Symmetric Liapunov center theorem

In this article, using an infinite-dimensional equivariant Conley index, we prove a generalization of the profitable Liapunov center theorem for symmetric potentials. Consider the system $\ddot{q}= -\nabla U(q),$ where $U(q)$ is a $\Gamma$-symmetric potential, where $\Gamma$ is a compact Lie group acting linearly on $\mathbb{R}^n$. If the system possess a non-degenerate orbit of stationary solutions $\Gamma(q_0)$ with trivial isotropy group, such that there exists at least one positive eigenvalue of the Hessian $\nabla^2 U(q_0)$, then in any neighbourhood of orbit $\Gamma(q_0)$ there is a periodic orbit of solutions of the system.


Introduction
One of the most famous theorems concerning the existence of periodic solutions of ordinary differential equations is the celebrated Liapunov center theorem [23]. Consider the second order autonomous systemq(t) = −∇U (q(t)), where U ∈ C 2 (R n , R), ∇U (0) = 0 and det ∇ 2 U (0) = 0. Let be σ(∇ 2 U (0)) the spectrum of the respective Hessian at 0, Liapunov's center theorem says that if σ(∇ 2 U (0)) ∩ (0, +∞) = {β 2 1 , . . . , β 2 m } for β 1 > . . . > β m > 0 and there is β j 0 satisfying β 1 /β j 0 , . . . , β j 0 −1 /β j 0 ∈ N, then there is a sequence {q k (t)} of periodic solutions of system q(t) = −∇U (q(t)), (1.1) with amplitude tending to zero and the minimal period tending to 2π/β j 0 . Proof of this theorem one can find in [29], see also [8,9,36]. Generalization of this theorem in two directions is due to Szulkin [36]. In the first direction Szulkin, using the infinite-dimensional Morse theory for strongly-indefinite functionals, proved Liapunov type center theorem for Hamiltonian systemṡ z(t) = J∇H(z(t)), (1.2) where H ∈ C 2 (R 2n , R), ∇H(0) = 0, det ∇ 2 H(0) = 0, and J = 0 −I I 0 is the standard simplectic matrix in R 2n , see Theorem 4.1 and Corollaries 4.2, 4.3 of [36]. The next important generalization was to consider system (1.2) with Hamiltonian H for which 0 ∈ R 2n is an isolated degenerate critical point of the Hamiltonian H with nontrivial Conley index i.e. ∇H(0) = 0, 0 ∈ R 2n is isolated in (∇H) −1 (0) and CI({0}, −∇H) = [ * , * ], where CI denotes the Conley index and * denotes a point. The Liapunov center type theorem for such system has also been proved in [36], see theorem 4.4 and corollary 4.5 of [36]. Taking the advantage of variational structure of system (1.2), usually the authors convert the problem of the existence of non-stationary periodic solutions in a neighborhood of a stationary one into a bifurcation problem. Finally, they apply the Morse theory or the Conley index theory to prove the existence of bifurcation of periodic solutions. In this way they obtain a local bifurcation (a sequence of solutions bifurcating from the family of trivial ones) which does not have to be global (a connected set of solutions bifurcating from the family of trivial ones), see [3,10,24,27,37] for discussions and examples.
Another generalization of the Liapunov center theorem is due to Dancer and the second author [14]. They considered system (1.2) with Hamiltonian H for which 0 ∈ R 2n is an isolated degenerate critical point of H with nontrivial Brouwer index i.e. 0 ∈ R 2n is isolated in (∇H) −1 (0) and deg B (∇H, B 2n α , 0) = 0, where α > 0 is sufficiently small and deg B (·) is the Brouwer degree. Note that since χ(CI({0}, −∇H)) = deg B (∇H, B n α , 0), the assumption considered in [14] implies that of [36], where χ(·) is the Euler characteristic. Under this stronger assumption they have proved that there is a connected set of non-stationary periodic solutions of system (1.2) emanating from the stationary solution u 0 ≡ 0. In order to prove this theorem they have applied the degree theory for S 1 -equivariant gradient maps, see [19].
There are also theorems giving estimations of the number of periodic orbits of system (1.2) on an energy level close to the non-degenerate critical point 0 ∈ R 2n of the Hamiltonian H due to Weinstein [39] and Moser [31]. For differential equations with first integral there are similar results due to Dancer and Toland [15], Marzantowicz and Parusiński [28].
It can happen that the stationary solutions of system (1.1) are not isolated critical points of the potential U and the set of stationary solutions consists of the orbits of a compact Lie group Γ. For example the Lennard-Jones potential U : Ω → R is Γ = SO(2)-invariant and (∇U ) −1 (0) ∩ Ω consist of Γ-orbits i.e. the stationary solutions of system (1.1) are not isolated, see [12,13]. It is worth pointing out that one can not apply the theorems mentioned above to the study of non-stationary periodic solutions of system (1.1).
The main result of this article is the following.
To prove this theorem we apply the infinite-dimensional version of the (Γ × S 1 )-equivariant Conley index theory due to Izydorek [25]. We emphasize that if the group Γ is trivial then the above theorem is the classical Liapunov center theorem.
After this introduction our article is organized as follows. In the first part of Section 2 we consider 2π-periodic solutions of system (1. is injective, see Theorem 2.3 and Corollary 2.1. Section 3 is devoted to the computations of the G-equivariant Conley index CI G (G(x 0 ), −∇ϕ) of a non-degenerate orbit G(q 0 ) of critical points of an invariant potential ϕ ∈ C 2 G (Ω, R). First of all we are interested in finding relation between the equivariant Conley index of a nondegenerate orbit and the equivariant Conley index of a non-degenerate critical point of the potential restricted to the space orthogonal to this orbit. Such relation is proved in Theorem 3.1. This relation allows us to distinguish the equivariant Conley index of non-degenerate orbits analyzing only the potentials restricted to the orthogonal spaces to these orbits, see Corollaries 3.1, 3.2, 3.3. Finally in Theorem 3.2 we distinguish equivariant Conley index of so called special non-degenerate orbits.
Our main results are proved in Section 4, where we consider system (1.1) with Γ-symmetric potential U and study periodic solutions of this system in a neighborhood of a non-degenerate orbit Γ(q 0 ) of critical points of U i.e. we have proved the Symmetric Liapunov center theorem (Theorem 1.1). This Theorem is a natural generalization of the classical Liapunov center theorem. The basic idea is to consider periodic solutions of system (1.1) as critical orbits of G = (Γ × S 1 )-invariant family of functionals, see equation (4.2). In other words we have converted the problem of the existence of periodic solutions of system (1.1) in a neighborhood of Γ(q 0 ) into G-symmetric, infinite-dimensional and variational bifurcation problem.
To prove the existence of bifurcation we use the infinite-dimensional equivariant Conley index due to Izydorek [25]. First we prove a technical lemma 4.1 which yields information on the S 1 -equivariant Conley indices of critical points of functionals restricted to the orthogonal space to the orbit Γ(q 0 ) ⊂ H 1 2π . Next we prove Theorem 1.1. Finaly in Section 5 we consider two simple examples coming from celestial mechanics just to show the strength of our main result, and how we can use it. Specifically we analyze a couple of generic galactic type potentials, and show how to find periodic orbits on them.

Preliminaries
In this section we give a brief exposition of material on functional analysis which we will need in the rest of this article. Our purpose is summarize without proofs the relevant tools on equivariant topology used along this paper.
Fix an open set Ω ⊂ R n and consider the following system of second order equations where U ∈ C 2 (Ω, R).
We define 2π is a separable Hilbert space with a scalar product given by the formula where (·, ·) and · are the usual scalar product and norm in R n , respectively. It is easy to show that H 1 2π , ·, · H 1 2π is an orthogonal representation of the group S 1 with an S 1 -action given by shift in time. It is clear that H 1 2π (Ω) is S 1 -invariant. Let be {e 1 , . . . , e n } ⊂ R n be the standard basis in R n . Define H 0 = R n , H k = span{e i cos kt, e i sin kt : i = 1, . . . , n} and note that and that the finite-dimensional spaces H k , k = 0, 1, . . . are orthogonal representations of S 1 .
In other words the gradient ∇Φ : H 1 2π (Ω) → H 1 2π is an S 1 -equivariant C 1 -operator in the form of a compact perturbation of the identity. It is known that solutions of system (2.1) are in one to one correspondence with S 1 -orbits of solutions of ∇Φ(q) = 0. From now on we assume that q 0 ∈ (∇U ) −1 (0).
Consider the linearization of the system (2.1) at q 0 of the form   . The corresponding functional Ψ : H 1 2π → R is defined as follows where L : H 1 2π → H 1 2π is a linear, self-adjoint, S 1 -equivariant and compact operator. It is clear that ∇Ψ(q) = q − Lq + ∇ 2 U (q 0 )q 0 .
Let G be a compact Lie group. 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 = gH ′ g −1 . The conjugacy is an equivalence relation on sub(G). The class of H ∈ sub(G) will be denoted by (H) G and the set of conjugacy classes will be denoted by sub [G]. Denote by ρ : G → O(n, R) a continuous homomorphism. The space R n with the G-action defined by G × R n ∋ (g, x) → ρ(g)x ∈ R n is said to be a real, orthogonal representation of G which we write V = (R n , ρ). To simplify notations we write gx instead of ρ(g)x.
If x ∈ R n then a group G x = {g ∈ G : gx = x} ∈ sub(G) is called the isotropy group of x and G(x) = {gx : g ∈ G} is the orbit through x. Note the orbit G(x) is a smooth G-manifold Two orthogonal representations of G, say V = (R n , ρ), V ′ = (R n , ρ ′ ), are equivalent (briefly V ≈ G V ′ ) if there is an equivariant linear isomorphism L : V → V ′ i.e. the isomorphism L satisfies L(gx) = gL(x) for any g ∈ G, x ∈ R n . Put D(V) = {x ∈ V : x ≤ 1}, S(V) = ∂D(V) and S V = D(V)/S(V). Since the representation V is orthogonal, these sets are G invariant.
Denote by R[1, m], m ∈ N, a two-dimensional representation of the group S 1 with an action of S 1 given by (Φ(e φ ), (x, y)) −→ (Φ(e φ )) m (x, y) T , where Φ(e iφ ) = cos φ − sin φ sin φ cos φ . For k, m ∈ N we denote by R[k, m] the direct sum of k copies of R[1, m], we also denote by R[k, 0] the k-dimensional trivial representation of S 1 . The following classical result gives a complete classification (up to an equivalence) of finite-dimensional S 1 -representations, in [1] you can find a proof of it.
Moreover, the equivalence class of V is uniquely determined by sequences {k i }, {m i } satisfying 2.5.
Assume that H ∈ sub(G). Let Y be a H-space. The product G × Y carries H-action (h, (g, y)) → (gh −1 , hy). The orbit space of H-action is denoted by G × H Y and called the twisted product over Let Y be a pointed H-space with a base point * . Denote by G + the group G with disjoint G-fixed base point * added. Define the smash product of G + and Y by G The orbit space is denoted by G + ∧ H Y and called the smash over H, see [16].
Below we introduce the notion of an admissible pair of compact Lie groups. Such pairs will play crucial role in computations of the equivariant Conley index of a non-degenerate orbit of critical points of invariant potentials.
Let G be a compact Lie group such that G ∈ sub( G) and H ∈ sub(G). If the pair (G, H) is not admissible then the pair ( G, H) is not admissible. (2), we observe that By remark 2.1 we obtain that the pair (SO(n), H), n ≥ 4, is not admissible.
(23)} ∈ sub(T). Note that H is commutative. We claim that the pair (T, H) is not admissible. Indeed, fix g = (123) ∈ T and note that for proves that the pair (T, H) is not admissible. Moreover, by remark 2.1 we obtain that the pair (SO(n), H), n ≥ 3, is not admissible.
Remark 2.2. If G is commutative then for any H ∈ sub(G) the pair (G, H) is admissible.
Proof. Suppose, contrary to our claim, that there is H ∈ sub(O(2)) and K 1 , Since the cardinalities of adjoint groups are equal and (SO (2) We denote by F * (G) the set of finite pointed G-CW-complexes and by F * [G] the set of Ghomotopy types of elements of F * (G). Note that S V ∈ F * (G). By [X] G ∈ F * [G] we denote the G-homotopy class of X ∈ F * (G). Let F be the free abelian group generated by the elements of with an additive and multiplicative structures given by (2.6), (2.7), respectively, is a commutative ring with unit I = χ G (G/G + ).
We call (U (G), +, ⋆) the Euler ring of a compact Lie group G.
We claim that χ S 1 S W is invertible in the Euler ring U (S 1 ). Indeed by formula (2.9) and (2.10) obtain which completes the proof.
The principal significance of the following theorems is that they allow us to express the G-equivariant Euler characteristic of a G-CW-complex G + ∧ H X in terms of a H-equivariant Euler characteristic of the H-CW-complex X. Later on, using these relations, we will distinguish equivariant Conley indices of non-degenerate orbits of critical points of invariant potentials.
Proof. First of all note that consists of s equivariant cells and the j-th cell is of dimension k j ∈ N∪{0} and orbit type (K j ) H . By proposition (II.1.13) of [16] and formula (2.11) we obtain that which completes the proof.
We can significantly simplify theorem 2.2 assuming that the pair (G, H) is admissible. In the theorem below we consider this case. Theorem 2.3. We fix H ∈ sub(G) in such that it satisfies that the pair (G, H) is admissible and is also a H-CW-complex X. [16] for more details).

The equivariant Conley index
In this section we express the equivariant Conley index of a non-degenerate critical orbit of an invariant potential in terms of the equivariant Conley index of an isolated critical point of this potential restricted to the space orthogonal to this orbit. This relation allows us to distinguish Conley indices of two non-degenerate critical orbits. Let be G a compact Lie group. We fix k ∈ N ∪ {∞} and an open and G-invariant subset Ω ⊂ R n .
The rest of this section is dedicated to the study of the G-equivariant Conley index, see [6,17,19,35], of the isolated invariant set G(x 0 ) considered as a G-orbit of stationary solutions of the equationẋ(t) = −∇ϕ(x(t)). Note that since the orbit G(x 0 ) is non-degenerate, In the following, for simplicity of notation, we write H instead of G x 0 . Since R n is an orthogonal representation of G, R). Since the orbit G(x 0 ) is non-degenerate, ∇φ(x 0 ) = 0 and ∇ 2 φ(x 0 ) is non-degenerate. That is why, by the Morse lemma, x 0 is an isolated critical point of the potential φ.
Note that (N, ǫ (x 0 ), S − ǫ (x 0 )) is a strong H-deformation retract of (N, L), we obtain the following equality Let us compute the G-equivariant Conley index of the isolated invariant set G(x 0 ) considered as a G-orbit of stationary solutions of the equationẋ(t) = −∇ϕ(x(t)). Note that if (X, A) is a pair of H-spaces and (X 0 , A 0 ) is a strong H-deformation retract of (X, A), then (G × H X 0 , G × H A 0 ) is a strong G-deformation retract of (G × H X, G × H A), see [16,26] for other properties of the twisted product over H.
) is a strong G-deformation retract of (G × H N, G × H L). Consequently we obtain the following equality In the theorem below we present the relation between the equivariant Conley indices given by formulas (3.3) and (3.4) i.e. we express the G-equivariant Conley index of a non-degenerate orbit G(x 0 ) ⊂ (∇ϕ) −1 (0) in terms of the H-equivariant Conley index of a non-degenerate critical point x 0 ∈ (∇φ) −1 (0) of the restricted functional φ. In the following corollary we show how to distinguish G-equivariant Conley indices of nondegenerate orbits of critical points of a potential ϕ ∈ C 2 G (Ω, R) considering the restrictions of this potential to orthogonal spaces to these orbits. .
In the corollary below we assume that the pair (G, H ν ), ν ∈ { ′ , ′′ }, is admissible. It allows us to control the relation between the equivariant Euler characteristics It follows that the assumption , which completes the proof. The following corollary is a consequence of Corollary 3.2. The point of the corollary is that its assumptions are expressed in terms of the representation theory of compact Lie groups. (1) H is nontrivial and connected, dim T ⊥ (1) Applying Theorem 3.2 of [34] we obtain . The rest of the proof is a direct consequence of Corollary 3.2.
(2) The proof of this case is literally the same as the proof of the previous one with Theorem 3.2 of [34] replaced by Theorem 3.1 of [34].

Proof of the Symmetric Liapunov center theorem
In this section, using the equivariant Conley index defined in [25], we prove the main result of this article, the Symmetric Liapunov center theorem for second order differential equations with symmetric potentials stated in Theorem1.1.
We consider R n as an orthogonal representation of a compact Lie group Γ and denote by ρ : Γ → O(n, R) the representation homomorphism. Denote by Ω ⊂ R n an open and Γ-invariant subset. Fix U ∈ C 2 Γ (Ω, R) and q 0 ∈ Ω a critical point of the potential U. It is clear that Γ(q 0 ) ⊂ (∇U ) −1 (0) i.e. the orbit Γ(q 0 ) consists of critical points of U. In this section we study periodic solutions of system (1.1) in a neighborhood of the orbit Γ(q 0 ).
We observe that the orbit Γ(q 0 ) is Γ-homeomorphic to Γ/Γ q 0 = Γ, for this reason it can happen that elements of this orbit are not isolated. For example if Γ = SO(2) acts freely on Ω then the orbit Γ(q 0 ) is SO(2)-homeomorphic to Γ/Γ q 0 = Γ/{e} = SO(2) ≈ S 1 . Note that if the group Γ is trivial then we obtain the classical Liapunov center theorem, see [8,9,29,36] and references therein.
Before we begin with the proof of Theorem 1.1 we will prove one technical lemma. This lemma will be the key ingredient in the proof of our main result. Note that the study of periodic solutions of system (1.1) of any period is equivalent to the study of 2π-periodic solutions of the following family   q (t) = −λ 2 ∇U (q(t)) q(0) = q(2π) q(0) =q(2π) . (4.1) The 2πλ-periodic solution of system (1.1) corresponds to 2π-periodic solutions of (4.1). Since Γ(q 0 ) ⊂ (∇U ) −1 (0), for every λ > 0 the orbit Γ(q 0 ) consists stationary solutions of system (4.1). Periodic solutions of system (4.1) can be considered as critical orbits of G = (Γ × S 1 )-invariant potential of the class C 2 defined on H 1 2π (Ω) × (0, +∞). It is easy to show that H 1 2π , ·, · H 1 2π is an orthogonal representation of the group G with a G-action defined by . are orthogonal representations of G, see formula (2.2) and the text below it. Define a It is well known that the solutions of equation are in one to one correspondence with the solutions of system (4.1). Since q 0 ∈ H 1 2π (Ω) is a constant function, G(q 0 ) = Γ(q 0 ) ⊂ H 0 = R n ⊂ H 1 2π solve equation To proof Theorem 1.1 we will study solutions of equation (4.2). More precisely, we will apply theorems of equivariant bifurcation theory to prove the existence of local bifurcation of solutions of equation (4.2) from the family T . As a topological tool we will use the equivariant Conley index, see [6,17,19,35] and its infinite-dimensional generalization [25].
We claim that bifurcation of solutions of equation (4.2) from the trivial family T can occur only at degenerate levels. Below we characterize these levels.
Without loss of generality we can assume that J(∇ 2 U (q 0 )) = ∇ 2 U (q 0 ) i.e. ∇ 2 U (q 0 ) is in the Jordan normal form. Since the Hessian ∇ 2 U (q 0 ) is symmetric, it is diagonal. This assumption will simplify arguments in the rest of the proof without loss of generality.
In the rest of the proof we show that these conditions are fulfilled.
Since H 1 = span{e i cos t, e i sin t : i = 1, . . . , n} and the action of the group S 1 on H 1 is given by shift in time, the spaces H + 1,± are representations of the group S 1 such that H + 1,± ≈ S 1 R[r ± , 1]. Hence by formula (2.10) we have It follows that r − = r + , a contradiction.

Applications
In order to show the strength of our main result 1.1 in this section we apply it to a couple of special class of galactic potentials. We point out that our goal here is not the analysis of specific galaxies and their dynamics, we just want take a kind of generic galactic potentials to show how to find periodic orbits on them. Must of the work on galactic potentials is numeric, we will show here, just in a couple of simple cases, the way to obtain periodic orbits in an analytic way. The target is that it could be used as a started point for people working in the field.
Since many galactic potentials defined in the plane are of the form U (x 2 , y 2 ) (see for instance [2,32]), or more generally must of them are considered as a perturbation of an harmonic oscillator (see [11] for more details) having the form U (x, y) = ω 2 (x 2 + y 2 ) + εV (||(x, y)|| 2 ) and since we have studied symmetric potentials along this paper, we will apply the symmetric Liapunov center theorem to potentials of the form U (||(x, y)|| 2 ), which can be seen as a special class of galactic potentials.
We believe that generalizations of important results as the Liapunov center theorem for symmetric potentials could be useful in some applications (see for instance the generalization of the famous Weierstrass model for homogeneous potentials [7]) Example 5.1. Assume that a galaxy on the plane is moving under the influence of the potential U (x) = −2 x 4 + 5 3 x 6 − 1 4 x 8 . Consider the polynomial ϕ : R → R defined by ϕ(t) = −2t 2 + 5 3 t 3 − 1 4 t 4 . We observe that ϕ ′ (t) = −t(t − 1)(t − 4) and U (x) = ϕ( x 2 ). Since the gradient ∇U : R 2 → R 2 is given by ∇U (x) = 2ϕ ′ ( x 2 )x, we obtain Taking into account that ∇ x i U (x) = 2ϕ ′ ( x 2 )x i we compute the Hessian .