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 on W . 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