Isotropy of Angular Frequencies and Weak Chimeras With Broken Symmetry
IISOTROPY OF ANGULAR FREQUENCIES AND WEAKCHIMERAS WITH BROKEN SYMMETRY
CHRISTIAN BICK
Abstract.
The notion of a weak chimeras provides a tractable defini-tion for chimera states in networks of finitely many phase oscillators.Here we generalize the definition of a weak chimera to a more gen-eral class of equivariant dynamical systems by characterizing solutionsin terms of the isotropy of their angular frequency vector—for coupledphase oscillators the angular frequency vector is given by the averageof the vector field along a trajectory. Symmetries of solutions auto-matically imply angular frequency synchronization. We show that thepresence of such symmetries is not necessary by giving a result for theexistence of weak chimeras without instantaneous or setwise symmetriesfor coupled phase oscillators. Moreover, we construct a coupling func-tion that gives rise to chaotic weak chimeras without symmetry in weaklycoupled populations of phase oscillators with generalized coupling. Introduction
The emergence of collective dynamics in networks of coupled oscillatoryunits is a fascinating phenomenon observed in science and technology [1,2]. Symmetric phase oscillator networks provide paradigmatic models tounderstand collective dynamics in the weak coupling limit [3, 4, 5, 6]. Suchdynamical systems with symmetry are equivariant with respect to the actionof a group [7, 8, 9], that is, the vector field commutes with the group actionon phase space. Equivariance implies that any solution of the system ismapped to another solution by the action of the symmetry group and ittypically constrains the dynamics, for example by giving rise to dynamicallyinvariant subspaces. The solutions themselves may (but do not have to) havenontrivial symmetry, that is, there may be nontrivial elements elements ofthe symmetry group that keep the solution fixed, either pointwise or asa set. For example, for globally coupled identical oscillators the solutioncorresponding to full synchrony, where the states of all oscillators are equal,has full symmetry itself. Of course, there may be other solution with lesssymmetry relative to the symmetries of the system.Recently, the observation of “symmetry breaking” in symmetrically cou-pled phase oscillator systems, i.e., the observation of solutions with localizedsynchronous dynamics coexisting with localized incoherence, has sparked alot of interest. Such solutions, commonly known as chimera states—see [10]for a recent review—were first observed in symmetric rings of coupled phaseoscillators [11, 12]. In the limit of infinitely many oscillators, they correspondto stationary or periodic patterns of the phase density distribution [13, 14].By contrast, it was not until recently that Ashwin and Burylko [15] gave
Date : September 25, 2018. a r X i v : . [ m a t h . D S ] F e b CHRISTIAN BICK a testable mathematical definition for chimera states, a weak chimera , fornetworks of finitely many phase oscillators whose phases ϕ k ∈ T = R / π Z , k = 1 , . . . , n , evolve according to(1) d ϕ k d t = ˙ ϕ k = ω + 1 n n (cid:88) j =1 H kj g ( ϕ k − ϕ j ) . Here the H kj determine the network topology (respecting a subgroup Γof the group S n of permutations of n symbols acting transitively on theindices of the oscillators) and g : T → R is the generalized coupling (orphase interaction) function. Weak chimeras are defined in terms of partialangular frequency synchronization on trajectories. More precisely, if ˆ ϕ is acontinuous lift of a solution ϕ of (1) with initial condition ϕ to R n definethe asymptotic angular frequency of oscillator k as(2) Ω k ( ϕ ) = lim T →∞ ˆ ϕ ( T ) T .
According to [15], a compact, connected, chain-recurrent, and dynamicallyinvariant set A ⊂ T n is a weak chimera if there are distinct oscillators j, k, (cid:96) such that Ω j ( ϕ ) = Ω k ( ϕ ) (cid:54) = Ω (cid:96) ( ϕ ) for all ϕ ∈ A .Weak chimeras and angular frequency synchronization relate to symme-try. Assuming that all limits (2) exist, we have a frequency vectorΩ( ϕ ) = (Ω ( ϕ ) , . . . , Ω n ( ϕ )) ∈ R n . The group S n also acts on R n by permuting indices. If A is a weak chimeraas above and τ kj ∈ S n denotes the transposition swapping indices k and j then τ kj Ω( ϕ ) = Ω( ϕ ). That is, τ kj is a symmetry of the angular frequencyvector Ω( ϕ ). While weak chimeras have provided a suitable frameworkto derive for example existence results [16], there are two shortcomings.First, while chimera states have also been reported in more general oscillatormodels [17, 18] the definition above applies to phase oscillators only. Second,the symmetries of the angular frequency vector Ω may be different fromthe symmetries of the system. As a consequence, if A is a weak chimerathen τ kj A may not be a weak chimera or even a solution of the system atall. Interestingly, while it has been argued that chimera states are relevantdue to their nature of solutions with broken symmetry [12], their propertieshave never been phrased in terms of symmetries of the dynamical system.The contribution of this paper is twofold: first, we give a definition of aweak chimera in the language of equivariant dynamical systems and, second,we show that symmetries of the solution are not necessary for the occurrenceof weak chimeras. More precisely, we define weak chimeras in terms of theisotropy of the angular frequency vector which can be stated for more generaloscillator systems. We observe that, in a suitable setup, asymptotic angularfrequencies are averages of equivariant observables. Therefore, symmetriesof solutions translate directly into symmetries of the angular frequencies.Thus, the presence of symmetries of solutions facilitates the emergence ofweak chimeras and, in fact, most weak chimeras that have been constructedexplicitly [15, 19, 16] are solutions with (instantaneous) symmetries. Is itpossible to construct weak chimeras without instantaneous or setwise sym-metries for which the angular frequencies have symmetries that are not a HIMERAS AND SYMMETRY 3 property of the solution itself? This question motivates the second contri-bution. Extending recent persistence results [16] that rely on constructinggeneralized coupling functions between oscillators, we prove a persistenceresult for weak chimeras with trivial symmetry in weakly coupled popula-tions of phase oscillators. Moreover, we present an explicit example of a C ∞ coupling function that gives rise to a chaotic weak chimera withoutinstantaneous or setwise symmetries in a nontrivially coupled system.This paper is organized as follows. In Section 2 we review some terminol-ogy on equivariant dynamics that is needed in the subsequent sections. InSection 3 we then apply these notions to general oscillator systems with sym-metry which yields a new definition of a weak chimera in terms of symmetriesof the angular frequency vector. As we show in Section 4 this definition iscompatible with previous definitions. In Section 5 we prove a persistenceresult for weak chimeras without instantaneous or average symmetries. Fi-nally, we present an explicit example of a coupling function which gives riseto chaotic weak chimeras with trivial symmetries in Section 6 and finishwith some concluding remarks.2. Preliminaries
Quasi-regular points.
Let X be a compact differentiable manifoldwith a flow Φ t : X → X , t ∈ R . A point x ∈ X is quasi-regular if the limitlim T →∞ T (cid:90) T f (Φ t ( x ))d t exists for all continuous functions f : X → R . Theorem 1 ([20, 21]) . The set of points which are not quasi-regular haszero measure with respect to every finite measure on X that is invariantunder the flow Φ t . Equivariant dynamical systems.
Let F : X → T X be a smoothvector field on X where T X denotes the tangent bundle. Suppose that agroup Γ acts on X . The vector field F is Γ-equivariant if(3) F ( γx ) = ˆ γF ( x )for all γ ∈ Γ where ˆ γ is the induced action on the tangent space. A Γ-equivariant vector field defines a Γ -equivariant dynamical system (4) ˙ x = F ( x )on X [8, 9]. For a set A ⊂ X define the set of instantaneous symmetries T ( A ) = { γ ∈ Γ | γx = x for all x ∈ A } (5)and the set of symmetries on average (or setwise symmetries)Σ( A ) = { γ ∈ Γ | γA = A } . (6)Clearly, T ( A ) ⊂ Σ( A ). If Γ x = { γ ∈ Γ | γx = x } denotes the stabilizer or isotropy subgroup of x ∈ X we have T ( A ) = (cid:84) x ∈ A Γ x .Note that if γA ∩ A = ∅ for all γ ∈ Γ (cid:114) { id } then Σ( A ) = { id } . The converseholds only under additional assumptions [22]. Henceforth, let Φ t : X → X , t ∈ R , denote the flow defined by the differential equation (4). A set A ⊂ X CHRISTIAN BICK is (forward) flow-invariant or dynamically invariant if Φ t ( A ) ⊂ A for all t ≥
0. Moreover, A is stable if for every neighborhood U of A there existsan open neighborhood V ⊂ U of A such that Φ t ( V ) ⊂ U for all t ≥
0. Acompact stable set A is an attractor if A = ω ( x ) is the ω -limit set of somepoint x ∈ X .For attractors and the action of the orthogonal group O ( n ) on R n there isthe following dichotomy [23] that characterizes the symmetries on average. Proposition 1.
Let Γ ⊂ O ( n ) be a finite subgroup. For an attractor A ⊂ R n we have for any γ ∈ Γ either γA = A or γA ∩ A = ∅ .Remark . The same statement holds for repellers—dynamically invariantsets that are attractors when time is reversed. However, it does not necessar-ily hold for dynamically invariant sets of saddle type, sets that are attracting(or repelling) in a more general sense, or heteroclinic attractors.For δ > B δ ( A ) denote an (open) δ -neighborhood of A . Corollary 1.
Let Γ ⊂ O ( n ) be a finite subgroup and let A ⊂ R n be compactattractor for the flow defined by (4) . If Σ( A ) = { id } then there exists a δ > such that Σ( D ) = { id } for any D ⊂ B δ ( A ) .Proof. Suppose that Σ( A ) = { id } . By Proposition 1 we have γA ∩ A = ∅ forany γ (cid:54) = id. Since A is closed there exists a δ > γB δ ( A ) ∩ B δ ( A ) = ∅ . Therefore, Σ( D ) = { id } for any D ⊂ B δ ( A ). (cid:3) Equivariant observables.
Suppose that Γ acts on both X and R m for some m ∈ N (cid:114) { } . Definition 1.
A continuous Γ-equivariant map O : X → R m is an observ-able .Given a solution x ( t ) of (4) with initial condition x (0) = x , the limit(7) K O ( x ) = lim T →∞ T (cid:90) T O ( x ( t ))d t (if it exists) is an average of O along the trajectory x (integrate compo-nentwise if m > x and henceforth we will always assume that x ∈ X isquasi-regular when averages (7) are evaluated.Suppose that A ⊂ X is dynamically invariant and supports a Φ t -invariantergodic probability measure µ . Write(8) K µ O ( A ) = (cid:90) A O ( x )d µ. By the Birkhoff ergodic theorem [24, Theorem 4.1.2] we have(9) K O ( x ) = K µ O ( A ) . for µ -almost every x ∈ A . In particular, the limit (7) exists for µ -almostevery x ∈ A . For ease of notation, we will simply write K O ( A ) = K µ O ( A )unless the choice of measure is important. Of course, not every ergodic in-variant measure is “physically relevant” since µ may be singular with respectto the Lebesgue measure. If an attractor A supports a Sinai–Ruelle–Bowen(SRB) measure µ [25, 24] there is a neighborhood W of A such that (9) HIMERAS AND SYMMETRY 5 holds for Lebesgue-almost every x ∈ W . Thus the average (8) is observedfor “typical” initial conditions with respect to the Lebesgue measure.Now K O ( A ) has an isotropy group Γ K O ( A ) and a simple calculation [8]shows that(10) Σ( A ) ⊂ Γ K O ( A ) , that is, any symmetry on average is contained in the isotropy group of theobservation.The converse does not hold for general observables. Detectives [8, 26, 27,28] are an important class of observables for which the isotropy is genericallyequal to the symmetries on average. Given a suitably large m ∈ N (cid:114) { } , anobservable is an (ergodic) detective if for any ω -limit set A there exists anopen dense set of near-identity Γ-equivariant diffeomorphisms ψ : R m → R m such that Γ K O ( ψ ( A )) = Σ( A ). Hence, detectives are particular observablesto “detect” the symmetries of attractors.3. Weak Chimeras and Symmetries on Average
Isotropy of angular frequencies and weak chimeras.
The sym-metry point of view now allows to define weak chimeras in terms of their sym-metries as solutions relative to the symmetries of the system itself. Write i = √−
1. Let Γ ⊂ S n be a subgroup that acts transitively on C n • := ( C (cid:114) { } ) n by permuting coordinates and suppose that F : C n • → C n is Γ-equivariant.The map F = ( F , . . . , F n ) determines a dynamical system on C n • where theevolution of z = ( z , . . . , z k ) is given by(11) ˙ z k = z k F k ( z ) . For k ∈ { , . . . , n } define(12) C k ( z ) = z k | z k | . In the following we assume that F is such that (a) the dynamics of (11)are well defined on C n • , (b) we have ω ( z ) ⊂ C n • for all z ∈ C n • and (c) thederivative C (cid:48) k ( z ( t )) := dd t C k ( z ( t )) exists for any trajectory z ( t ). These as-sumptions are easy to work with but can be relaxed as one typically onlyneeds well-defined dynamics on a neighborhood of T n ⊂ C n • .Note that C k ( z , . . . , z n ) projects onto the unit circle in the k th coordi-nate. Let γ T denote the parametrized curve in C n • determined by a solution z ( t ) of (11) for t ∈ [0 , T ]. The change of argument of z k along γ T is givenby(13) ∆ arg C k ( γ T ) = 1 i (cid:90) T C (cid:48) k ( z ( t )) C k ( z ( t )) d t = (cid:90) T Im( F k ( z ( t )))d t. Thus, we obtain the average angular frequency in the k th coordinate (equiv-alent to the average winding number when multiplied by 2 π )(14) Ω k ( z ) := lim T →∞ T (cid:90) T Im( F k ( z ( t )))d t One can think of each complex variables z k representing phase and amplitude of anoscillator. CHRISTIAN BICK along a trajectory z ( t ) with initial condition z . Definition 2.
The vectorΩ( z ) = (Ω ( z ) , . . . , Ω n ( z ))is the angular frequency vector of the trajectory with initial condition z ∈ C n • .Since S n also acts on R n by permuting indices, Im( F ) : C n • → R n is aΓ-equivariant observable for (11) andΩ( z ) = K Im( F ) ( z ) , that is, the angular frequency vector is the observation of Im( F ) along atrajectory. For a compact and invariant set A ⊂ C n • with an unique ergodicinvariant measure we write Ω( A ) = K Im( F ) ( A ) for the angular frequencyvector of A .The symmetries of the system (11) now allow to phrase angular frequencysynchronization in terms of the isotropy of the angular frequency vector. Anobservation of Im( F ) has isotropy subgroup Γ Ω( A ) ⊂ Γ. This motivates adefinition of a weak chimera [15]—originally limited to networks of phaseoscillator—to more general oscillator systems (11).
Definition 3.
A compact, connected, chain-recurrent, and dynamically in-variant set A ⊂ C n • is a weak chimera for (11) if { id } (cid:40) Γ Ω( ϕ ) (cid:40) Γfor all ϕ ∈ A . If a weak chimera A supports an SRB measure then it iscalled observable and we have { id } (cid:40) Γ Ω( A ) (cid:40) Γ . Remark . Asymptotic winding (or rotation) numbers can be defined in amore general setting: they quantify how trajectories of a given flow windaround a topological space X ; cf. [20, 29] for details. These winding numbersare defined for continuous maps f : X → S = { z ∈ C | | z | = 1 } . For spaceswith finitely generated homology, it suffices to evaluate winding numbers formaps f k corresponding to a basis of the first cohomology [20].Here we have X = C n • and the maps C k defined above correspond tothe generators of the homology of C n • . Since we consider flows given by aΓ-equivariant differential equation, we characterize weak chimeras by thesymmetry properties of the asymptotic winding numbers. In the languageof asymptotic cycles, these are solutions where for certain “directions” thewinding behavior is the same while for other directions it is distinct. Thissuggests that the notion can be further extended to equivariant dynamicalsystems on more general X with nontrivial homology.Note that the weak chimeras of Definition 3 are defined solely in termsof the symmetry properties of the system. Moreover, the definition extendsbeyond the weak coupling limit of interacting limit cycle oscillators [30]:systems of the form (11) describe dynamical systems close to a Hopf bi-furcation [31] or more general oscillator models where “amplitude-mediatedchimeras” have been observed [17]. Moreover, the next proposition asserts HIMERAS AND SYMMETRY 7 that the change of argument of z k along C (cid:96) ( γ T ) cannot be bounded to obtainnontrivial winding numbers; such dynamics are observed for “pure ampli-tude chimeras” [18] and thus our definition is sufficiently general to providea rigorous framework for such chimeras. Proposition 2.
Suppose that z ( t ) is a solution of (11) such that there are j (cid:54) = (cid:96) , M, R > such that ∆ arg C (cid:96) ( γ T ) < M and ∆ arg C j ( γ T ) > RT forall T . Then Γ Ω( A ) (cid:40) Γ .Proof. Immediate from Ω k ( z ) = lim T →∞ T ∆ arg C k ( γ T ). (cid:3) Definition 3 is compatible with the action of the symmetry group on C n • . Proposition 3. If A ⊂ T n is a weak chimera, so is γA for any γ ∈ Γ .Proof. The assertion follows directly from Γ-equivariance of F . (cid:3) This implies in particular that the isotropy of the angular frequency vec-tors are conjugate if weak chimeras are related by symmetry. If γ ∈ Σ( A )then Ω( γA ) = Ω( A ), that is, the angular frequency vectors (and thereforethe isotropy) are identical.3.2. Symmetries imply frequency synchronization.
Intuitively speak-ing, a weak chimera A consists of solutions of (11) along which the averageangular frequencies have some symmetries but not too many. Inclusion (10)implies(15) { id } ⊂ T ( A ) ⊂ Σ( A ) ⊂ Γ Ω( A ) ⊂ Γ ⊂ S n . Consequently, if a solution has nontrivial instantaneous symmetry then thecorresponding angular frequency vector has nontrivial isotropy. Similarly,the angular frequency vector of dynamically invariant sets with nontrivialsetwise symmetry has nontrivial isotropy. For invariant sets with nontrivial(setwise or instantaneous) symmetry, (15) implies that one condition of Def-inition 3 is automatically satisfied. In that sense the presence of symmetries“facilitates” the occurrence of weak chimera states.More generally speaking, symmetries of the system give sufficient condi-tions for angular frequency synchronization [32]. These are not necessaryas there may be other dynamically invariant subspaces where oscillators arephase and frequency locked which are not induced by symmetry but ratherby balanced polydiagonals of colored graphs [33].4.
Weak Chimeras for Networks of Phase Oscillators
Definition 3 relates to the original definition of a weak chimera for net-works of coupled phase oscillators [15]. We will not restrict ourselves tosystems (1) but consider a more general setup that may include, for exam-ple, nonpairwise interactions [31, 34]. More precisely, let X = T n and letΓ ⊂ S n act transitively on T n by permuting indices. A smooth Γ-equivariantvector field Y : T n → R n now defines a Γ-equivariant dynamical system(16) ˙ ϕ = Y ( ϕ ) . that describes the evolution of n phase oscillators where the state of oscil-lator k is given by ϕ k ∈ T . CHRISTIAN BICK
Write z k = exp( iϕ k ) and identify initial conditions ϕ ∈ T n with z ∈ C n • .The dynamics of (16) can be embedded in C n • as(17) ˙ z k = z k ( iY k ( ϕ )) . Therefore(18) Ω k ( ϕ ) := K Y k ( z ) = lim T →∞ T (cid:90) T Y k ( ϕ ( t ))d t and if A ⊂ T n is compact, dynamically invariant supporting an SRB mea-sure then Ω( ϕ ) = K Y ( A )is the angular frequency vector for A . Moreover, with (16) we have(19) (cid:90) T Y k ( ϕ ( t ))d t = (cid:90) T ˙ ϕ k ( t )d t = ˆ ϕ k ( T ) − ˆ ϕ k (0)where ˆ ϕ is a continuous lift of the trajectory ϕ ( t ) to R n . Thus,Ω k ( ϕ ) = lim T →∞ ˆ ϕ ( T ) T as given by (2). Note also that Ω k ( A ) correspond to the average frequencydefined in [32] and relates to the rotation vector for torus maps [35].Compared to the original definition of a weak chimera in [15], Definition 3is more restrictive. More precisely, for A we require that frequency synchro-nization is only relevant for a weak chimera if the oscillators are related bysymmetry. By contrast, the original definition considers the set(20) Θ( A ) = { γ ∈ S n | γ Ω( A ) = Ω( A ) } rather than the isotropy Γ Ω k ( A ) . Note that Θ( A ) may be strictly largerthan Γ Ω( A ) . For example if Z n = Z /n Z ⊂ S n denotes the cyclic groupand X is Z n equivariant but not S n -equivariant and ϕ = · · · = ϕ n (forexample a nonlocally coupled ring of phase oscillators [11]) is a solution of˙ ϕ = X ( ϕ ) then Θ (cid:0) (cid:8) ϕ (cid:9) (cid:1) = S n (cid:41) Z n .5. Persistence of Weak Chimeras without Symmetry onAverage for Diffusively Coupled Phase Oscillators
The inclusions (15) in Section 3.2 imply that any (nontrivial) instan-taneous or average symmetry of a dynamically invariant set gives non-trivial isotropy of the angular frequency vector. This is the case for theweak chimeras constructed in [15, 19, 16]. In this section we constructweak chimeras with trivial average symmetries for systems consisting of twoweakly interacting populations of phase oscillators.5.1.
Coupling function separability for symmetric diffusively cou-pled phase oscillators.
For φ, ψ ∈ T n define X = ( X , . . . , X n ) by(21) X k ( φ, ψ ) := 1 n n (cid:88) j =1 g ( φ k − ψ j ) . HIMERAS AND SYMMETRY 9
The dynamics of a fully symmetric network of n phase oscillators with cou-pling function g is given by the S n -equivariant dynamical system on T n where(22) ˙ ϕ k = Y k ( ϕ ) := X k ( ϕ, ϕ )describes the evolution of the k th oscillator . We may assume g (0) = 0 bygoing to suitable co-rotating reference frame, ϕ k (cid:55)→ ϕ k − ωt . If the choiceof coupling function g is important we write Y ( g ) or X ( g ) to highlight thedependency. Reducing the continuous T symmetry of (22) allows to set ϕ = 0. Because of the S n -equivariance, the canonical invariant region (23) C := { ϕ ∈ T n | ϕ < · · · < ϕ n < π } is dynamically invariant. It is bounded by hypersurfaces corresponding tocluster states with ϕ k = ϕ k +1 and there is a residual Z n symmetry on C [30,36].For a compact flow invariant set A ⊂ T n define(24) Ξ( A ) = (cid:91) k (cid:54) = j { ϕ k − ϕ j | ϕ ∈ A } . Note that Ξ( γA ) = Ξ( A ) for all γ ∈ S n . Definition 4.
Two sets A , A ⊂ C are coupling function separated if thereare open intervals Q A , Q A ⊂ T with Ξ( A (cid:96) ) ⊂ Q A (cid:96) , (cid:96) = 1 ,
2, and Q A ∩ Q A = ∅ where the bar denotes topological closure.For Q ⊂ T define Ξ − ( Q ) := { ϕ ∈ T | Ξ( { ϕ } ) ⊂ Q } and W ( g ) ( Q ) := (cid:104) min k ∈{ ,...,n } inf ϕ ∈ Ξ − ( Q ) Y ( g ) k ( ϕ ) , max k ∈{ ,...,n } sup ϕ ∈ Ξ − ( Q ) Y ( g ) k ( ϕ ) (cid:105) . Lemma 1. (1) If A ⊂ C with Ξ( A ) ⊂ Q is dynamically invariant for thedynamics of (22) with coupling function g then Ω k ( ϕ ) ∈ W ( g ) ( Q ) .(2) Suppose that A (cid:96) ⊂ C , (cid:96) = 1 , , are compact and coupling functionseparated with separating sets Q A (cid:96) . Then for any η ≥ we can finda coupling function ˆ g such that B η (cid:16) W (ˆ g ) ( Q A ) (cid:17) ∩ B η (cid:16) W (ˆ g ) ( Q A ) (cid:17) = ∅ . (3) Let A (cid:96) be as above and let A (cid:48) , A (cid:48) ⊂ C be dynamically invariant forthe dynamics of (22) with Ξ( A (cid:48) (cid:96) ) ⊂ Q A (cid:96) . Then there is a couplingfunction ˆ g such that A (cid:48) , A (cid:48) are dynamically invariant for the dy-namics of (22) with ˆ g and Ω k (cid:0) ϕ A (cid:48) (cid:1) (cid:54) = Ω j (cid:0) ϕ A (cid:48) (cid:1) for all k, j and ϕ A (cid:48) (cid:96) ∈ A (cid:48) (cid:96) . We obtain (22) by setting H kj = 1 for all k, j in (1). Proof.
To prove (1) note first that invariance of C implies that Ω j ( ϕ ) =Ω k ( ϕ ) for all k, j and ϕ ∈ A . Standard integral estimate for (18) yieldΩ k ( ϕ ) ∈ (cid:104) inf ϕ ∈ A Y ( g ) k ( ϕ ) , sup ϕ ∈ A Y ( g ) k ( ϕ ) (cid:105) ⊂ W ( g ) ( Q ) . To prove (2) consider coupling functions ˆ g with ˆ g ( φ ) = g ( φ ) + a (cid:96) for all φ ∈ Q A (cid:96) , (cid:96) = 1 ,
2. Since Y (ˆ g ) k ( ϕ ) = 1 n n (cid:88) j =1 ( g ( φ k − ψ j ) + a (cid:96) ) = a (cid:96) + Y ( g ) k ( ϕ )for all ϕ with Ξ( { ϕ } ) ⊂ Q A (cid:96) we have W (ˆ g ) ( Q A (cid:96) ) = (cid:104) inf W ( g ) ( Q A (cid:96) ) + a (cid:96) , sup W ( g ) ( Q A (cid:96) ) + a (cid:96) (cid:105) . For a given η ≥ a , a such that B η (cid:16) W ( g ) ( Q A ) (cid:17) ∩ B η (cid:16) W ( g ) ( Q A ) (cid:17) = ∅ to obtain the desired coupling function ˆ g .Note that replacing g by ˆ g as above preserves dynamically invariant sets A with Ξ( A ) ⊂ Q A (cid:96) . Claim (3) now follows from (1) and (2) with η = 0. (cid:3) Remark . The notion of function coupling separability and Lemma 1 gen-eralize to a finite number of sets A , . . . , A r . The function g − ˆ g can typicallybe chosen to be C ∞ .5.2. Relative equilibria with trivial symmetry.
We now show thatchoosing the coupling function appropriately in an arbitrarily small neigh-borhood of zero gives rise to asymptotically stable relative equilibria withtrivial symmetry for (22).Let 0 = α < · · · < α n < π . The function(25) ϕ (cid:63) ( t ) = ( α + tω (cid:63) , . . . , α n + tω (cid:63) ) ∈ C with ω (cid:63) = n (cid:80) nj =2 g ( − α j ) is a relative equilibrium of (22) for any couplingfunction g such that(26) 1 n (cid:88) j (cid:54) = k ( g ( α k − α j ) − g ( − α j )) = 0for all k = 2 , . . . , n . For a relative equilibrium we have(27) Ξ( { ϕ (cid:63) } ) = (cid:91) k (cid:54) = j { α k − α j } ⊂ [ − α n , α n ] ⊂ T . In particular, we have a relative equilibrium if the coupling function g van-ishes on Ξ( { ϕ (cid:63) } ). Since α n can be chosen arbitrarily small, the relativeequilibrium can be chosen arbitrarily close to the fully synchronized solu-tion ϕ = · · · = ϕ n . We have Ω( { ϕ (cid:63) } ) = ( ω (cid:63) , . . . , ω (cid:63) ).Stability of the relative equilibrium is determined by the linearization(28) ∂Y k ∂ϕ j = (cid:40) n (cid:80) l (cid:54) = k g (cid:48) ( α k − α l ) if k = j, − n g (cid:48) ( α k − α j ) otherwise.By choosing the coupling function appropriately on Ξ( A ) the relative equi-librium will be asymptotically stable. For example, if g (cid:48) ( φ ) = 0 for φ < HIMERAS AND SYMMETRY 11 and g (cid:48) ( φ ) < φ > ϕ (cid:63) isasymptotically stable. Lemma 2.
Let ϕ (cid:63) ( t ) be a relative equilibrium of (22) as defined in (25) .If | α n | < π then T ( { ϕ (cid:63) } ) = Σ( { ϕ (cid:63) } ) = { id } .Proof. It suffices to show that Σ( { ϕ (cid:63) } ) = { id } . Assume that γ ∈ Σ( { ϕ (cid:63) } )with γ (cid:54) = id. Then there exists a τ ≥ α γk = α k + τ ω (cid:63) mod 2 π for all k . Recall that 0 = α < · · · < α n < π . Since γ (cid:54) = id, γ permutessome indices. Assume without loss of generality that α γ > α and α γ < α .We have α γ − α = α γ − α = τ ω (cid:63) mod 2 π . But since α γ − α > α γ − α < m > α γ − α − α γ + α = 2 mπ .This is a contradiction since | α γ − α − α γ + α | ≤ α n < π . (cid:3) Weak chimeras with Σ( A ) = { id } in weakly coupled populationsof phase oscillators. Chaotic weak chimeras have many features associ-ated with classical chimera states including positive maximal Lyapunov ex-ponents. Hence, rather than using a hyperbolicity argument to constructnonchaotic weak chimeras as in [15], we aim to construct weak chimeraswith Σ( A ) = { id } in a more general setup which allows for positive maximalLyapunov exponents. To this end, we extend recent results from [16] withrespect to the instantaneous and setwise symmetries of the constructed sets.Coupling two populations of n oscillators, whose uncoupled dynamics aregiven by (22), defines a dynamical system on T n . More explicitly, write ϕ = ( ϕ , ϕ ) ∈ T n × T n = T n , ϕ (cid:96) = ( ϕ (cid:96), , . . . , ϕ (cid:96),n ) and consider theproduct system(29) ˙ ϕ = Y ( g,ε )1 ( ϕ , ϕ ) = Y ( g ) ( ϕ ) + εX ( g ) ( ϕ , ϕ ) , ˙ ϕ = Y ( g,ε )2 ( ϕ , ϕ ) = Y ( g ) ( ϕ ) + εX ( g ) ( ϕ , ϕ ) , with Y ( g ) , X ( g ) as in (22), (21). Observe that for ε = 0 the system decouplesinto two identical groups of n oscillators—both of which with nontrivialdynamics (22). For ϕ ∈ T n we denote the asymptotic angular frequencyof the oscillator with phase ϕ (cid:96),k by Ω (cid:96),k ( ϕ ) = Ω ( g,ε ) (cid:96),k ( ϕ ).Let Γ = S n (cid:111) S where (cid:111) is the wreath product. The system (29) is Γ-equivariant [37]; we have Γ = S n (cid:111) S = ( S n ) (cid:111) S where the elements of S n permute the oscillators within each group of n oscillators and the action of S permutes the two groups. Observe that this is only a semidirect product (cid:111) as the two sets of permutations do not necessarily commute. The oscillatorsare indistinguishable as this group acts transitively on the oscillators.Weak chimeras in the product system persist for weak coupling 0 ≤ ε (cid:28) A is sufficiently stable if isthere is an open neighborhood of A on which a Lyapunov function is defined.The persistence theorem for weak chimeras [16, Theorem 4] generalizes tocoupling function separated sets that are sufficiently stable. Theorem 2.
Suppose that g is a coupling function such that A , A ⊂ C are compact, forward invariant, coupling function separated and sufficiently stable sets for the dynamics of (22) with Y ( g ) . Then for any sufficientlysmall δ > there exist a smooth coupling function ˆ g and ε > such thatfor any ≤ ε < ε the weakly coupled product system (29) with g replacedby ˆ g has a sufficiently stable weak chimera A ( ε ) with A ( ε ) ⊂ B δ ( A × A ) .Proof. First consider the coupling function separated sets A , A ⊂ T n asdynamically invariant sets for (22), a factor of the uncoupled system. Sup-pose that Q A , Q A are the separating sets and for a coupling function ˜ g define M (˜ g ) := max ( ϕ ,ϕ ) ∈ T n (cid:12)(cid:12) X (˜ g ) ( ϕ , ϕ ) (cid:12)(cid:12) < ∞ . Now choose a couplingfunction ˆ g according to Lemma 1(2) for η = 1 and fix ε := M (ˆ g ) − . Forany 0 ≤ ε < ε we have(30) B εM (ˆ g ) ( W (ˆ g ) ( Q A )) ∩ B εM (ˆ g ) ( W (ˆ g ) ( Q A )) = ∅ . since εM (ˆ g ) < ε M (ˆ g ) = 1.Now consider the product system (29). For any sufficiently small δ > ε > A ( ε ) ⊂ B δ ( A × A ) for all0 ≤ ε < ε as in [16]. Set ε < min { ε , ε } . By restricting δ appropriately,the sets A ( ε ) are weak chimeras.To show that Γ Ω( ϕ ) (cid:54) = { id } , assume that δ is so small that B δ ( A × A ) ⊂C . This implies that the phase ordering within each population is preserved.Hence, for given (cid:96) = 1 , (ˆ g,ε ) (cid:96),k ( ϕ ) = Ω (ˆ g,ε ) (cid:96),j ( ϕ ) =: Ω (ˆ g,ε ) (cid:96), ∗ ( ϕ )for all k, j and any ϕ ∈ A ε . Thus Γ Ω( ϕ ) (cid:54) = { id } .It remains to be shown that Γ Ω( ϕ ) (cid:54) = Γ. Let A ( ε ) (cid:96) denote the projec-tion of A ( ε ) onto ϕ (cid:96) . Now assume that δ is sufficiently small such that A ( ε ) ⊂ B δ ( A × A ) implies that Ξ (cid:0) A ( ε ) (cid:96) (cid:1) ⊂ Q A (cid:96) for all 0 ≤ ε < ε . Since Y (ˆ g,ε )1 ( ϕ , ϕ ) = Y (ˆ g ) ( ϕ ) + εX ( ϕ , ϕ ) integral estimates as in Lemma 1(1)imply that Ω (ˆ g,ε ) (cid:96), ∗ ∈ B ε M (cid:0) W (ˆ g ) ( Q A (cid:96) ) (cid:1) for all 0 ≤ ε < ε . By choice of ˆ g , Equation (30) now implies Ω (ˆ g,ε )1 , ∗ ( ϕ ) (cid:54) =Ω (ˆ g,ε )2 , ∗ ( ϕ ) for all ϕ ∈ A ( ε ) . Thus Γ Ω( ϕ ) (cid:54) = Γ and A ( ε ) is a weak chimera. (cid:3) The following statement asserts that trivial symmetries in the factorscarry over to the product dynamics.
Lemma 3.
Let A , A ⊂ T n be coupling function separated attractors for (22) with Σ( A ) = Σ( A ) = { id } . Then Σ( A × A ) = { id } for the product sys-tem (29) .Proof. Write S = { id , τ } . For any γ ∈ ( S n ) we have ( γ, id)( A × A ) ∩ ( A × A ) = ∅ in T n by assumption. Since A and A are couplingfunction separated, we have A ∩ A = ∅ in T n . Write Γ V = (cid:83) γ ∈ Γ γV , V ∈ { A , A } . The fact that Ξ( γA ) = Ξ( A ) implies Γ A ∩ Γ A = ∅ in T n . Since ( γ, τ )( A × A ) ⊂ Γ A × Γ A for any γ ∈ ( S n ) we have( γ, τ )( A × A ) ∩ ( A × A ) = ∅ and by Proposition 1 the claim follows. (cid:3) Combining the perturbation result Theorem 2 with the symmetry con-siderations, we can now state the main theorem of this section. In order to
HIMERAS AND SYMMETRY 13 apply Theorem 2 we make a slightly stronger assumption concerning stabil-ity of the relative periodic orbits.
Theorem 3.
Suppose that g is a coupling function such that for the S n -equivariant dynamics of (22) with Y ( g ) the set A coh = { ϕ (cid:63) } is a sufficientlystable relative equilibrium and A inc ⊂ T n is sufficiently stable attractor thatare coupling function separated and Σ( A coh ) = Σ( A inc ) = { id } . Then forany sufficiently small δ > there is a coupling function ˆ g and ε > suchthat for any ≤ ε < ε there is a weak chimera A ( ε ) ⊂ B δ ( A coh × A inc ) with Σ( A ( ε ) ) = { id } for the S n (cid:111) S -equivariant dynamics of (29) with ˆ g .Proof. Suppose that Σ( A coh ) = Σ( A inc ) = { id } ⊂ S n . By Lemma 3 we haveΣ( A coh × A inc ) = { id } ⊂ S n (cid:111) S . Since A coh × A inc is assumed to be anattractor, Corollary 1 implies that there exists a δ > A ⊂ B δ ( A coh × A inc ) we have Σ( A ) = { id } ⊂ S n (cid:111) S .For sufficiently small 0 < δ < δ Theorem 2 yields an ε > A ( ε ) ⊂ B δ ( A coh × A inc ) for all 0 ≤ ε < ε . Since δ < δ theargument above implies that Σ( A ( ε ) ) = { id } for all such ε . (cid:3) Remark . (1) In fact, the condition that A inc is a relative equilibrium isnot necessary. Theorem 3 holds for any compact, sufficiently stableattractor A coh ⊂ C that is coupling function separated from A inc .Moreover, the same statement holds for (sufficiently unstable) re-pellers.(2) Even if the weak chimera A (0) = A coh × A inc is observable, extraassumptions on the persistence of SRB measures are needed to provethat A ( ε ) is an observable weak chimera.6. A Numerical Example of a Chaotic Weak Chimera withTrivial Symmetry
We now give an explicit example of a coupling function such that thedynamics of the product system (29) give rise to a chaotic weak chimerawith trivial symmetry for ε > n = 4 oscillators give rise to chaoticattractors A with Σ( A ) = { id } [38, 39]. Define(31) g ( φ ) = (cid:88) r =0 c r cos( rφ + ξ r )with c = − c = − c = −
1, and c = − .
88. For ξ = η , ξ = − η , ξ = η + η , and ξ = η + η with η = 0 . η = 0 . g give rise to a chaoticattracting set A inc ⊂ C with positive maximal Lyapunov exponents andΣ( A inc ) = { id } . For A inc we have Ξ( A inc ) ⊂ [0 . , π − .
4] as shown inFigure 1.A suitable local perturbation of the coupling function g yields bistabilitybetween A inc and a relative equilibrium with trivial symmetry in the system − −
303 0 π π π π g ( φ ) Phase differences φ Figure 1.
The attracting sets A inc and A coh of the dynam-ics given by (22) with coupling function ˆ g are coupling func-tion separated. The coupling function ˆ g as defined in (33) isdepicted by a gray line. The values of g on Ξ( A inc ) are indi-cated by filled circles and on Ξ( A coh ) by 12 hollow circles.defined by (22). Let(32) ˜ g ( φ ) = (cid:88) r =6 a r cos( rφ + ζ r )with parameters a r , ζ r as given in Appendix A. Moreover, define β ( x ) := (cid:40) exp (cid:16) − − x (cid:17) if − < x < , a ∈ R , b ∈ (0 , π ) be parameters. Now define β ab ( φ ) := aβ (cid:0) φb (cid:1) with φ taken modulo 2 π with values in ( − π, π ] is a 2 π -periodic “bump function.”Fix a = 2 . b = 0 .
25. Define the C ∞ function(33) ˆ g := g + ˜ gβ ab . We have ˆ g ( φ ) = g ( φ ) for all φ ∈ [ b, π − b ]. Since Ξ( A inc ) ⊂ [ b, π − b ] we have Y (ˆ g ) ( ϕ ) = Y ( g ) ( ϕ ) for all ϕ ∈ A inc . Thus A inc ⊂ C is also a chaotic attractingset for the dynamics of (22) with coupling function ˆ g . In addition, there is astable relative periodic orbit ϕ (cid:63) ( t ) ≈ ( tω (cid:63) , . tω (cid:63) , . tω (cid:63) , . tω (cid:63) ). For A coh = { ϕ (cid:63) ( t ) | t ≥ } we have Ξ( A coh ) ⊂ [ − . , . A inc and A coh are coupling function separated; see Figure 1.Now consider two weakly coupled populations (29) of n = 4 oscillators.Since Σ( A coh ) = Σ( A inc ) = { id } we have that Σ( A coh × A inc ) = { id } for ε = 0and we expect dynamically invariant sets with trivial symmetry for small ε >
0. We integrated system (29) numerically and calculated the maximalLyapunov exponent from the variational equations. The attracting set A ( ε ) for ε = 0 .
01 close to A coh × A inc with trivial setwise symmetries and positivemaximal Lyapunov exponent is shown in Figure 2; the absolute value of thelocal order parameter R (cid:96) ( t ) = (cid:12)(cid:12) (cid:80) j =1 exp( iϕ (cid:96),j ) (cid:12)(cid:12) gives information aboutthe synchronization of each population: it is equal to one if all oscillatorswithin the populations are phase synchronized. Integration was carried out in MATLAB using the variable order scheme ode113 with aadaptive time step ∆ t ≤ − subject to conservative relative and absolute error tolerancesof 10 − and 10 − respectively. HIMERAS AND SYMMETRY 15 ϕ , ϕ , ϕ , ϕ , ϕ , ϕ , ϕ , ϕ , − . O s c ill a t o r ˙ ϕ ℓ , k ( t ) λ m a x Time t (a) Phase evolution − − . . . R y , y , R − − . . . R y , y , R (b) Projected dynamics of each population
Figure 2.
Chaotic weak chimeras with trivial setwise sym-metries appear in the S (cid:111) S -equivariant system (29) withtwo populations of n = 4 oscillators for ε = 0 .
01. Panel ( a )shows the phase evolution: the phase of the oscillators (pe-riodic color scale, ϕ (cid:96),k ( t ) = 0 in black and ϕ (cid:96),k ( t ) = π inwhite) in a co-rotating frame at the speed of the first oscilla-tor is shown at the top, the instantaneous frequencies ˙ ϕ (cid:96),k ( t )in the middle, and convergence of the maximal Lyapunov ex-ponent at the bottom. Panel ( b ) shows the dynamics on theattracting set for each population in the Z -equivariant pro-jections y (cid:96) = (sin( ϕ (cid:96), − ϕ (cid:96), ) , sin( ϕ (cid:96), − ϕ (cid:96), )) where Z arethe permutations within populations that preserve the phaseordering.For increasing coupling parameter ε (while keeping the initial conditionfixed) the symmetries of the attracting chaotic weak chimeras A ( ε ) change;cf. Figure 3. We integrated the system for T = 2 · time units to cal-culate both the maximal Lyapunov exponents and detect the presence ofnontrivial symmetries. For A ( ε ) ⊂ C we have to check for permutationsof oscillators that preserve the ordering of the phases within each popu-lation to determine the symmetry of the attractor. To this end, we cal-culated the ergodic average S (cid:96) ( ε ) = (cid:82) T sin( ϕ (cid:96), ( t ) − ϕ (cid:96), ( t ))d t along the − − . . . R y , y , R − − . . . R y , y , R (a) Attractor in the S -equivariant projection y (cid:96) − . . .
02 0 .
04 0 .
06 0 .
08 0 . − λ m a x ( ε ) R a n g e o f ˙ ϕ ℓ , k Inter-population coupling ε (b) Maximal Lyapunov exponents and symmetries with varying ε Figure 3.
Increasing ε yields chaotic weak chimeras thatundergo symmetry increasing bifurcations. Panel ( a ) showsa trajectory for ε = 0 . A ( ε ) with Σ( A ( ε ) ) (cid:54) = { id } . Panel ( b ) shows maximal Lyapn-uov exponents obtained by integrating (29) from a fixedinitial condition on A (0) for T = 2 · time units. Themarker indicates the symmetry of the attractor A ( ε ) ⊂ T n :“ • ” for Σ( A ( ε ) ) = { id } , “ (cid:5) ” if Σ( A ( ε ) ) ? = { id } , and “ ◦ ”if Σ( A ( ε ) ) (cid:54) = { id } . The shaded regions show the intervals[min k,t ˙ ϕ (cid:96),k ( t ) , max k,t ˙ ϕ (cid:96),k ( t )] for (cid:96) = 1 (dark gray) and (cid:96) = 2(light gray)—where these do not overlap, there is no fre-quency synchronization between the two populations andhence a weak chimera.trajectory which converges zero if Σ( A ( ε ) ) (cid:54) = { id } . Note that if symmet-ric copies of attractors merge in a symmetry increasing bifurcation [40],these ergodic averages may converge very slowly. Previous numerical in-vestigations of the chaotic attractor in the uncoupled system [39] showedthat attractors with trivial symmetry are confined to one quadrant underthe projection y (cid:96) = (sin( ϕ (cid:96), − ϕ (cid:96), ) , sin( ϕ (cid:96), − ϕ (cid:96), )). Thus, the number ofquadrants Q (cid:96) ( ε ) that the projected trajectory enters being greater than oneindicates that a symmetry increasing bifurcation may have occurred; com-pare also Figures 2(b) and 3(a). Consequently, we conclude Σ( A ε ) = { id } if | S ( ε ) | > − —but we write Σ( A ε ) ? = { id } if Q ( ε ) > A ε ) (cid:54) = { id } otherwise. HIMERAS AND SYMMETRY 17
Further numerical investigation shows that there is multistability for ε ≥
0; the attracting sets A ( ε ) for ε > Discussion
If a dynamical system has permutational symmetry Γ, what are the sym-metry properties of the asymptotic angular frequencies which describe howtrajectories wind around phase space? For the dynamical systems on C n • considered in Section 3, the asymptotic angular frequencies are given byaverages of Γ-equivariant observables. This observation yields a natural re-formulation of the notion of a weak chimera in terms of the isotropy of thevector of asymptotic angular frequencies. Our definition is not only compat-ible with the action of Γ but also goes beyond phase oscillators in the weakcoupling limit: it applies to more general oscillator models where chimerastates have been reported [17, 18]. With a rigorous definition in place, itwould be desirable to prove the existence of weak chimeras in such systemsand show that the dynamics observed are persistent phenomena. These ideasequally apply to more general spaces X with a symmetry group acting on it;here asymptotic winding numbers of asymptotic cycles describe the rotationof a trajectory with respect to topological properties of X [20, 41, 29]. Pre-cisifying the notion of a weak chimera for general topological spaces X withsymmetry is beyond the scope of the current paper and will be addressed infuture work.Using coupling functions that give rise to relative equilibria with trivialsymmetry, we showed that for symmetric phase oscillator systems that thereare indeed weak chimeras that have symmetries in the frequencies that arenot present in the solutions. This motivates some further symmetry relatedquestions. For example, what are the possible isotropy groups of the an-gular frequency vector for a Γ-equivariant system that do not arise fromthe symmetries of the solutions themselves? (These are obviously restrictedto subgroups of the symmetry group.) Which symmetry increasing bifur-cations happen as the inter-population coupling ε is increased (Figure 3)?While chaotic dynamics do persist up to ε ≈ . ε ≈ . ε close to zero. Thus, are therechaotic weak chimeras with trivial symmetry for “strongly coupled” popu-lations of phase oscillators? Moreover, in general there will be more thanone ergodic invariant measure supported on a weak chimera. For each ofthese measures we obtain asymptotic angular frequency vectors that poten-tially have different isotropy. While the set of all measures supported onthe invariant set of interest [42] yields bounds of the asymptotic angularfrequencies (see also [16]), a more detailed understanding what the specificisotropy subgroups for the invariant measures are and how they bifurcatewould be desirable.It is also worth noting that asymptotic angular frequencies as averagesand their isotropy may still be well be defined if the permutational symme-try of the system is broken due to a (small) perturbation. However, care has to be taken to extend the notion of a weak chimera to nearly symmet-ric systems since symmetry breaking can have drastic effects on frequencysynchronization [43].“Classical” chimera states were first observed on rings of nonlocally cou-pled phase oscillators [11]. A finite-dimensional approximation yields a dy-namical system that is equivariant with respect to the action of the dihedralgroup [30]. Roughly speaking, classical chimeras on finite dimensional ringsare trajectories that show characteristic angular frequency synchronizationfor some finite time as they exhibit pseudo-random drift along the ringbefore converging to the fully synchronized state [44, 45]. These are notweak chimeras in the sense defined above. By contrast, initial conditionsin the (dynamically invariant) fixed point spaces of a reflection symmetryyield symmetric solutions that eventually converge to the fully synchronizedstate [46], resembling a transient weak chimera. Interestingly, the chaoticweak chimeras constructed here share an important feature with these “clas-sical” chimera states: the isotropy of the angular frequency vector may belarger than the symmetry of the solution itself as the oscillators in the “co-herent” region are never perfectly phase synchronized. Thus, clarifying therelationship between classical chimera states on rings and the symmetryof the system further—also with respect to the symmetries of the systemthat describes the continuum limit—provides exciting directions for futureresearch. Acknowledgements
The author would like to thank the anonymous referees for their feed-back which significantly helped to improve the presentation of the results.Moreover, the author would like to thank Peter Ashwin, Erik Martens, andOleh Omel’chenko for stimulating discussions and critical feedback on themanuscript. The research leading to these results has received funding fromthe People Programme (Marie Curie Actions) of the European Union’s Sev-enth Framework Programme (FP7/2007–2013) under REA grant agreementno. 626111.
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Appendix A. A trigonometric polynomial coupling function
The Fourier coefficients of (32) in Section 6 are given by a = − . ζ = 0 . a = 0 . ζ = 0 . a = 0 . ζ = 0 . a = − . ζ = 0 . a = − . ζ = 0 . a = − . ζ = 0 . a = 0 . ζ = 0 . a = − . ζ = 0 . a = 0 . ζ = 0 . a = − . ζ = 0 . a r = 0, ζ r = 0 for all other r ∈ N ..