HOPF BIFURCATION WITH S N -SYMMETRY

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1 HOPF BIFURCATIO WITH S -SYMMETRY AA PAULA S. DIAS AD AA RODRIGUES Abstract. We study Hof bifurcation with S -symmetry for the standard absolutely irreducible action of S obtained from the action of S by ermutation of coordinates. Stewart Symmetry methods in collisionless many-body roblems, J. onlinear Sci obtains a classification theorem for the C-axial subgrous of S S. We use this classification to rove the existence of branches of eriodic solutions with C-axial symmetry in systems of ordinary differential euations with S - symmetry osed on a direct sum of two such S -absolutely irreducible reresentations, as a result of a Hof bifurcation occurring as a real arameter is varied. We determine the generic conditions on the coefficients of the fifth order S S -euivariant vector field that describe the stability and criticality of those solution branches. We finish this aer with an alication to the cases = 4 and = 5.. Introduction The general theory of Hof Bifurcation with symmetry was develoed by Golubitsky and Stewart [3] and by Golubitsky, Stewart, and Schaeffer [6]. Golubitsky and Stewart [4] alied the theory of Hof bifurcation with symmetry to systems of ordinary differential euations having the symmetries of a regular olygon this is, with D n -symmetry. They studied the existence and stability of symmetry-breaking branches of eriodic solutions in such systems. Finally, they alied their results to a general system of n nonlinear oscillators, couled symmetrically in a ring, and describe the generic oscillation atterns. Since the develoment of the theory, some examles were studied with detail see for examle, [],[7]-[9],[0, Chater 5],[],[],[7], [9],[],[3]. In this aer we study one of the few classic roblems in the theory of Hof bifurcation with symmetry that has not been comletely investigated: Hof bifurcation with S - symmetry. This roblem is relevant to, for examle, the behaviour of all-to-all couled nonlinear oscillators. See for examle the grou theoretic work done by Aronson et al. [] on eriod doubling with S -symmetry and its alication to couled arrays of Josehson junctions; and the work of Ashwin and Swift [4] on the analysis of networks of identical dissiative oscillators weakly couled. The basic existence theorem for Hof bifurcation in the symmetric case is the Euivariant Hof Theorem, which involves C-axial isotroy subgrous of S S in this case, this is, isotroy subgrous with two-dimensional fixed-oint subsace. Stewart [] obtains a classification theorem for C-axial subgrous of S S. We use this classification and the Euivariant Hof Theorem to rove the existence of branches of eriodic solutions in systems of ordinary differential euations with S -symmetry taking the restriction of the standard action of S on C onto a S -simle sace. Moreover, we determine generic conditions on the coefficients of the fifth order S S -euivariant Date: January 6, Mathematics Subject Classification. 37G40, 37C3, 34C5. Key words and hrases. Hof bifurcation, eriodic solution, satio-temoral symmetry. The second author was suorted by FCT Grant SFRH/BD/863/004. Research of APSD and AR suorted in art by Centro de Matemática da Universidade do Porto CMUP financed by FCT through the rogrammes POCTI and POSI, with Portuguese and Euroean Community structural funds.

2 AA PAULA S. DIAS AD AA RODRIGUES vector field that describe the stability of the different tyes of bifurcating eriodic solutions. Consider the natural action of S on C where σ S acts by ermutation of coordinates: σz,..., z = z σ,..., z σ where z,..., z C. Observe the following decomosition of C subsaces for this action: into invariant C = C,0 V where and C,0 = {z,..., z C : z + + z = 0} V = {z,..., z : z C} = C. The action of S on V is trivial and the sace C,0 is S -simle: where S acts absolutely irreducibly on C,0 = R,0 R,0 R,0 = {x,..., x R : x + + x = 0} = R. We say that a reresentation of a grou Γ on a vector sace V is absolutely irreducible, or the sace V is said to be absolutely irreducible, if the only linear maings on V that commute with Γ are the scalar multiles of the identity. If we have a local Γ-euivariant Hof bifurcation roblem, generically the centre subsace at the Hof bifurcation oint is Γ-simle see [6, Proosition XVI.4]. We make that assumtion here. Thus we consider a general S -euivariant system of ordinary differential euations ODEs dz dt = fz, λ, where z C,0, λ R is the bifurcation arameter and f : C,0 R C,0 is smooth and commutes with the restriction of the natural action of S on C to the S -simle sace C,0. Observe that f0, λ 0 since Fix C,0S = {0}. We study Hof bifurcation of from the trivial euilibrium, say, at λ = 0, and so we assume that df 0,0 has urely imaginary eigenvalues ±i after rescaling time if necessary. Thus if we denote the eigenvalues of df 0,λ by σλ ± iρλ then σ0 = 0, ρ0 = see [6, Lemma XVI.5] and we make the standard hyothesis of the Euivariant Hof Theorem: σ 0 0. Under the above hyothesis, we can assume that the action of S on the centre sace C,0 of df 0,0 that can be identified with the exonential of df 0,0 is given by multilication by e iθ : 3 θz,..., z = e iθ z,..., z for θ S, z,..., z C,0.

3 HOPF BIFURCATIO WITH S -SYMMETRY 3 Structure of the Paer. In Section we recall the key oints for Hof bifurcation theory of symmetric systems. In Section 3 we describe the classification of the C-axial subgrous of S S acting on C,0 given by Stewart []. There are two tyes of C-axial subgrous of S S : I, and II Theorem 3.. We also obtain the isotyic decomosition of C,0 for the action of each of these grous Table 4. In Section 4 we use the Euivariant Hof Theorem to rove the existence of branches of eriodic solutions with these symmetries of by Hof bifurcation from the trivial euilibrium at λ = 0 for a bifurcation roblem with symmetry Γ = S. The main result of this aer is Theorem 4. determining the directions of branching and the stability of eriodic solutions guaranteed by the Euivariant Hof Theorem. For solutions with symmetry II the terms of the degree three truncation of the vector field determine the criticality of the branches and also the stability of these solutions near the origin. However, for solutions with symmetry I,, although the criticality of the branches is determined by the terms of degree three, the stability of solutions in some directions is not. Moreover, in one articular direction, we show that even the degree five truncation is too degenerate. In Section 5 we resent our results for the cases = 4 and = 5. While for = 4 we only need the degree three truncation of the vector field in order to determine the branching euations and the stability of the eriodic solutions guaranteed by the Euivariant Hof Theorem, for = 5 the degree five truncation is necessary and sufficient. We determine exlicitly the directions of branching and the stability of these solutions in both cases. Furthermore, we observe the generic existence of eriodic solutions with submaximal symmetry for the = 4 case. In Section 6 we rove the main result of this aer on the stability of the eriodic solutions guaranteed by the Euivariant Hof Theorem, Theorem 4.. In Aendix A we state the main technical details for the comutation of the fifth order truncation of the vector field euivariant under S S defined on C,0. Finally, in Aendix B we resent the bifurcation diagrams for the eriodic solutions with maximal isotroy for Hof bifurcation with S 4 S -symmetry.. Background In this section we review some key oints related to Hof bifurcation theory of symmetric systems. For the basics of euivariant bifurcation theory see, for examle, Golubitsky et al.[6, Chater XVI]. Consider a system of ODEs 4 dx dt = fx, λ, f0, 0 = 0, where x V, λ R is the bifurcation arameter, f : V R V is a smooth C maing and f0, λ 0 for all λ R. We say that 4 undergoes a Hof Bifurcation at λ = 0 if df 0,0 has a air of urely imaginary eigenvalues. Here, df 0,0 denotes the n n Jacobian matrix of the derivatives of f with resect to the variables x j, evaluated at x, λ = 0, 0. Suose that Γ is a comact Lie grou with a linear action on V = R n and f commutes with Γ or it is Γ-euivariant, this is, fγx, λ = γfx, λ for all γ Γ, x V, λ R. We are interested in branches of eriodic solutions of 4 occurring by Hof bifurcation from the trivial solution x, λ = 0, 0. We say that a reresentation V of Γ is Γ-simle if either V = W W, where W is absolutely irreducible for Γ, or V is irreducible, but not absolutely irreducible for Γ. Suose that df 0,0 has urely imaginary eigenvalues ±iω. Then, generically, the corresonding real generalized eigensace of df 0,0 is Γ- simle see [6, Proosition XVI.4]. Assuming these conditions and suosing that R n is Γ-simle, after an euivariant change of coordinates and a rescaling of time if

4 4 AA PAULA S. DIAS AD AA RODRIGUES necessary, we can assume that df 0,0 has the form 0 Im df 0,0 = I m 0 = J where I m is the m m identity matrix and m = n/. This is due to the fact that if we assume that R n is Γ-simle, the maing f is Γ-euivariant and df 0,0 has i as an eigenvalue, then the eigenvalues of df 0,λ consist of a comlex conjugate air σλ ± iρλ, each with multilicity m. Moreover, σ and ρ are smooth functions of λ and there is an invertible linear ma S : R n R n, commuting with Γ, such that df 0,0 = SJS see [6, Lemma XVI.5]. We define the isotroy subgrou of x V in Γ as x = {γ Γ : γx = x} Γ and the fixed-oint sace of a subgrou Γ is the subsace of V defined by Fix = {x V : γx = x, γ }. For any Γ-euivariant maing f and any subgrou Γ we have ffix R Fix. Identify the circle S with R/πZ and suose that xt is a eriodic solution of 4 in t of eriod π. A symmetry of xt is an element γ, θ Γ S such that γxt = xt θ. The set of all symmetries of xt forms a subgrou xt = {γ, θ Γ S : γxt = xt θ}. Take the natural action of Γ S on the sace C π of π-eriodic functions from R into V, given by γ, θ xt = γxt + θ. Thus, the action of Γ on C π is induced through its satial action on V and S acts by hase shift. This way, the initial definition of symmetry of the eriodic solution xt may be rewritten as γ, θ xt = xt and with resect to this action, xt is the isotroy subgrou of xt. So if we assume 4 where f commutes with Γ and df 0,0 = L has urely imaginary eigenvalues, we can aly a Liaunov-Schmidt reduction, reserving symmetries, that will induce a different action of S on a finite-dimensional sace, which can be identified with the exonential of L Ei acting on the imaginary eigensace E i of L. The reduced function of f will commute with Γ S see [6, Chater XVI Section 3]. Consider the system of ODEs 4, where f : R n R R n is smooth and commutes with a comact Lie grou Γ. Assume the generic hyothesis that R n is Γ-simle and df 0,0 has i as eigenvalue. Thus, after a change of coordinates, we can assume that df 0,0 = J, where m = n/. The eigenvalues of df 0,λ are σλ ± iρλ each with multilicity m. Therefore σ0 = 0 and ρ0 =. Furthermore, assume that σ 0 0, that is, the eigenvalues of df 0,λ cross the imaginary axis with nonzero seed. Let Γ S be an isotroy subgrou such that dimfix =. Then by the Euivariant Hof Theorem see [6, Theorem XVI 4.] there exists a uniue branch of small-amlitude eriodic solutions to 4 with eriod near π, having as their grou of symmetries. The basic idea in the Euivariant Hof Theorem is that small amlitude eriodic solutions of 4 of eriod near π corresond to zeros of a reduced euation ϕx, λ, τ = 0 where τ is the eriod-erturbing arameter. To find eriodic solutions of 4 with

5 HOPF BIFURCATIO WITH S -SYMMETRY 5 symmetries is euivalent to find zeros of the reduced euation with isotroy and they corresond to the zeros of the reduced euation restricted to Fix. The main tool for calculating the stabilities of the eriodic solutions including those guaranteed by the Euivariant Hof Theorem is to use a Birkhoff normal form of f: by a suitable coordinate change, u to any given order k, the vector field f can be made to commute with Γ and S in the Hof case. This result is the euivariant version of the Poincaré-Birkhoff normal form Theorem. Suose that the vector field f in 4 is in Birkhoff normal form. Then it is ossible to erform a Liaunov-Schmidt reduction on 4 such that the reduced euation ϕ has the exlicit form ϕx, λ, τ = fx, λ + τjx, where τ is the eriod-scaling arameter see [6, Theorem XVI 0.]. Let x 0, λ 0, τ 0 be a solution to ϕ = 0 with isotroy, and let xt be the corresonding solution of 4. Then xt is orbitally stable if the n d where d = dimγ + dim eigenvalues of dϕ x0,λ 0,τ 0 which are not forced to be zero by the grou action have negative real arts see [6, Corollary XVI 0.]. Thus, the assumtions of Birkhoff normal form imlies that we can aly the standard Hof Theorem to ẋ = fx, λ restricted to Fix R. In this case, exchange of stability haens, so that if the trivial steady-state solution is stable subcritically, then a subcritical branch of eriodic solutions with isotroy subgrou is unstable. Suercritical branches may be stable or unstable deending on the signs of the real art of the eigenvalues on the comlement of Fix. Call the system ẏ = Ly + g y + + g k y the kth order truncated Birkhoff normal form. The dynamics of the truncated Birkhoff normal form are related to, but not identical with, the local dynamics of the system 4 around the euilibrium x = 0. On the other hand, in general, it is not ossible to find a single change of coordinates that uts f into normal form for all orders. And if it is, then there is the roblem of the first tail. When discussing the stability of the solutions found using the Euivariant Hof Theorem we suose that the kth order truncation of f commutes also with S. Thus we are ignoring terms of higher order that do not commute necessarily with S and that can change the dynamics. However, we use a result that we state below that guarantees that the stability results for the eriodic solutions hold even when f is of the form fx, λ + o x k, where f commutes with Γ S but o x k commutes only with Γ, rovided k is large enough see [6, Theorem XVI.]. Here we use hx = o x k to mean that hx/ x k 0 as x 0. Before we state the result, we recall the definition of -determined stability of an isotroy subgrou Γ S. Suose that dim Fix =. Following [6, Definition XVI.], we say that has -determined stability if all eigenvalues of d f x0,λ 0 + τ 0 J, other than those forced to zero by, have the form on a eriodic solution xs of µ j = α j a m j + oa m j 5 ẋ = fx, λ such that xs = a, where α j is a C-valued function of the Taylor coefficients of terms of degree lower or eual in f. We exect that the real arts of the α j to be generically

6 6 AA PAULA S. DIAS AD AA RODRIGUES nonzero: these are the nondegeneracy conditions on the Taylor coefficients of f at the origin that are obtained when comuting stabilities along the branches. In this case, we say that f is nondegenerate for. Suose that the hyotheses of the Euivariant Hof Theorem hold, and the isotroy subgrou Γ S has -determined stability. Let k and assume that fx, λ = fx, λ + o x k where f commutes with Γ S and is nondegenerate for. Then for λ sufficiently near 0, the stabilities of a eriodic solution of ẋ = fx, λ with isotroy are given by the same exressions in the coefficients of f as those that define the stability of a solution of the truncated Birkhoff normal form ẋ = fx, λ with isotroy subgrou see [6, Theorem XVI.]. As it has been said there always exists a olynomial change utting f in the form fx, λ + o x k. Thus, if the -determined stability condition holds, the stability analysis for f is comleted. 3. C-Axial Subgrous of S S Recalling and 3, we consider the following action of S S on C,0 : 6 σ, θz,..., z = e iθ z σ,..., z σ where σ, θ S S and z,..., z C,0. In order to aly the Euivariant Hof Theorem we reuire information on the C-axial isotroy subgrous this is, on the isotroy subgrous with two-dimensional fixed-oint subsace of S S for this action. In this section we recall the classification obtained by Stewart [] of the C-axial isotroy subgrous of S S. We use then the form of such isotroy subgrous to obtain the isotyic decomosition of C,0 under the action of each of these grous. We recall that if S S, we can decomose C,0 into isotyic comonents 7 C,0 = U U r where each U j is the isotyic comonent of tye V j for the action of on C,0. Here V,..., V r are distinct -irreducible saces. Thus if W is a Γ-invariant subsace of C,0 and -isomorhic to V j then W U j. These decomositions will lay an imortant role at the calculation of the stability of the eriodic solutions guaranteed by the Euivariant Hof Theorem 3.. C-Axial Subgrous of S S. Isotroy subgrous of S S acting on the S -simle sace C,0 are of the tye H θ = {h, θh : h H} where H S and θ : H S is a grou homomorhism see [6, Definition XVI 7., Proosition XVI 7.]. Also the C-axial subgrous are maximal with resect to fixing a comlex line Cz = {µz : µ C}, where z 0. A vector z such that the isotroy subgrou z in S S fixes only Cz is called an axis. Stewart [] comutes the C-axial isotroy subgrous, u to conjugacy, by describing the axes, that is, the orbit reresentatives. Theorem 3. Stewart []. Suose that. Then the axes of S S acting on C,0 have orbit reresentatives as follows: Tye I Let = k + where k,, 0. Let ξ = e πi/k and set 8 z =,..., ; ξ,..., ξ; ξ }{{}}{{},..., ξ ;... ; ξ }{{} k,..., ξ k ; 0,..., 0. }{{}}{{}

7 HOPF BIFURCATIO WITH S -SYMMETRY 7 Tye II Let = +, < / and set 9 z = where a = /. Proof. See Stewart [, Theorem 7].,..., ; a,..., a }{{}}{{} ext we consider the corresonding isotroy subgrous as in []. For tye I we have C-axial subgrous H θ = z where 0 z = S Z k S def = I,. Here denotes the wreath roduct see Hall [8,. 8] and the tilde indicates that Z k is twisted into S. Let K = kerθ = S S k S, where S j is the symmetric grou on B j = {j +,..., j} and S is the symmetric grou on B 0 = {k +,..., }. ow we have the action of the twist in each of the k blocks of elements. Let α =, +, +,..., k +. Then I, is generated by α, π/k and K. For the tye II, the isotroy subgrou is z = S S def = II where the resective factors are the symmetric grous on {,..., } and { +,..., }. Table lists the C-axial isotroy subgrous of S S acting on C,0 and the corresonding fixed-oint subsaces. Remark 3.. In terms of all-to-all couled nonlinear cells systems of ordinary differential euations, one interretation is that solutions with I,-symmetry have k grous, each grou comrising cells with the same waveform and the same hase, and one grou of cells also with the same waveform and the same hase. Cells from the k grous oscillate identically excet for hase shifts of jt/k, for j = 0,,..., k and T is the eriod, between each grou. Cells in the grou of cells oscillate with kth the freuency of the cells of the other grous. Solutions with II corresond to two grous consisting of and cells. At each grou, cells have the same waveform and same hase. Each grou oscillates with a different wave form. Examle 3.3. We aly Theorem 3. to the cases = 4 and = 5. i For = 4, note that the number of artitions of an element of C 4,0 into k blocks of eual elements each, lus a grou of null elements with 4 = k + where k 4, and 0 is four: thus we obtain, u to conjugacy, four isotroy subgrous of tye I. ow if 4 = + where < then =, = 3 and so we get, u to conjugacy, one isotroy subgrou of tye II. See Table for the C-axial isotroy subgrous of S 4 S, the corresonding generators and the fixed-oint subsaces. Observe that we have used Table with the trilets, k, and the airs, corresonding to each isotroy subgrou of a given form as follows: 3 : = k =, = 0; : =, k =, = ; 3 : =, k = 3, = ; 4 : =, k = 4, = 0; 5 : =, = 3.

8 8 AA PAULA S. DIAS AD AA RODRIGUES Isotroy Subgrou I, = S Z k S = k +, k,, 0 Fixed-Point Subsace z,...; ξz }{{},...;... ; ξ k z }{{},...; 0,..., 0 : z }{{}}{{} C II = S S = +, < z,..., z ; }{{} z,..., z : z C }{{} Table. C-axial isotroy subgrous of S S acting on C,0 and fixed-oint subsaces. Here ξ = e πi/k. Isotroy Generators Orbit Fixed-Point Subgrou Reresentative Subsace = S Z 43, π, 34, π,,, {z, z, z, z : z C} = Z S 34,, π,, 0, 0 {z, z, 0, 0 : z C} 3 = Z 3 4 = Z 4 3, π 3 34, π, ξ, ξ, 0 { z, ξz, ξ z, 0 : z C }, i,, i {z, iz, z, iz : z C} 5 = S 3 3, 4, 3, 3, 3 { z, 3 z, 3 z, 3 z : z C } Table. C-axial isotroy subgrous of S 4 S acting on C 4,0, generators, orbit reresentatives and fixed-oint subsaces. Here ξ = e πi/3. ii For = 5 we have five isotroy subgrous of tye I and two of tye II. See Table 3. Secifically, we have that i, i =,..., 5 are of the form I, and 6, 7 are of the form with: II 4 : = k =, = ; : =, k =, = 3; 3 : =, k = 3, = ; 4 : =, k = 4, = ; 5 : =, k = 5, = 0; 6 : =, = 3; 7 : =, = 4.

9 HOPF BIFURCATIO WITH S -SYMMETRY 9 Isotroy Generators Fixed-Point Subgrou Subsace = S Z, 34, 34, π {z, z, z, z, 0 : z C} = Z S 3 34, 35,, π {z, z, 0, 0, 0 : z C} 3 = Z 3 S 45, 3, π { 3 z, ξz, ξ z, 0, 0 : z C }, ξ = e πi/3 4 = Z 4 34, π {z, iz, z, iz, 0 : z C} 5 = Z { 5 345, π 5 z, ξz, ξ z, ξ 3 z, ξ 4 z : z C }, ξ = e πi/5 { 6 = S S 3, 34, 35 z, z, 3 z, 3 z, 3 z : z C } 7 = S 4 3, 4, 5 { z, 4 z, 4 z, 4 z, 4 z : z C } Table 3. C-axial isotroy subgrous of S 5 S acting on C 5,0, generators and fixed-oint subsaces. 3.. Isotyic decomosition of C,0 under the action of the C-Axial Subgrous of S S. For the two tyes of isotroy subgrous I, and II, we decomose C,0 into subsaces, each of which is invariant under a different reresentation of the corresonding isotroy subgrou. The isotyic comonents for the action of I, and II on C,0 are listed in Table 4. Secifically, for I, = S Z k S we form the isotyic decomosition k 5 C,0 = W 0 W W W 3 where W 0 = Fix I,, W and the k subsaces P j, j =,, k are comlex one-dimensional subsaces, invariant under I,. Moreover, W and W 3 are comlex invariant subsaces of dimension resectively and k. ote that if = 0 we have I, = S Z k and then W, W do not occur in the isotyic decomosition of C,0 for the action of I,. Moreover, we only have the occurrence of W in the isotyic decomosition if. Furthermore, we only have the isotyic comonent W 3 if and P j if k 3. For II = S S we form the isotyic decomosition 6 C,0 = W 0 W W where W 0 = Fix II and W, W are comlex invariant subsaces of dimension resectively and that are the sum of two isomorhic real absolutely irreducible reresentations of dimension resectively and of II. j= P j

10 0 AA PAULA S. DIAS AD AA RODRIGUES Tye of Isotroy Subgrou Isotyic comonents I, W 0 = {z,..., z }{{} ; ξz,..., ξz ;... ; ξ k z }{{},..., ξ k z ; 0,..., 0 : z }{{}}{{} C} = k + k, 0 W = {z,..., z }{{} ; k z,..., k z : z C} if k }{{} W W 3 = {0,..., 0; z,..., z, z z : z,..., z C} if }{{} = {z,..., z, z ;... ; z }{{} k +,..., z k, z k ; 0,..., 0} if }{{}}{{} P j = {z,...; ξ j z }{{},...;... ; ξ jk z }{{},...; 0,..., 0 : z }{{}}{{} C} if k 3 for j =,..., k II = + < W 0 = z,..., z ; }{{} z,..., z : z C }{{} W = {z,..., z, z z, 0,..., 0 : z,..., z C} if W = {0,..., 0, z +,..., z, z + z : z +,..., z C} if Table 4. Isotyic comonents of C,0 for the action of I, and II. Here, in W 3 we have z = z z,..., z k = z k + z k and z,..., z,..., z k +,..., z k C. Examle 3.4. We return to the cases = 4 and = 5. i Recalling Table, euation 3 and Table 4, for the three isotroy subgrous i, for i =, 3, 4, of S 4 S, the isotyic decomosition takes, resectively, the form C 4,0 = W 0 W W, C 4,0 = W 0 W P, C 4,0 = W 0 P P 3 where W 0 = Fix i, W, W, P and P 3 are the comlex one-dimensional isotyic comonents for the action of i on C 4,0. For and 5 we obtain that C 4,0 = W 0 W 3 and C 4,0 = W 0 W, where W 3, W are comlex two-dimensional invariant subsaces. See Table 5. ii For = 5, we recall Table 3 for the C-axial subgrous of S 5 S, euation 4 and Table 4. We have that i, for i =,..., 5 are of the form I, and 6, 7 are of the form II. We decomose C 5,0 into isotyic comonents for the action of each isotroy subgrou i, see Table 6.

11 HOPF BIFURCATIO WITH S -SYMMETRY Isotroy subgrou and Isotyic comonents of C 4,0 Orbit Reresentative = S Z W 0 = Fix = {z, z, z, z : z C} z = z, z, z, z W 3 = {z, z, z, z : z, z C} = Z S W 0 = Fix = {z, z, 0, 0 : z C} z = z, z, 0, 0 W = {z, z, z, z : z C} W = {0, 0, z, z : z C} 3 = Z 3 W 0 = Fix 3 = { z, ξz, ξ z, 0 : z C } z = z, ξz, ξ z, 0 W = {z, z, z, 3z : z C} P = { z, ξ z, ξz, 0 : z C } 4 = Z 4 W 0 = Fix 4 = {z, iz, z, iz : z C} z = z, iz, z, iz P = {z, z, z, z : z C} P 3 = {z, iz, z, iz : z C} 5 = S 3 W 0 = Fix 5 = { z, 3 z, 3 z, 3 z : z C } z = z, 3 z, 3 z, 3 z W = {0, z, z 3, z z 3 : z, z 3, C} Table 5. Isotyic decomosition of C 4,0 for the action of each of the isotroy subgrous listed in Table. Here ξ = e πi/3. Remark 3.5. Ashwin and Swift [4] resent a framework for analysing arbitrary networks of identical dissiative oscillators assuming weak couling. When every oscillator is connected to every other one with eual couling, the network has S -symmetry. Assume a S -symmetric network of identical oscillators, each having an asymtotically stable limit cycle, with couling arameter ɛ. For ɛ = 0 the dynamics of the system reduces to a linear flow on an -torus T which is normally hyerbolic, and hence ersists for ɛ 0. For weak couling the euations can be averaged, leading to a S T -euivariant flow on the torus T. In [4], the authors classify the ossible satio-temoral symmetry grous of any eriodic oscillation by comuting all isotroy subgrous for the action of S T on T. These grous u to conjugacy are in oneto-one corresondence with the ways of writing = mk + +k l with integers m and k k k l. They also comute the stability and redict the generic bifurcations of some of the eriodic orbits which can have submaximal symmetry. ow the action of S T on T can be seen as a linear action of S T on C, restricted to T. It follows then that some of the grous described by [4], not necessarily maximal, are in corresondence with the maximal grous I,0 for k =, l =, = m and II for m =, k =, k = = of S S action on C,0 obtained by Stewart []. Moreover, for those grous, the descrition of the stability atterns that is made by [4] using the symmetry roerties is euivalent to the one we obtain considering the decomosition of the sace into isotyic comonents. In future work we lan to look for solutions with submaximal symmetry, where we hoe that some of work done in [] can be used.

12 AA PAULA S. DIAS AD AA RODRIGUES Isotroy subgrou and Isotyic comonents of C 5,0 Orbit Reresentative = S Z W 0 = Fix = {z, z, z, z, 0 : z C} z = z, z, z, z, 0 W = {z, z, z, z, 4z : z C} W 3 = {z, z, z, z, 0 : z, z C} = Z S 3 W 0 = Fix = {z, z, 0, 0, 0 : z C} z = z, z, 0, 0, 0 W = { z, z, 3 z, 3 z, 3 z : z C } W = {0, 0, z, z, z z : z C} 3 = Z 3 S W 0 = Fix 3 = { z, ξz, ξ z, 0, 0 : z C } z = z, ξz, ξ z, 0, 0 W = { z, z, z, 3 z, 3 z : z C } W = {0, 0, 0, z, z : z C} ξ = e πi/3 P = { z, ξ z, ξ 4 z, 0, 0 : z C } 4 = Z 4 W 0 = Fix 4 = { z, ξz, ξ z, ξ 3 z, 0 : z C } z = z, ξz, ξ z, ξ 3 z, 0 W = {z, z, z, z, 4z : z C} ξ = i P = { z, ξ z, ξ 4 z, ξ 6 z, 0 : z C } P 3 = { z, ξ 3 z, ξ 6 z, ξ 9 z, 0 : z C } 5 = Z 5 W 0 = Fix 5 = { z, ξz, ξ z, ξ 3 z, ξ 4 z : z C } z = z, ξz, ξ z, ξ 3 z, ξ 4 z P = { z, ξ z, ξ 4 z, ξ 6 z, ξ 8 z : z C } ξ = e πi/5 P 3 = { z, ξ 3 z, ξ 6 z, ξ 9 z, ξ z : z C } P 4 = { z, ξ 4 z, ξ 8 z, ξ z, ξ 6 z : z C } 6 = S S 3 W 0 = Fix 6 = { z, z, 3 z, 3 z, 3 z : z C } z = z, z, 3 z, 3 z, 3 z W = {z, z, 0, 0, 0 : z, z, C} W = {0, 0, z, z, z z : z, z, C} 7 = S 4 W 0 = Fix 7 = { z, 4 z, 4 z, 4 z, 4 z : z C } z = z, 4 z, 4 z, 4 z, 4 z W = {0, z, z 3, z 4, z z 3 z 4 : z, z 3, z 4, C} Table 6. Isotyic decomosition of C 5,0 for the action of each of the isotroy subgrous listed in Table Periodic Solutions with Maximal Isotroy Consider the system of ODEs dz dt = fz, λ, where f : C,0 R C,0 is smooth, commutes with Γ = S and df 0,λ has eigenvalues σλ ± iρλ with σ0 = 0, ρ0 = and σ 0 0. Our aim is to study eriodic solutions of 7 obtained by Hof bifurcation from the trivial euilibrium. ote that we are assuming that f satisfies the conditions of the Euivariant Hof Theorem. By Theorem 3. we have u to conjugacy the C-axial subgrous of S S. See Table. Therefore, we can use the Euivariant Hof Theorem to rove the existence of

13 HOPF BIFURCATIO WITH S -SYMMETRY 3 eriodic solutions with these symmetries for the bifurcation roblem 7 with symmetry Γ = S. As stated in Section, eriodic solutions of 7 of eriod π/ + τ are in one-toone corresondence with the zeros of gz, λ, τ, the reduced function obtained by the Lyaunov-Schmidt rocedure where τ is the eriod-erturbing arameter. Assuming that f commutes with Γ S, gz, λ, τ has the exlicit form 8 gz, λ, τ = fz, λ + τiz. see [6, Theorem XVI 0.]. Throughout denote by νλ = µλ + τi. If zt is a eriodic solution of 7 with λ = λ 0 and τ = τ 0, and z 0, λ 0, τ 0 is the corresonding solution of 8, then there is a corresondence between the Flouet multiliers of zt and the eigenvalues of dg z0,λ 0,τ 0 such that a multilier lies inside resectively outside the unit circle if and only if the corresonding eigenvalue has negative resectively ositive real art see [6, Corollary XVI 0.]. So, we determine the orbital stability of each tye of bifurcating eriodic orbit by calculating the eigenvalues of dg z0,λ 0,τ 0 to the lowest order in z that are not forced by the symmetry to be zero. As g commutes with Γ S, it mas Fix into itself where is either of tye I, or II described in Table. By the Euivariant Hof Theorem, for each of the conjugacy classes I, and II, we have a distinct branch of eriodic solutions of 7 that are in corresondence with the zeros of g with isotroy I, and II. These zeros are found by solving g Fix I, = 0 and g Fix II = 0 and Fix I,, Fix II are two-dimensional. ote that to find the zeros of g, it suffices to look at reresentative oints on Γ S orbits. Let z0 Γ be the isotroy subgrou of z 0. Then, for σ z0 we have dg z0 σ = σdg z0. That is, dg z0 commutes with the isotroy subgrou of z 0. For the two tyes of isotroy subgrous I, and II, it is ossible to ut the Jacobian matrix dg z0 into block diagonal form. We do this using the isotyic decomosition of C,0 for the action of each C-axial I, and II on C,0 obtained in Section 3. and listed in Table 4. We recall that each isotyic comonent is invariant under a different reresentation of the corresonding isotroy subgrou z0 and so it is left invariant by dg z0. See for examle [6, Theorem XII 3.5]. We organize the rest of this section in the following way. Below we resent the general form of a S S -euivariant bifurcation roblem, u to degree 5. Using that we obtain the form of g in 8. Then we describe the branching euations and the stability of the eriodic solutions of 7 obtained by Hof bifurcation from the trivial euilibrium guaranteed by the Euivariant Hof Theorem Theorem 4.. General Form of a S S -euivariant Hof Bifurcation Problem. We resent the general form of the S S -euivariant bifurcation roblem 7, u to degree 5, leaving the details to Aendix A. Our stability results stated in Theorem 4. below show that the degree 5 terms of f in 7 are necessary to describe the stability of some of the eriodic solutions guaranteed by the Euivariant Hof Theorem. If we suose that the Taylor series of degree five of f around z = 0 commutes also with S, then we can write 9 fz = f z, f z,..., f z = f z,..., z, λ, f z, z,..., z, λ,..., f z, z,..., z, λ

14 4 AA PAULA S. DIAS AD AA RODRIGUES where and f z,..., z, λ = µλz + f 3 z,..., z, λ + f 5 z,..., z, λ + f 3 ] z,..., z, λ = A [ z z k= z k z k + A z k= z k + A 3 z k= z k f 5 ] z,..., z, λ = A 4 [ z 4 z k= z k 4 z k + A 5 z i= z i 4 + A 6 z i= z i j= z j + A 7 z i= z i j= z j + A 8 [z j= z j z j ] i= z i j= z j z j + A 9 [z 3 j= z j ] k= z3 k j= z j + [ A 0 z i= z i ] j= z j + A z i= z i zi + A [z j= z3 j ] i= z i j= z3 j + [ A 3 z k= z k z k i= z i ] j= z j z j + A 4 [ z z k= z k i= z ] i z i j= z j + A 5 [ z z k= z k i= z ] i z i j= z j with z = z z. The coefficients A i, for i =,..., 5 are comlex smooth functions of λ, µ0 = i and Reµ 0 0. Suose that Reµ 0 > 0. Rescaling λ if necessary we can suose that Reµλ = λ + where + stands for higher order terms in λ. Thus the trivial solution of 7 is stable for λ negative and unstable for λ ositive near zero. Throughout, subscrits r and i on the coefficients A,..., A 5 refer to real and imaginary arts. Stability Result. The main result of this aer is the following theorem which we rove in Section 6: Theorem 4.. Consider the system 7 where f is as in 9 and 4. Assume that Reµ 0 > 0, such that the trivial euilibrium is stable if λ < 0 and it is unstable if λ > 0 near the origin. For each tye of the isotroy subgrous of the form I, and II listed in Table, consider the corresonding isotyic decomosition of C,0 resented in Table 4. Let 0,..., r be the functions of A,..., A 5 evaluated at λ = 0 and listed in: Table 8; Table 9 if the isotroy subgrou has tye I, for k =, k = 3 or II ; Table 0 if the isotroy subgrou has tye I,, for 3 < k. Then: For each i the corresonding branch of eriodic solutions is suercritical if 0 < 0 and subcritical if 0 > 0. Table 7 lists the branching euations. For each i, if j > 0 for some j = 0,..., r, then the corresonding branch of eriodic solutions is unstable. If j < 0 for all j, then the branch of eriodic solutions is stable near λ = 0 and z = 0. The alication of this result to study Hof bifurcation with S -symmetry for a secific value of consists mainly into the following stes: i To consider the general form of the Hof bifurcation roblem with S S - symmetry 7 where f is given by 9. ii To enumerate the C-axial subgrous of S S using Table.

15 HOPF BIFURCATIO WITH S -SYMMETRY 5 Isotroy Subgrou Branching Euations I,, < k λ = A r + ka 3r z + = k +,, 0 I,, k = λ = [A r + A r + A 3r ] z + = +,, 0 ] II λ = A r [ z = +, A r + A 3r + z + < Table 7. Branching euations for S Hof bifurcation. Subscrit r on the coefficients refer to the real art and + stands for higher order terms. Isotroy Subgrou 0 I,, < k = k +,, 0 A r + ka 3r I,, k = A r + A r + A 3r = +,, 0 ] II A r [ + A r + A 3r + = +, < Table 8. Stability for S Hof bifurcation in the direction of W 0 = Fix. For each grou, the corresonding branch of eriodic solutions is suercritical if 0 < 0 and subcritical if 0 > 0. iii For each C-axial subgrou : c.i To describe the isotyic decomosition for its action on C,0 according Table 4. c.ii Using Table 7, to obtain the euation of the branch of eriodic solutions with -symmetry for the Hof bifurcation roblem guaranteed by the Euivariant Hof Theorem. The criticality of the branch is given by the sign of 0 listed in Table 8: it is suercritical if 0 < 0 and subcritical if 0 > 0. c.iii To enumerate the functions of the coefficients A,..., A 5 evaluated at λ = 0 of f that determine the stability of the corresonding eriodic solutions near the origin: using Table 8, Table 9 if the isotroy subgrou has tye I, for k =, k = 3 or II and Table 0 if the isotroy subgrou has tye

16 6 AA PAULA S. DIAS AD AA RODRIGUES I,, for 3 < k. Observe that in Tables 9 and 0, we consider only the exressions determining the stability in the directions of the isotyic comonents that occur in the isotyic decomosition of C,0 under the action of. If j < 0 for all j, then the branch of eriodic solutions is stable near λ = 0 and z = 0. If for some j we have j > 0 then the branch of eriodic solutions is unstable. In articular, if the branch is subcritical j > 0, then the solutions are unstable. 5. Two examles: = 4 and = 5 In this section we aly the results of sections 3 and 4 to study Hof bifurcation with S -symmetry for the secial cases = 4 and = 5. When = 4, we show that the directions in which we need the fifth degree truncation of the vector field do not aear in the isotyic decomosition for the action of each C-axial grou on C 4,0. This means this is the case in fact the only one for values of 4 where the degree three truncation of the vector field determines generically the stability of the solutions guaranteed by the Euivariant Hof Theorem. When = 5, we have that the directions in which we need the degree five truncation of the vector field are resent in the isotyic decomosition for some of the C-axial isotroy subgrous. Moreover, the degree five terms determine comletely the stability of the eriodic solutions guaranteed by the Euivariant Hof Theorem. 5.. S 4 Hof Bifurcation. We consider the action of S 4 S on C 4,0 given by 6 for = 4. We study Hof bifurcation with S 4 -symmetry by considering dz 0 = fz, λ, dt where f : C 4,0 R C 4,0 is smooth, commutes with S 4 and df 0,λ has eigenvalues σλ ± iρλ with σ0 = 0, ρ0 = and σ 0 0. Assuming that f also commutes with S we obtain the general form of f given by 9 with = 4. Examle 3.3 describes the C-axial subgrous of S 4 S : we obtain four isotroy subgrous of the tye I, and one of tye II see Table. Moreover, recall the corresondence between the notation of Table and the C-axial subgrous i of S 4 S given by 3. ow in Table 5, we have the isotyic decomosition of C 4,0 for the action of each of these isotroy subgrous. The Euivariant Hof Theorem guarantees that for each C-axial grou we have a branch of eriodic solutions with that symmetry of 0 obtained by Hof bifurcation from the trivial euilibrium since we are assuming that f satisfies the conditions of the Euivariant Hof Theorem. For each value of, and k if alicable associated with i, using Table 7, we get the branching euations listed on Table. We comute now from Tables 8, 9 and 0 the criticality and the stability of the solutions guaranteed by the Euivariant Hof Theorem. The exressions for 0 for each isotroy subgrou follow from 3 and Table 8. We obtain now the exressions for,..., r. For the grous,, 3 and 5 in Table 5, we aly Table 9: note for examle that is I, with = k = and = 0. This corresonds to an isotroy subgrou I, with k = on Table 9. Since = 0 we get, from the last two exressions associated with the stability in the directions of the isoyic comonent W 3. ow is I, with =, k =, =, the grou 3 is I, with =, k = 3, =, and 5 is II with = and = 3. For the grou 4 in Table 5, which is I, with =, k = 4, = 0, we use now Table 0. ote that as =, we have that W 3 does not occur in the isotyic decomosition of C 4,0 for its action. See Table for the comlete stability analysis

17 HOPF BIFURCATIO WITH S -SYMMETRY 7 Isotroy Isotyic,..., r Subgrou Comonent I,, k = W = +, 0 W W 3 4 A r A r, 4 A A + A + A A r A r, A + A A if A r A r, A A A + A if if I,, k = 3 W 6 A r, if 6 A + A = 3 + W { Ar, A if, 0 W ReA A if P { Ar + 6A r A + 6A 3 A II W = + W + 3 A r + A r + 3 A A r + A r A + A A A if + A + A + + A if Table 9. Stability for S Hof bifurcation in the directions of each isotyic comonent. and Aendix B for the bifurcation diagrams for the eriodic solutions with C-axial symmetry. Remark 5.. i The stability of the eriodic solutions guaranteed by the Euivariant Hof Theorem obtained in Table deends only on the coefficients A, A, A 3 of the degree three truncation of the vector field f given by 9 for = 4.

18 8 AA PAULA S. DIAS AD AA RODRIGUES Isotroy Isotyic,..., r Subgrou Comonent I,, 3 < k W = k +, 0 W W 3 P P k P j j = 3,..., k k A r, if k A + A A r, A if { the fifth degree truncation is too degenerate to determine the stability in the directions in W 3 if k 4, A r A k A r + ka r, A A + ka A if k 4 ReA ξ + ReA A 4 + ka A 4 if k 5 ReA ξ + ReA A 4 + ka A 4 if k 6 Table 0. Stability for S Hof bifurcation in the directions of each isotyic comonent. Here ξ = A 4 + 3kA + k k A3 + ka 4 +k k A4 +k A 5 and ξ = ξ 3kA ka 4. ii Observe that eriodic solutions with 3 -symmetry are always unstable since generically = A > 0. If A r > 0, then solutions with symmetry 4 are unstable and if A r < 0, then solutions with symmetry are also unstable. Periodic solutions with submaximal isotroy. We look now for ossible branches of eriodic solutions that can bifurcate for the system 0 with = 4 with submaximal isotroy. We have that the grous Z and S listed in Table 3 are submaximal isotroy subgrous of S 4 S. In fact, using the results of Ashwin and Podvigina [3] see remark below, these are the only isotroy subgrous of S 4 S with fixed-oint subsace of comlex dimension As it was stated before, when f is suosed to commute also with S, then the roblem of finding eriodic solutions of ż = fz, λ can be transformed to the roblem of finding the zeros of gz, λ, τ = 0 where g = f + τiz. However, for the branches of eriodic solutions with submaximal isotroy that are found here, we can no longer guarantee that they exist for 0 if f commutes only with S 4 even with the third order Taylor series commuting with S. These solutions branches are guaranteed only for the third order truncation with which we work from now on. Consider the truncation of f as in 9

19 HOPF BIFURCATIO WITH S -SYMMETRY 9 Isotroy Subgrou Branching Euations λ = A r + 4A r + 4A 3r z + λ = A r + A r + A 3r z + 3 λ = A r + 3A 3r z + 4 λ = A r + 4A 3r z + 5 λ = A r + 4A r + 4A 3r z + Table. Branching euations for S 4 Hof bifurcation. Subscrit r on the coefficients refer to the real art and + stands for higher order terms. with = 4 of degree three and the resective reduced vector field g = f + τiz of the same degree. Let = Z. We study g Fix. Consider the normalizer of in S 4 S, S4 S, defined by S4 S = {γ S 4 S : γ γ = }. ow S4 S is the largest subgrou of S 4 S acting on Fix. See for examle Chossat and Lauterbach [5, Lemma..9]. Thus g Fix is S4 S -euivariant. Easy comutations show that S4 S = D 4 S. One way of roving this is the following. We recall that the isotroy subgrous Γ S are always of the form G θ = {g, θg Γ S : g G} where G Γ and θ : G S is a grou homomorhism. Denote by K = Kerθ. ow by Golubitsky and Stewart [5, Lemma.5], we have that Γ S G θ = CG, K S where CG, K = {γ Γ : γgγ g K, g G}. Taking G θ = = Z = {Id, 34, π}, the rojection of G θ into S 4 is the grou G = {Id, 34}, and θ is the homomorhism θ : G S with trivial kernel such that θ34 = π. It follows then that for this case CG, K = {Id, 4, 34, 43, 34, 34, 43, 3} = D 4. When we restrict g to Fix Z = {z, z, z, z : z, z C}, we obtain the following D 4 S -euivariant system: ż = z λ + iω + A z + z + B z + Cz z ż = z λ + iω + A z + z + B z + Cz z where z, z C, A = A 3, B = A + A and C = A. This is the normal form for the generic Hof bifurcation roblem with symmetry D 4 studied by Swift [3]. The nontrivial solutions in the sace Fix Z with maximal isotroy are the solutions with symmetry S Z, Z 4, Z S recall Table, corresonding, resectively, to zeros of tye z = z, z = iz and z = 0. ote that for solutions corresonding to the

20 0 AA PAULA S. DIAS AD AA RODRIGUES Isotroy 0,..., r Subgrou A r 4A r A r + 4A r + 4A 3r A 4A A + 4A A r + A r + A 3r A r A r A + A A A r 4 A A + A A r 3 A r + 3A 3r A A r + 6A r A + 6A 4 A A r 4 A r + 4A 3r A A r + 8A r A + 8A A 5A r A r A r + 4A r + 4A 3r 5A + A A + A Table. Stability for S 4 Hof bifurcation. For each i, the corresonding branch of eriodic solutions is suercritical if 0 < 0 and subcritical if 0 > 0. If j > 0 for some j = 0,..., r, then the corresonding branch of eriodic solutions is unstable. If j < 0 for all j, then the solutions are stable near λ = 0 and z = 0. ote that solutions with 3 -symmetry are always unstable. Isotroy Subgrou Generators Fixed-Point Subsace = Z 34, π {z, z, z, z : z, z C} = S 3 {z, z, z, z z : z, z C} Table 3. Generators and fixed-oint subsaces corresonding to the isotroy subgrous of S 4 S with fixed-oint subsaces of comlex dimension two.

21 HOPF BIFURCATIO WITH S -SYMMETRY isotroy subgrou Z S we have that z, 0, z, 0 is conjugated to z, z, 0, 0. Their stability roerties are studied in [4], [6] and [3]. By [3], in addition to these eriodic solutions, there can be a fourth branch of eriodic solutions to with z z and z z 0. Thus, these corresond to Z -symmetric solutions of 0 where f is as in 9 with = 4 truncated to the third order. Moreover, this solution branch exists if and the solutions are generically unstable. Re[A + A A ] < A < A + A Remark 5.. In [3], Ashwin and Podvigina considered Hof bifurcation with the grou O of rotational symmetries of the cube. The grou O is isomorhic to S 4 and it has two non-isomorhic real irreducible reresentations of dimension three. In [3] they consider the irreducible reresentation of O corresonding to rotational symmetries of a cube in R 3 = W. When studying Hof bifurcation, they take two coies of this irreducible reresentation. Secifically, they consider the action of O S on W W generated by: ρ z, z, z 3 = z, z 3, z ρ 00 z, z, z 3 = z, z, z 3 γ θ z, z, z 3 = e iθ z, z, z 3 θ S. Although the ermutation grou S 4 is isomorhic to the grou of rotations of a cube, the action of O on W and the natural action of S 4 on R 4,0 are not isomorhic. Recall that R 4,0 = {x, x, x 3, x 4 R 4 : x +x +x 3 +x 4 = 0}. However, the action of O S on W W and the action of S 4 S on C 4,0 are isomorhic see [3]. Thus from the oint of view of Hof bifurcation, the two non-isomorhic actions of S 4 give rise to the same results. In [3], the isotroy lattice for the action of O S on C 3 is obtained and the isotroy subgrous with fixed-oint subsaces of comlex dimension two have normalizers given, resectively, by D 4 S and D S. These are in corresondence with the normalizers of Z and S for the action of S 4 S on C 4,0 considered here. 5.. S 5 Hof Bifurcation. We consider the action of S 5 S on C 5,0 given by 6 for = 5. We study Hof bifurcation with S 5 -symmetry by considering 3 dz dt = fz, λ, where f : C 5,0 R C 5,0 is smooth, commutes with S 5 and df 0,λ has eigenvalues σλ ± iρλ with σ0 = 0, ρ0 = and σ 0 0. Assuming that f also commutes with S we obtain the general form of f given by 9 with = 5. Examle 3.3 describes the C-axial subgrous of S 5 S acting on C 5,0, together with their generators and fixed-oint subsaces, given by Table 3: we have used Table where i, i =,..., 5 are of the form I, and 6, 7 are of the form II. The corresondence between the notation of Table and the C-axial subgrous i of S 5 S is given by 4. Table 6 gives the isotyic decomosition of C 5,0 for each of the isotroy subgrous i listed in Table 3. Proceeding the same way that we did for = 4 we get Tables 4 and 5, which give resectively the branching euations and the stability for Hof bifurcation with S 5 -symmetry. ote that in articular it follows that solutions with 3 and with 4 - symmetry are always unstable.

22 AA PAULA S. DIAS AD AA RODRIGUES Isotroy Subgrou Branching Euations λ = A r + 4A r + 4A 3r z + λ = A r + A r + A 3r z + 3 λ = A r + 3A 3r z + 4 λ = A r + 4A 3r z + 5 λ = A r + 5A 3r z + 6 λ = A r + 0A r + 0A 3r z + 7 λ = A r + 5A r + 5A 3r z + Table 4. Branching euations for S 5 Hof bifurcation. Subscrit r on the coefficients refer to the real art and + stands for higher order terms. 6. Proof of Theorem 4. In this section we rove the main result of this aer, Theorem 4.. We consider the system 7 where f is as in 9. Assume that Reµ 0 > 0, such that the trivial euilibrium is stable if λ < 0 and it is unstable if λ > 0 near the origin. Thus f satisfies the conditions of the Euivariant Hof Theorem. It follows that for each C-axial subgrou of S S in Table, we have a branch of eriodic solutions to 7 with that symmetry obtained by Hof bifurcation from the trivial euilibrium. Moreover, since we are assuming that f in 9 commutes also with S, as stated in Section, eriodic solutions of 7 of eriod π/ + τ are in one-to-one corresondence with the zeros of gz, λ, τ, where 4 gz, λ, τ = fz, λ + τiz. is the exlicit form of the reduced function obtained by the Lyaunov-Schmidt rocedure. Here τ is the eriod-erturbing arameter. Throughout denote by νλ = µλ +τi. In order to determine the stability of such solutions, we recall that there is a corresondence between the Flouet multiliers of zt and the eigenvalues of dg z0,λ 0,τ 0, if zt is a eriodic solution of 7 with λ = λ 0 and τ = τ 0, and z 0, λ 0, τ 0 is the corresonding solution of 8: a multilier lies inside resectively outside the unit circle if and only if the corresonding eigenvalue has negative resectively ositive real art. So, we determine the stability of each tye of bifurcating eriodic orbit by calculating the eigenvalues of dg z0,λ 0,τ 0 to the lowest order in z. Let z0 Γ be the isotroy subgrou of z 0. ow for each C-axial subgrou z0 of S S in Table, we have that g mas Fix z0 into itself since g commutes with S S. Periodic solutions of 7 with z0 -symmetry are in corresondence with the zeros of g with isotroy z0 which are obtained by solving g Fixz0 = 0. ote that to find the zeros of g, it suffices to look at reresentative oints on S S orbits. We obtain Table 7.

23 HOPF BIFURCATIO WITH S -SYMMETRY 3 Isotroy 0,..., r Subgrou A r + 4A r + 4A 3r 3 5 A r 4A r 3 5 A 4A 5 A + 4A A r 4A r A 4A A + 4A 5 A r A r A r + A r + A 3r 5 A A 3 5 A + A A r A r A + A A 3 A r + 3A 3r A r A + 6A 5 A A A r + 6A r A r A A 4 A r + 4A 3r A r A r A + 8A A A r + 8A r A A r 5 A r + 5A 3r Re[A ξ ξ ] A r + 0A r A + 0A A A r 0A r 3 A r + 0A r + 0A 3r 3 3 A 0A A A 3 A r 8A r 3 3 A 8A A + 0A A r + 5A r + 5A 3r Re A 5 4 A A 5 4 A 6 A A Table 5. Stability for S 5 Hof bifurcation. Here ξ = A 4 + 0A 4 and ξ = A 4 +5A +5A 4. For each i, the corresonding branch of eriodic solutions is suercritical if 0 < 0 and subcritical if 0 > 0. If j > 0 for some j = 0,..., r, then the corresonding branch of eriodic solutions is unstable. If j < 0 for all j, then the solutions are stable near λ = 0 and z = 0. ote that solutions with 3 and 4 symmetry are always unstable.

24 4 AA PAULA S. DIAS AD AA RODRIGUES The Jacobian dg z0 commutes with z0. ow using the isotyic decomosition of C,0 for the action of each C-axial z0 obtained in Table 4, we have that each isotyic comonent is left invariant by dg z0. Thus it is ossible to ut the Jacobian matrix dg z0 into block diagonal. That is, the stability of the eriodic solutions is determined by the restrictions of dg z0 to each of the isotyic comonents of C,0 for the action of z0. Also note that as the grou action forces some of the Flouet multiliers to be eual to one, it also forces the corresonding eigenvalues of dg z0,λ 0,τ 0 to be eual to zero. Recall [6, Theorem XVI 6.]. The eigenvectors associated with these eigenvalues are the tangent vectors to the orbit of S S through z 0. If the solution z 0 has symmetry z0, then the grou orbit has the dimension of Γ S / z0 and so the number of zero eigenvalues of dg z0,λ 0,τ 0 forced by the grou action is d z0 = dim z0 since dim S S =. The grous z0 = I, and z0 = II are discrete, then there is one eigenvalue forced by the symmetry to be zero this is, we get d z0 =. To comute the eigenvalues it is convenient to use the comlex coordinates. We take co-ordinate functions on C : z, z, z, z,..., z, z. These corresond to a basis B for C with elements denoted by b, b, b, b,..., b, b. Recall that an R-linear maing on C R has the form 5 ω αω + βω where α, β C. The matrix of this maing in these coordinates, α β 6 M =, β α has trm = Reα, detm = α β. The eigenvalues of this matrix are trm trm ± detm. If one eigenvalue is zero, then detm = 0 and the sign of the other eigenvalue if it is not zero is given by the sign of the real art of α. If M has no zero eigenvalues, then the eigenvalues have negative real art if and only if the determinant is ositive and the trace is negative. I, = S Z k S, where = k +, k,, 0 The fixed-oint subsace of I, = S Z k S is Fix, I = {z,..., z ; ξz,..., ξz;... ; ξ }{{}}{{} k z,..., ξ k z; 0,..., 0 : z C} }{{}}{{} where ξ = e πi/k. Using the euation 8 where f is as in 9, after dividing by z we have if k νλ + A + ka 3 z + = 0 where + denotes terms of higher order in z and z, and taking the real art of this euation, we obtain, λ = A r + ka 3r z +