Numerical bifurcation of Hamiltonian relative periodic

orbits

Claudia Wulff and Frank Schilder

Department of Mathematics

University of Surrey

Guildford GU2 7XH, United Kingdom

email: c.wulff@surrey.ac.uk, f.schilder@surrey.ac.uk

July 23, 2008

Abstract

Relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems andoccur for example in celestial mechanices, molecular dynamics and rigid body motion. RPOsare solutions which are periodic orbits of the symmetry-reduced system. In this paperwe analyze certain symmetry-breaking bifurcations of Hamiltonian relative periodic orbitsand show how they can be detected and computed numerically. These are turning pointsof RPOs, relative period-doubling and relative period-halving bifurcations along branchesof RPOs. In a comoving frame the latter correspond to symmetry-breaking/symmetry-increasing pitchfork bifurcations or to period doubling/period-halving bifurcations. Weapply our methods to the family of rotating choreographies which bifurcate from the fa-mous Figure Eight solution of the three body problem as angular momentum is varied.We find that the Eights rotating around the e2-axis bifurcate to the family of rotatingEights that connect to the Lagrange relative equilibrium. Moreover, we find several rela-tive period-doubling bifurcations and a turning point of the planar rotating choreographywhich bifurcates from the Figure Eight solution when the third component of the angularmomentum vector is varied.

Contents

1 Introduction 2

2 Persistence and numerical continuation of transversal RPOs 3

2.1 Continuation of transversal Hamiltonian relative equilibria . . . . . . . . . . . . . 32.1.1 Persistence of transversal relative equilibria . . . . . . . . . . . . . . . . . 42.1.2 Numerical computation of transversal relative equilibria . . . . . . . . . . 72.1.3 Numerical path-following of relative equilibria . . . . . . . . . . . . . . . . 8

2.2 Persistence of transversal RPOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Numerical continuation of transversal RPOs . . . . . . . . . . . . . . . . . . . . . 142.4 Numerical path-following of RPOs . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Turning points of relative equilibria and RPOs . . . . . . . . . . . . . . . . . . . 20

2.5.1 Turning points of Hamiltonian relative equilibria . . . . . . . . . . . . . . 202.5.2 Turning points of RPOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.3 Detection and computation of turning points of relative equilibria and RPOs 21

1

3 Hamiltonian relative period doubling bifurcations 22

3.1 A theorem on relative period doublings of RPOs . . . . . . . . . . . . . . . . . . 223.2 Detection of relative period doublings . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Computation of relative period doubling points and branch switching . . . . . . . 253.4 Detection and computation of relative period halving bifurcations . . . . . . . . . 26

4 Application to rotating choreographies 27

4.1 N-body problems and their symmetries . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Rotating choreographies of the three body problem . . . . . . . . . . . . . . . . . 284.3 Flip up pitchfork bifurcation of the type II rotating eight . . . . . . . . . . . . . 294.4 Turning points and relative period doubling bifurcations of type III rotating chore-

ographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Conclusion 32

1 Introduction

Relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems. They areperiodic orbits of the symmetry reduced system. In the original phase space they represent aperiodic vibrational dynamics superimposed with a drift along the symmetry group, e.g. su-perimposed with a rotation or translation. In recent years a lot of progress has been madein the bifurcation theory of Hamiltonian RPOs, see [14, 16, 22, 20]. But a general theory ofgeneric bifurcations of RPOs so far only exists for dissipative systems [10, 21]. The additionalstructure of symmetric Hamiltonian systems changes the generic behaviour compared to generalsystems dramatically. As a result of this, a general bifurcation theory of Hamiltonian RPOsand the parallel development of numerical methods for the detection and computation of thosebifurcations are an open problem. Recent progress in the continuation of normal periodic orbitsof symmetric Hamiltonian systems has been made by Munoz-Almaraz et al [4, 15]. Chencineret al [3] continue rotating choreographies of the three-body problems which bifurcate from thefamous Figure Eight solution [2].

In [20] we developed a persistence theory for nondegenerate RPOs with generic drift mo-mentum pair (see Section 2 below for definitions of these terms). In [25] we have extended thenumerical methods for the continuation of periodic orbits of general symmetric systems from[23] to Hamiltonian systems based on the theoretical persistence results from [20].

In this paper we prove a persistence result for transversal Hamiltonian RPOs of compactgroup actions with generic drift-momentum pair. These include turning points of RPOs in energyor momentum. Moreover, we prove a theorem on relative period doubling of RPOs with regulardrift momentum pair. We then present underdetermined nonlinear systems of equations thatare satisfied by Hamiltonian relative equilibria and RPOs respectively, have regular derivativeat their solutions and are therefore amenable to standard numerical path-following methods.Moreover, we develop algorithms for the computation of turning points, relative period doublingand relative period halving bifurcations of Hamiltonian RPOs. We continue in a conservedquantity of the system, which is either the energy or a component of the momentum map.The list of bifurcations of Hamiltonian RPOs which we study is by no means exhaustive. Inparticular, in this paper we assume that the spatial symmetry of the RPO is trivial (by reducingthe dynamics on the fixed point space of the spatial symmetry). Hence we do not deal withbifurcations breaking the spatial symmetry at all. Although we present new theoretical results,the emphasis in this paper is the “translation” of theoretical persistence and bifurcation resultsinto efficient algorithms for the numerical path-following and the numerical computation ofbifurcations.

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The paper is structured as follows: The topic of Section 2 is the continuation of transversalrelative equilibria and RPOs. First, in Section 2.1, we review the notion of transversal relativeequilibria and their persistence and present a numerical method for their computation. InSection 2.2 we introduce introduce so-called transversal RPOs. These generalize the conceptof transversal relative equilibria introduced in [17]. The nondegeneracy condition which werequired in earlier works [24, 20, 25] is a more restrictive condition. We then generalize ourearlier numerical continuation methods [25] for nondegenerate RPOs to transversal RPOs ofcompact group actions in Sections 2.3, 2.4. In Section 2.5 we numerically analyze turning pointsof relative equilibria and RPOs which are continued in some component of the momentum orin energy (in the case of RPOs). These occur at transversal, but degenerate relative equilibriarsp. RPOs. We present numerical methods for their detection and computation.

In Section 3 we present a theorem on relative period doubling bifurcations of nondegenerateRPOs with generic drift momentum pair. In a comoving frame these correspond to period-doubling or symmetry breaking pitchfork bifurcations (as analyzed in [5] for non-Hamiltoniansystems with discrete symmetries). We then present numerical methods for the detection andcomputation of relative period doubling and relative period halving bifurcations of Hamiltonianrelative periodic orbits.

In Section 4 we apply our results to rotating choreographies of the three-body problemwhich bifurcate from the famous Figure Eight solution of Chenciner and Montgomery [2]. It iswell-knnown that one of the non-planar rotating Figure Eight families to the Lagrange relativeequilibrium (for example, see the discussion in [3]). We find that the other non-planar familyof rotating Figure Eights bifurcates in a relative period halving bifurcation to the family ofrotating choreographies that connect to the Lagrange relative equilibrium. Moreover, we findseveral relative period-doubling bifurcations and a turning point of the planar rotating choreog-raphy which bifurcates from the Figure Eight solution when the third component of the angularmomentum vector is varied.

2 Persistence and numerical continuation of transversal

RPOs

In this section we present methods for the continuation of transversal RPOs extending resultsof [20, 23]. We start with the simpler case of continuation of transversal relative equilibria inSection 2.1. Then we prove a persistence result for transversal RPOs (Section 2.2. In Sections2.3 and 2.4 we develop a method for the path-following of transversal RPOs, and in Section 2.5we show how to detect and compute turning points of relative equilibria and RPOs.

2.1 Continuation of transversal Hamiltonian relative equilibria

We consider a Hamiltonian system

x = fH(x) = J∇H(x) (2.1)

with Hamiltonian (energy) H(x) on a finite-dimensional symplectic vector space X = R2d withsymplectic structure matrix J (i.e., J is skew-symmetric and invertible). Let

Ω(v, w) = 〈J−1v, w〉

be the symplectic form generated by J. Let Φt(x0) denote the flow of (2.1) by , i.e., x(t) = Φt(x0)is a solution of (2.1) with initial value x(0) = x0. Then the energy H(x) is a conserved quantityof (2.1): H(Φt(x0)) = H(x0) for all x0, t. We assume that a finite-dimensional compact Lie

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group Γ acts on X symplectically symplectically (i.e., Ω is Γ-invariant) and that the HamiltonianH is Γ-invariant, which implies that (2.1) is Γ-equivariant, i.e., fH(γx) = γfH(x), x ∈ X , γ ∈ Γ.So whenever x(t) is a solution of (2.1) then so is γx(t). We call the elements of Γ the symmetriesof (2.1). Let g = TidΓ denote the Lie algebra of Γ. By Noether’s theorem locally there is aconserved quantity Jξ of (2.1) for each continuous symmetry ξ ∈ g of the system uch that Jξ isthe Hamiltonian for the symplectic flow x→ exp(ξt)x, see, e.g., [1, 12]. The map Jξ(x) = J(x)(ξ)is linear in ξ, so that J is a map from a neighbourhood of each x ∈ X to g∗, called a momentummap. Let Adγ , γ ∈ Γ, denote the adjoint action of Γ on g: Adγξ = γξγ−1, ξ ∈ g, γ ∈ Γ. Thenthe coadjoint action of Γ on g∗ is given by

γµ = (Ad∗γ)−1µ, γ ∈ Γ. (2.2)

We assume throughout the paper that J is defined on the whole of X and is Γ-equivariantwith respect to the Γ-action on X and the coadjoint action on g∗. Moreover, we choose anAd-invariant inner product on g such that the adjoint action on g is by orthogonal matrices andthe adjoint and coadjoint actions can be identified.

As usual (c.f. [7]), for an action of a group Γ on a space X we define the isotropy group of x ∈ Xas Γx = γ ∈ Γ, γx = x. For any subgroup K or element γ ∈ Γ of Γ we define the fixed pointspace of K rsp. γ as FixX (K) = x ∈ X , γx = x ∀γ ∈ K and FixX (γ) = x ∈ X , γx = xrespectively. We denote by N(K) the normalizer of the subgroup K of Γ. For any group Γdefine Γid to be the identity component of Γ.

Example 2.1 In the case of rotational symmetries Γ = SO(3) the space of momenta is g∗ =so(3)∗ ≡ R3 and J : X → R3 is the angular momentum, see Section 4 below for an examplefrom celestial mechanics. In this g = so(3) ' R3 and the adjoint and coadjoint actions are justthe usual multiplication by matrices in SO(3). Here the identification so(3) ' R3 is given by themap

ξ = (ξ1, ξ2, ξ3) →

0 ξ3 −ξ2

−ξ3 0 ξ1ξ2 −ξ1 0

. (2.3)

The Lie bracket becomes [ξ, η] = ξ × η, where ξ, η ∈ R3 ' so(3).

A point x ∈ X lies on a relative equilibrium Γx if there is some ξ ∈ g such that ξx = fH(x),i.e., the relative equilibrium through x is an equilibrium of the Hamiltonian system (2.1) in aframe moving with velocity ξ. We call ξ the drift velocity of the relative equilibrium at x.

Let x lie on a relative equilibrium with drift velocity ξ and momentum µ = J(x) at x. Thereis a simple relation between the drift velocity and momentum of a relative equilibrium:

ad∗ξ µ = 0, (2.4)

which is implied by momentum conservation: µ = J(x) = J(Φt(x)) = J(exp(tξ)x) = Ad∗exp(−tξ)µ,

c.f. [17, 24]. This relation is crucial for the problem of persistence to nearby momentum values,as we will see below.

2.1.1 Persistence of transversal relative equilibria

Before we come to the numerical continuation of transversal RPOs we first present a persistenceresult for transrversal relative equilibria with regular velocity-momentum pair and show how tocontinue them numerically.

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Definition 2.2 [17, 24]

(i) We call pairs (ξ, µ) ∈ g × g∗ satisfying (2.4) velocity-momentum pairs and denote thespace of velocity-momentum pairs by

(g ⊕ g∗)c := (ξ, µ) ∈ g ⊕ g∗, ad∗ξµ = 0. (2.5)

(ii) We define an action of Γ on the space of velocity-momentum pairs as follows:

γ(ξ, µ) = (Adγξ, (Ad∗γ)−1µ), γ ∈ Γ, (ξ, µ) ∈ (g ⊕ g∗)c.

For later purposes we define the isotropy subgroup Γ(ξ,µ) of (ξ, µ) ∈ (g ⊕ g∗)c with respectto this action as

Γ(ξ,µ) = γ ∈ Γ, γ(ξ, µ) = (ξ, µ),

denote its Lie algebra by g(ξ,µ) and let r(ξ,µ) = dimg(ξ,µ). Moreover, we define the isotropysubgroup of ξ ∈ g as Γξ = Γ(ξ,0) = γ ∈ Γ, Adγξ = ξ and the momentum isotropysubgroup of µ ∈ g∗ by Γµ = Γ(0,µ) = γ ∈ Γ, (Ad∗

γ)−1µ = µ, denote their Lie algebras bygξ and gµ respectively, and define rξ = dimgξ, rµ = dimgµ.

(iii) We call a velocity-momentum pair (ξ, µ) ∈ (g⊕g∗)c regular if dimg(ξ,µ) is locally constantin the space of velocity-momentum pairs (2.5).

(iv) We call ξ ∈ g regular if rξ is locally constant in g.

(v) We call µ ∈ g∗ regular if rµ is locally constant in g∗.

Remark 2.3 As shown in [17, 24], (ξ, µ) ∈ (g ⊕ g∗)c is regular if and only if g(ξ,µ) is theLie algebra of a maximal torus. In particular, for a regular velocity-momentum pair (ξ, µ) theisotropy subalgebra g(ξ,µ) is abelian. Regular velocity-momentum pairs (ξ, µ) are generic in thespace of velocity-momentum pairs (g⊕g∗)c. Regular µ ∈ g∗ are generic in g∗ and regular ξ ∈ g

are generic in g. The velocity of a regular velocity momentum pair is generically regular. In thiscase g(ξ,µ) = gξ holds. Similarly, the momentum µ of a regular velocity momentum pair (ξ, µ)is generically regular. In this case g(ξ,µ) = gµ holds. These statements are needed later on.

Next, we describe the form that (2.1) takes, in symmetry-adapted coordinates. This is neededfor the derivation of the persistence result and the numerical continuation method for relativeequilibria Denote by N a normal space transverse to Γx at x, i.e., X = TxΓx ⊕N . Then N isa model for the space of group orbits X/Γ near x. Moreover there are a choice of normal spaceN and coordinates x ' (γ, v), γ ∈ Γ, v ∈ N , near Γx such that x ' (id, 0), and the dynamics inthese coordinates takes the form [18]:

γ = γfΓ(ν, w) = γDνh(ν, w),

ν = fN0(ν, w) = ad∗

Dνh(ν,w)ν,

w = fN1(ν, w) = JN1

Dwh(ν, w).

(2.6)

Here N = N0 ⊕ N1 and v ∈ N is decomposed as v = (ν, w), ν ∈ N0, w ∈ N1. The spaceN1 = kerDJ(x)∩N is symplectic with symplectic structure matrix JN1

and is called symplecticnormal space. Let nµ be a Γµ-invariant complement to gµ in g∗. Then the annihilator anng∗(nµ)of nµ in g is a Γµ-invariant section transverse to the momentum group orbit Γµ at µ in g∗ andN0 ' g∗

µ ' ann(nµ). Moreover h(ν, w) is the Hamiltonian in the coordinates x ' (γ, ν, w) Theoriginal relative equilibrium corresponds to the equilibrium (ν, w) = 0 of the (ν, w)-subsystem

5

of (2.6), i.e., Dwh(0) = 0. Moreover fΓ(0, 0) = ξ where ξ is the drift velocity of the relativeequilibrium at x. The momentum map in these coordinates takes the form

j(γ, ν, w) = γ(µ+ ν). (2.7)

With respect to the decomposition X = TxΓx⊕N0⊕N1, the linearization A = D(fH(x)− ξ)at the relative equilibrium in a frame moving with its drift velocity ξ is given by

A =

−adξ D2νh(0) D2

νwh(0)0 ad∗

ξ |g∗

µ0

0 JN1D2

ν,wh(0) JN1D2

wh(0)

, (2.8)

see [18]. Decompose ν ∈ g∗µ as ν = χ+ ζ where χ ∈ ker ad∗

ξ |g∗

µ' g∗

(ξ,µ)and ζ ∈ anng

∗

µ(g(ξ,µ)).

Here we use that ker ad∗ξ |g∗

µ= anng

∗

µ(adξgµ), that g(ξ,µ) = ker adξ |gµ and that

ker adξ|gµ ⊕ adξgµ = gµ. (2.9)

The latter is true because Γ is compact and so adξ is semisimple.

We see from (2.8) that Jordan blocks of A = D(fH(x) − ξ) to the eigenvalue 0 of A corre-sponding to the kernel g∗

(ξ,µ)of A0 := ad∗

ξ |g∗

µand the kernel of A1 = DwfN1

(0) occur in the

matrix A10 := DνfN1(0) and may result in [A10, A1] having full rank even if A1 does not have

full rank. This motivates the following definition:

Definition 2.4 A relative equilibrium Γx with regular velocity-momentum pair is called transver-sal if the matrix

[DχfN1(ν, w),DwfN1

(ν, w)]|(ν,w)=0 (2.10)

has full rank.

As in [24] we call a relative equilibrium nondegenerate if DwfN1(0) is invertible. Our definition

of a transversal relative equilibrium is equivalent to the definition: By [17, Theorem 4] a relativeequilibrium of a compact group action with regular velocity-momentum pair is transversal if inour notation the matrix DwfN1

(0) is either invertible or has a semisimple eigenvalue 0 and ifDχfN1

(0), χ ∈ g∗(ξ,µ)

, ν = (χ, ζ) ∈ g∗µ, maps g∗

(ξ,µ)onto the kernel of DwfN1

(0). Equation (5) of

[17] gives, in our notation,A1A10 = −A10A0.

This implie that the image of DχfN1(0) lies in the kernel of DwfN1

(0). Therefore the definitionof transversality of [17] is equivalent to the condition (2.10) of Definition 2.4.

Remark 2.5 Let Sµ be a nonlinear slice transverse to Γx in J−1(µ) where µ = J(x). ThenTxSµ ' N1 and Sµ is called the Marsden-Weinstein reduced phase space. Then the relativeequilibrium Γx is a critical point of H |Sµ , i.e., DH |Sµ(x) = 0. If µ is regular then gµ hasconstant dimension for µ ≈ µ and we can choose Sµ to depend smoothly on µ. In this case thetransversality condition (2.10) is equivalent to the condition that Dχ,xDH |Sµ+ν (x) has full rank.

As shown in [17], near a transversal relative equilibrium Γx of a compact group action withregular velocity momentum pair (ξ, µ) as defined in Definition 2.4, there is an r-dimensionalmanifold of relative equilibria, with r = dimg(ξ,µ):

Theorem 2.6 Let x lie on a transversal relative equilibrium Γx of (2.1) with regular velocity-momentum pair (ξ, µ) ∈ (g⊕g∗)c. Let r = r(ξ,µ). Then there is an r-dimensional family Γx(s),s ∈ Rr, of relative equilibria near Γx such that x(0) = x, with momentum µ+χ(s), χ(s) ∈ g∗

(ξ,µ)

where χ(0) = 0, and with drift velocity ξ(s) ∈ g(ξ,µ) close to ξ(0) = ξ. If the relative equilibriumΓx is nondegenerate then we can choose s = χ ∈ g∗

(ξ,µ).

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Proof. For sake of completeness and to see the connection with the numerical continuationmethod of Section 2.1.2 we include a proof which is only valid for compact groups and differentfrom the proof given in [17].

We see from (2.8) that x is a transversal relative equilibrium if we only reduce by the group

Γ = Γξ. Denote by N = N0 ⊕ N1 a normal space transverse to the group orbit such that in the

coordinates x ' (γ, ν, w), γ ∈ Γ the vectorfield (2.1) takes the form (2.6). Since g(ξ,µ) is abelianby Remark 2.3 we have f eN0

(ν, w) ≡ 0 where ν ∈ g∗(ξ,µ)

, and [Dνf eN1(0),Dwf eN1

(0)] has full rank.

So there is a manifold of equilibria (ν(s), w(s)) of f eN , s ∈ Rr, which gives a manifold of relativeequilibria x(s) of (2.1).

2.1.2 Numerical computation of transversal relative equilibria

Let Γx be a transversal relative equilibrium with regular velocity-momentum pair (ξ, µ) at x.Let e1ξ , . . . , e

rξ denote a basis of g(ξ,µ). Identify (ξ1, . . . , ξr) ∈ Rr with ξ =

∑ri=1 ξie

iξ. Let

Ji(x) := Jeiξ(x). Because of Remark 2.3 typically the drift velocity ξ of the relative equilibrium

is regular, and we have gξ = g(ξ,µ).The manifold of relative equilibria from Theorem 2.6 can then be computed by solving the

underdetermined system

F (x, ξ, λµ) = fH(x) −r∑

i=1

λµ,i∇Ji(x) −r∑

i=1

ξieiξx = 0, (2.11)

where F : X × R2r → X :

Theorem 2.7 Let x lie on a transversal relative equilibrium Γx with regular velocity ξ ∈ g andmomentum µ and let r = dimg(ξ,µ). Then the derivative DF (y) of (2.11) has full rank at anysolution y = (x, ξ, λµ) of F = 0 close to (x, ξ, 0), and any such solution satisfies λµ = 0 and,hence, is a relative equilibrium of (2.1).

Proof. From (2.8) we see that, to take account of the symmetry-induced kernel vectors of

DfH(x) − ξ, it suffices to reduce by Γ := Γξ, as in the proof of Theorem 2.6. From

DF (x, ξ, 0) =(D(fH(x) − ξ), e1ξx, . . . , e

rξx,∇J1(x), . . . ,∇Jr(x)

),

we see that DF (x, ξ, 0) has full rank if and only if [Dνf eN1(0),Dwf eN1

(0)] has image N1 (as definedin the proof of Theorem 2.6. This condition is satisfied if and only if the relative equilibrium istransversal.

The solution manifold of (2.11) has dimension 2r. By Theorem 2.6 the points (γx(s), ξ(s), 0),where γ ∈ Γid

(ξ,µ), are solutons of F (x, ξ, 0) = 0 with F from (2.11), and this set is also 2r-

dimensional. Therefore we have λµ = 0 in any solution of (2.11) near (x, ξ, 0).

Under the assumptions of Theorem 2.7, the underdetermined system (2.11) is amenable tostandard numerical methods; for example, the Gauss-Newton method applied to (2.11) convergesfor initial values y = (x, ξ, λµ) close to (x, ξ, 0).

Remark 2.8 In Theorem 2.6 we only assume that the relative equilibrium Γx has a regularvelocity-momentum pair (ξ, µ) and not necessarily a regular velocity ξ, so that in general gξ 6=

g(ξ,µ). In this case let e1ξ , . . . , eqξ denote a basis of gξ such that span(er+1

ξ , . . . , eqξ) = nµ ∩ gξ and

span(e1ξ , . . . , erξ) = g(ξ,µ). As before, identify ξ = (ξ1, . . . , ξq) ∈ Rq with

∑qi=1 ξie

iξ.

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Because of Remark 2.3 we typically have q = r. In the case that q > r at x then thederivative DF (x, ξ, 0) of (2.11) does not have full rank, and convergence of the Gauss-Newtonmethod applied to (2.11) is expected to be slow, also at relative equilibria nearby. Instead of(2.11) it is then advantageous to solve

F (x, ξ, λµ) =

fH(x) −∑r

i=1 λµ,i∇Ji(x) −∑q

i=1 ξieiξx

Jr+1(x)...Jq(x)

= 0, (2.12)

where F : X × Rq × Rr → X × Rq−r. Note that µ(ejξ) = 0 for j = r + 1, . . . , q since µ ∈ g∗

(ξ,µ).

Because of (2.8) the derivative of (2.12)

DF (x, ξ, 0) =

D(fH(x) − ξ) e1ξx . . . eqξx ∇J1(x) . . . ∇Jr(x)

DJr+1(x) 0 . . . 0 0 . . . 0... 0 . . . 0 0 . . . 0DJq(x) 0 . . . 0 0 . . . 0

has full rank, and so the solution manifold of (2.12) is 2r-dimensional. Moreover, all solutionsclose to the original relative equilibrium satisfy ξi = 0, i = r+ 1, . . . , q. To see this note that, asin the proof of Theorem 2.7, the points (γx(s), ξ(s), 0, 0), s ∈ Rr, from Theorem 2.6, γ ∈ Γid

(ξ,µ),

form a 2r-dimensional manifold of solutions of (2.12) as their momentum values µ(s) = J(x(s))satisfy µ(s) − µ ∈ g∗

(ξ,µ)⊆ ann(nµ) and hence µj(s) = 0, j = r + 1, . . . q.

Remark 2.9 Munoz-Almaraz et al (see equation (6.9) of [15]) call a relative equilibrium Γx of(2.1) with drift velocity ξ normal if in our notation

R(D(fH (x) − ξ)) + zx = kerDJ(x) (2.13)

for some subgroup Z which lies in the centre of Γid and satisfies ξ ∈ z. Here z is the Lie algebraof Z. In particular they require ξ to commute with all of g. Under this assumption they provein [15, Theorem 16] that the x persists to an m-dimensional manifold of points x on relativeequilibria which lie in a section N transverse to Γx at x. Here m = dimZ. They show thatthis manifold of relative equilibria can be computed numerically by solving a system of the form(2.11) with (in our notation) ξ ∈ z, λµ ∈ g∗ ' Rg, and with g = dim Γ phase conditions. Notethat by projecting (2.13) to N1 we see that normal relative equiibria are nondegenerate.

2.1.3 Numerical path-following of relative equilibria

Let, as before, x lie on a transversal relative equilibrium Γx with regular velocity momentumpair (ξ, µ). Let ei

ξ, i = 1, . . . , g be a basis of g such that spaneiξ, i = q + 1, . . . , g = adξg,

and spaneiξ, i = 1, . . . , q = gξ , spanei

ξ, i = 1, . . . , r = g(ξ,µ), as in (2.9). For µ ∈ g∗ letµj = µ(ξj), j = 1, . . . , g. We fix g − 1 momentum components, without loss of generality, thecomponents µ[ = (µ2, . . . µg), g = dim Γ, to get a one-parameter family. We solve

F µ[

(x, ξ, λµ) =

F (x, ξ, λµ)J2(x) − µ2

...Jr(x) − µr

= 0. (2.14)

8

Then for any solution y = (x, ξ, 0) of (2.14) the x-component lies in

X µ[

= x ∈ X , Jj(x) = µj , j = 2, . . . , g.

To see this, first note that by identifying g with g∗ by a Γ-invariant inner product we getµ ∈ g(ξ,µ) and so µj = 0, j = r + 1, . . . , g. For ξ ∈ gξ with ξ ≈ ξ we have gξ ⊆ gξ, g∗

ξ ⊆ g∗ξ.

By construction ξ ∈ gξ at any solution y = (x, ξ, 0) of (2.14). As stated before (2.4), we havead∗

ξµ = 0 for the drift velocity ξ and momentum µ = J(x) of the relative equilibrium givenby y = (x, ξ, 0). From this we conclude that Jj(x) = 0, j = q + 1, . . . , g, where (x, ξ, 0) solves(2.14). If q 6= r then µj = 0, j = r+ 1, . . . , q are subequationns of F = 0, see (2.12). Hence, thesolutions of (2.14) satisfy J[(x) = µ[. Moreover, we have:

Corollary 2.10 Let Γx be a transversal relative equilibrium with regular velocity-momentumpair (ξ, µ) and define χ(s) as in Theorem 2.6. Then the (r − 1, r − 1)-matrix Q with Qij =∂si+1

χj+1(s)|s=0 is invertible if and only if [Dχ1fN1

(0),DwfN1(0)] has full rank and if and only

DF µ[

(x, ξ, 0) has full rank. In this case there is a path x(ε) of points on relative equilibria in

X µ[

with x(0) = x which solves (2.14). We say that the relative equilibrium Γx is transversalwith respect to C(x) := J1(x).

Under the above assumption, (2.14) can be solved by standard numerical methods for un-derdetermined systems, for example by the Gauss-Newton method, for initial values close toy = (x, ξ, 0). For tangential continuation methods, we choose nontrivial continuation tangent

t = t(y) in a solution point y in the kernel of DF µ[

(y) which is orthogonal to the group orbit,

i.e., 〈t, tξi〉 = 0, i = 1, . . . , r. Here tξi ∈ ker(DF µ[

(y)) is given by tξi = (tξix , t

ξi

ξ , tξi

λµ), tξi

x = eξi x,

tξi

ξ = 0, tξi

λµ= 0, i = 1, . . . , r.

Remark 2.11 In the case of dissipative ODEs

x = f(x, λ) (2.15)

which depend on a parameter λ ∈ R and are equivariant under a compact group Γ the bundleequations (2.6) take the more general form

γ = γfΓ(v, λ), v = fN (v, λ) (2.16)

and the relative equilibrium through x = (id, 0) at parameter λ satisfies fN (0, λ) = 0 andfΓ(0, λ) = ξ. A relative equilibrium of a general non-Hamiltonian Γ-equivariant system (2.15)is called transversal if the matrix [DvfN (0, λ),DλfN (0, λ)] has image N and nondegenerate ifDvfN (0), λ) is invertible. Near a transversal relative equilibrium there is a path of equilibria v(s)with parameter λ(s) of the slice equation (the v-equation of (2.16)) which corresponds to a pathx(s) ' (id, v(s)) of relative equilibria of (2.15). The path of relative equilibria (x(s), λ(s)) canbe computed by solving the following underdetermined system which is the analogue of (2.11)in the non-Hamiltonian case:

F (x, ξ, λ) = fH(x) −

q∑

i=1

ξieiξx (2.17)

where q = dimgξ. Typically the drift velocity ξ of the relative equilibrium at x is regular inwhich case q = r is the rank of Γ (the dimension of maximal tori of Γ).

9

2.2 Persistence of transversal RPOs

A point x ∈ X lies on a relative periodic orbit (RPO) if there exists t > 0 such that Φt(x) ∈ Γx.The infimum τ of such t is called the relative period of the RPO and the element σ ∈ Γ suchthat σΦτ (x) = x is called a phase-shift symmetry, reconstruction phase or drift symmetry of theRPO. The relative periodic orbit P itself is given by

P = γΦθ(x), γ ∈ Γ, θ ∈ R.

We assume that τ > 0 so that P is a proper RPO (i.e., not a relative equilibrium). For α ∈ Γ,ξ ∈ g let Z(α) = γ ∈ Γ, γα = αγ and Z(ξ) = γ ∈ Γ, Adγξ = ξ denote the centralizer ofα and ξ respectively.

An RPO of an compact group action becomes periodic in a comoving frame as the followinglemma shows:

Lemma 2.12 [21, 20, 25]

a) Any element σ of a compact group Γ can be decomposed as

σ = α exp(−ξ),

for some ξ ∈ g and α ∈ Γ such that

α` = id for some ` ∈ N, Adαξ = ξ,

andZ(σ) = Z(α) ∩ Z(ξ).

b) For any relative periodic orbit with drift-momentum pair (σ, µ) ∈ (Γ×g∗)c, trivial isotropyK and relative period τ there is a frame moving with velocity ξ ∈ g(σ,µ), called drift velocityof the RPO with respect to x, and some integer ¯ such that in this comoving frame theRPO becomes a periodic orbit of period T = `τ and drift symmetry α ∈ Γµ. Moreoverσ = α exp(−τ ξ).

Remark 2.13 Note that the decomposition σ = α exp(−ξ) in Lemma 2.12 is in general notunique: assume that the group C generated by σ is continuous. Let η be an infinitesimal rotationin the Lie algebra of C which generates the rotation group exp(φη) = Rφ ∈ C, φ ∈ [0, 2π]. Thenother possible choices for α and ξ would be α = R2πj/`α where j ∈ Z, gcd(`, j) = 1, and

ξ = ξ + 2π(n+ j` )η, n ∈ N.

Let x lie on a relative periodic orbit with drift symmetry σ, momentum µ = J(x) at x, andrelative period τ . The following simple relation between the drift symmetry and momentum ofan RPO

σµ = µ, (2.18)

is analogous to the corresponding relation (2.4) for relative equilibria, and is crucial for theproblem of persistence to nearby momentum values, see [20] and the sections below. It simplyfollows from the fact that J is preserved by the flow, and so σµ = σJ(x) = J(σx) = J(Φ−τ (x)) =J(x) = µ.

We will need the following definitions, which are analogous to the corresponding definitions2.2 for relative equilibria.

10

Definition 2.14 [20], [25]

(i) We call pairs (σ, µ) ∈ Γ×g∗ satisfying (2.18) drift-momentum pairs and denote the spaceof drift-momentum pairs by

(Γ × g∗)c := (γ, µ) ∈ Γ × g∗, γµ = µ. (2.19)

(ii) We define an action of Γ on the space of drift-momentum pairs as follows:

γ(σ, µ) = (γσγ−1, (Ad∗γ)−1µ), γ ∈ Γ, (σ, µ) ∈ (Γ × g∗)c.

For later purposes we define the isotropy subgroup Γ(σ,µ) of (σ, µ) ∈ (Γ×g∗)c with respect tothis action as Γ(σ,µ), denote its Lie algebra by g(σ,µ) and let r(σ,µ) = dim g(σ,µ). Moreoverwe define the isotropy subgroup of σ ∈ Γ as Γσ = Γ(σ,0), denote its Lie algebra by gσ andlet rσ = dimgσ.

(iii) We call a drift-momentum pair (σ, µ) ∈ (Γ×g∗)c regular if r(σ,µ) is locally constant in thespace of drift-momentum pairs (2.19).

(iv) We call σ ∈ Γ regular if rσ is locally constant.

Remark 2.15 [20, 25] Regular drift-momentum pairs (σ, µ) are generic in the set of drift-momentum pairs (Γ×g∗)c. Regular elements σ ∈ Γ are generic in Γ. The drift symmetry σ of aregular drift-momentum pair (σ, µ) is typically regular in which case gσ = g(σ,µ), and typicallythe momentum µ of a regular drift-momentum pair (σ, µ) is regular in which case gµ = g(σ,µ).Moreover the following conditions are equivalent:

(i) (σ = α exp(−ξ), µ) is a regular drift momentum pair;

(ii) g(σ,µ) is the Lie algebra of a Cartan subgroup;

(iii) g(σ,µ) = gα ∩ g(ξ,µ) is abelian.

Let x lie on an RPO P with relative period τ . We assume without loss of generality thatthe isotropy K of the RPO is finite (if not, we restrict the dynamics to the fixed point spaceFix(K) of K so that K is trivial). We call a K-invariant section S := Sx that is transverse to theRPO at x, i.e., transverse to gx⊕ span(fH(x)), a Poincare section at x. As usual, we define thePoincare map Π : S → S as follows. For x ∈ S close to x there are unique γ(x) ∈ Γ, γ(x) ≈ σ,and τ(x) ≈ τ such that γ(x)Φτ(x)(x) ∈ S (this follows from the implicit function theorem sincewe assume that the isotropy K of the RPO is trivial). Now we set

Π(x) = γ(x)Φτ(x)(x). (2.20)

Let E = H(x) and µ = J(x) be the energy and momentum of the RPO at x and let S E,µ ⊆ X E,µ

be a Poincare-section transverse to the time orbit and Γµ-orbit through x within the energymomentum level set

X E,µ = x ∈ X , E = H(x), µ = J(x).

Denote the corresponding Poincare-map by ΠE,µ : SE,µ → SE,µ. For the definition of transver-sality of an RPO we need the following lemma:

Lemma 2.16 There is a choice of Poincare-section S near an RPO at x with momentumµ = J(x), energy E, drift symmetry σ, and finite isotropy K such that S E,µ = S ∩ X E,µ

and the following hold true:

11

a) The tangent space TxS to the Poincare-section S at x takes the form

TxS ' N = N0⊕N1⊕N2, where N0 ' g∗µ, N1 ' TxS∩kerDJ(x)∩kerDH(x), N2 ' R.

Here N1 = TxSE,µ is a symplectic vectorspace, N0 is isomorphic to a Γµ-invariant sectiontransverse to the momentum group orbit Γµ in g∗ and N2 parametrizes the energy level.

b) Let Π : S → S be the Poincare map. Then there is a choice of coordinates x ' (ν, w,E) ∈ Swhere ν ∈ N0, w ∈ N1 and E ∈ N2, and the point x is identified with ν = 0, w = 0, E = E,such that the Poincare map takes the form

Π(ν, w,E) = (ΠN0(ν, w,E),ΠN1

(ν, w,E),ΠN2(ν, w,E)).

Here

ΠN0(ν, w,E) = γ(ν, w,E)ν, ΠN2

(ν, w,E) = E, ΠN1(0, w, E) = ΠE,µ(w),

γ(ν, w,E) ∈ Γµ, γ(0, 0, E) = σ, and the map w → ΠN1(ν, w,E) is symplectic. In these

coordinates the momentum map restricted to N takes the form

j|N (ν, w) = µ+ ν.

Proof. Part a) is contained in [22, Theorem 3.1] and most of part b) is implicitly contained in[20]:

Theorem 3.3 and Remark 3.4 of [22] implies that a Γ-invariant neighbourhood U is sym-plectomorphic to (Γ × R/2πZ × N )/Zn where N can be decomposed as specified above. TheZn -action is generated by the drift symmetry α of the RPO P in the comoving frame andacts as (γ, v, θ) → (γα−1, Qv, θ + 1) where Q acts v = (ν, w, e) ∈ N = N0 ⊕ N1 ⊕ N2 asQ(ν, w,E) = (Ad∗

αν,QN1w, e) and QN1

has order n. The symplectic form is Zn-invariant andgiven by

ωΓ×N×T ∗S1 = ωΓ×g∗

µ+ ωN1

+ ωT ∗S1 , (2.21)

on Γ × R/2πZ × N . Here T ∗S1 ' R/2π × N2, ωT ∗S1 is the symplectic form on T ∗S1, ωΓ×g∗

µ

is the restriction to Γ × g∗µ ' Γ × ann(nµ) ⊆ Γ × g∗ ' T ∗Γ of the symplectic form on T ∗Γ and

ωN1is the symplectic form on N1. As before, nµ is Γµ-invariant complement to gµ in g. The

energy of (γ, ν, w, e, θ) is E = h(ν, w, θ) + e and its momentum is J(γ, ν, w, e, θ) = γ(µ+ ν), see[22, Remark 3.4d)]. The map (γ, ν, w, e, θ) → (γ, ν, w,E, θ) is a symplectic tramsformation.

In these coordinates the Poincare-section S is given by S = (γ, v, θ) ∈ (Γ×N )/Z, θ = 0,and the Poincare-map Π(x) = γ(x)Φτ(x)(x) becomes a map Π : N → N which decomposes asform Π = (Π0,Π1,Π2), where Π0 maps into N0, Π1 into N1 and Π2(ν, w,E) into N2. Due to theform of the momentum map in these coordinates and the conservation of momentum we haveΠ0(ν, w,E) = γ(ν, w,E)(µ + ν) − µ. Since ann(nµ) is Γµ-invariant and ν ∈ ann(nµ), ν ≈ 0, wehave γ(ν, w,E) ∈ Γµ. Energy conservation implies that Π2(ν, w,E) ≡ E. Moreover the mapΠ1(ν, ·, E) : N1 → N1 is symplectic.

The relation between the linearization of the Poincare-map and the full linearization σDΦτ (x)at the RPO through x is given in follow proposition:

Proposition 2.17 [22, Propositions 4.3 and 4.4] Let x = σΦτ (x) ∈ X lie on an RPO Pof a compact group action Γ with relative period τ , finite isotropy subgroup and momentum

12

µ = J(x), and let M = σDΦτ (x). Then the matrix M has the following structure with respectto the decomposition TxX = TxΓx⊕ span(fH(x)) ⊕N :

M =

Adσ 0 DN

0 1 ΘN

0 0 MN

, (2.22)

and the block MN in (2.22) has the following block structure with respect to the decompositionN = N0 ⊕N1 ⊕N2:

MN =

M0 0 0M10 M1 M1,2

0 0 1

,

where M0 = Ad∗σ−1 |g∗

µ. Note that MN = D Π(x). Moreover

DN =

(D0 D1 D2

0 0 0

), ΘN = (Θ0,Θ1,Θ2).

and

Ad−1σ =

((Adσ)−1|gµ 0

0 PnµAdσ |nµ

).

Here Pgµ is a projection on gµ with kernel and nµ and Pnµ = idg −Pgµ a projection onto nµ.

Remark 2.18 In [22, Proposition 4.3] we have ΘN = 0 in (2.22) and τ = 1 since time isreparametrized such that τ(x) ≡ 1 in the definition of the Poincare map Π. The matrix DN isthe drift-derivative with respect to the variables v ∈ N .

As before let x lie on an RPO with drift-momentum pair (σ, µ) and let e1ξ , . . . , erξ be a basis

of g(σ,µ) and let µi = µ(eiξ), Ji = Jei

ξ, i = 1, . . . , r. We then have

Fixg∗

µ(Ad∗

σ) = anng∗

µ((Ad∗

σ − id)gµ) ' g∗(σ,µ),

and a complement of g∗(σ,µ) in g∗

µ is given by anng∗

µ(g(σ,µ)). We decompose ν ∈ g∗

µ as ν = (χ, ζ)

where χ ∈ g∗(σ,µ) and ζ ∈ anng∗

µ(g(σ,µ)).

The following transversality condition for RPOs is a direct extension of Definition 2.4 toRPOs.

Definition 2.19 A relative periodic orbit of a Hamiltonian system (2.1) through x ' (0, 0, E) ∈N is called transversal if it is not a relative equilibrium and if

D(χ,w,E)(ΠN1(0, 0, E) − idN1

)

has full rank where χ ∈ g∗(σ,µ).

We call an RPO through x ' (0, 0, E) ∈ N nondegenerate if DwΠN1(0, 0, E) = DxΠE,µ(x)

has no eigenvalue 1.

Remark 2.20 If the RPO through x has a regular momentum µ = J(x) at x then gµ and hencealso SE,µ+ν has locally constant dimension and we can choose SE,µ+ν to depend smoothly onE and ν. Moreover the RPO is transversal if

D(x,E,ν)(ΠE,µ+ν(x) − idSE,µ+ν )|(E=E,x=x,ν=0) (2.23)

has full rank. If the momentum µ of the RPO is nongeneric then SE,µ changes dimension forµ 6= µ, µ ≈ µ, and this equivalence does not hold. The situation is analogous in the case ofrelative equilibria as we have seen in Remark 2.5.

13

Example 2.21 In the case of the symmetry group Γ = SO(2), every momentum µ ∈ so(2)∗ isregular. Hence (2.23) can be applied to check whether an RPO is transversal.

Note that the RPO is called nondegenerate in [20, 25] if DxΠE,µ : SE,µ → SE,µ = DwΠN1(0)

does not have an eigenvalue 1 at a point x of the relative periodic orbit. The following theoremis an extension of the persistence result [20, Theorem 4.2] for nondegenerate RPOs to transversalRPOs.

Theorem 2.22 Let Γ be compact. Let x lie on a transversal RPO P of (2.1) with relative periodτ and regular drift-momentum pair (σ, µ) ∈ (Γ×g∗)c. Let r = r(σ,µ), decompose σ = α exp(−τ ξ)as before, and let E = H(x) be the energy of the RPO. Then

a) there is an (r + 1)-dimensional family P(s), s ∈ Rr+1, of RPOs near P = P(0) such thatx(s) ∈ P(s) is smooth, x(0) = x, with

energy E(s) close to E(0) = E,momentum µ+ χ(s), χ(s) ∈ g∗

(σ,µ), where χ(0) = 0,

relative period τ(s) close to τ(0) = τ ,drift symmetry σ(s) close to σ(0) = σ, anddrift velocity ξ(s) ∈ gµ close to ξ(0) = ξ ∈ g(σ,µ)

where σ(s) = α exp(−τ(s)ξ(s)), Adαξ(s) = ξ(s).

b) If the RPO P is nondegenerate then we can choose s = (E,χ).

Note that by the above theorem transversal RPOs of compact group actions persist to nearbyenergy-level sets and to those nearby momenta which are fixed by the drift symmetry σ of theRPO at x and lie in a section transverse to the momentum group orbit Γµ (the last condition isonly needed in order to guarantee that the RPOs parametrized by s are not symmetry related).During the continuation the momentum µ(s), drift symmetry σ(s) and drift velocity ξ(s) of theRPOs P(s) vary, but the drift symmetry α of the RPOs in the comoving frame is fixed.

Proof. Since Adσ does not have Jordan blocks, we can conclude from Proposition 2.17 that theRPO is still transversal when considered as an RPO for the symmetry group Γ = Γσ = Z(σ).From Remark 2.15 the isotropy algebra g(σ,µ) ⊆ g(α,µ) is abelian. This implies, by Lemma 2.16 b)

applied to the symmetry group Γ, that ΠN0(ν, w) ≡ ν. Here N0 ' g∗

(σ,µ), (ν, w) ∈ N = N0⊕N1

and N is the slice to the action of Γ from Lemma 2.16. Due to the definition of transversality(Definition (2.19)) we can solve ΠN1

(ν, w, E) = w for (ν, w, E)(s), s ∈ Rr+1, by the implicitfunction theorem with ν(s) = χ(s) ∈ g∗

(σ,µ). This gives an (r + 1)-dimensional family of RPOs

of (2.1) with the properties required.

2.3 Numerical continuation of transversal RPOs

In this section we show that the numerical methods presented in [25] for nondegenerate RPOs ofcompact symmetry group actions with regular drift-momentum pair also converge for transversalRPOs of noncompact algebraic group actions. We restrict attention to the case of single shooting.

Note that by Remark 2.15 generically the drift symmetry σ of an RPO is regular, and weassume this in the following theorem.

14

Theorem 2.23 Let Γ be compact, let x lie on a transversal RPO P with regular drift symmetryσ ∈ Γ, and momentum µ, and decompose σ = α exp(−τ ξ) as in Lemma 2.12. We set r = r(σ,µ)

and denote a basis of g(σ,µ) by e1ξ , . . . , erξ . For ξ ∈ g(σ,µ) let ξ =

∑ri=1 ξie

iξ and identify

ξ = (ξ1, . . . , ξr) ∈ Rr. As before, let Ji = Jeiξ, i = 1, . . . , r, and define

x = fH(x) + λE∇H(x) +

r∑

i=1

λµ,i∇Ji(x) −r∑

i=1

ξieiξ. (2.24)

Denote the flow of (2.24) by Φt(x; ξ, λE , λµ) . Then the derivative DF (y) of

F (x, T, ξ, λE , λµ) = αΦT`(x; ξ, λE , λµ) − x = 0, (2.25)

whereF : X × R

2+2r → X ,

has full rank at any solution y = (x, T, ξ, λE , λµ) of F = 0 close to (x, T , ξ, 0, 0). Moreover anysuch solution satisfies λE = 0, λµ = 0, and, hence, determines an RPO of (2.1).

Proof. As in the proof of Theorem 2.22 we replace Γ by Γσ = Z(σ). Since gσ = g(σ,µ), we lookfor RPOs with drift velocity ξ ∈ gσ.

We have

DF (x, T , ξ, 0, 0) = [DxF,DTF,DξF,DλEF,DλµF ]

=(σDΦτ (x) − id, fH(x), e1ξx, . . . , e

rξx,DλEF,DλµF

).

The (r + 1, r + 1)-matrix B with

Bij = DJi(x)Dλµ,jF (y) = DJi(x)αDλµ,j Φτ (x; ξ, 0, λµ)|λµ=0,

i, j = 1, . . . , r,

Bi,r+1 = DJi(x)DλEF (y) = DJi(x)αDλE Φτ (x; ξ, λE , 0)|λE=0,

i = 1, . . . , r,

Br+1,i = DH(x)Dλµ,iF (y) = DH(x)αDλµ,i Φτ (x; ξ, 0, λµ)|λµ=0,

i = 1, . . . , r,

Br+1,r+1 = DH(x)DλEF (y) = DH(x)αDλE Φτ (x; ξ, λE , 0)|λE=0

has full rank. This was shown in [25]. We sketch the proof: Since (σ, µ) is regular the isotropyalgebra g(σ,µ) is abelian by Remark 2.15. Consequently, Φt(·; ξ) = exp(−t

∑ri=1 ξie

iξ)Φ

t(·) con-serves the momenta Jj , j = 1, . . . , r, and as shown in [25], we have

DH(x)αDλE Φτ (x; ξ) =

∫ τ

0

‖DH(Φs(x; ξ))‖2ds,

DJi(x)αDλµjΦτ (x; ξ) =

∫ τ

0

〈∇Ji(Φs(x; ξ)),∇Jj(Φ

s(x; ξ))〉ds,

DJi(x)αDλE Φτ (x; ξ)) =

∫ τ

0

〈DH(Φs(x; ξ)),DJi(Φs(x; ξ)〉ds,

DH(x)αDλµiΦτ (x; ξ) =

∫ τ

0

〈DH(Φs(x; ξ)),DJi(Φs(x; ξ))〉ds.

15

Hence any c = (cµ,1, . . . , cµ,r, cE) with cTBc = 0 satisfies cEDH(x) +∑r

i=1 cµ,iDJi(x) = 0,which contradicts the assumption that x does not lie on a relative equilibrium. Therefore B hasfull rank and is positive definite.

From Proposition 2.17 and the above we see that DF (x, T , ξ, 0, 0) has full rank iff the matrixPN1

DxF has image N1. Here PN1is a projection on N1 with kernel N0 and gx. Note that

PN1DxF = [M10,MN1

− idN1,M12] = [DνΠN1

(0),DwΠN1(0) − id,DEΠN1

(0)]

where we used the notation of Proposition 2.17. Therefore, by the definition of transversality(Definition 2.19), the matrix DF (x, T , ξ, 0, 0) has full rank and F = 0 has a (2+2r)-dimensionalfamily of solutions.

By Theorem 2.22 there is an (r + 1)-dimensional manifold P(s) of RPOs of (2.1) near xand x(s) ∈ P(s) has drift symmetry σ(s) ∈ Γ(σ,µ) and an average drift velocity ξ(s) ∈ g(σ,µ)

which commutes with α. Therefore y = (x(s), T (s) = `τ(s), ξ(s), 0, 0), is a solution of F = 0for s ≈ 0. Moreover, since, by Remark 2.15, the group Γid

(σ,µ) is abelian, for every γ ∈ Γ(σ,µ),

γ ≈ id and t ≈ 0, the point y = (γΦt(x(s)), T (s), ξ(s), 0, 0) is a solution of F = 0. This gives an(2r + 2)-dimensional manifold of solutions of F = 0. Hence the (2r + 2)-dimensional solutionmanifold of F = 0 consists of RPOs of (2.1) near x and satisfies λE = λµ = 0.

Under the assumptions of Theorem 2.23, the underdetermined system (2.25) is amenable tostandard numerical methods; for example, the Gauss-Newton method applied to (2.25) convergesfor initial values y = (x, T, ξ, λE , λµ) close to y.

Remark 2.24 In Theorem 2.22 we only assume that the RPO P has a regular drift-momentumpair (σ, µ) and not necessarily a regular drift symmetry σ so that in general gσ 6= g(σ,µ).

In this case let e1ξ , . . . , eqξ denote a basis of gσ such that span(er+1

ξ , . . . , eqξ = nµ ∩ gσ and

span(e1ξ , . . . , erξ = g(σ,µ). Let, as before, Jei

ξ= Ji, i = 1, . . . , q. Note that µ(ej

ξ) = 0, j =

r + 1, . . . , q. Because of Remark 2.15 we generically have q = r, but when q > r at the RPO Pthen the derivative DF (y) of (2.25) does not have full rank, and convergence of a Gauss-Newtonmethod applied directly to (2.25) is expected to be slow. In a manner similar to the case ofrelative equilibria, see Remark 2.8, it is advantageous to solve the system

F (x, T, ξ, λE , λµ) =

αΦT`(x; ξ, λE , λµ) − x

Jr+1(x)...Jq(x)

= 0, (2.26)

whereF : X × R

2+q+r → X × Rq−r

and now Φt(x; ξ, λE , λµ) is the flow of

x = fH(x) + λE∇H(x) +

r∑

i=1

λµ,i∇Ji(x) −

q∑

i=1

ξieiξ.

Then

DF (x, T , ξ, 0, 0) = [DxF,DTF,DξF,DλEF,DλµF ]

=

σDΦτ (x) − id fH(x) e1ξx . . . eqξx DλEF DλµF

DJr+1(x) 0 0 0 0 0 0... 0 0 0 0 0 0DJq(x) 0 0 0 0 0 0

16

has full rank and so the solution manifold of (2.26) is (2r + 2)-dimensional. As in the proofof Theorem 2.23, for γ in the abelian group Γid

(σ,µ), and t ∈ R, the points γΦt(x(s)) have

momentum µ(s) = J(γΦt(x(s))) satisfying µ(s) − µ ∈ g∗(σ,µ) ⊆ ann(nµ). Therefore the set

y = (γΦt(x(s)), T (s), ξ(s), 0, 0), γ ∈ Γid(σ,µ), t ∈ R defines an (2r + 2)-dimensional solution

manifold of (2.26), too. From this λE = 0, λµ = 0 follows.

Remark 2.25 In order to continue RPOs in numerically delicate situations, that is, when thesingle shooting method is ill-conditioned, we use multiple shooting rather than single shooting,cf. [25] and references in there: we compute k points on the RPO in the comoving frame bysolving the underdetermined equation

F (x1, . . . , xk , T, ξ, λE, λµ) = 0, F : X k × R2+2r → X k. (2.27)

Here xj ∈ X , j = 1, . . . , k, T, λE ∈ R, λµ, ξ ∈ Rr. Moreover 0 = s1 < . . . < sk+1 = 1 is apartition of the unit interval, ∆si = si+1 − si for i = 1, . . . k, and

Fi(x1, . . . , xk, T, ξ, λE , λµ) =

Φ

∆siT

` (xi; ξ, λE , λµ) − xi+1 for i = 1, . . . , k − 1,

αΦ∆skT

` (xk ; ξ, λE , λµ) − x1 for i = k.(2.28)

It is well-known, see, e.g., [25] and references therein, that the derivative DF of (2.27) has fullrank in an RPO if and only if the corresponding derivative of the single shooting equation (2.25)has full rank. Therefore Theorem 2.23 can readily be applied to the multiple shooting context.

Remark 2.26 The relationship

DxF (x, `τ , ξ, 0, 0) = σDΦτ (x) − id

between the derivative of the Poincare-map on one hand, and the shooting equations (2.25) onthe other hand will play an important role in the computation of bifurcations, see Sections 2.5and 3. In the multiple-shooting case the matrix σDΦτ (x) can be obtained from the linearizationof (2.27), for example see [25].

Remark 2.27 Munoz-Almaraz et al [15] call a periodic orbit of (2.1) with period T through xnormal if, in our notation

Im(DxΦT (x) − id) + span(fH(x)) = kerDJ(x) ∩ kerDH(x), (2.29)

see [15, Definition 4]. By projecting both sides of (2.29) on N1 ⊆ kerDJ(x) ∩ kerDH(x) wesee that a normal periodic orbit is nondegenerate and in particular transversal. In the case oftrivial symmetry Γ = id, a periodic orbit of (2.1) is normal if and only if it is nondegenerate.For nontrivial symmetry groups a nondegenerate periodic orbit need not be normal as for anondegenerate periodic orbit gµx lies in DJ(x) ∩ kerDH(x), but might not lie in the image

of (DxΦT (x) − id). Galan et al [4, 15] prove continuation results for normal periodic orbitsand design numerical continuation methods which are similar to our shooting equations (2.25)in the case of trivial drift symmetry σ, see [15, Theorem 13]. However they only computeperiodic orbits, not RPOs and they require the solution to lie in a Poincare-section to theperiodic orbit by adding phase conditions. Moreover, they study persistence of periodic orbitswhen external parameters are varied, see [15, Theorems 7, 14], whereas we restrict attentionto continuation in the conserved quantities, energy and momentum, of the system. In [15,Theorem 7] they continue normal periodic orbits in external parameters, in [15, Theorem 14]they instead require that the eigenvalue 1 of the linearization M = DΦT (x) of the periodic

17

orbit has geometric multiplicity dim Γ + 1, i.e., that dim ker(M − id) = dim Γ + 1. Note thatdim ker(M− id) = codim range(M− id). We see from Proposition 2.17 (with ξ = 0, α = id), thatDH(x),DJeξ

i∈ codim range(M − id), where i = 1, . . . , r, r = dim Γµ. Because of σ = id and the

structure of DN (see Proposition 2.17) we have codim range(M − id) ≥ dim Γ+1. Therefore thecondition dim ker(M − id) = dim Γ + 1 implies that the matrix [M10,M1,M12] has full rank sothat the periodic orbit is transversal. But the condition dim ker(M − id) = dim Γ + 1 from [15,Theorem 14] is stronger than transversality as conditions on ΘN and DN have to be imposedtoo.

2.4 Numerical path-following of RPOs

As before let x lie on an RPO with drift-momentum pair (σ, µ) and let ξ1, . . . , ξr be a basis ofg(σ,µ). For µ ∈ g∗ let µj = µ(ξj), j = 1, . . . , g.

We can fix the momentum value µ and continue in energy, i.e., solve the equation

F µ(x, T, ξ, λE , λµ) =

F (x, T, ξ, λE , λµ)J2(x) − µ1

...Jr(x) − µr

. (2.30)

The x-component of the solution y = (x, T, ξ, 0, 0) of (2.31) then lies in X µ = J−1(µ) which canbe shown as in Section 2.1.3: First, note that by identifying g with g∗ by a Γ-invariant innerproduct we get µj = 0, j = r + 1, . . . , g. For σ ∈ gσ close to σ we have gσ ⊆ gσ , g∗

σ ⊆ g∗σ.

By construction, ξ ∈ gσ at any solution y = (x, T, ξ, 0, 0) of (2.31). Since Ad∗σµ = µ for the

drift symmetry σ = α exp(−τξ), τ = T/`, and momentum µ = J(x) of the RPO given byy = (x, T, ξ, 0, 0) (see (2.18)), we conclude that Jj(x) = 0, j = q+1, . . . , g. If q 6= r then µj = 0,j = r + 1, . . . , q is one of the equations of F , see (2.26). Hence the solutions of (2.30) satisfyJ(x) = µ.

Alternatively, We can fix the energy and g − 1 out of the first g momentum components toget a one-parameter family, without loss of generality the g − 1 components µ[ = (µ2, . . . , µg)g = dim Γ, by solving

F E,µ[

(x, ξ, T, λE , λµ) =

F (x, ξ, λE , λµ)J2(x) − µ2

...Jr(x) − µr

H(x) − E

. (2.31)

As above, we see that the solutions y = (x, T, ξ, 0, 0) of (2.30) satisfy

x ∈ X E,µ[

= x ∈ X , H(x) = E, Jj(x) = µj , j = 2, . . . , g.

Corollary 2.28 Let x lie on transversal RPO with regular drift-momentum pair (σ, µ) andenergy E, define the Poincare-map Π as before and choose coordinates as in Lemma 2.16. Thenthe matrix

[DEΠN1(0),DwΠN1

(0) − idN1] (2.32)

has full rank if and only if the (r, r)-matrix ∂sχ(s)|s=0 from Theorem 2.22 is invertible. In thiscase there is a path x(ε) ∈ J−1(µ) of points on RPOs P(ε) with energy E(ε) such that x(0) = x,P(0) = P, E(0) = E. If the RPO P is nondegenerate, we can choose ε = E. MoreoverDF µ(x, T , ξ, 0, 0) has full rank and there is a two-dimensional solution manifold of (2.30) inX µ, which consists of points on the family P(ε).

18

If (2.32) is satisfied we say that the RPO P is transversal with respect to C(x) := H(x).Under this assumption, (2.30) can be solved by standard numerical methods, for example bythe Gauss-Newton method, for initial values close to y = (x, T , ξ, 0, 0).

Corollary 2.29 Let x lie on transversal RPO with regular drift-momentum pair (σ, µ) andenergy E, define the Poincare-map Π as before and choose coordinates as in Lemma 2.16. Thenthe matrix

[Dχ1ΠN1

(0),DwΠN1(0) − idN1

] (2.33)

has full if and only if the (r, r)-matrix

∂s(E,χ2, . . . , χr)(s)|s=0

from Theorem 2.22 is invertible. In this case there is a path x(ε) ∈ X E,µ[

of points on RPOsP(ε) with energy E(ε) such that x(0) = x, P(0) = P, µ1(0) = µ1. If the RPO P is nondegenerate

then we can choose ε = µ1. Moreover DF E,µ[

(y) has full rank, and there is a two-dimensionalsolution manifold of (2.31), which consists of points on the family P(ε)

If (2.33) is satisfied we say that the RPO P is transversal with respect to C(x) := J1(x)(analogously to the case of relative equilibria, see Section 2.1.3). nder this assumption, (2.31)can be solved by standard numerical methods, for example by the Gauss-Newton method, forinitial values close to y = (x, T , ξ, 0, 0).

For continuation we use the tangent t(y) to the solution manifold of F E,µ[

= 0 or F µ = 0 at

y that lies in the kernel of DF E,µ[

(y) or DF µ(x, T , ξ, 0, 0) and has an x-component tx = tx(y)orthogonal to gσx and fH(x).

Remark 2.30 In the actual implementation it is more convenient to add the continuationparameter C = E rsp. C = µ1 to the vector of unknowns y = (x, T, ξ, λE , λµ, C) and to addthe additional equation C(x) − C = 0 to F . Then tC = DC(x)tx as required, and computingboundary points in C is greatly simplified.

Remark 2.31 In the multiple shooting context, let F µ : X k × R2r+2 → X k × Rr rsp. F E,µ[

:X k × R2r+2 → X k × Rr be given by (2.27) with r conserved quantities fixed as in (2.30) and(2.31)). At a transversal RPO the kernel of DF (y) is (r + 1)-dimensional by Corollaries 2.28rsp. 2.29. We write vectors in the kernel as t = (t1, . . . , tk, tT , tξ , tλE , tλµ), where tj ∈ X ,j = 1, . . . , k, tξ ∈ Rq, tλµ ∈ Rr. Then the kernel of DF (y) contains the vectors tf , tξ1 , . . . , tξg

wheretfi = f(xi), i = 1, . . . , k, tfT = 0 = tfλE

= 0, tfξ = 0, tfλµ= 0, (2.34)

andtξj

i = ejξxi, i = 1, . . . , k, t

ξj

T = tξj

λE= 0, t

ξj

ξ = 0, tfξj= 0. (2.35)

We define the continuation tangent t = (t1, . . . , tk, tT , tξ , tλE = 0, tλµ = 0) as the element of thekernel which is orthogonal to tf and tξ1 , . . . , tξr , as in [25] for nondegenerate RPOs.

Remark 2.32 In the case of dissipative parameter dependent Γ-equivariant ODEs (2.15) wecall an RPO through x with parameter λ transversal if [DxΠ(x, λ) − id,DλΠ(x, λ)] has rank Nand nondegenerate if DxΠ(x has no eigenvalues one. Near a transversal RPO there is a path(x(s), λ(s)) of fixed points of Π corresponding to RPOs of (2.15). Numerically, this path canbe computed by solving the underdetermined system F (x, T, ξ, λE , λµ) = αΦT

`(x; ξ, λ) − x = 0

where Φt(x; ξ, λ) is the flow of

x = f(x, λ) −

q∑

i=1

ξieiξ.

19

Here q = dim gσ and σ is the drift symmetry at the RPO through x. This case is similar to, butmuch simpler than in the Hamiltonian case, see (2.24), (2.25). Note that typically r = dimgσ

is locally constant, see Remark 2.15.

2.5 Turning points of relative equilibria and RPOs

In this section we deal with the most simple situation of a critical relative equilibriaum RPO,namely we consider a transversal relative equilibrium/RPO with regular velocity-momentumpair / drift-momentum pair which ceases to be non-degenerate. As before, we restrict to thecase of free group actions.

2.5.1 Turning points of Hamiltonian relative equilibria

Let x lie on a transversal relative equilibrium and regular velocity-momentum pair (ξ, µ) and as-sume that the relative equilibrium is degenerate. By Theorem 2.6 it persists to an r-dimensionalfamily x(s) of relative equilibria nearby with velocity-momentum pairs (ξ(s), µ(s) = µ + χ(s)),χ ∈ g∗

(ξ,µ)' Rr. Since the relative equilibrium is dengenerate the matrix JN1

D2wh(0) has an

eigenvalue 0. Therefore we see from the proof of Theorem 2.6 that the (r, r)-matrix ∂χ∂s (0) is

singular. The eigenvalue 0 of JN1D2

wh(0) is of algebraic multiplicity two since JN1D2

wh(0) isinfinitesimally symplectic, and, generically, of geometric multiplicity one, see [13]. As discussedabove, see (2.14), we numerically compute a one-parameter family of relative equilibria by fixingall components of the momentum map except for the first component, C(x) := J1(x).

Proposition 2.33 Assume that the relative equilibrium through x is degenerate, but transversal

with respect to the conserved quantity C(x) := J1(x) of (2.1). Let x(ε) ∈ X µ[

be the path ofrelative equilibria from Corollary 2.10, c(ε) = C(x(ε)). Then generically x is a turning point inc(ε) = C(x(ε)), i.e., c′(0) = 0 and c′′(0) 6= 0, and the pair of eigenvalues of the linearization ofthe relative equilibria, which collide at 0 at the turning point, lies on the imaginary axis beforethe turning point is passed, and on the real axis after the turning point, or vice versa.

Proof. As we see from the proof of Theorem 2.6, the relative equilibria x(s) ' (ν(s), w(s)) ∈N0 ⊕ N1 correspond to equilibria of the ν-dependent Hamiltonian system dw

dt = fN1(ν, w) after

symmetry reduction by Γ = Γξ. Here, ν = χ and so νj = 0, j = 2, . . . , r, ν1 = c. Definethe parameter-dependent Hamiltonian vectorfield fN1

(c, w) by choosing ν in this way. By as-sumption DwfN1

(0, 0) has a one-dimensional kernel and [DcfN1(0, 0),DwfN1

(0, 0)] has full rank.Denote the solutions of fN1

(c, w) = 0 corresponding to x(ε) as (c(ε), w(ε)) where c = c(0). SinceDwfN1

(c, 0)w′(0) + DcfN1(0, 0)c′(0) = 0, we infer from the above that DwfN1

(0, 0)w′(0) = 0,DcfN1

(c, 0) 6= 0, and so c′(0) = 0. Moreover, applying well-known results on turning points ofHamiltonian systems on the Hamiltonian vectorfield fN1

(c, w), we see that typically c′′(0) 6= 0and that the pair of eigenvalues of DfN1

(c(ε), w(ε)), which collide at 0 at ε = 0, is on the imag-inary axis for ε < 0 (ε > 0) and on the real axis for ε > 0 (ε < 0).

2.5.2 Turning points of RPOs

Let x lie on a transversal RPO with relative period τ and regular drift momentum pair (σ, µ)and energy E, and assume that the RPO is degenerate. By Theorem 2.22 it persists to an (r+1)-dimensional family x(s) of RPOs with drift-momentum pairs (σ(s), µ(s) = µ+χ(s)) and energyE(s), where χ ∈ g∗

(σ,µ) ' Rr. Since the RPO is dengenerate the matrix DwΠN1(0) = DxΠE,µ(x)

has an eigenvalue 1. Therefore we see from the proof of Theorem 2.22 that the (r+1, r+1)-matrix∂∂s (E,χ)(0) is singular.

20

As discussed above, see (2.31) and (2.30), we numerically compute a one-parameter familyof RPOs and continue RPOs with respect to a component of the momentum map or the energy.Then we have:

Proposition 2.34 Assume that the RPO through x is degenerate, but transversal with respectto the conserved quantity C(x) := J1(x) or C(x) := H(x) of (2.1). Denote by x(ε) ∈ X µ or

x(ε) ∈ XE,µ[

the path of RPOs from Corollary 2.28 and Corollary 2.29, and let c(ε) = C(x(ε)).Then generically x is a turning point in c(ε) = C(x(ε)), i.e., c′(0) = 0 and c′′(0) 6= 0, and thepair of eigenvalues of the linearization of the RPOs, which collide at 1 at the turning point, lieson the unit circle before the turning point is passed, and on the real axis after the turning point,or vice versa.

Proof. From the proof of Theorem 2.22 we see that x lies on a turning point in the conservedquantity C(x) if DΠE,µ(x) = DΠN1

(0) has rank defect one, i.e., has a double eigenvalue 1with geometric multiplicity 1, and if D(w,C)ΠN1

(0) has full rank. Then c′(0) = 0. This can beseen as follows: Since x is degenerate, but transversal, we have (DwΠN1

(0) − id)w′(0) = 0 andDCΠN1

(0) 6= 0. Then (DwΠN1(0) − id)w′(0) + DCΠN1

(0)c′(0) = 0 implies c′(0) = 0.From the proof of Theorem 2.22 we see that the path of RPOs x(ε) ' (ν(ε), w(ε), E(ε))

corresponds to fixed points of the (ν, E)-dependent symplectic map Π eN1(ν, w, E) = w after

reduction by Γ = Γσ such that x(0) = x ' (0, 0, E). Here ν = χ ∈ g∗(σ,µ), so that in the

case C(x) = H(x) we have ν ≡ 0 and in the case C(x) = J1(x) we have νj ≡ 0, j = 2, . . . , r,E(ε) = E. So we obtain a symplectic map Π eN1

(c, w) which depends on one parameter c, andat c = C(x), w = 0, the derivative DwΠ eN1

(0, 0) has an eigenvalue 1 with geometric multiplicityone and algebraic multiplicity two such that [DcΠ eN1

(0, 0),DwΠ eN1(0, 0)] has full rank.

Applying the well-known results on turning points of parameter-dependent symplectic maps,see, e.g., [13], on the symplectic map Π eN1

then proves the proposition.

2.5.3 Detection and computation of turning points of relative equilibria and RPOs

Turning points along paths of RPOs continued in the conserved quantity C(x) can be detectedby a sign change of

u(y) = 〈tx,∇C(x)〉 (2.36)

between two consecutively computed solutions y(0) = (x(0), T (0), ξ(0)) and y(1) = (x(1), T (1), ξ(1))of (2.31) or (2.30) respectively. Here tx is the x-component of the continuation tangent t(y) aty = (x, T, ξ). Analogously, turning points of paths of relative equilibria which are continuedin C(x) = J1(x) are detected by a sign change of (2.36) between two consecutively computedsolutions of (2.14).

Once detected, turning points can be computed by a combination of Hermite interpolationand subdivision of (2.36) along the solution path of (2.31) or (2.30), see, e.g., [23] and referencestherein.

Remark 2.35 In the case of non-Hamiltonian systems, see Remark 2.11 and 2.32, turningpoints are detected and computed in the same way, only that now c = λ and u(y) = tλ is theλ-component of the continuation tangent ty at y.

21

3 Hamiltonian relative period doubling bifurcations

In this section we first present a theorem on relative period doubling bifurcations of RPOs withregular drift momentum pair (Section 3.1). Then, in Sections 3.2 and 3.3, we show how to detectand compute relative period doubling bifurcations along branches of RPOs. In Section 3.4 wethen deal with the detection and computation of relative period halving bifurcations duringnumerical path-following of RPOs.

3.1 A theorem on relative period doublings of RPOs

Let x lie on a nondegenerate RPO P of a free group action with regular drift momentumpair (σ, µ) and energy E. We say that x is a relative period doubling bifurcation point ifDΠE,µ(x) = DΠN1

(0, 0, 0) has an eigenvalue −1. This eigenvalue has algebraic multiplicity ≥ 2since, by Lemma 2.16 b), the map ΠN1

is symplectic. We make the generic assumption that theeigenvalue −1 of DΠN1

(0) has algebraic multiplicity 2 and geometric multiplicity 1 and denotethe eigenvector of DΠN1

(0) to the eigenvalue −1 by w.Let x(E,χ), χ ∈ g∗

(σ,µ), be the family of RPOs through x = x(E, 0) whose existence was

proven in Theorem 2.22. Denote its coordinates on N as (ν(χ,E), w(χ,E)). Let λ1(E,χ)and λ2(E,χ) be the eigenvalues of DwΠN1

(ν(E,χ), w(E,χ)) which collide at the bifurcation:λ1(E, 0) = λ2(E, 0) = −1 and denote the generalized eigenspace of DwΠN1

(ν(E,χ), w(E,χ), E)to the eigenvalues λ1(E,χ) and λ2(E,χ) by Y(E,χ). Then Y(E,χ) and the matrices

M1(E,χ) := DwΠN1(ν(E,χ), w(E,χ), E)|Y(E,χ)

are symplectic (2, 2)-matrices and depend smoothly on (E,χ). If

ψ(E,χ) := tr(M(E,χ)) + 2 = 0, (3.1)

then M(E,χ) has a double eigenvalue −1. We assume that

D(E,χ)ψ(E, 0) 6= 0.

In [13] it is shown that this condition is generically satisfied in the space SP(2) of symplectic(2, 2)-matrices. Under this condition the equation ψ(E,χ) = 0 determines a smooth hypersurfaceof codimension 1 in (E,χ)-space. We choose coordinates such that D(E,χ1)ψ(E, 0) 6= 0.

Remark 3.1 At a transverse passing of the hypersurface ψ(E,χ) = 0 in SP(2) the eigenvaluesare either on the unit circle before the collision and on the real axis after or vice versa: tosee this let λ1(E,χ) = −1 + δ(E,χ) where δ(E, 0) = 0. Then λ2(E,χ) = 1/(−1 + δ(E,χ)) =−1−δ(E,χ)−δ2(E,χ)+h.o.t., and so ψ(E,χ) = −δ2(E,χ)+h.o.t.. As ψ(E,χ) changes sign atthe relative period-doubling hypersurface determined by (3.1), we see that δ(E,χ) is real beforebifurcation and complex after bifurcation or vice versa.

In the following theorem we make the generic assumption that DEψ(E, 0) 6= 0. We deal withthe case DEψ(E, 0) = 0, Dχ1

ψ(E, 0) 6= 0 in Remark 3.3.

Theorem 3.2 Let x be a relative period doubling bifurcation point with a regular drift-momentumpair (σ, µ) and energy E. Assume that (σ2, µ) is also a regular drift-momentum pair and thatr = r(σ,µ)(Γ) = r(σ2 ,µ)(Γ). Then, generically, a family x(ε, χ), χ ∈ g∗

(σ,µ), of points on RPOs

P(ε, χ) with relative period τ(ε, χ), drift symmetry σ(ε, χ), momentum µ(ε, χ) = µ + χ, andenergy E(ε, χ) bifurcates from x such that

x(0, 0) = x, τ (0, 0) = 2τ , σ(0, 0) = σ2, µ(0) = µ, E(0) = E.

22

Moreover,

σ(ε, χ) = α2 exp(τ (ε, χ)ξ(ε, χ)), with ξ(s) ∈ g(σ,µ), Adαξ(ε, χ) = ξ(ε, χ).

We have Dεx(0, 0) = w, and, with x(−ε, χ) = Π(x(ε, χ)), we obtain a smooth manifold of RPOswith relative period τ(−ε, χ) = τ(ε, χ), energy E(−ε, χ) = E(ε, χ), momentum µ(−ε, χ) = µ(ε, χ)and drift symmetry σ(−ε, χ) = σ(ε, χ).

Proof. We reduce by Γ = Γσ only. Since Γ is compact the matrices Adσ and Ad∗σ|g∗

µdo not have

Jordan blocks. Therefore we conclude from (2.22) that the RPO through x is nondegeneratewhen considered as an RPO for the symmetry group Γ = Γσ. Then, as before, since g(σ,µ)

is abelian by Remark 2.15, we have ΠN0(ν, w, E) ≡ ν, where ν = χ ∈ g∗

(σ,µ) ' Rr. Here

N0 ' g∗(σ,µ), (ν, w) ∈ N = N0 ⊕ N1 and N is the slice to the action of Γ from Lemma 2.16.

Moreover, if we reduce by Γ only, then DwΠN1(0, 0, 0) still has an eigenvalue −1 of multiplicity

two, and the symplectic map Π eN1(ν, ·, E) undergoes a period-doubling bifurcation at E = E,

ν = 0.On the manifold of fixed points w(E, ν) of ΠN1

(ν, w, E), corresponding to RPOs P(E,χ),χ = ν, of (2.1) there is a codimension one manifold w(χ) with parameters ν = χ and E(χ) suchthat DwΠN1

(ν, w, E)(χ)) has an eigenvalue −1, where χ ∈ Rr. Then, generically, the parameterdependent equation

Π2eN1

(ν, w, E) = w

has a second solution manifold w(ε, χ) to the parameters ν(ε, χ) = χ and E = E(ε, χ) such thatw(0, χ) = w(χ) [13].

Hence, we obtain an (r + 1)-dimensional family of fixed points v(E,χ) := (ν, w, E)(ε, χ) ofΠ2

eN, which gives rise to a family x(ε, χ) of points on RPOs P(ε, χ) with relative period τ (ε, χ)

such that τ(0, 0) = 2τ , drift symmetry σ(ε, χ) with σ(0, 0) = σ2, momentum µ(ε, χ) = µ + χ,and energy E(ε, χ) such that E(0, 0) = E.

From σ(ε, χ)µ(ε, χ) = µ(ε, χ) (as we saw in (2.18)) we conclude that σ(ε, χ) ∈ Γµ(ε,χ). Since

N0 ' g∗(σ,µ) is invariant under Γµ we know that Γµ(ε,χ) ⊆ Γµ = Γ(σ,µ) and, therefore, σ(ε, χ) ∈

Γ(σ,µ). By Remark 2.15 we know that Γid(σ,µ) is abelian and that g(σ,µ) = gα ∩ g(ξ,µ). So we

can decompose σ(ε, χ) = α2 exp(τ (ε, χ)ξ(ε, χ)) and ξ(ε, χ) ∈ g(σ,µ), Adαξ(ε, χ) = ξ(ε, χ), andAd∗

αχ = χ.Moreover, from [13] we have Dεw(0, 0) = w, and with w(−ε, χ) := ΠN1

(ν(ε, χ), w(ε, χ)), weobtain a smooth manifold of RPOs at ε = 0 and see that E(ε, χ) = E(−ε, χ), τ (ε, χ) = τ (−ε, χ).Also, σ(ε, χ) = γ(x(−ε, χ))γ(x(ε, χ)) equals σ(−ε, χ) = γ(x(ε, χ))γ(x(−ε, χ)) as γ(x(±ε, χ)) ∈αΓid

(σ,µ) and g(σ,µ) = g(σ,µ) ⊂ gα. Furthermore, χ = ν(−ε, χ) = αν(ε, χ).

Remark 3.3 If DEψ(E, 0) = 0, Dχ1ψ(E, 0) 6= 0, then Theorem 3.2 remains valid if we change

the parametrization from (ε, χ) to s = (ε, χ2, . . . , χr, e) where e = E − E. In this case thecomponent ν(s) ∈ g∗

(σ,µ) of the bifurcating RPOs through x(s) ' (ν(s), w(s), E(s)) on the slice

N and the momentum µ(s) of the RPO through x(s) depend nonlinearly on s and E(s) = E+e.

Remark 3.4 In the comoving frame ξ a relative period doubling bifurcation becomes a flip-doubling bifurcation or a flip-pitchfork bifurcation of the corresponding T = `τ -periodic orbit[5, 23]. The drift symmetry of the bifurcating periodic orbit in the comoving frame is α = α2.If ` is even then its period in the comoving frame is T ≈ T and its spatio-temporal symmetrygroup is Z˜ with ˜= 2`. This scenario is called a flip-pitchfork bifurcation in [5]. If ` is odd then

its period in the comoving frame is T ≈ 2T and its spatio-temporal symmetry group remains Z ˜

with ˜= `. This scenario is called flip-doubling bifurcation in [5].

23

Remark 3.5 The assumption r(σ,µ) = r(σ2,µ) implies that the block M0 = Ad∗σ−1 |g∗

µin the

linearization M = σDΦτ (x) of the RPO at the bifurcation point does not have an eigenvalue−1, see Proposition 2.17. Therefore, the eigenvalue −1 of DΠ(x) has algebraic multiplicity two.Generically, the drift symmetry σ2 of the bifurcating RPO at the bifurcation point is regular.In this case, σ is regular too, and the above assumption reads rσ2 = rσ . Then Adσ does nothave an eigenvalue −1 either, and the eigenvalue −1 of M has algebraic multiplicity two, too.If σ2 is not regular then M might have additional eigenvalues −1.

If we continue in energy while fixing the value of the momentum map as in (2.30), we getthe following corollary:

Corollary 3.6 Under the assumptions of Theorem 3.2, if DEψ(E, 0) 6= 0, there is a smoothpath x(ε) ∈ X µ with x(0) = x, ε ≥ 0, such that x(ε) lies on an RPO P(ε) of (2.1) with relativeperiod τ(ε), with drift symmetry σ(ε) at x(ε), energy E(ε) and momentum µ where σ(0) = σ2,E(0) = E, σ′(0) = 0, E′(0) = 0.

Similarly, if we continue in a momentum component while fixing the other momentum compo-nents and the energy as in (2.31), we get the following corollary:

Corollary 3.7 Under the assumptions of Theorem 3.2, if Dχ1ψ(E, 0) 6= 0, there is a smooth

path x(ε) ∈ X E,µ[

with x(0) = x, such that x(ε) lies on an RPO P(ε) of (2.1), ε ≥ 0, withrelative period τ (ε), drift symmetry σ(ε) at x(ε), energy E and momentum µ(ε) = (µ1(ε), µ

[)where σ(0) = σ2, µ1(0) = µ1, µ

′1(0) = 0, σ′(0) = 0.

3.2 Detection of relative period doublings

Assume as before that at x there is a relative period doubling bifurcation and that DwΠN1(0)

has an eigenvalue −1 with multiplicity 2. Let P(c) be a branch of RPOs with C(x) = J1(x) orC(x) = H(x) as in Corollaries 3.6, 3.7 above. Choose coordinates on the Poincare section S atx as in Lemma 2.16 and let x(c) ' (ν(c), w(c), E(c)) ∈ N such that x(c) = x.

Lemma 3.8 Generically, at a relative period doubling bifurcation point x ∈ P along a path ofRPOs x(c) ∈ P(c), the determinant det(DwΠN1

((ν, w,E)(c)) + idN1) changes sign.

Proof. Let M1(c) := DwΠN1((ν, w,E)(c)). Under the above assumptions, the pair λ1,2(c)

of eigenvalues of M1(c) with λ1,2(c) = −1 generically lies on the unit circle before collisionand on the real axis after collision or vice versa as we saw in Remark 3.1. Let B(c) =DwΠN1

((ν, w,E)(c))|Y(c) and let Y(c) be the generalized eigenspace of M1(c) to the eigen-values λ1,2(c) with λ1,2(c) = −1. Then, λ1(c) = −1 + a(c), λ2(c) = 1/(−1 + a(c)) for c < cwith a(c) ∈ R, a(c) = 0, and λ1,2(c) = e±iφ(c) with φ(c) ∈ R, φ(C) = π or vice versa. Then,det(B(c) + id2) = −a(c)2 +O(a3(c)) < 0 for c < c and

det(B(c) + id2) = (eiφ(c) + 1)(e−iφ(c) + 1) = 2(1 + cosφ(c)) > 0 for c > c,

or vice versa. Since the other eigenvalues of M1(c) are bounded away from −1 near c = c, weconclude that det(M1(c) + idN ) also changes sign.

Under the assumptions of Theorem 3.2 also det(DΠ(x(c)) + idN ) changes sign at c = c, as wesaw in Remark 3.5; hence, in a manner similar to the case of dissipative systems with discretesymmetry groups [23], relative period doubling bifurcations can be detected by a sign change of

d(y) = det(PN αDxΦτ (x; ξ)PN + idN ). (3.2)

24

Here PN is the projection from the phase space X to the tangent space N of the Poincaresection at x with kernel TxP . Recall that αDxΦτ (x; ξ) is related to the derivative of the shootingequations, see Remark 2.26. The projection P eN1

can be computed numerically as an orthogonalprojection to gσx and fH(x) and kerDJi(x), i = 1, . . . , r, r = r(σ,µ).

Remark 3.9 Note that the trace condition (3.1) can not be used to detect relative period-doubling bifurcations in general (of course this test works if dimX = 2). The determinanttest based on (3.2) also works if instead of a projection on N a projection on in (3.2) is used.Since the eigenvalue −1 generically has algebraic multiplicity two at bifurcation, see Remark3.5, we could remove the projection PN completely from the test function d(y) or replace it a

the projection onto N1 with kernel N0 ⊕ gσx ⊕ span(fH(x)). Here N = N0 ⊕ N1 ⊕ N2 is thenormal space for the symmetry group Γ = Γσ. From a theoretical point of view, a projectionon N1 is possible as well. However, the dimension of N1 may vary along the branch of RPOs,and this may cause numerical instability. Note that the dimension of N is constant along thebranch of RPOs as the group is assumed to act freely. Moreover, since µ is a regular momentumfor the group Γ = Γσ, the dimension of N1 is also locally constant.

3.3 Computation of relative period doubling points and branch switch-

ing

Relative period doubling bifurcations can be computed similarly to period doubling bifurcationsin the dissipative case, see, e.g., [9, 23], by subdivision of (3.2) along a family of solutions y(c)of (2.30) or (2.31) respectively.

Once a relative period doubling point (x, τ , ξ) on the original branch has been found, thestarting point y = (x, T , ξ) for the bifurcating branch has to be computed. We set T = T at thebifurcation point for a relative flip pitchfork bifurcation (as defined in Remark 3.4) and T = 2Totherwise, and ξ = ξ.

In a manner similar to the case of dissipative systems with finite symmetry [23], we computethe tangent t for the bifurcating branch as follows: At a relative period doubling bifurcationthe matrix M in (2.22) has an eigenvalue −1 which is generically of algebraic multiplicity two(Remark 3.5) and M = σDΦτ (x) has an eigenvector w to the eigenvalue −1 corresponding tothis eigenvalue of M1. The fact that w lies in the kernel of M +id follows from Proposition 2.17,the form of DN and the fact that A0 = Adσ|gµ has no eigenvalue −1. We compute that

DH(x)w = DH(x)Mw = −DH(x)w = 0.

Moreover,DJej

ξ(x)w = 0, j = 1, . . . , rµ,

since Ad∗σ|g∗

µhas no eigenvalue −1. So, for the computation of the branch down direction we

only have to project w orthogonal to f(x) and gx to obtain a projected vector w which lies inN1. This vector w is an eigenvector of M1 to the eigenvalue −1. Let t = (tx, tT , tξ, tλE , tλµ) bethe continuation tangent for the bifurcating branch. Clearly, tλE = 0, tλµ = 0. By Theorem

3.2 the bifurcating family of RPOs P(ε, χ) satisfies Dετ (0) = 0 and Dεξ(0) = 0 so that tT = 0,tξ = 0. Moreover Dεx(0) = w so that tx = w.

Remark 3.10 In the multiple shooting context, see Remark 2.31, it is natural to doublethe number of multiple-shooting points on the bifurcating branch. The second half of x =(x1, . . . , x2k) is computed as

xi = xi for i = 1, . . . , k, xi+k = α−1xi for i = 1, . . . , k.

25

The continuation tangent t = (tx, 0, 0, 0, 0, 0) of the bifurcating branch is computed in a similarway: we compute the eigenvector w to the eigenvalue −1 of M = σDΦτ (x) (see Remark 2.26)and set

t1 = w, ti+1 = Giti, i = 1, . . . , k − 1, ti+k = −α−1ti, i = 1, . . . , k − 1.

Here the matrices

Gi = DxΦ∆siT

` (xi;λ), i = 1, 2, . . . , k − 1, Gk = αDxΦ∆skT

` (xk ;λ)

are available as derivative of (2.27) if for example a Gauss-Newton method is used to solve(2.27). Let F : X k × Rr+q+2 → X k × Rq , be given by (2.30) or (2.31). Define tf , tξ1 , . . . , tξr asin (2.34) and (2.35), see Remark 2.31. Define tf , tξj ∈ X 2k × Rr+1, j = 1, . . . r, as

tfi = tfi , tξj

i = tξj

i , i = 1, . . . , k, tfi+k = α−1tfi , tξj

i+k = α−1tξj

i , i = 1, . . . , k,

and set tfT = 0, tξj

T = 0, tfξ = 0, tξj

ξ = 0. Projecting t orthogonally to tf , tξ1 , . . . , tξr we obtaina continuation tangent for the bifurcating branch.

3.4 Detection and computation of relative period halving bifurcations

In this section we show how relative period halving bifurcations, which occur along branches ofRPOs defined by (2.30) or (2.31) and satisfy the assumptions of Corollary 3.6 or Corollary 3.7,respectively, can be detected and computed numerically.

As in the case of dissipative systems with finite symmetry group discussed in [23], we computeall choices of α such that α2 = α, and at each solution y = (x, T, ξ) we compute for each choiceof α,

u(y) := αΦT2` (x; ξ) − x.

A relative period halving bifurcation is detected, if

〈u(0), u(1)〉 < 0, u(0) = u(y(0)), u(1) = u(y(1)), (3.3)

between two consecutively computed solutions y(0) = (x(0), T (0), ξ(0)) and y(1) = (x(1), T (1), ξ(1))of (2.30) or (2.31), respectively. This can be seen as follows: by Theorem 3.2, the functionsx(ε), ξ(ε) and τ(ε) = T (ε)/` are smooth in ε. Moreover at the relative period halving point,u(y(ε))|ε=0 = 0 and Dεu(y(ε))|ε=0 = −2w 6= 0.

By Corollary 3.6 or Corollary 3.7 at the relative period halving bifurcation point we haveDεC(x(ε))|ε=0 = 0. Therefore, once a relative period halving bifurcation has been detectedbetween y(0) and y(0), an initial guess y for the bifurcation point can be computed by usinginterpolation (y(τ), c(τ)) between (y(0), C(x(0)) and (y(1), C(x(1)) to find a critical point of c(τ).Then the bifurcation point can be computed by switching to the branch of RPOs with halvedrelative period and using the methods for relative period doubling bifurcations from the previoussection on this branch (similarly to case of dissipative systems with finite symmetry groups [23]).

Since we assume (see Theorem 3.2) that the relative period halving point is an RPO withregular drift momentum pair (σ, µ), by Remark 2.15 the subalgebra g(σ,µ) is abelian. Therefore

Γid(σ,µ) is isomorphic to Tn and is a subgroup of Z(α). Here, Tn = (S1)n is an n-dimensional

torus. The group L = Γ(σ,µ) ∩ Z(Γid(σ,µ))/Γ

id(σ,µ) is finite because Γ is assumed to be compact.

Since r(σ2,µ) = r(σ,µ) in Theorem 3.2, all possible choices of square roots α of α lie in Z(Γid(σ,µ)).

So after reduction by Γid(σ,µ), we are back to the case of finite groups L treated in [23].

Assume that α has a square root α ∈ L: α2 = α. Then α has indeed two square roots inZ2`(α) ⊆ L, namely α1 = α and α2 = α`+1. In the case of continuous symmetry groups Γ the

26

test (3.3) for relative period halving bifurcations has to be used for all α = α1,2 exp(ξ) whereξ ∈ tn is such that ξj = 0 or ξj = π, j = 1, . . . , n. Here tn is the Lie algebra of Tn. For example,if Γid

(σ,µ) = SO(2), and Rφ is a rotation by φ, then exp(ξ) = id or exp(ξ) = Rπ.

Remark 3.11 In the case of non-Hamiltonian systems, see Remark 2.32, relative period dou-bling points are detected and computed in the same way, with c = λ in Section 3.3. In the caseof relative period halving, the possible drift symmetries α in the comoving frame for the relativeperiod halving bifurcation point can be found as in the Hamiltonian case above, if we assumethat both, the drift symmetry σ of the RPO on the branch with halved relative period, and thedrift symmetries of the RPOs on the original branch near the bifurcation are regular and thatrσ = rσ2 .

4 Application to rotating choreographies

In this section we apply our methods for numerical bifurcations of RPOs to rotating choreogra-phies of the three-body problem. Using the software package SYMPERCON [19] we find thatthe type II rotating choreography undergoes a symmetry-increasing flip pitchfork bifurcation inthe corotating frame to the type I rotating choreographies. We also report on several relativeperiod doubling bifurcations and a turning point of the planar (type III) rotating choreographies.

4.1 N-body problems and their symmetries

We consider the motion of N identical bodies of mass 1 in R3 subject to internal forces theyexert on each other. We assume that these forces are given by 1

2N(N − 1) identical copies of apotential energy function V (one for each pair of bodies), which depends only on the distancebetween the bodies. Writing pj for the momenta conjugate to the positions qj , q = (q1, . . . , qN ),p = (p1, . . . , pN), the Hamiltonian is

H(q, p) =1

2

N∑

j=1

|pj |2 +

∑

i<j

V (rij), where rij = |qi − qj | and V (r) = −1

r. (4.1)

Excluding collisions, the configuration space Q is

Q = q = (q1, . . . , qN ) ∈ R3(N−1), qi 6= qj for i 6= j

and the phase space is Q× R3N ⊂ R6N . The equations of motion are

qj = pj , pj =∑

i6=j

qi − qjr3ij

, j = 1, . . . , N, (4.2)

and the angular momentum is J(q, p) =∑N

j=1 qj ∧ pj . Without loss of generality, the centre ofmass of the systems can be assumed to be fixed at 0 restricting the configuration space to

Q0 = q ∈ Q :

N∑

j=1

qj = 0

with corresponding phase space X = Q0 × R3(N−1) ⊆ R6(N−1).The N -identical-body Hamiltonian (4.1) has the following symmetries:

27

1. Rotations and reflections of R3: These form the orthogonal group O(3) which acts diago-nally on the positions and velocities:

R(q1, . . . , qN , p1, . . . , pN ) = (Rq1, . . . , RqN , Rp1, . . . , RpN), R ∈ O(3), qj , pj ∈ R3.

We define the symmetry axis of a rotational symmetry to be its usual rotation axis andthat of a reflectional symmetry to be the axis perpendicular to the reflection plane. In thefollowing let κi ∈ O(3) be the reflection with symmetry axis ei, i.e., let κi be such thatκiei = −ei, κie

j = ej for j 6= i, i, j = 1, 2, 3. We denote by Rj(φ) a rotation around theej axis by the angle φ.

2. Permutations of identical bodies: Because we assume that all the bodies are identical theHamiltonian is also invariant under the action of SN , the group of all permutations of theintegers 1, . . . , N :

π(q1, . . . , qN , q1, . . . , qN ) = (qπ(1), . . . , qπ(N), pπ(1), . . . , pπ(N)) π ∈ SN , qj , pj ∈ R3.

We frequently use the notation π = (π(1), . . . , π(N)).

Taken together these three symmetry groups give an action of

Γ = O(3) × SN

on X and leaves the Hamiltonian (4.1) invariant.

Remark 4.1 We call a matrix ρ ∈ GL(n) of a general Hamiltonian system (2.1) a time-reversingsymmetry of (2.1) if

H(ρx) = H(x), x ∈ X , ρJ = −Jρ.

This implies that fH(ρx) = −ρfH(x), x ∈ X , and so with x(t) also ρx(−t) is a solution of (2.1).In addition to the symmetries listed above, the N -body system (4.2) has the time-reversingsymmetry

ρ(q1, . . . , qN , p1, . . . , pN ) = (q1, . . . , qN ,−p1, . . . ,−pN) qj , pj ∈ R3

which generates a group Z2(ρ) of order 2. A periodic orbit through x with drift symmetry α,so that αΦτ (x) = x is transformed into an orbit through ρx = ραρ−1Φ−τ (ρx) and has driftsymmetry ρα−1ρ−1 = α−1; see [11] for a discussion of reversible RPOs.

4.2 Rotating choreographies of the three body problem

As in [3] we define:

Definition 4.2 A periodic orbit of (4.2) is a choreography if all the bodies follow the same pathin R3, separated only by a phase shift. This is equivalent to requiring that the spatio-temporalsymmetry group L of the periodic orbit contains an order N cyclic permutation π ∈ SN whichcan always be taken to act on Q by πq = (q2, q3, . . . , qN , q1). Similarly a relative periodic orbitof (4.2) with angular velocity ξ is a rotating choreography if it is a choreography in coordinatesrotating with velocity ξ.

The famous Figure Eight of Chenciner and Montgomery [2] is a choreography of the N -identical-body system (4.2) with N = 3.

Let e1, e2, e3 be a fixed orthogonal set of axes in R3 and assume that the eight lies in theplane perpendicular to e3 aligned along the e2 axis with both e2 axis and e1 axis as symmetry

28

axis. As before, for i = 1, 2, 3 let κi denote the (time-preserving) reflection with reflection axisei. The purely spatial symmetry group of the Figure Eight choreography is the group

K = Z2 = 〈κ3〉

generated by κ3, a reflection about the (x1, x2)-plane containing the Figure Eight. The driftsymmetry α := κ1(231) of the eight is a reflection in the e1, e2-plane composed with a cyclicpermutation of the bodies and has order ` = 6.

As shown by Chenciner et al. [3] three families of rotating choreographies bifurcate from theFigure Eight when momentum is switched on:

I. The type I family of rotating eights PI(E, ν) rotates around the e1 axis. Its drift symmetryin the rotating frame is αI = κ1(231) and has order `I = 6. The relative periods τI(E, ν)of the bifurcating RPOs are close to the relative period of the original Figure Eight,τI(0, 0) = τ .

II. The type II rotating eights PII(E, ν) rotate around the e2-axis. Its drift symmetry in therotating frame is αII = κ1κ3(231) and has order `II = 6. The relative period τII (E, ν) ofthe bifurcating RPOs satisfies τII(0, 0) = τ .

III. The type III rotating eights PIII(E, ν) are planar, i.e., have spatial symmetry K = 〈κ3〉.The drift symmetry in the rotating frame is αIII = (312) and has order `III = 3. Therelative period τIII(E, ν) of the family of bifurcating RPOs has doubled at the bifurcationpoint: τIII(0, 0) = 2τ .

It is well-known that the type I rotating choreography ends at a Lagrange relative equilibrium,see, e.g., the discussion in [3].

4.3 Flip up pitchfork bifurcation of the type II rotating eight

Along the branch of rotating eights of type II with drift symmetry σ(s) = exp(τ(s)ξ(s))α, whereα = αII = (231)R2(π) is of order ` = 6 and the energy is fixed to the value E = −1.287of the original Figure Eight, using SYMPERCON we found a symmetry-increasing pitchforkbifurcation in a corotating frame. The emanating RPOs P(s) have halved relative period anddrift symmetry σ(s) = exp(τ (s)ξ(s))α such that, at the bifurcation point, σ2(0) = σ, α2 =α and τ(0) = τ /2. Here, α = (312)κ2R2(π/2) has order ˜ = 12. This corresponds to abifurcation increasing the spatio-temporal symmetry Z6(αII) in the corotating frame to Z12(α).The bifurcation, marked as ”PD” in the second plot of Figure 1, occurs at

q1 = (0.5545,−0.3628, 0.2530), p1 = (0.3616, 0.1919,−0.7436),

q2 = (−0.4921, 0.3094, 0.3679), p2 = (0.5237, 0.3138, 0.6587),

the period in the frame rotating with velocity ωrot = 0.7305 around the e2-axis is T = 9.626, thesecond component of the angular momentum is µ2 = 1.576, and µ1 = µ3 = 0. The linearizationσDΦτ (x) at the bifurcation point has six eigenvalues 1 and the eigenvalues

λ1 = −2.10, λ2 = −0.477, λ3,4 = 0.453± i 0.892, λ5,6 = −0.553± i 0.833.

The first plot of Figure 1 shows the continuation of the Figure Eight at zero momentum up tothe bifurcation point. For each computed rotating eight solution the motion of the first body inthe (q1,1, q1,3)-plane is depicted in the respective corotating frame in which the solution becomesperiodic. The Figure Eight trajectory of the first body lies in the (q1,1, q1,2)-plane, therefore it

29

-1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0-0.6

-0.4

-0.2

-0.0

0.2

0.4

0.6

+ + + + + + + + + + ++

+

+

+

+

+

+

++++ + ++

+

++++++

+ +++

+

+

+++

+

+

+++

+

+

+

+

+

+++

+

+

+

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.0

-0.8

-0.6

-0.4

-0.2

-0.0

+ + + + + + ++

+

+

+

+

+q1,3

q1,1

q1,2

LC

PD

Fig.8,II

Fig.8,I

LC

PD

µ2

Figure 1: Symmetry-increasing bifurcation of the type II rotating eights.

corresponds to a line in the plot. We can see from the picture that at the bifurcation point thesolution is invariant under rotations by π/2.

As indicated by the label in Figure 1, the bifurcating family lies on the type I family PI(E, ν)of rotating eights. The branch ends at a relative Lyapounov centre bifurcation in the Lagrangerelative equilibrium at µ2 = 1.869, marked as ”LC” in the second plot of Figure 1. At therelative Lyapounov centre bifurcation the relative period τLC

I of the type I family and therelative period τLC of the bifurcating family of RPOs are identical, but the periods T LC

I andTLC in their respective corotating frames satisfy TLC = 2TLC

I , and the rotating frames anddrift-symmetries of both rotating choreographies in their corotating frames are also different. Ifwe continue the bifurcating family backwards to momentum µ2 = 0 then we obtain the originalFigure Eight solution (denoted ”Fig.8 I” in the second plot of Figure 1), but with nonvanishingrotation frequency and period 2T in the corotating frame. Here τ , T are the relative period andthe period of the Figure Eight solution at momentum µ = 0.

To understand this note that relative periods of RPOs are independent of the coordinateframe, but the period in the corotating frame and the drift-symmetry in the corotating framediffer for different corotating frames (Remark 2.13).

The drift-symmetry αI = κ1(231) of the type I family PI(E, ν) of rotating eights in itscorotating frame is transformed into the drift symmetry α = R2(π/2)κ2(312) of the fam-ily P(s) of rotating choreographies bifurcating from the type II rotating eight by conjuga-tion with the reversing symmetry ρ and, thus, inverting αI (as discussed in Remark 4.1 and[11]), by conjugating with the symmetry element R3(π/2), so that αI becomes κ2(312), andthen by a change of the rotation frequency ωrot

I (E, ν) of the RPO PI(E, ν) to the rotation fre-quency ωrot(E, ν) such that (ωrot(E, ν)−ωrot

I (E, ν))τI (E, ν) = π/2. This transforms the periodTI(E, ν) = `IτI(E, ν) = 6τI(E, ν) of the type I rotating eight PI(E, ν) in its corotating frameto T (E, ν) = ˜τI(E, ν) = 12τI(E, ν) = 2TI(E, ν) and explains the above observations.

If we continue the branch of type I rotating choreographies from the original Figure Eightin the momentum component µ1 fixing the energy, this connection between the type I and typeII rotating eights corresponds to a relative period-doubling and a symmetry-breaking pitchforkbifurcation in the corotating frame at µ1 = 1.576. In this case the bifurcating branch PII(s) hasdrift symmetry αII = α2

I = (312) of order ˜II = 3 in the corotating frame. If we continue the

bifurcating branch PII(s) back to the Figure Eight solution at µ1 = 0, then the Figure Eighton this branch (denoted ”Fig.8 II” in the second plot of Figure 1) has nonvanishing rotationfrequency and period T /2 = ˜

II τ in this corotating frame. Denote the rotation frequencyof PII(s) by ωrot

II (s). Applying the above transformation on the branch of type I rotatingchoreographies amounts to conjugating αII with ρ and changing the rotation frequency ωrot

II (s)of the RPO PII(s) to ωrot

II (s) where (ωrotII (s) − ωrot

II (s))τII (s) = π. Hence, we retrieve the drift

30

symmetry αII of the type II rotating eight in the original corotating frame, as expected.

4.4 Turning points and relative period doubling bifurcations of type

III rotating choreographies

As reported in [25], the type III rotating eights can be continued to negative momentum valuesµ3 < 0 at fixed energy E = −0.1287 and, after coming close to a collision, this family undergoesa relative period doubling bifurcation at momentum µ3 = −6.6383. A more detailed investi-gation revealed that along the primary branch further relative period-doubling bifurcations (atmomentum µ3 = −6.4136 and µ3 = −6.5550) and a turning point (at µ3 = −6.6627) occurbefore the relative period doubling bifurcation point at µ3 = −6.6383 takes place, see Figure 2.

At the turning point (marked ”TP” in the first panel of Figure 2) through

q1 = (1.509,−0.1663), p1 = (−9.321, 5.939), q2 = (0.017, 0.0999), p2 = (0.1616, 0.2359),

the RPO has period T = 299.02 in the cororating frame rotating with frequency ωrot = 0.0029.In addition to the multipliers λ1,2,3,4,5,6 = 1 its linearization in the corotating frame σDΦτ (x)has eigenvalues λ7 = −3.712, λ8 = −0.2694.

+++

+

+

+

++

+++++

++

++

+

++

++ + + + + ++

+++++++++++++++ + + + + + + + + + + + + + + + + ++

++++

+

++

+

++++++++

-6.6 -6.5 -6.4 -6.3 -6.2 -6.1 -6.0

-12.5

-12.0

-11.5

-11.0

-10.5

-10.0

-9.5

+ + + + + + +

+

+

+

+

+

+

+

-12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

+ + + + + + + + + + + +

+

+

+

+

+

+

+

+

+

+

+

-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

+ + + + + + + + + +

+

+

+

+

+

+

+

+

+

q1,2

q1,1

q1,3

TP

µ3

PD1

PD2

PD3

q1,2

q1,1

Figure 2: Turning points and relative period doubling bifurcation of the type III rotating chore-ographies; the second picture shows the primary solution on the lower branch, the third picturethe primary solution on the upper branch.

The type III rotating choreography, as continued from the Figure Eight, is on the lowerpart of the (unlabelled) primary solution branch in the first plot of Figure 2. The second plotof Figure 2 shows this solution at momentum µ3 = −6.2, and the third plot shows it afterthe turning point, on the upper branch of the bifurcation diagram, at momentum µ3 = −6.2.Note that the linearization at the RPO in the second plot has, in addition to four eigenvaluesλ1,2,3,4 = 1, one pair on the unit circle and two real negative eigenvalues.

The linearization at the RPO in the third plot has, in addition to four eigenvalues λ1,2,3,4 = 1,two real positive eigenvalues and one pair on the unit circle.

When the turning point is approached from the lower part of the primary branch then onepair of eigenvalues of the linearization switches from a pair on the unit circle to a pair on thereal line in a collision at λ5,6 = 1, cf. Proposition 2.34.

The first relative period doubling bifurcation (the emanatating branch is marked ”PD1” inthe first plot of Figure 2) is at momentum µ3 = −6.4136, and the RPO passes through

q1 = (2.023, 0.1397), p1 = (−9.928, 6.417), q2 = (−0.0084, 0.1079), p2 = (0.1635, 0.2083)

and has period T = 225.51 in a frame rotating with frequency ωrot = 0.0166. In addition tofour eigenvalues λ1,2,3,4 = 1 and two eigenvalues λ5,6 = −1 its linearization has the eigenvalues

31

-12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

+ + + + + + + + + + +

+

+

+

+

+

+

+

+

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-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0

-10.0

-5.0

0.0

5.0

10.0

+ + + + + + + + + + +

+

+

+

+

+

-12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

+ + + + + + + + + + + ++

+

+

+

+

+

+

+

+

+

+

q1,3

q1,2

q1,3

µ3

q1,2

q1,1

Figure 3: RPOs bifurcating from the type III rotating choreographies.

λ7 = −7.362, λ8 = −0.1358, and the pair of eigenvalues −1 passes from the unit circle to thereal axis as µ3 is decreased.

The first plot of Figure 3 shows the solution on branch PD1 at momentum µ3 = −6.2. Inaddition to four eigenvalues λ1,2,3,4 = 1, the linearization at this solution has four real positiveeigenvalues.

The second relative period doubling bifurcation (the emerging branch is marked ”PD2” inthe first plot of Figure 2) is at momentum µ3 = −6.555, and the RPO passes through

q1 = (1.766,−0.0071), p1 = (−9.684, 6.265), q2 = (−0.0025, 0.1019), p2 = (0.1646, 0.2206)

and has period T = 245.92 in a frame rotating with frequency ωrot = 0.0120. In addition tofour eigenvalues λ1,2,3,4 = 1 and two eigenvalues λ5,6 = −1 its linearization has the eigenvaluesλ7 = −8.265, λ8 = −0.1210, and the pair of eigenvalues −1 passes from the real axis to the unitcircle as µ3 is decreased.

The second plot of Figure 3 shows the solution on branch PD2 at momentum µ3 = −6.2. Inaddition to four eigenvalues λ1,2,3,4 = 1, the linearization at this solution has one positive realpair of eigenvalues and one pair on the unit circle.

The third relative period doubling bifurcation point (the emanating branch is marked ”PD3”in the first plot of Figure 2) at momentum µ3 = −6.6383, which we already reported in [25],passes through

q1 = (1.482,−0.3477), q2 = (−9.178, 5.833),

p1 = (0.0285, 0.1116), p2 = (0.1592, 0.2366),

T = 325.86, ωrot = −0.00049.

The Floquet eigenvalues are

λ1,2,3,4 = 1, λ5,6 = −1, λ7 = 20.17, λ8 = 0.0496.

The third plot of Figure 3 shows the solution on branch PD3 at momentum µ3 = −6.2.In addition to four eigenvalues λ1,2,3,4 = 1, the linearization at this solution has four negativeeigenvalues.

5 Conclusion

In this paper we presented methods for detecting and computing certain critical points includingsymmetry breaking and symmetry increasing bifurcations of Hamiltonian relative periodic orbitswith regular drift-momentum pair in the case of compact symmetry groups. The bifurcations weanalyzed occur generically during the pathfollowing of RPOs under the assumption that isotropygroups are trivial. We applied our results to rotating choreographies in the three-body problem.

32

Note that a systematic theory for all generic bifurcations of Hamiltonian RPOs does notyet exist. Consequently, a systematic numerical treatment of all such bifurcations is yet to bedeveloped. For dissipative differential equations generic bifurcations of symmetric periodic orbitshave been classified by Lamb and Melbourne [10] and a general bifurcation theory for RPOs ofdissipative systems has been developed in [21].

A next step would be to develop numerical methods for the computation of generic symmetrychanging bifurcations of RPOs of dissipative systems, which break spatial as well as spatiotem-poral symmetries, and then to extend these methods to bifurcations of Hamiltonian RPOs whichbreak discrete spatial symmetries. Note that bifurcations of Hamiltonian RPOs breaking con-tinuous isotropy are much more difficult to analyze as the momentum map J is in general notsurjective near such points. Preliminary results on bifurcations of Hamiltonian relative equilibriahave been obtained for example in [8, 16], see also references therein.

In the bifurcations that we analyze in this paper we assume that the RPO at the bifurcationhas a generic drift-momentum pair. The numerical treatment of bifurcations from RPOs withsingular drift-momentum pair is as yet an open problem.

Acknowledgements

The authors thank Mark Roberts for stimulating discussions and comments. CW thanks theFreie Universitat Berlin for their hospitality during the preparation of the manuscript, andacknowledges funding by by the Leverhulme Foundation. FS and CW were supported by EPSRCgrant EP/D063906/1.

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