The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems
We consider the existence of the periodic solutions in the neighbourhood of equilibria for equivariant Hamiltonian vector fields. If the equivariant symmetry acts antisymplectically and , 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.
We first give some definitions for our problem. A matrix is called (anti)symplectic if . Consider a vector field and the system Let be a diffeomorphism of into itself. If , we call that the system (1.1) is -equivariant or the vector field is -equivariant. Denote by the identity matrix. In this paper, satisfies . When the system (1.1) is -equivariant, if is a solution, then is also a solution. An orbit is called symmetric if it is -invariant; that is, .
Problems. Consider a system , with being an matrix and a vector function. Suppose that the system has a nondegenerate integral. Suppose that has a pair of purely imaginary eigenvalues and no other eigenvalues of the form , . That is, the eigenvalues 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 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  obtained an infinite number of periodic solutions in arbitrarily small neighborhoods of the origin for Hamiltonian systems with convex potential. Weinstein  obtained the Liapunov Center families with no eigenvalue assumptions when the equilibrium point is a nondegenerate minimum. In , 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 , 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  studied this bifurcation and classified the periodic solutions that are spawned by this resonance. Using singularity theory with a distinguished parameter developed in , Bridges  obtained the periodic solutions in a Hamiltonian-Hopf bifurcation.
Moreover, reversible systems have been studied for many years. Devaney  first proved a Liapunov Center theorem for reversible vectors fields. Vanderbauwhede  and Sevryuk  also studied reversible vectors fields. In , 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  obtained the group theoretical classification of linear reversible equivariant systems, there has been an increasing interest for reversible equivariant systems. Later, Hoveijn et al.  obtained the linear normal form and unfolding theory of reversible equivariant linear systems.
In , 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 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 . The results in  are as follows. If 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 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 , Sternberg theorem for equivariant Hamiltonian vector fields was considered.
Motivated by , 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 acts antisymplectically and . 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 equivariant Hamiltonian vector field , with the equivariant symmetry acting antisymplectically and . Assume that the Jacobian matrix has two pairs of purely imaginary eigenvalues and no other eigenvalues of the form , . 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 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 as they approach the equilibrium. Furthermore, there are no other periodic solutions with period close to 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 .
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 be a linear Hamiltonian vector field. Then, it follows that . If is -equivariant, we have . If is (anti)symplectic, we get .
Since we are interested in (partially) elliptic equilibria, we assume that has a pair of purely imaginary eigenvalues and . Moreover, if the eigenvector of has , then is an eigenvector for .
Since is both Hamiltonian and -equivariant, this implies that if the eigenvector of has the eigenvalue , then is also an eigenvector for the eigenvalue .
Hoveijn et al.  considered the linear normal form theory which is based on the construction of minimal -invariant subspaces. By , we are only interested in minimal invariant subspaces on which is semisimple; that is, is diagonalizable over . Here, the type of minimal invariant subspace depends on whether acts symplectically or antisymplectically.
Lemma 3.1. Consider a linear -equivariant Hamiltonian vector field with acting (anti)symplectically and . Let be a minimal -invariant subspace on which has purely imaginary eigenvalues. Then , and have the following normal forms.(1)If acts symplectically, it follows that , and (2)If acts antisymplectically, it follows that and
Proof. Let be a 2-dimensional symplectic subspace on which has purely imaginary eigenvalues. By standard Hamiltonian theory and multiplication of time by a scalar, and can take the same normal form on . If the equivariant symmetry acts symplectically, we have and . Let and be the eigenvectors of . By , a minimal invariant subspace is obtained by choosing . Since on , we have on .
If acts antisymplectically, the dimension of the minimal invariant subspace is not two. A 2-dimensional subspace is defined as above. Assume that . If , by , it follows that . This is converse that acts antisymplectically. So, we have and a minimal invariant subspace is given by . So, . Moreover, we get and . Since , it follows that .
Remark 3.2. Now, we give the examples for the system (1.1) whether acts symplectically or antisymplectically, where and here are defined as and in Lemma 3.1. If acts symplectically, the system (1.1) can be written as where the Hamiltonian function is , satisfies , and is calculated the same as in Lemma 3.1. If acts antisymplectically, the system (1.1) can be written as where the Hamiltonian function is , satisfies , and is calculated the same as in Lemma 3.1.
Remark 3.3. When acts antisymplectically, under the base , we obtain , and have the forms of , and in Lemma 3.1, respectively. However, when acts antisymplectically, under the base , we have In this case, 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 acts antisymplectically or symplectically, by Lemma 3.1, has a single pair or double pairs of purely imaginary eigenvalues . Moreover, these purely imaginary eigenvalues of are nonresonant; that is, has no other eigenvalues of the form with . This condition is clearly generic (codimension zero). We want to find the families of periodic solutions in the neighbourhood of the equilibrium point.
Assume that a vector field has an equivariant symmetry group , which implies the existence of representations such that , for all . Define by where is the space of maps and is the space of maps . The map is by the “-lemma,” that is, in Section 2.4 of . Clearly, the solutions of correspond to -periodic solutions of (1.1).
Now, define an action or in by where is an element of , , and .
By the -equivariance of , we have that is -equivariant
Assume that . The derivative of at is , where with . Moreover, , where .
By (4.3), is also -equivariant such that . Then, preserves and Range .
Below, we will obtain that and are also -invariant. We have Here, the orthogonal complement is taken in and by where and is a normalized Haar measure for . Note that for all . So, and are -invariant.
By and are -invariant, we can obtain that the projections commute with . If , let , where and . Then, and .
Next, define a map with by solving for using the implicit function theorem.
Moreover, we can prove that commutes with . Define by . Note that . We have . Then, is also the solution of (4.8). Moreover, . By uniqueness, we get . So, commutes with . Then,
Then, solutions of the equation are given by , where is a solution of the bifurcation equation
Now, we will obtain that is also -equivariant.
Lemma 4.1. If is -equivariant, then the bifurcation map is -equivariant
Proof. Since -equivariance of , and , it is easy to obtain this result.
Lemma 4.1 indicates that how the symmetry enters the bifurcation equation . Below, we will also consider the relation between the symmetry and the Hamiltonian function of .
Since the vector field is Hamiltonian, it follows that holds for all . Define the map by Since the vector field is Hamiltonian with the Hamiltonian function , we have that the implicit constraint satisfies This condition is rephrased as saying that the map , regarded as a (parameter-dependent) vector field on and is Hamiltonian with respect to the weak symplectic form and with the Hamiltonian function
If the actions of are antisymplectic, we have In this case, is anti-invariant; that is, and the Hamiltonian function satisfies
By Theorem 6.2 in , it follows that if then the bifurcation map is also a Hamiltonian vector field with Hamiltonian , and and the function have the same invariance properties as the given Hamiltonian .
In this paper, is finite dimensional and (4.21) holds. So, the bifurcation equation is a Hamiltonian vector field with Hamiltonian and if the actions of are antisymplectic, then The corresponding symplectic form is the restriction of to . Moreover, in this paper, we have and .
5. Proof of Theorem 2.1
In this section, we prove Theorem 2.1. 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 symmetry acts antisymplectically. By Lemma 3.1, in this case, there are two pairs of purely imaginary eigenvalues. Without loss of generality, , and take the (normal) forms of Lemma 3.1, where is the four-dimensional eigenspace of for .
Since , it follows that So, the bifurcation equation is denoted by Since is Hamiltonian, let with the Hamiltonian function where takes the (normal) form of Lemma 3.1. By (4.22), it follows that is -invariant, with acting on as and is -anti-invariant, with
By -invariance, it follows that
Then, by the -anti-invariance, we have that is equivalent to is real for all and which satisfy and . If is not real for some and which satisfy and , by the -anti-invariance, we have . Thus, without generality, since is real, we obtain where .
We have that is generated by
Then, the symplectic form for the reduced bifurcation equation satisfies where takes the (normal) form of Lemma 3.1. So, for all .
Using the similar calculation and by (4.17), it follows that the -dependence of the lowest (quadratic) order of has the form
We first give Lemma 5.1, which is used for finding the symmetric periodic orbits near the equilibrium point.
5.1. Symmetric Periodic Solutions
Since , then the symmetric periodic solutions of the bifurcation equation lie in the level set . In fact, if , we get . Then, by (4.9), the symmetric solutions lying in correspond to symmetric -periodic solutions for (4.1). Below, we prove there exist the symmetric solutions for the equation .
Lemma 5.1. In the neighbourhood of the equilibrium , there exist the symmetric solutions for the equation .
Proof. Since , the symmetric solutions lie in . Then, the equation is equivalent to or . By (5.8) and (5.12), we have . Using the implicit function theorem for , there exists a function for all sufficiently small that corresponds to a family of symmetric periodic solutions of the system (1.1) with period .
Remark 5.2. Apparently, the symmetric periodic solutions should be twoparameters. But since the vector field 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 Lemma 5.1, 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 as the solutions tend to the equilibrium point.
Proof. The proof can be divided into three cases: , and and.
and: let . It follows that is equivalent to . By (5.8), can be written as
By (5.8), and multiplying the first equation of (5.13) by and the second equation of (5.13) by , we have where is the partial derivative of with respect to , , , and .
Now, we consider the real parts of (5.14) and (5.15). Subtracting the real part of (5.14) from (5.15), we obtain So, By (5.17), we have where is a constant. Let us take (5.18) into and verify . Then, the first equation becomes If , it follows that By (5.20), we get But (5.21) does not hold in this case. Therefore, we find no small solutions of the bifurcation equation for , and .
and: in this case, the equation is equivalent to Moreover, . Using the implicit function theorem for (5.22), there exists a function with for all sufficiently small . 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.
and: similarly to the above case, for , there exists a function with for all sufficiently small . 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 -image of the family with and .
Remark 5.4. We now consider the case that acts symplectically. In this case, purely imaginary eigenvalues pairs typically arise isolated. Since on the two-dimensional eigenspace of for 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].
The authors would like to thank the referees for their valuable comments and suggestions.
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