smooth in ν. Now, for ² ≥ 0 small, we deﬁne a G-invariant C∞ partition of unity function on V by   0 if v ∈ U²,ν θ (v) = (0, 1) if v ∈ U ²,ν  2²,ν 1 elsewhere and we deﬁne the following G-invariant perturbation of φν:

Φ²,ν(v) = (1 − θ²,ν(v))Ψν(v) + θ²,ν(v)φν(v).

k ∗ where Ψ : V → R is a smooth G-invariant mapping such that Ψν = φν on Vν . Then, for ² > 0, ∗ Φ²,ν is smooth on V since Vν is entirely contained in U²,ν on which θ²,ν ≡ 0. Moreover, Φ0,ν = φν. We are thus led to consider the following family of G-equivariant bifurcation problems

² ˜ g1(w1, w2, ν) = f1(w1, w2, Φ²,ν(w1, w2), ν)

² 0 ² where g1 is smooth for ² > 0 and g1 = g1. The solution sets of g1 = 0 for small ² > 0 are all ² 0 G-diﬀeomorphic. By continuity, as ² → 0 the solution set of g1 = 0 converges to a subset B of the solution set of g1 = 0. Therefore, a subset of g1 = 0 is G-homeomorphic to the zero set of a smooth G-equivariant mapping h1.

Remark 7.8 Although Theorem 3.12 cannot guarantee smooth equivalence between the zero sets of g1 and h1, in many situations one finds that they are in fact G-diffeomorphic. In particular, if φν is smooth, like in Examples 3.9, 3.15, 3.17. On the other hand, counterexamples exist, see Example 7.4. It is an interesting question whether necessary and/or sufficient conditions for the existence of smooth equivalence can be formulated and whether this nonsmoothness can give rise to additional branches. At present we have no such example. Proof of Theorem 3.6 By Theorem 3.12, there exists a set B0, G-homeomorphic to the persistent bifurcation set of a smooth G-equivariant bifurcation problem with no parameter symmetry (since G acts trivially on W2). Let B0 be the set of equilibria in a small neighborhood of (x, λ) = (0, 0) with isotropy subgroups 0 0 Σ containing no reversing symmetries and define B0 = B ∩ B0. Since s(Σ) = 0 if Σ does not have 0 reversing symmetries, then the dimension of the branches in B0 and B0 are the same. 0 0 Consider now the bifurcating branches in B \ B0. These bifurcating branches have isotropy subgroups Σ which have reversing symmetries. Since W2 ⊂ FixW2 (G), we have

s(Σ) = dim FixW2 (Σ) − dim FixW2,σ (Σ) = dim W2, for each isotropy subgroup Σ which has a reversing symmetry. Therefore each branch with 0 0 isotropy subgroup Σ obtained in B \ B0 locally has s(Σ) = dim W2 additional dimensions in the 0 0 0 dim W2 full G-reversible-equivariant bifurcation problem and we deﬁne Br = (B \ B0) × R . If B is the set of equilibria in a small neighborhood of (x, λ) = (0, 0) we now show that all equilibria with isotropy subgroup Σ 6= G in B come from the bifurcating branches B0. 40

Equilibria in B with isotropy subgroup Σ 6= G may come from B0 or from the nondegenerate local zero set. Now, since s(G) ≥ s(Σ) for all isotropy subgroups Σ, this implies dim EG ≥ dim EΣ for all isotropy type (Σ) (see Proposition 6.6). Hence EG is never in the boundary of EΣ for all Σ. Thus the nondegenerate local zero set at (x, λ) = (0, 0) has only points with isotropy subgroup G. If the dim W1 subset obtained in Corollary 7.3 is smooth (i.e. w2 = φλ(w1) with φλ smooth) then the proof of Theorem 3.12 shows that all bifurcating solutions are included in B0. That is, 0 0 0 B0 = B0 and B = B0 ∪ Br. If the dim W1 subset obtained in Corollary 7.3 is not smooth, then the proof of Theorem 3.12 0 0 0 cannot rule out additional branches outside of B so that B ⊃ B0 ∪ Br

Acknowledgements The authors would like to thank Mike Field for his invaluable advice, suggestions and comments, in particular regarding G-transversality theory. This research has been supported by the UK Engi- neering and Physical Sciences Research Council (EPSRC) (PLB,JSWL), the Nuﬃeld Foundation (JSWL), NSERC Postdoctoral Fellowship (PLB) and NSERC Discovery Grants from J. B´elair and C. Rousseau (PLB). PLB and JSWL gratefully acknowledge the hospitality and support of the Department of Mathematics, Imperial College London, the Mathematics Institute, University of Warwick, and the CRM, Montreal, during visits at which part of the research was carried out.

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