The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems

and Applied Analysis 3 Here, we assume that the equivariant symmetry S acts antisymplectically and S2 I. Now, we also consider the symmetric property of periodic solutions. This property was not studied for Hamiltonian vector fields without the other structure previously. 2. Main Results Theorem 2.1. Consider an equilibrium 0 of a C∞ equivariant Hamiltonian vector field f , with the equivariant symmetry S acting antisymplectically and S2 I. Assume that the Jacobian matrix Df 0 has two pairs of purely imaginary eigenvalues ±i and no other eigenvalues of the form ±ki, k ∈ Z. Then, the equilibrium is contained in a two-dimensional flow-invariant surface that consists of a one-parameter family of symmetric periodic solutions whose period tends to 2π as they approach the equilibrium. Moreover, the equilibrium is also contained in two smooth two-dimensional flowinvariant manifolds, each containing a one-parameter family of nonsymmetric periodic solutions whose period tends to 2π as they approach the equilibrium. Furthermore, there are no other periodic solutions with period close to 2π in the neighbourhood of 0. Remark 2.2. Here, the existence and the symmetric property of periodic solutions near the equilibrium point are all considered. The main idea is similar to 16 . 3. Linear Equivariant Hamiltonian Vector Field with Purely Imaginary Eigenvalues We now consider the persistent occurrence of purely imaginary eigenvalues in equivariant Hamiltonian vector fields. Let A0 be a linear Hamiltonian vector field. Then, it follows that A0J −JA0 . If A0 is S-equivariant, we have A0S SA0. If S is anti symplectic, we get SJ ±JS. Since we are interested in partially elliptic equilibria, we assume that A0 has a pair of purely imaginary eigenvalues λ and −λ. Moreover, if the eigenvector e1 of A0 has λ, then e1 is an eigenvector for λ. Since A0 is both Hamiltonian and S-equivariant, this implies that if the eigenvector e of A0 has the eigenvalue λ, then Se is also an eigenvector for the eigenvalue λ. Hoveijn et al. 15 considered the linear normal form theory which is based on the construction of minimal 〈J, S〉-invariant subspaces. By 15 , we are only interested in minimal invariant subspaces on which A0 is semisimple; that is, A0 is diagonalizable over C. Here, the type of minimal invariant subspace depends on whether S acts symplectically or antisymplectically. Lemma 3.1. Consider a linear S-equivariant Hamiltonian vector field A0 with Sacting (anti)symplectically and S2 I. Let V be a minimal A0, J, S -invariant subspace on which A0 has purely imaginary eigenvalues. Then A0|V , J |V and S|V have the following normal forms. 1 If S acts symplectically, it follows that dimV 2, and S|V ( 1 0 0 1 ) , J |V ( 0 −1 1 0 ) , A0|V ( 0 −1 1 0 ) . 3.1 4 Abstract and Applied Analysis 2 If S acts antisymplectically, it follows that dimV 4 and S|V ⎛ ⎜⎜⎜⎜⎝ 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 ⎞ ⎟⎟⎟⎟⎠, J |V ⎛ ⎜⎜⎜⎜⎝ 0 −1 0 0 1 0 0 0 0 0 0 1 0 0 −1 0 ⎞ ⎟⎟⎟⎟⎠, A0|V ⎛ ⎜⎜⎜⎜⎝ 0 −1 0 0 1 0 0 0 0 0 0 −1 0 0 1 0 ⎞ ⎟⎟⎟⎟⎠. 3.2 Proof. Let W be a 2-dimensional symplectic subspace on which A0 has purely imaginary eigenvalues. By standard Hamiltonian theory and multiplication of time by a scalar, A0 and J can take the same normal form on W . If the equivariant symmetry S acts symplectically, we have SA0 A0S and SJ JS. Let e1 and e1 be the eigenvectors of A0. By SA0 A0S, a minimal invariant subspace is obtained by choosing Se1 e1. Since J A0 on W , we have SJ JS onW . If S acts antisymplectically, the dimension of the minimal invariant subspace is not two. A 2-dimensional subspace W is defined as above. Assume that S W W . If S W W , by SA0 |W A0S |W , it follows that SJ |W JS |W . This is converse that S acts antisymplectically. So, we have S W W ′ / W and a minimal invariant subspace is given by V W ⊕W ′. So, dimV 4. Moreover, we get J |W ′ S−1JS |W − S−1SJ |W −J |W and A0|W ′ S−1A0S |W S−1SA0 |W A0|W . Since A0|W J |W , it follows that J |W ′ −A0|W ′ . Remark 3.2. Now, we give the examples for the system 1.1 whether S acts symplectically or antisymplectically, where J and S here are defined as J |V and S|V in Lemma 3.1. If S acts symplectically, the system 1.1 can be written as ( ẋ1 ẏ1 ) ( 0 −1 1 0 )( Hx1 Hy1 ) ⎛ ⎝−y1 − 3y2 1 x1 3x2 1 ⎞ ⎠ f x , 3.3 where the Hamiltonian function is H x1, y1 1/2 x2 1 y 2 1 x 3 1 y 3 1, f satisfies fS Sf , and A0 df 0 is calculated the same as A0|V in Lemma 3.1. If S acts antisymplectically, the system 1.1 can be written as ⎛ ⎜⎜⎜⎜⎝ ẋ1 ẏ1 ẏ2 ẋ2 ⎞ ⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎝ 0 −1 0 0 1 0 0 0 0 0 0 1 0 0 −1 0 ⎞ ⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎝ Hx1 Hy1 Hy2 Hx2 ⎞ ⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎝ −y1 − x1y2 x1 y1y2 − x2y2 −x2 − x1y2 y2 − x1y1 x1x2 ⎞ ⎟⎟⎟⎟⎟⎠ f x , 3.4 where the Hamiltonian function isH x1, y1, y2, x2 1/2 x2 1 y 2 1 − 1/2 x2 2 y2 2 x1y1y2− x1x2y2, f satisfies fS Sf , and A0 df 0 is calculated the same as A0|V in Lemma 3.1. Abstract and Applied Analysis 5and Applied Analysis 5 Remark 3.3. When S acts antisymplectically, under the base e1, e2, Se1, Se2 , we obtain S, J and A0 have the forms of S|V , J |V and A0|V in Lemma 3.1, respectively. However, when S acts antisymplectically, under the base e1, Se2, e2, Se1 , we have S|V ⎛ ⎜⎜⎜⎜⎝ 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 ⎞ ⎟⎟⎟⎟⎠, J |V ⎛ ⎜⎜⎜⎜⎝ 0 0 −1 0 0 0 0 −1 1 0 0 0 0 1 0 0 ⎞ ⎟⎟⎟⎟⎠, A0|V ⎛ ⎜⎜⎜⎜⎝ 0 0 −1 0 0 0 0 1 1 0 0 0 0 −1 0 0 ⎞ ⎟⎟⎟⎟⎠. 3.5 In this case, J |V is the standard form. However, for convenience, we use the forms of Lemma 3.1 in this paper. 4. Liapunov-Schmidt Reduction In this paper, when S acts antisymplectically or symplectically, by Lemma 3.1,A0 has a single pair or double pairs of purely imaginary eigenvalues ±i. Moreover, these purely imaginary eigenvalues of A0 are nonresonant; that is, A0 has no other eigenvalues of the form ±ki with k ∈ Z. This condition is clearly generic codimension zero . We want to find the families of periodic solutions in the neighbourhood of the equilibrium point. In this section, we introduce the main technique which is a Liapunov-Schmidt reduction. The Liapunov-Schmidt reduction here is similar to the one in 16, 18, 19 . Assume that a C∞ vector field f : O ⊂ RN → RN has an equivariant symmetry group G, which implies the existence of representations ρ : G → O N such that fρ γ ρ γ f , for all γ ∈ G. Define F : C1 2π × R → C0 2π by F u, τ 1 τ du ds − f u , 4.1 whereC1 2π is the space ofC 1 maps u : S1 → RN andC0 2π is the space ofC0 maps v : S1 → RN . The map F is C∞ by the “Ω-lemma,” that is, in Section 2.4 of 20 . Clearly, the solutions of F u, τ 0 correspond to 2π/ 1 τ -periodic solutions of 1.1 . Now, define an action T : G̃ × C0 2π → C0 2π or in C1 2π by ( Tgu ) t ρ ( γ ) u t θ , 4.2 where g γθ is an element of G̃, G̃ G × S1, γ ∈ G and θ ∈ S1. By the G-equivariance of f , we have that F is G̃-equivariant F ( Tgu, τ ) TgF u, τ , ∀g γθ ∈ G̃. 4.3 Assume that f 0 0. The derivative of F at u 0 is L, where Lv dF 0, 0 · v v′ −A0v, 4.4 with A0 Df 0 . Moreover, kerL span {Re ev0 , Im ev0 } {Re zev0 | z ∈ C}, where A0v0 iv0. 6 Abstract and Applied Analysis By 4.3 , L is also G̃-equivariant such that LTg TgL. Then, Tg preserves kerL and RangeL. Below, we will obtain that kerL ⊥ and Range L ⊥ are also Tg-invariant. We have C1 2π kerL ⊕ kerL ⊥, C0 2π Range L ⊕ ( Range L )⊥ . 4.5 Here, the orthogonal complement is taken in C0 2π and C 1 2π by


Introduction
We first give some definitions for our problem. A 2n × 2n matrix T is called anti symplectic if T T JT ±J. Consider a C ∞ vector field f : O ⊂ R N → R N and the system d dt x f x .

1.1
Let S be a diffeomorphism of R N into itself. If Sf fS, we call that the system 1.1 is S-equivariant or the vector field f is S-equivariant. Denote by I the identity matrix. In this paper, S satisfies S 2 I. When the system 1.1 is S-equivariant, if x t is a solution, then Sx t is also a solution. An orbit x t is called symmetric if it is S-invariant; that is, Sx t x t .
Problems. Consider a systemẋ Ax g x , x ∈ R n with A being an n×n matrix and g O x 2 a C ∞ vector function. Suppose that the system has a nondegenerate integral. Suppose that A has a pair of purely imaginary eigenvalues ±i and no other eigenvalues of the form ±ki, k ∈ Z. That is, the eigenvalues ±i are nonresonant with the other ones. Then, the well-known Liapunov Center theorem tells us that there exists a one-parameter family of periodic orbits emanating from the equilibrium point with the period being close to 2π as they approach to the equilibrium. We call such families the Liapunov Center families.
This result can be used easily to Hamiltonian systems and obtain existence of periodic solutions. Later, many mathematicians were dedicated to study periodic solutions of Hamiltonian systems and tried to generalize the result. Gordon 1 obtained an infinite number of periodic solutions in arbitrarily small neighborhoods of the origin for Hamiltonian systems with convex potential. Weinstein 2 obtained the Liapunov Center families with no eigenvalue assumptions when the equilibrium point is a nondegenerate minimum. In 3 , Moser proved that the integral manifold contains at least one periodic solution whose period is close to that of a periodic solution of the linearized system near the equilibrium point. In 4 , Weinstein proved that a Hamiltonian system possesses at least one periodic solution on each energy surface, provided that this energy surface is compact, convex, and contains no stationary point of the vector field.
Bifurcation theory describes how the dynamics of systems change as parameters varied. The study of the Hamiltonian Hopf bifurcation has a long history. The Hamiltonian-Hopf bifurcation involves the loss of linear stability of a fixed point by the collision of two pairs of imaginary eigenvalues of the linearized flow and their subsequent departure off the imaginary axis. van der Meer 5 studied this bifurcation and classified the periodic solutions that are spawned by this resonance. Using Z 2 singularity theory with a distinguished parameter developed in 6 , Bridges 7 obtained the periodic solutions in a Hamiltonian-Hopf bifurcation.
Moreover, reversible systems have been studied for many years. Devaney 8 first proved a Liapunov Center theorem for reversible vectors fields. Vanderbauwhede  In 16 , Buzzi and Lamb obtained a Liapunov Center theorem for purely reversible Hamiltonian vector fields that are both Hamiltonian and reversible at the same time. They obtained the existence of periodic solutions in the neighbourhood of elliptic equilibria when the reversing symmetry R acts symplectically or antisymplectically. Previously, the symmetric property of periodic solutions is not considered, and the existence of additional periodic solutions is not ruled out. But these problems were considered in 16 . The results in 16 are as follows. If R acts antisymplectically, generically purely imaginary eigenvalues are isolated, and the equilibrium is contained in a local two-dimensional invariant manifold containing a one-parameter family of symmetric periodic solutions. If R acts symplectically, generically purely imaginary eigenvalues are doubly degenerate, and the equilibrium is contained in two two-dimensional invariant manifolds, each containing a one-parameter family of nonsymmetric periodic solutions, and a three-dimensional invariant manifold containing a two-parameter family of symmetric periodic solutions. In 17 , Sternberg theorem for equivariant Hamiltonian vector fields was considered.
Motivated by 16 , in this paper, we consider a Liapunov Center theorem for equivariant Hamiltonian vector fields that are both Hamiltonian and equivariant at the same time. Here, we assume that the equivariant symmetry S acts antisymplectically and S 2 I. Now, we also consider the symmetric property of periodic solutions. This property was not studied for Hamiltonian vector fields without the other structure previously.

Main Results
Theorem 2.1. Consider an equilibrium 0 of a C ∞ equivariant Hamiltonian vector field f, with the equivariant symmetry S acting antisymplectically and S 2 I. Assume that the Jacobian matrix Df 0 has two pairs of purely imaginary eigenvalues ±i and no other eigenvalues of the form ±ki, k ∈ Z. Then, the equilibrium is contained in a two-dimensional flow-invariant surface that consists of a one-parameter family of symmetric periodic solutions whose period tends to 2π as they approach the equilibrium. Moreover, the equilibrium is also contained in two smooth two-dimensional flowinvariant manifolds, each containing a one-parameter family of nonsymmetric periodic solutions whose period tends to 2π as they approach the equilibrium. Furthermore, there are no other periodic solutions with period close to 2π in the neighbourhood of 0.

Linear Equivariant Hamiltonian Vector Field with Purely Imaginary Eigenvalues
We now consider the persistent occurrence of purely imaginary eigenvalues in equivariant Hamiltonian vector fields. Let A 0 be a linear Hamiltonian vector field. Then, it follows that Since we are interested in partially elliptic equilibria, we assume that A 0 has a pair of purely imaginary eigenvalues λ and −λ. Moreover, if the eigenvector e 1 of A 0 has λ, then e 1 is an eigenvector for λ.
Since A 0 is both Hamiltonian and S-equivariant, this implies that if the eigenvector e of A 0 has the eigenvalue λ, then Se is also an eigenvector for the eigenvalue λ.
Hoveijn et al. 15 considered the linear normal form theory which is based on the construction of minimal J, S -invariant subspaces. By 15 , we are only interested in minimal invariant subspaces on which A 0 is semisimple; that is, A 0 is diagonalizable over C.
Here, the type of minimal invariant subspace depends on whether S acts symplectically or antisymplectically. 1 If S acts symplectically, it follows that dim V 2, and Abstract and Applied Analysis 2 If S acts antisymplectically, it follows that dim V 4 and where the Hamiltonian function is H x 1 , y 1 1/2 x 2 1 y 2 1 x 3 1 y 3 1 , f satisfies fS Sf, and A 0 df 0 is calculated the same as A 0 | V in Lemma 3.1. If S acts antisymplectically, the system 1.1 can be written as where the Hamiltonian function is H x 1 , y 1 , y 2 , x 2 1/2 x 2 1 y 2 1 − 1/2 x 2 2 y 2 2 x 1 y 1 y 2 − x 1 x 2 y 2 , f satisfies fS Sf, and A 0 df 0 is calculated the same as A 0 | V in Lemma 3.1.
Abstract and Applied Analysis 5 Remark 3.3. When S acts antisymplectically, under the base e 1 , e 2 , Se 1 , Se 2 , we obtain S, J and A 0 have the forms of S| V , J| V and A 0 | V in Lemma 3.1, respectively. However, when S acts antisymplectically, under the base e 1 , Se 2 , e 2 , Se 1 , we have In this case, J| V is the standard form. However, for convenience, we use the forms of Lemma 3.1 in this paper.

Liapunov-Schmidt Reduction
In this paper, when S acts antisymplectically or symplectically, by Lemma 3.1, A 0 has a single pair or double pairs of purely imaginary eigenvalues ±i. Moreover, these purely imaginary eigenvalues of A 0 are nonresonant; that is, A 0 has no other eigenvalues of the form ±ki with k ∈ Z. This condition is clearly generic codimension zero . We want to find the families of periodic solutions in the neighbourhood of the equilibrium point.
In this section, we introduce the main technique which is a Liapunov-Schmidt reduction. The Liapunov-Schmidt reduction here is similar to the one in 16, 18, 19 . Assume that a C ∞ vector field f : O ⊂ R N → R N has an equivariant symmetry group G, which implies the existence of representations ρ : where C 1 2π is the space of C 1 maps u : S 1 → R N and C 0 2π is the space of C 0 maps v : S 1 → R N . The map F is C ∞ by the "Ω-lemma," that is, in Section 2.4 of 20 . Clearly, the solutions of F u, τ 0 correspond to 2π/ 1 τ -periodic solutions of 1.1 .

Now, define an action
where g γθ is an element of G, G G × S 1 , γ ∈ G and θ ∈ S 1 .
Abstract and Applied Analysis By 4.3 , L is also G-equivariant such that LT g T g L. Then, T g preserves ker L and Range L.
Below, we will obtain that ker L ⊥ and Range L ⊥ are also T g -invariant. We have Here, the orthogonal complement is taken in C 0 2π and C 1 2π by where u, v 2π 0 u t t v t dt and μ is a normalized Haar measure for G. Note that T g u, T g v u, v for all g ∈ G. So, ker L ⊥ and Range L ⊥ are T g -invariant. By Range L and Range L ⊥ are T g -invariant, we can obtain that the projections where v ∈ Range L ⊥ and w ∈ Range L. Then, T g P u T g w P T g w P T g v T g w P T g u and T g I − P u T g v I − P T g v I − P T g v T g w I − P T g u . Next, define a C ∞ map ω : ker L × R → ker L ⊥ with ω 0, 0 0 by solving PF k ω, τ 0, 4.8 for ω ω k, τ using the implicit function theorem. Moreover, we can prove that ω commutes with T g . Define ω g : ker L×R → ker L ⊥ by ω g k, τ T g −1 ω T g k, τ . Note that T g k ∈ ker L. We have PF k ω g k, τ , τ PF T g −1 T g k ω T g k, τ , τ T g −1 PF T g k ω T g k, τ , τ 0. Then, ω g is also the solution of 4.8 . Moreover, ω g 0, 0 ω 0, 0 0. By uniqueness, we get ω g k, τ ω k, τ . So, ω commutes with T g . Then, Then, solutions of the equation F u, τ 0 are given by u k ω k, τ , where k is a solution of the bifurcation equation Now, we will obtain that ϕ k, τ is also G-equivariant.
Proof. Since G-equivariance of I − P , F and ω k, τ , it is easy to obtain this result. 7 Lemma 4.1 indicates that how the symmetry enters the bifurcation equation ϕ. Below, we will also consider the relation between the symmetry and the Hamiltonian function of ϕ.
Since the vector field f X H is Hamiltonian, it follows that holds for all v ∈ R 2n . Define the map Abstract and Applied Analysis then the bifurcation map ϕ is also a Hamiltonian vector field with Hamiltonian h k H k ω k , and ϕ and the function h have the same invariance properties as the given Hamiltonian H.
In this paper, ker L is finite dimensional and 4.21 holds. So, the bifurcation equation ϕ is a Hamiltonian vector field with Hamiltonian h k, τ H k ω k , τ and if the actions of G are antisymplectic, then The corresponding symplectic form is the restriction of Ω to ker L. Moreover, in this paper, we have G Z 2 and G Z 2 × S 1 ∼ O 2 .

Proof of Theorem 2.1
In this section, we prove Theorem 2. Note that dim ker L 4 and 4.21 holds. By Theorem 6.2 in 18 , the bifurcation equation ϕ is also a Hamiltonian vector field. We now proceed to apply the Liapunov-Schmidt reduction of Section 4.
Since ker L ∼ R 4 ∼ C 2 , it follows that So, the bifurcation equation is denoted by Since ϕ is Hamiltonian, let ϕ 2J∇ z h with the Hamiltonian function where J takes the normal form of Lemma 3.1. By 4.22 , it follows that h is S 1 -invariant, h • θ h with θ ∈ S 1 acting on C 2 as and h is S-anti-invariant, h • S −h with S z 1 , z 2 z 2 , z 1 .
We have that ker L is generated by for all k 1 , k 2 ∈ ker L. Using the similar calculation and by 4.17 , it follows that the τ-dependence of the lowest quadratic order of h has the form Using 5.8 and 5.12 , we have ψ 1 0, 0, τ τ/2. We first give Lemma 5.1, which is used for finding the symmetric periodic orbits near the equilibrium point.

Symmetric Periodic Solutions
Since Proof. Since S z 1 , z 2 z 2 , z 1 , the symmetric solutions lie in Fix S { z, z ∈ C 2 }. Then, the equation ϕ 2J∇ z h 0 is equivalent to z 0 or ψ z, τ 0. By 5.8 and 5.12 , we have ∂ψ/∂τ 0, 0 1/2 / 0. Using the implicit function theorem for ψ 0, there exists a function τ τ |z| 2 for all sufficiently small |z| that corresponds to a family of symmetric periodic solutions of the system 1.1 with period 2π/ 1 τ .
Remark 5.2. Apparently, the symmetric periodic solutions should be twoparameters. But since the vector field f is the equivariant Hamiltonian vector field, then the symmetric periodic solutions are in fact one-parameter.

Nonsymmetric Periodic Solutions
By 5.8 , and multiplying the first equation of 5.13 by z 1 and the second equation of 5.13 by z 2 , we have where ψ j i is the partial derivative of ψ j with respect to X i , X 1 |z 1 | 2 |z 2 | 2 , X 2 |z 1 z 2 | 2 , and X 3 z 1 z 2 . Now, we consider the real parts of 5.14 and 5.15 . Subtracting the real part of 5.14 from 5.15 , we obtain X 1 ψ I 2  .

5.22
Moreover, ∂r/∂τ | |z 2 | 2 ,τ 0,0 1/2. Using the implicit function theorem for 5.22 , there exists a function τ τ |z 2 | 2 with τ 0 0 for all sufficiently small |z 2 |. Correspondingly, we have a one-parameter family of nonsymmetric periodic solutions of the system 1.1 contained in a local smooth two-dimensional invariant manifold. z 1 / 0 and z 2 0: similarly to the above case, for ϕ 0, there exists a function τ τ |z 1 | 2 with τ 0 0 for all sufficiently small |z 1 |. Correspondingly, there is another oneparameter family of nonsymmetric periodic solutions of the system 1.1 filling out a local smooth two-dimensional invariant manifold. This family of nonsymmetric periodic solutions are the S-image of the family with z 1 0 and z 2 / 0. Remark 5.4. We now consider the case that S acts symplectically. In this case, purely imaginary eigenvalues pairs typically arise isolated. Since S on the two-dimensional eigenspace V of ±i for A 0 is the identity matrix as in Lemma 3.1, then the symmetric property of the periodic solutions is insignificant. However, we can prove the existence of a Liapunov Center family using the fact that the flow is Hamiltonian 18, 21 .