Periodic Solutions of Symmetric Hamiltonian Systems

This paper is devoted to the study of periodic solutions of a Hamiltonian system z˙(t)=J∇H(z(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{z}(t)=J \nabla H(z(t))$$\end{document}, where H is symmetric under an action of a compact Lie group. We are looking for periodic solutions in a neighborhood of non-isolated critical points of H which form orbits of the group action. We prove a Lyapunov-type theorem for symmetric Hamiltonian systems.


Introduction
Consider a first-order systeṁ on R 2N , where J = 0 I −I 0 is the standard symplectic matrix and H : R 2N → R is a Hamiltonian of the class C 2 .
The existence of periodic orbits in Hamiltonian dynamics is an important and widely studied problem. In 1895 Lyapunov [20] proved his center theorem, i.e. the existence of a one-parameter family of periodic solutions of (HS) tending to a non-degenerate equilibrium. The next important result of Weinstein [33] shows the existence of at least N geometrically distinct periodic solutions at any energy level of the Hamiltonian H . The further development of the Weinstein theorem was performed by Moser [23]. In 1978 Fadell and Rabinowitz [8] proved the lower bound for the number of small nontrivial solutions of (HS) depending on the period. See [27] for the general overview of the results up to 1982. The results of Weinstein and Moser were generalized by Bartsch in 1997, [2]. The problem of the existence of periodic solutions of (HS) in a case of a degenerate equilibrium was also studied by Szulkin [32] and Dancer with Rybicki [6], who generalized the classical result of Lyapunov.
For the two other versions see Theorems 5.2 and 5.3.
Furthermore, we show that the Lyapunov-type theorem of Dancer and Rybicki (Theorem 5.5) is generalized by the main result of this paper: Theorem 4.1. In the last part of this section we reformulate the second-order Newtonian system to the Hamiltonian one. Then the two symmetric versions of the Lyapunov center theorem, Theorem 5.6 proven in [24] and Theorem 5.7 proven in [25], are also the consequences of the results proven in this paper.
The last section is devoted to an application of the abstract results of this paper. We study the existence of quasi-periodic motions of the satellite in a nearby of a geostationary orbit of an oblate spheroid. In order to do this we consider a gravitational motion in the rotating frame where the corresponding Hamiltonian is given by formula (6.2). It is SO (2) invariant and possesses a critical point which represents the geostationary orbit in the original coordinates. Theorem 5.4 will be directly applied in this problem to prove the existence of trajectories with arbitrarily small deviations from the geostationary ones.

Preliminaries
In this section we recall the basic material on equivariant topology from [7], [17] and prove some preliminary results. Throughout this section G stands for a compact Lie group.

Groups and Their Representations
We denote by sub(G) the set of all closed subgroups of G. Two subgroups H, H ∈ sub(G) are said to be conjugate in G if there is g ∈ G such that H = g H g −1 . The conjugacy is an equivalence relation on sub(G). The class of H ∈ sub(G) we denote by (H ) G and the set of conjugacy classes will be denoted by sub [G].
If x ∈ R n then G(x) = {gx : g ∈ G} is called the orbit through x and a group G x = {g ∈ G : gx = x} ∈ sub(G) is said to be the isotropy group of x. It is known that if G(x 1 ) = G(x 2 ) then (G x 1 ) G = (G x 2 ) G i.e. the isotropy groups of the elements of common orbit are conjugate. Moreover, the orbit G(x) is a smooth G-manifold G-diffeomorphic to G/G x . An open subset Ω ⊂ R n is said to be G-invariant if G(x) ⊂ Ω for every x ∈ Ω.
Below we recall the notion of an admissible pair, which was introduced in [24], where one can find some examples and properties. Definition 2.1. Fix H ∈ sub(G). A pair (G, H ) is said to be admissible if for any K 1 , K 2 ∈ sub(H ) the following condition is satisfied: Note that if Γ is a compact Lie group, then the pair (Γ × S 1 , {e} × S 1 ) is admissible; see Lemma 2.1 of [24]. This property will play a crucial role in the proof of the main result, Theorem 4.1.
Recall that a unitary group U (N ) is defined by Let ρ : G → U (N ) be a continuous homomorphism. The space R 2N with the G-action defined by G × R 2N (g, x) → ρ(g)x ∈ R 2N is said to be a real, unitary representation of G which we write V = (R 2N , ρ). To simplify notation we write gx instead of ρ(g)x and R 2N instead of V if the homomorphism is given in general.
Two unitary representations of G,

G-Equivariant Maps
for every g ∈ G and x ∈ Ω. The set of G-invariant C k -potentials will be denoted by C k G (Ω, R).
Fix ϕ ∈ C 2 G (Ω, R) and denote by ∇ϕ, ∇ 2 ϕ the gradient and the Hessian of ϕ, respectively. For x 0 ∈ Ω denote by m − (∇ 2 ϕ(x 0 )) the Morse index of the Hessian of ϕ at x 0 i.e. the sum of the multiplicities of negative eigenvalues of the symmetric matrix ∇ 2 ϕ(x 0 ). Similarly, by the m + (∇ 2 ϕ(x 0 )) we denote the sum of the multiplicities of positive eigenvalues of ∇ 2 ϕ(x 0 ).

Equivariant Conley Index
We denote by F * (G) the category of finite pointed G-CW-complexes (see [7] for definition and examples), where morphisms are continuous G-equivariant maps preserving a base points; we denote by F * [G] the set of G-homotopy types of elements of F * (G), where [X] G ∈ F * [G] (or [X] when no confusion can arise) denotes a G-homotopy type of the pointed G-CW complex X ∈ F * (G). If X is a G-CW-complex without a base point, then we denote by X + a pointed G-CWcomplex X + = X ∪ { * }.
Now we briefly recall the definition of the equivariant version of the classical Conley index, see [1,4,9,10,30] for the details. Consider a finite-dimensional unitary G-representation (V, ·, · ) and U ⊂ V × R. Let η : U → V be a G-flow i.e. a flow which is equivariant under the G-action on V.
Let S be isolated η-invariant set.
L is the set of exit points from the set N i.e. if x ∈ N and t 1 > 0 are such that η( It is known that for any isolated invariant set there exists a G-index pair contained in its isolating neighborhood. Moreover, for any two G-index pairs (N 1 , L 1 ), (N 2 , L 2 ) for the set S two G-homotopic types [N 1 /L 1 ] G , [N 2 /L 2 ] G are equal. Therefore we are able to define the equivariant Conley index as the G-homotopic type of the pointed space Recall that CI G (S, η) ∈ F * [G], see [10]. Definition of the classical Conley index (without G-action) coincides with the above construction with G = {e}.
Example 2.6. Consider R n as an orthogonal representation of a group G. Let η be a flow generated by a gradient vector field −∇ F, where F ∈ C 2 G (R n , R), 0 is an isolated critical point of F and the hessian ∇ 2 F(0) is an isomorphism. It is known that {0} is an isolated η-invariant set and by Hartman-Grobman theorem the flow η is locally homeomorphic to the flow generated by the linearized vector field y → −∇ 2 F(0)y. Denote by R n − and R n + the generalized eigenspaces of ∇ 2 F(0) corresponding to the negative and positive eigenvalues. Then for sufficiently small Since the action of G is orthogonal, these sets are G-invariant. Moreover, the pairs (N , L) and (D R n − (0, ε), ∂ D R n − (0, ε)) have the same homotopy type. Therefore Note that an index pair (N , L) can be chosen without defining any isolated η-invariant set. Moreover, the Conley index is defined only by the sets N , L. Therefore, it is convenient to consider the Conley index of an isolating neighborhood CI G (N , η) = CI G (I nv (N , η), η).
The most important properties of the Conley index are given in the following theorem: Below we present the infinite-dimensional extension of the equivariant Conley index to Hilbert spaces due to Izydorek [16]. The construction is similar to the developing of the Leray-Schauder degree from the Brouwer degree by finitedimensional approximations. However, the index is not constant for sufficiently large approximations but only stabilize in the sense described below. Therefore, the construction requires the notation of equivariant spectra, see also [11,29]. Let ξ = (V n ) ∞ n=0 be a sequence of finite-dimensional orthogonal G-representations.
The last property tells about some stabilization of the spectrum in the sense of a homotopy equivalence of spaces. The set of G-spectra of type ξ is denoted by G S(ξ ).
where n 1 max(n 1 (E(ξ )), n 1 (E (ξ ))), such that 1. f n ∈ Mor G (E n , E n ) for n n 1 , 2. G-maps f n+1 • ε n and ε n • S V n f n are G-homotopic for every n n 1 , where S V f n denotes a suspension of f n .
Two G-maps f, g : E(ξ ) → E (ξ ) are G-homotopic if there exists n 1 n 0 such that f n , g n : E → E are G-homotopic for n n 1 . Following this definition in a natural way we understand a G-homotopy equivalence of two spectra E(ξ ), E (ξ ). The G-homotopy type of a G-spectrum E(ξ ) will be denoted by [E(ξ )] G (or shorter [E(ξ )]) and the set of G-homotopy types of G-spectra by [G S(ξ )] or simply [G S] when ξ is fixed or is not known yet.
We can consider G-spectra as a direct extension of G-CW-complexes if we consider a constant spectrum (each space is a given G-CW-complex). Now we define a infinite-dimensional generalization of equivariant Conley index given by Izydorek [16] in the case we deal with. Let (H, ·, · ) be an infinitedimensional orthogonal Hilbert representation of a compact Lie group G. Let R be given by Φ n = Φ |H n and ϑ n denotes the G-flow generated by ∇Φ n . Note that ∇Φ n (x) = Lx + P n • ∇ K (x), i.e. it is the n-th approximation of the original flow. Choose sufficiently large n 0 such that for n n 0 the set O n := O ∩ H n is an isolating G-neighborhood for the flow ϑ n . Then the set I nv(O n , ϑ n ) admits a G-index pair (Y n , Z n ) and we are able to define the Conley index We define the spectrum E(ξ ) We will write a vector field and an isolated invariant set or a flow and and isolating neighborhood synonymously i.e. CI G (N , ∇Φ) ≡ CI G (O, ϑ). The equivariant Conley index defined above inherits the properties of the finite-dimensional Conley index described in Theorem 2.7. The second of this properties (known as a continuation) provides the suitable local bifurcation theorem. Since we are going to prove the existence of a connected branch of solutions we need to apply a bifurcation theorem in some degree theory, therefore in the next section we briefly introduce an equivariant Euler characteristic.

Equivariant Euler Characteristic
Let (U (G), +, )) be the Euler ring of G, see [7] for the definition and more details. Let us briefly recall that the Euler ring U (G) is commutative, generated by is the universal additive invariant for finite pointed G-CWcomplexes known as the equivariant Euler characteristic. Remark 2.11. Below we present some properties of the Euler characteristic χ G (·).
For the proof of this fact see Lemma 3.4 in [18].
For the trivial group G = {e} the equivariant Euler characteristic χ {e} is the well-known Euler characteristic and U ({e}) = Z.
There is a natural extension of the equivariant Euler characteristic for finite pointed G-CW-complexes to the category of G-equivariant spectra due to GoŁȩbiewska and Rybicki [14]. Since a spectrum does not have to be constant from some point but only stabilizes, to define some element of Euler ring as an equivariant Euler characteristic of a spectrum we need to utilize this kind of stabilization.

Remark 2.13.
Note that a finite pointed G-CW-complex X can be considered as a constant spectrum E(ξ ), where E n = X for all n 0 and ξ is a sequence of trivial, one-point representations.
). Therefore we can treat CI G and Υ G as natural extensions of CI G and χ G , respectively.
In theorem 3.5 of [14] we find a very important formula connecting an equivariant Conley index, an equivariant Euler characteristic and a degree for equivariant gradient maps, defined in [13].

Theorem 2.14. Denote by η a local G-LS-flow generated by −∇Φ. Let O be an isolated η-invariant G-set. Then
By the above result and Theorem 3.1 of [14] we obtain the following product formula: Theorem 2.15. If N 1 , N 2 are isolated G-invariant sets for the local G-LS flows generated by ∇Ψ 1 and ∇Ψ 2 , respectively, then The next theorem is one of the most important fact in our reasoning. It allows us to simplify the distinguishing of the infinite-dimensional equivariant Conley indexes, significantly. In the view of theorem 2.14 and good properties of the degree for equivariant gradient maps it will provide the existence of global bifurcation in the proof of the main result.
Let H = ∞ n=0 H n be a representation of the compact Lie group G. Consider two functionals ϕ 1 , H) is completely continuous for i = 1, 2, which satisfy the conditions (B.1)-(B.3) described previously in Section 2.3. Note that T ⊥ x G(x), a space orthogonal to the orbit, is a representation of the isotropy group G x and if ϕ is The proof of the theorem above is based on a concept of smash product over group. One can find more details in [25], especially Definition 2.4.2, Theorem 2.4.1 and Theorem 2.4.2.

Equivariant Splitting Lemma
Let K be a compact Lie group and let (V, ·, · ) be an orthogonal Hilbert representation of K with an invariant scalar product ·, · . Assume additionally that dim V K < ∞. Here and subsequently, Ω ⊂ V stands for an open and invariant subset of V such that 0 ∈ Ω.
Consider a functional Ψ ∈ C 2 K (Ω, R) given by the formula which satisfies the following assumptions: Denote by ker A and im A the kernel and the image of ∇ 2 Ψ (0) = A, respectively. Notice that both, ker A and im A, are orthogonal representations of K . Moreover, ker A is finite dimensional and trivial representation of The following theorem (known as the splitting lemma) proves the existence of equivariant homotopy which allows us to study the product (splitted) flow (the proof of this theorem one can find in [25] (Theorem 2.5.2)): 3) and satisfies assumptions (F.1)-(F.5). Then, there exists ε 0 > 0 and K -equivariant gradient homotopy ∇H : satisfying the following conditions: There exists an K -equivariant, gradient mapping ∇ϕ :

Remark 2.18. The homotopy H is given by
Moreover, from the proof of Theorem 2.17 it follows that the potential ϕ : Remark 2.19. Note that we don't assume that ker A = {0}. In the case of trivial kernel the homotopy given in Theorem 2.17 provides a linearization of functional.

Variational Formulation for Hamiltonian Systems
Recall that we are interested in the existence of periodic solutions with any period of the system (HS). In order to find them we are going to study 2π -periodic solutions of the parameterized systeṁ which is in one-to-one correspondence to 2πλ-periodic solutions of the system (HS).
To prove the existence of solutions of the Hamiltonian system (HS-P) we are going to the study critical points of a corresponding functional.
Define the Sobolev space of 2π -periodic R 2N -valued functions The space E with inner product given by where · denotes the standard scalar product, is a Hilbert space usually denoted by H 1/2 (S 1 , R 2N ). Since we consider R 2N as a unitary representation of the compact Lie group Γ , E is a unitary G = Γ × S 1 -representation with the action given by and E k is a unitary G-representation for any k 1. Indeed, and therefore the action proposed in (3.2) is given on E k by the product of unitary Remark 3.1. Since we are going to study the Hamiltonian system (HS-P) is a neighborhood of the orbit of critical points Γ (z 0 ), without loss of generality we can assume that Hamiltonian H satisfies the following growth restriction: Indeed, we may chooseH such that ∇H is bounded (i.e. s = 1) andH (z) = H (z) in a neighborhood of the orbit Γ (z 0 ).
It is known (see [21]) that periodic solutions of the system (HS-P) are in one to one correspondence with S 1 -orbits of critical points of a potential Φ : E×(0, ∞) → R of a class C 1 defined by where Note that Φ(·, λ) acts on the subspace of constant functions E 0 as (3.6) see [12], the formula (3.3).
Since we consider R 2N as a unitary representation of a group Γ and H is Γ -invariant, the potential Φ is Γ -invariant. Moreover, it is S 1 -invariant since it acts on 2π -periodic functions.
Recall that since the Hamiltonian H is Γ -invariant, the solutions of the system (HS-P) form Γ -orbits i.e. if z 0 is a solution on (HS-P) then γ z 0 solves (HS-P) for any γ ∈ Γ . Therefore we are going to study G = Γ × S 1 -orbits of critical points of the corresponding G-invariant potential Φ i.e. we are interested in solutions of the system which has a form of (HS-P) with H (z) Taking into account the scalar product in E given by (3.1) and the formula (3.8), we obtain and, as a consequence, This means that ∇Φ L (z, λ) acts on E k = {a cos(kt) + b sin(kt) : a, b ∈ R 2N } for k 1 as a linear map Proof. Let (z, λ) = (a 0 + ∞ k=1 a k cos(kt) + b k sin(kt), λ) = (const., λ) be a critical point of Φ L and let k be such that |a k | 2 + |b k | 2 = 0. Then, in particular, It is easy to see that equation

Main Result
In this section we prove our main result of this paper i.e. the global bifurcation of periodic solutions of the system (HS) in the most general version. We emphasize our assumptions:   H (z(t)) emanating from the stationary solution z 0 (i.e. with amplitudes tending to 0) such that minimal periods of solutions in a small neighborhood of z 0 are close to 2π/β j 0 .

Remark 4.2.
The assumption (A7) is very general and laborious to verify. We will change and simplify them in some specific cases. However, it does not follow directly from the structure of a Hamiltonian system in general situation as we obtained in a study of Newtonian systems, see [24], the proof of Lemma 4.1.  Put Λ = { k β j : k ∈ N, iβ j ∈ σ (J ∇ 2 z H (z 0 ))}. In the theorem below we prove the necessary condition for the existence of bifurcation from the orbit G(z 0 )×{λ 0 }.

Theorem 4.4. (Necessary condition) If G(z 0 )×{λ 0 } is an orbit of global bifurcation of solutions of the equation (3.7) then ker
By a reasoning given in the proof of Theorem 3.2.1 in [25] we obtain ker To complete the proof we have to prove that it implies λ 0 ∈ Λ. The study of the kernel of ∇ 2 z Φ(z 0 , λ 0 ) is equivalent to the study of the linearized system (HS-L) where A = ∇ 2 z H (z 0 ). Therefore, by Lemma 3.2 we obtain the thesis.
Recall that the space perpendicular to the orbit at z 0 is an G z 0 -representation. Since z 0 is a constant function and by the assumption (A2) G z 0 = {e} × S 1 , H is an unitary S 1 -representation.
Put Ψ ± : H → R by Ψ ± (z) = Φ(z, λ ± ). Note that since H is an S 1 -representation, the potenatial Ψ ± is S 1 -invariant. Moreover, z 0 is an isolated critical point of Ψ ± . Since G(z 0 ) = Γ (z 0 ) ⊂ E 0 we have the following decomposition: In order to prove the main result of this paper we prove the existence of global bifurcation from the orbit G(z 0 ) × {λ 0 }; i.e. we need to prove formula (4.1). In the theorem below we simplify this formula to the study of potentials defined on the orthogonal section T ⊥ z 0 G(z 0 ).

Lemma 4.6. Under the above assumptions if
Proof. Since the pair (Γ × S 1 , {e} × S 1 ) is admissible (because S 1 is abelian), see Definition 2.1, and both ∇Φ(·, λ − ), ∇Φ(·, λ + ) are in the form of a compact perturbation of the same linear operator L, we can apply Theorem 2.16 to obtain the thesis directly.
From now our goal is to prove formula (4.2). The next step is to transform a problem into the study of Conley indexes with simpler structure of flows.
We define H : Since Ψ ±|E 0 = 2πλ ± H and the orbits G(z 0 )×{λ + }, G(z 0 )× {λ − } do not satisfy the necessary condition for the existence of bifurcation we obtain ker ∇ 2 Ψ ±| (0) = ker ∇ 2H (0) so the kernel is independent on λ ± . Since ∇ 2Ψ ± (0) is self-adjoint we are able to decompose Note that the S 1 -invariant potential Π ± : R ∞ → R of the linear vector field The next theorem simplifies the proof of formula (4.2) to the study of Conley indexes of linear vector fields. In order to prove it we apply the splitting lemma (Theorem 2.17).

Lemma 4.7. Under the above assumptions the formula (4.2) holds true if and only if
(4.4)

Proof. It is clear that by the properties of Conley index we have
Since we are going to apply splitting lemma (Theorem 2.17), now we verify that Ψ ± satisfies conditions (F.1)-(F.5) given on the page 8 with In a fashion similar to as above, ker 0)z and both summands are compact and S 1 -equivariant, ∇ζ ± is also compact and S 1 -equivariant. (F.4) It is obvious due to formula given in (F.3). (F.5) Since λ ± / ∈ Λ, i.e. the orbits G(z 0 ) × {λ ± } do not satisfy the necessary conditions for the existence of bifurcations, the orbit G(z 0 ) is isolated in the set (∇Φ(·, λ ± )) −1 (0). Therefore 0 ∈ H is an isolated critical point ofΨ ± .
Applying Theorem 2.17 (splitting lemma) and Theorem 2.15 (product formula), we obtain Since R 0 is an invariant space of the linear map ∇ 2 Ψ ± (0) we are able to decompose the linear flow to obtain and combining the flows given on the E 0 ∩ H, we finally obtain (4.5) If we study the homotopy H (see Theorem 2.17 and Remark 2.18) acting on the subspace of constant function E 0 , we obtain By the homotopy invariance of the Conley index, and since λ − , λ + are both positive, we have Note that the space E 0 is finite-dimensional and consists of constant functions (elements invariant on S 1 action), therefore for sufficiently small ε > 0, where the last equality follows from Poincaré-Hopf theorem (see [31]). Since we finally have that (4.7) By the assumption (A5) deg(∇ H |T ⊥ z 0 Γ (z 0 ) , B(z 0 , ε), 0) = 0 and due to equation (4.7) we obtain that the formula (4.2) is equivalent to and the proof is completed.
To verify formula (4.4) we are going to study the equivariant Conley index and equivariant Euler characteristic by definitions. Note that the vector field −∇Π ± : R ∞ → R ∞ is linear and the decomposition R ∞ = ∞ k=1 E k satisfies conditions (B.1)-(B.3) given on the page 6. Recall that λ 0 = 1

Remark 4.8.
Note that the linearization of the variational functional Φ for the parameterized Hamiltonian system (HS-P) is equal to variational functional for the linearized system (HS-L) (we remove high order tenses in both cases). Therefore the action of the linear vector field ∇Π ± : R ∞ → R ∞ is given on E k by β j for any k ∈ N and β j = β j 0 (see assumption (A6)), matrices T k,λ for k 2 are nonsingular if λ varies. Therefore the spectral decomposition of E k for k 2 given by −∇ 2 Π ± does not depend on λ ± , i.e.
As a consequence the spectra E − , E + whose homotopy types are Conley indexes CI S 1 ({z 0 }, −∇Π − ) , CI S 1 ({z 0 }, −∇Π + ) are of the same type ξ = (E k,+ ) ∞ k=2 . Define P n = n k=2 E k,+ . Put R n = n k=1 E k and consider Π n ± = Π ±|R n : R n → R. By Remark 2.12 we obtain for n large enough. Now, to prove formula (4.9), it is enough to show that Since −∇Π n ± is a linear isomorphism, Conley indexes are very simple, i.e.
CI S 1 {z 0 }, −∇Π n ± = S E 1,λ ± ,+ ⊕P n = S E 1,λ ± ,+ ∧ S P n . (4.11) By the assumption (A7), we have Since E 1 is a non-trivial S 1 -representation, by Remark 2.11 we obtain that Combining formulas (4.11) and (4.12), we finally obtain which completes the proof.  R 2N ). Now we prove that corresponding periodic solutions of Hamiltonian system tend to z 0 ∈ R 2N in L ∞ -norm, i.e. the amplitudes of these solutions are tending to zero. Let z(t) = a 0 + ∞ k=1 a k cos(kt) + b k sin(kt) be a solution of (HS-P) for λ close to λ 0 . Firstly, Under the condition (3.3), the map z(t) → ∇ H (z(t)) is continuous from L 2 (S 1 ) to L 2 (S 1 ) (see Proposition B.1 in [28]). Let ε > 0 and choose 0 < δ < ε such Since z is a solution of (HS-P), we obtain Applying Sobolev inequality (see Proposition 1.1 in [21]), we obtain Since λ is bounded in the neighborhood of λ 0 the convergence of solutions z to z 0 in the norm of H 1 implies the convergence in L ∞ .
To finish the proof we have to study the minimal periods of new periodic solutions. These were obtained by the bifurcation from the orbit G(z 0 ) × {λ 0 }, where the parameter λ 0 comes from the change of variables and describes the period of solutions. More precisely, we have already obtained the connected branch of solutions of the system (HS) emanating from the stationary solutions z 0 with periods close to 2πλ 0 = 2π By the non-resonance condition for eigenvalues (i.e β j /β j 0 ∈ N for all j = j 0 ),  The assumption (A6) was used only in the proof of Theorem 4.9, i.e. in the last step of the proof of our main theorem. We are able to remove this assumption, but then in the proof of Theorem 4.9 we need to study λ-depending decompositions of not only E 1 but any E k such that 1 β j 0 = k β j for some j. It will cause a complicated notation and the proof will be less readable. However, the change of dim E k,λ,+ when λ varies we will obtain in the same way as for k = 1. Note that assumption (A6) is always satisfied for j 0 = 1 since β 1 is the maximum of β i .

Corollaries
In this section we study how it is possible to modify assumption (A7). Moreover, we show that the results of this paper are generalizations of some versions of the Lyapunov center theorem.
The following theorem was proven by Szulkin ([32], Proposition 3.6): Due to the theorem above we are able to formulate new versions of assumption (A7): where E j is the eigenspace of J ∇ 2 H (z 0 ) in C 2N corresponding to iβ j and Z j the invariant subspace of J ∇ 2 H (z 0 ) in R 2N corresponding to ±iβ j . Note that if ∇ 2 H (z 0 ) |Z j 0 is a definite matrix, then the condition (A7.1) is satisfied. Therefore we put that If we are not interested in the minimal period of new solutions but only in the study of its existence, the assumptions can be modified. The computation of invariant subspaces Z j we can change to the study of general invariant subspace of J ∇ 2 H (z 0 ) associated to all the eigenvalues of the form ±iβ k . Denoting this subspace by Z we formulate a new condition.
Under this condition the assumption (A7.3) is satisfied for some eigenvalue of J ∇ 2 H (z 0 ) and we do not know it precisely. Therefore we exclude assumption (A6).
In the theorem below we prove the existence of periodic solutions of the system (HS) without information about their minimal periods. According to the reasoning above, it is clear that Theorem 5.3 is a direct consequence of Theorem 4.1. (A7.4) there exists a connected family of non-stationary periodic solutions of the systemż(t) = J ∇ H (z(t)) emanating from the stationary solution z 0 such that periods (not necessarily minimal) of solutions in the small neighborhood of z 0 are close to 2π/β j where iβ j ,

Theorem 5.3. Under the assumptions (A1)-(A5) and
Looking on the T 1,λ (∇ 2 H (z 0 )) from the other point of view, we see that Therefore if m + (∇ 2 H (z 0 )) = N , then m − (T 1,λ (∇ 2 H (z 0 ))) changes at some λ ∈ (0, ∞). Recall that the levels λ where it can change is Λ (see Lemma 3.2). Therefore the change of m − (T 1,λ (∇ 2 H (z 0 ))) implies the existence of purely imaginary eigenvalue of J ∇ 2 H (z 0 ). As a consequence we can propose a new condition: and we are able to formulate the next theorem without assumption (A4). Below we present a way in where the theorems presented above generalize classical Lyapunov center theorem and an analogous theorem for Hamiltonian systems that has been proved by Dancer and Rybicki [6]. Moreover, two symmetric versions of the Lyapunov center theorem proposed in [24] and [25] are generalized in this paper. Proof. This theorem follows directly from Theorem 4.1 if we consider trivial group Γ = {e}. In this case T z 0 Γ (z 0 ) = R 2N + z 0 .
Since r = q 2 1 + q 2 2 > 0, by the first two equations we have Further, by the third equation, Since ω, c, d > 0, we obtain q 3 = 0. As a consequence, the equation ( The Hessian is obviously degenerate (see Remark 2.4). One can see that it possesses eigenvalues 1 + ω 2 (with eigenvector [0, 1, 0, −1/ω, 0, 0] T ]) and 1 (with eigenvector [0, 0, 0, 0, 0, 1] T ). Denote be λ 1 , λ 2 , λ 3 the other three eigenvalues. If we compute the characteristic polynomial w(t) of ∇ 2 H (Q) its coefficient of the term t (which is the additive inverse of the product of eigenvalues different from the one zero-eigenvalue we have already know) equals and substituting formulas (6.4), we obtain Hence one or three of λ i are negative. Therefore the Hessian ∇ 2 H (Q) has two or four positive eigevalues. This means that the assumption (A7.5) is satisfied. Moreover, the kernel of this Hessian is one-dimensional which provides that the orbit SO(2)(Q) is non-degenerate. Therefore the assumption (A5) is also satisfied (see the reasoning in the last paragraph of the previous section on the page 19).
To summarize, all assumptions of Theorem 5.4 are satisfied. This provides the existence of periodic solutions in a nearby of an equilibrium Q in the rotating frame. These solutions correspond to a motion in a neighborhood of the geostationary orbit.