# The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems

Authors: Jia Li; Yanling Shi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider the existence of the periodic solutions in the
neighbourhood of equilibria forC∞ equivariant Hamiltonian vector fields. If the equivariant symmetry S acts antisymplectically and S2=I, we prove that generically
purely imaginary eigenvalues are doubly degenerate and the equilibrium is contained
in a local two-dimensional flow-invariant manifold, consisting of a one-parameter family of symmetric periodic solutions and two two-dimensional flow-invariant manifolds
each containing a one-parameter family of nonsymmetric periodic solutions. The result is a version of Liapunov Center theorem for a class of equivariant Hamiltonian
systems.


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## Body

## 1. Introduction

We first give some definitions for our problem. A2n×2n matrix T is called (anti)symplectic if TTJT=±J. Consider a C∞ vector field f:O⊂ℛN→ℛN and the system(1.1)ddtx=f(x).
Let S be a diffeomorphism of ℛN 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 S2=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 system ẋ=Ax+g(x), x∈Rn with A being an n×n matrix and g=O(x2) 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 Z2 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 [9] and Sevryuk [10] also studied reversible vectors fields. In [11], Golubitsky et al. studied families of periodic solutions near generic elliptic equilibria for reversible equivariant systems. In [12, 13], Montaldi et al. considered families of periodic solutions near generic elliptic equilibria for reversible equivariant Hamiltonian systems that are both Hamiltonian and reversible at the same time. Since Lamb and Roberts [14] obtained the group theoretical classification of linear reversible equivariant systems, there has been an increasing interest for reversible equivariant systems. Later, Hoveijn et al. [15] obtained the linear normal form and unfolding theory of reversible equivariant linear systems.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 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 aC∞ 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 flow-invariant 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.LetA0 be a linear Hamiltonian vector field. Then, it follows that A0J=-JA0T. 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 thatA0 has a pair of purely imaginary eigenvalues λ and -λ. Moreover, if the eigenvector e1 of A0 has λ, then e̅1 is an eigenvector for λ̅.SinceA0 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 linearS-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)
IfS acts symplectically, it follows that dimV=2, and
(3.1)S|V=(1001),J|V=(0-110),A0|V=(0-110).(2)
IfS acts antisymplectically, it follows that dimV=4 and(3.2)S|V=(0010000110000100),J|V=(0-1001000000100-10),A0|V=(0-1001000000-10010).Proof.
LetW 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 e̅1 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.
IfS 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
(3.3)(ẋ1ẏ1)=(0-110)(Hx1Hy1)=(-y1-3y12x1+3x12)=f(x),
where the Hamiltonian function is H(x1,y1)=(1/2)(x12+y12)+x13+y13, 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
(3.4)(ẋ1ẏ1ẏ2ẋ2)=(0-1001000000100-10)(Hx1Hy1Hy2Hx2)=(-y1-x1y2x1+y1y2-x2y2-x2-x1y2y2-x1y1+x1x2)=f(x),
where the Hamiltonian function is H(x1,y1,y2,x2)=(1/2)(x12+y12)-(1/2)(x22+y22)+x1y1y2-x1x2y2, f satisfies fS=Sf, and A0=df(0) is calculated the same as A0|V in Lemma 3.1.Remark 3.3.
WhenS 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
(3.5)S|V=(0001001001001000),J|V=(00-10000-110000100),A0|V=(00-10000110000-100).
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, whenS 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 aC∞ 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:C2π1×ℛ→C2π0 by(4.1)F(u,τ)=(1+τ)duds-f(u),
where C2π1 is the space of C1 maps u:S1→ℛN and C2π0 is the space of C0 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 actionT:G̃×C2π0→C2π0 or in C2π1 by(4.2)(Tgu)(t)=ρ(γ)(u(t+θ)),
where g=γθ is an element of G̃, G̃=G×S1, γ∈G and θ∈S1.By theG-equivariance of f, we have that F is G̃-equivariant(4.3)F(Tgu,τ)=TgF(u,τ),∀g=γθ∈G̃.Assume thatf(0)=0. The derivative of F at u=0 is L, where(4.4)Lv=dF(0,0)⋅v=v′-A0v,
with A0=Df(0). Moreover, kerL=span{Re(eisv0),Im(eisv0)}={Re(zeisv0)∣z∈𝒞}, where A0v0=iv0.By (4.3), L is also G̃-equivariant such that LTg=TgL. Then, Tg preserves kerL and Range L.Below, we will obtain that(kerL)⊥ and (RangeL)⊥ are also Tg-invariant. We have(4.5)C2π1=ker⁡L⊕(ker⁡L)⊥,C2π0=RangeL⊕(RangeL)⊥.
Here, the orthogonal complement is taken in C2π0 and C2π1 by(4.6)[u,v]=∫G̃〈Tgu,Tgv〉dμ,
where 〈u,v〉=∫02π[u(t)]tv(t)dt and μ is a normalized Haar measure for G̃. Note that [Tgu,Tgv]=[u,v] for all g∈G̃. So, (kerL)⊥ and (RangeL)⊥ are Tg-invariant.ByRangeL and (RangeL)⊥ are Tg-invariant, we can obtain that the projections(4.7)P:C2π0⟶RangeL,(I-P):C2π0⟶(RangeL)⊥
commute with Tg. If u∈C2π0, let u=v+w, where v∈(RangeL)⊥ and w∈RangeL. Then, Tg(P(u))=Tgw=P(Tgw)=P(Tgv+Tgw)=P(Tgu) and Tg((I-P)(u))=Tgv=(I-P)Tgv=(I-P)(Tgv+Tgw)=(I-P)(Tgu).Next, define aC∞ map ω:kerL×ℛ→(kerL)⊥ with ω(0,0)=0 by solving(4.8)PF(k+ω,τ)=0,
for ω=ω(k,τ) using the implicit function theorem.Moreover, we can prove thatω commutes with Tg. Define ωg:kerL×ℛ→(kerL)⊥ by ωg(k,τ)=Tg-1ω(Tgk,τ). Note that Tgk∈kerL. We have PF(k+ωg(k,τ),τ)=PF(Tg-1(Tgk+ω(Tgk,τ)),τ)=Tg-1PF(Tgk+ω(Tgk,τ),τ)=0. Then, ωg is also the solution of (4.8). Moreover, ωg(0,0)=ω(0,0)=0. By uniqueness, we get ωg(k,τ)=ω(k,τ). So, ω commutes with Tg. Then,(4.9)Tgω(k,τ)=ω(Tgk,τ).Then, solutions of the equationF(u,τ)=0 are given by u=k+ω(k,τ), where k is a solution of the bifurcation equation(4.10)φ(k,τ)=(I-P)F(k+ω(k,τ),τ)=0.Now, we will obtain thatφ(k,τ) is also G̃-equivariant.Lemma 4.1.
Iff is G-equivariant, then the bifurcation map φ is G̃-equivariant(4.11)φ(Tgk,τ)=Tgφ(k,τ),∀g∈G×S1.Proof.
SinceG̃-equivariance of I-P, F and ω(k,τ), it is easy to obtain this result.Lemma4.1 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 fieldf=XH is Hamiltonian, it follows that(4.12)ω(XH(u),v)=〈dH(u),v〉,
holds for all v∈ℛ2n. Define the map(4.13)Φ:C2π1×R×C2π0⟶(C2π1)*,
by(4.14)Φ(u,τ,v)⋅U=∫02π{ω(v-(1+τ)duds,U)+dH(u)⋅U}ds.
Since the vector field f is Hamiltonian with the Hamiltonian function H, we have that the implicit constraint Φ satisfies(4.15)Φ(u,τ,F(u,τ))=0.
This condition is rephrased as saying that the map F, regarded as a (parameter-dependent) vector field on C2π1 and is Hamiltonian with respect to the weak symplectic form(4.16)Ω(u,v)=12π∫02πω(u(s),v(s))ds,
and with the Hamiltonian function(4.17)H(u,τ)=12π∫02π{12ω((1+τ)duds,u)-H(u)}ds.If the actions ofG are antisymplectic, we have(4.18)ω(γx,γy)=-ω(x,y).
In this case, Ω is G̃=G×S1 anti-invariant; that is,(4.19)Ω(gu,gv)=-Ω(u,v),g=γθ,γ∈G,θ∈S1,
and the Hamiltonian function ℋ satisfies(4.20)H(gu,τ)=-H(u,τ).By Theorem  6.2 in [18], it follows that if(4.21)ker⁡L=ker⁡L*,
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,kerL 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(4.22)h(gk,τ)=-h(k,τ).
The corresponding symplectic form is the restriction of Ω to kerL. Moreover, in this paper, we have G=𝒵2 and G̃=𝒵2×S1≅O(2).


## 5. Proof of Theorem2.1

In this section, we prove Theorem2.1. Theorem 2.1 can be proved by Lemmas 5.1 and 5.3. By Lemma 5.1, the symmetric periodic solutions are obtained. The existence of the nonsymmetric periodic solutions can be verified by Lemma 5.3.Below, we consider the case that the equivariant symmetryS acts antisymplectically. By Lemma 3.1, in this case, there are two pairs of purely imaginary eigenvalues. Without loss of generality, S|V, J|V and A0|V take the (normal) forms of Lemma 3.1, where V is the four-dimensional eigenspace of ±i for A0.Note that dim kerL=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.SincekerL≅ℛ4≅𝒞2, it follows that(5.1)R4∋(x1,y1,x2,y2)≅(x1+iy1,x2+iy2)=(z1,z2)∈C2.
So, the bifurcation equation is denoted by(5.2)φ:C2×R⟶C2.
Since φ is Hamiltonian, let φ=2J∇z̅h with the Hamiltonian function(5.3)h:C2×R⟶R,
where J takes the (normal) form of Lemma 3.1. By (4.22), it follows that h is S1-invariant, h∘θ=h with θ∈S1 acting on 𝒞2 as(5.4)θ(z1,z2)=(e-iθz1,e-iθz2),
and h is S-anti-invariant, h∘S=-h with(5.5)S(z1,z2)=(z2,z1).ByS1-invariance, it follows that(5.6)h=h(|z1|2,|z2|2,z1z̅2,z2z̅1,τ).Then, by theS-anti-invariance, we have that(5.7)h=(|z1|2-|z2|2)ψ(|z1|2+|z2|2,|z1z2|2,z1z̅2,z2z̅1,τ)
is equivalent to (∂nψ/∂(z1z̅2)n)|z1z̅2=0 is real for all n,z1 and z2 which satisfy n∈𝒵 and z1z̅2=0. If (∂nψ/∂(z1z̅2)n)|z1z̅2=0 is not real for some n,z1 and z2 which satisfy n∈𝒵 and z1z̅2=0, by the S-anti-invariance, we have h=(|z1|2-|z2|2)ψ(|z1|2+|z2|2,|z1z2|2,τ). Thus, without generality, since h is real, we obtain(5.8)h=(|z1|2-|z2|2)ψ(|z1|2+|z2|2,|z1z2|2,z1z̅2,z2z̅1,τ)=(|z1|2-|z2|2)(ψ1(|z1|2+|z2|2,|z1z2|2,τ))+2Re(z1z̅2ψ2(|z1|2+|z2|2,|z1z2|2,z1z̅2,τ)),
where ψ=ψ1+2Re(z1z̅2ψ2).We have thatkerL is generated by(5.9)v1=(sin⁡s,-cos⁡s,0,0)T,v2=(cos⁡s,sin⁡s,0,0)T,v3=(0,0,sin⁡s,-cos⁡s)T,v4=(0,0,cos⁡s,sin⁡s)T.Then, the symplectic formΩ for the reduced bifurcation equation satisfies(5.10)Ω(vn,vm)=12π∫02π〈vn,Jvm〉ds,
where J takes the (normal) form of Lemma 3.1. So,(5.11)Ω(k1,k2)=〈k1,Jk2〉,
for all k1,k2∈kerL.Using the similar calculation and by (4.17), it follows that the τ-dependence of the lowest (quadratic) order of h has the form(5.12)h=(|z1|2-|z2|2)τ2+l(z1,z2,τ).Using (5.8) and (5.12), we have ψ1(0,0,τ)=τ/2.We first give Lemma5.1, which is used for finding the symmetric periodic orbits near the equilibrium point.


### 5.1. Symmetric Periodic Solutions

Sinceh∘S=-h, then the symmetric periodic solutions of the bifurcation equation lie in the level set h=0. In fact, if h=0, we get φ=0. Then, by (4.9), the symmetric solutions lying in h=0 correspond to symmetric 2π-periodic solutions for (4.1). Below, we prove there exist the symmetric solutions for the equation h=0.Lemma 5.1.
In the neighbourhood of the equilibrium(z1,z2,τ)=(0,0,0),  there exist the symmetric solutions for the equation h=0.Proof.
SinceS(z1,z2)=(z2,z1), the symmetric solutions lie in Fix(S)={(z,z)∈𝒞2}. Then, the equation φ=2J∇z̅h=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 fieldf is the equivariant Hamiltonian vector field, then the symmetric periodic solutions are in fact one-parameter.
Next, we study the nonsymmetric periodic solutions in the neighbourhood of the equilibrium 0.


### 5.2. Nonsymmetric Periodic Solutions

Lemma 5.3.
Except the symmetric Liapunov Center family described in Lemma5.1, there are two nonsymmetric Liapunov Center families of periodic solutions, each contained in a local two-dimensional smooth manifold with the period of the periodic solutions converging to 2π as the solutions tend to the equilibrium point.Proof.
The proof can be divided into three cases:z1≠0,z2≠0andz1≠z2,z1=0andz2≠0, and z1≠0andz2=0.
z1≠0,z2≠0andz1≠z2: let I1=|z1|2-|z2|2. It follows that φ=0 is equivalent to ∇z̅h=0. By (5.8), ∇z̅h=0 can be written as
(5.13)(z1-z2)ψ+I1∇z̅ψ=0.
By (5.8), and multiplying the first equation of (5.13) by z̅1 and the second equation of (5.13) by z̅2, we have(5.14)|z1|2(ψ+I1(ψ11+|z2|2ψ21+2Re[z1z̅2(ψ12+|z2|2ψ22)]))+2I1(z2z̅1(ψ2+z1z̅2ψ32¯))=0,(5.15)|z2|2(-ψ+I1(ψ11+|z1|2ψ21+2Re[z1z̅2(ψ12+|z1|2ψ22)]))+2I1Re(z1z̅2(ψ2+z1z̅2ψ32))=0,
where ψij is the partial derivative of ψj with respect to Xi, X1=|z1|2+|z2|2, X2=|z1z2|2, and X3=z1z̅2.
Now, we consider the real parts of (5.14) and (5.15). Subtracting the real part of (5.14) from (5.15), we obtain
(5.16)X1ψ+I12ψ11+2I12Re(z1z̅2ψ12)=0.
So,
(5.17)X1ψ+I12∂ψ∂X1=0.
By (5.17), we have
(5.18)ψ=ce-X12/2I12,
where c is a constant. Let us take (5.18) into h=I1ψ and verify ∇z̅h=0. Then, the first equation ∇z̅1h=0 becomes
(5.19)ce-X12/2I12z1-I1ce-X12/2I12⋅4X1z1I12-4I1z1X124I14=0.
If ψ≠0, it follows that
(5.20)1=I1X1-X12I12.
By (5.20), we get
(5.21)|z1|4+|z2|4=-2|z2|4.
But (5.21) does not hold in this case. Therefore, we find no small solutions of the bifurcation equation for z1≠0, z2≠0and z1≠z2.
z1=0andz2≠0: in this case, the equation ∇z̅h=0 is equivalent to
(5.22)z2(ψ1+|z2|2ψ11)=0⟺r(|z2|2,τ)=ψ1+|z2|2ψ11=0.
Moreover, (∂r/∂τ)|(|z2|2,τ)=(0,0)=1/2. Using the implicit function theorem for (5.22), there exists a function τ=τ(|z2|2) with τ(0)=0 for all sufficiently small |z2|. 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.
z1≠0andz2=0: similarly to the above case, for φ=0, there exists a function τ=τ(|z1|2) with τ(0)=0 for all sufficiently small |z1|. Correspondingly, there is another one-parameter 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 z1=0 and z2≠0.Remark 5.4.
We now consider the case thatS acts symplectically. In this case, purely imaginary eigenvalues pairs typically arise isolated. Since S on the two-dimensional eigenspace V of ±i for A0 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].


## 5.1. Symmetric Periodic Solutions

Sinceh∘S=-h, then the symmetric periodic solutions of the bifurcation equation lie in the level set h=0. In fact, if h=0, we get φ=0. Then, by (4.9), the symmetric solutions lying in h=0 correspond to symmetric 2π-periodic solutions for (4.1). Below, we prove there exist the symmetric solutions for the equation h=0.Lemma 5.1.
In the neighbourhood of the equilibrium(z1,z2,τ)=(0,0,0),  there exist the symmetric solutions for the equation h=0.Proof.
SinceS(z1,z2)=(z2,z1), the symmetric solutions lie in Fix(S)={(z,z)∈𝒞2}. Then, the equation φ=2J∇z̅h=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 fieldf is the equivariant Hamiltonian vector field, then the symmetric periodic solutions are in fact one-parameter.
Next, we study the nonsymmetric periodic solutions in the neighbourhood of the equilibrium 0.


## 5.2. Nonsymmetric Periodic Solutions

Lemma 5.3.
Except the symmetric Liapunov Center family described in Lemma5.1, there are two nonsymmetric Liapunov Center families of periodic solutions, each contained in a local two-dimensional smooth manifold with the period of the periodic solutions converging to 2π as the solutions tend to the equilibrium point.Proof.
The proof can be divided into three cases:z1≠0,z2≠0andz1≠z2,z1=0andz2≠0, and z1≠0andz2=0.
z1≠0,z2≠0andz1≠z2: let I1=|z1|2-|z2|2. It follows that φ=0 is equivalent to ∇z̅h=0. By (5.8), ∇z̅h=0 can be written as
(5.13)(z1-z2)ψ+I1∇z̅ψ=0.
By (5.8), and multiplying the first equation of (5.13) by z̅1 and the second equation of (5.13) by z̅2, we have(5.14)|z1|2(ψ+I1(ψ11+|z2|2ψ21+2Re[z1z̅2(ψ12+|z2|2ψ22)]))+2I1(z2z̅1(ψ2+z1z̅2ψ32¯))=0,(5.15)|z2|2(-ψ+I1(ψ11+|z1|2ψ21+2Re[z1z̅2(ψ12+|z1|2ψ22)]))+2I1Re(z1z̅2(ψ2+z1z̅2ψ32))=0,
where ψij is the partial derivative of ψj with respect to Xi, X1=|z1|2+|z2|2, X2=|z1z2|2, and X3=z1z̅2.
Now, we consider the real parts of (5.14) and (5.15). Subtracting the real part of (5.14) from (5.15), we obtain
(5.16)X1ψ+I12ψ11+2I12Re(z1z̅2ψ12)=0.
So,
(5.17)X1ψ+I12∂ψ∂X1=0.
By (5.17), we have
(5.18)ψ=ce-X12/2I12,
where c is a constant. Let us take (5.18) into h=I1ψ and verify ∇z̅h=0. Then, the first equation ∇z̅1h=0 becomes
(5.19)ce-X12/2I12z1-I1ce-X12/2I12⋅4X1z1I12-4I1z1X124I14=0.
If ψ≠0, it follows that
(5.20)1=I1X1-X12I12.
By (5.20), we get
(5.21)|z1|4+|z2|4=-2|z2|4.
But (5.21) does not hold in this case. Therefore, we find no small solutions of the bifurcation equation for z1≠0, z2≠0and z1≠z2.
z1=0andz2≠0: in this case, the equation ∇z̅h=0 is equivalent to
(5.22)z2(ψ1+|z2|2ψ11)=0⟺r(|z2|2,τ)=ψ1+|z2|2ψ11=0.
Moreover, (∂r/∂τ)|(|z2|2,τ)=(0,0)=1/2. Using the implicit function theorem for (5.22), there exists a function τ=τ(|z2|2) with τ(0)=0 for all sufficiently small |z2|. 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.
z1≠0andz2=0: similarly to the above case, for φ=0, there exists a function τ=τ(|z1|2) with τ(0)=0 for all sufficiently small |z1|. Correspondingly, there is another one-parameter 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 z1=0 and z2≠0.Remark 5.4.
We now consider the case thatS acts symplectically. In this case, purely imaginary eigenvalues pairs typically arise isolated. Since S on the two-dimensional eigenspace V of ±i for A0 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].



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*Source: 530209-2012-01-18.xml*