On the dynamics of the Euler equations on so(4)

This paper deals with the Euler equations on the Lie Algebra so(4). These equations are given by a polynomial differential system in . We prove that this differential system has four 3-dimensional invariant manifolds and we give a complete description of its dynamics on these invariant manifolds. In particular, each of these invariant manifolds are fulfilled by periodic orbits except in a zero Lebesgue measure set.

Equation (1) can have a very complicated dynamics, due to the complexity of the system: nonlinear, high dimension and many parameters. From the integrability point of view, in the paper [7] it is proved that the Euler equations on the Lie algebra so(4) with a diagonal quadratic Hamiltonian either satisfy the Manakov condition, or have at most four functionally independent polynomial first integrals.
In the present paper, our first interest was about the existence or non-existence of periodic orbits for system (1). As we will see in what follows, we have proved that there exist three-dimensional invariant manifolds for the flow of system (1) that are fulfilled by periodic orbits.
CONTACT Claudio A. Buzzi claudio.buzzi@unesp.br In our first result, Theorem 1.1, we prove the existence of four invariant manifolds and as a consequence, in Corollary 1.2, we show that the restriction of system (1) to these invariant manifolds is completely integrable. In order to state our result we define the following manifolds: ( 2 ) and the functions F 1 (x, y, z) = x 2 + y 2 + z 2 and F 2 (x, y, z) = αx 2 + βy 2 + γ z 2 are first integrals of system (2).
In our second theorem we provide the dynamics on the invariant manifolds W i , these results are well-known properties of the rigid body dynamics, see [1,9]. Although both works describe the qualitative dynamical behaviour of system (2), we present a new proof using the symmetry property of the vector field. Up to the best of our knowledge, this approach has not been reported in the literature. Theorem 1.3: Consider system (2) with α = β, α = γ and β = γ (without loss of generality we can assume α < β < γ ). For each r > 0 the sphere S r = {x 2 + y 2 + z 2 = r} is an invariant manifold by the flow of system (2), and the dynamics on these sphere is described in Figure 1 The paper is organized as follows. In Section 2 we present a brief description of the reversible and equivariant differential systems, which are the main tool used in our work. In Sections 3 and 4 we give the proofs of Theorem 1.1 and 1.3.  (2) with α < β < γ on the sphere S r . This figure also appears at the cover of reference [9].

Reversible and equivariant systems
We consider a vector field f : R n → R n , generating a dynamical system, i.e. the flow of Let R be an involution, i.e. a diffeomorphism of R n to itself in such a way that is a solution too. In this case we say that R is a reversing symmetry of the vector field f. Analogously, considering S another involution, f is called S-equivariant if S * f = f • S, in this case every solution x(t) implies the existence of a solution Sx(t), and we say S is a symmetry of f. For more details on reversible and equivariant systems see [6]. The sets of fixed points Fix(R) = {x ∈ R n : R(x) = x} and Fix(S) = {x ∈ R n : S(x) = x} play important roles. One of these roles is given by the next lemma. The fixed set Fix(S) is invariant under the flow generated by f, that is, a trajectory with a point in the fixed set will remain in it for all time.

Lemma 2.1 (See Section 3.1 of [6]): If f is an S-equivariant vector field, then Fix(S) is an invariant manifold under the flow of f.
Another important property is the following.

Lemma 2.2:
Let R 1 and R 2 be two involutions from R n to R n such that S = R 1 • R 2 is also an involution. If the vector field f is reversible with respect to the involutions R 1 and R 2 , then it is S-equivariant.
Next lemma gives additional properties of the reversible systems that will be useful in our work. Lemma 2.3: Let R be a reversing symmetry of the vector field f.

(i) If P is an equilibrium point of f such that P /
∈ Fix(R), then P = R(P) = P is also an equilibrium point of f. In this case, if λ is an eigenvalue of df P , then −λ is an eigenvalue of df P . In particular, if P is a hyperbolic saddle then P also is a hyperbolic saddle.
t ∈ R} is a heteroclinic orbit connecting P and P .
Proof: It is clear that if P / ∈ Fix(R), then P = R(P) = P. On the other hand if f (P) = 0, then f (P ) = f (R(P)) = −dR P (f (P)) = −dR P (0) = 0. Derivating the expression of the reversibility we obtain dR P • df P • (dR P ) −1 = −df P , so it is clear that if λ is an eigenvalue of df P then −λ is an eigenvalue of df P . The first part of statement (ii) follows by the continuity of the involution R. And the second part follows by the Existence and Uniqueness Theorem of ODE's, because both solutions x(t) and R(x(−t)) coincide at t = 0.

Proof of Theorem 1.1 and Corollary 1.2
This section is devoted to give the proof of Theorem 1.1 and Corollary 1.2.
Proof of Theorem 1.1: According to the previous section it is easy to see that system (1) is R i -reversible with respect to the involutions R i : R 6 → R 6 , i = 1, . . . , 8, given by Now for i, j = 1, . . . , 8 we use the notation S ij to denote the composition S ij = R i • R j . So, by Lemma 2.2, we have that system (1) is S ij -equivariant with respect to these involutions:

From Lemma 2.1 we obtain the following 3-dimensional invariant manifolds
for system (1).

Remark 3.1:
We observe that the sets are two-dimensional vector subspaces filled by equilibrium points of system (1). The occurrence and stability of these equilibriums were studied in [3].
Proof of Corollary 1.2: The proof is very similar for each one of the invariant manifolds W i . We prove it only for the invariant manifold W 1 . System (1) restrict to W 1 is given bẏ and obviously it is a system (2). Finally, by considering F 1 (x, y, z) = x 2 + y 2 + z 2 and F 2 (x, y, z) = αx 2 + βy 2 + γ z 2 , for system (2) we have that for all (x, y, z) ∈ R 3 and i = 1, 2. Thus, F 1 and F 2 are first integrals of system (2).

Proof of Theorem 1.3
Proof of Theorem 1.3: Since F 1 (x, y, z) = x 2 + y 2 + z 2 is a first integral of system (2), then the sphere S r is an invariant manifold for this system. In what follows we will prove statements (i), (ii) and (iii). For statement (i) it is easy to show that (0, 0, ±r), (0, ±r, 0) and (±r, 0, 0) are equilibrium point of system (2). Now we need to analyse the dynamics of the system in a neighbourhood of these equilibrium points. For this purpose consider the parametrizations z = ± r − x 2 − y 2 , y = ± √ r − x 2 − z 2 and x = ± r − y 2 − z 2 to understand the dynamics in the hemispheres z > 0 (z < 0), y > 0 (y < 0) and x > 0 (x < 0) respectively.
We started with z = r − x 2 − y 2 . Then system (2) restricted to the surface {x 2 + y 2 + z 2 = r, z > 0} can be written aṡ Doing the change of the time t given by dt = r − x 2 − y 2 dτ in the previous differential system in a neighbourhood of the equilibrium point (0, 0, r) its dynamics is equivalent to the one of the following system where the prime denotes derivative with respect to the new time τ .
Analogously, using a convenient time re-parametrization the dynamics of system (2) restricted to the surface {x 2 + y 2 + z 2 = r, y > 0} in a neighbourhood of the equilibrium point (0, r, 0) is equivalent to the dynamics of the systeṁ Again by the linearity of these system and the hypothesis α < β < γ , the equilibrium point (0, r, 0) is saddle. The symmetry ensures that the equilibrium point (0, −r, 0) is also saddle.
Lastly system (2) using a convenient time re-parametrization the dynamics of system (2) restricted to the surface {x 2 + y 2 + z 2 = r, x > 0} in a neighbourhood of the equilibrium point (r, 0, 0) is equivalent to the dynamics of the systeṁ Since α < β < γ and system (7) is linear, the equilibrium point (r, 0, 0) is a centre, and by the symmetry the equilibrium point (−r, 0, 0) is also centre. This concludes the proof of statement (i).
To show statement (ii) and (iii), denote by f the vector field defined by system (2), and consider the involution R y (x, y, z) = (x, −y, z). Then the vector field f is R y -reversible.  We take now the quotient betweenż andẋ in system (6), that is Integrating this differential equation we obtain the hyperbolas because α < β < γ . See Figure 3.
Since f is R y -reversible by Lemma 2.3 the unstable separatrices coming from the saddle (0, −r, 0) until the plane {y = 0} connect with the stable separatrices of the saddle (0, r, 0). Thus we have two heteroclinic connections.
Again being f R y -reversible, by Lemma 2.3 each hyperbola of the southern hemisphere connects with a hyperbola of the northern hemisphere, producing a periodic orbit of some of the centres (±r, 0, 0) and (0, 0, ±r), see Figure 1. This completes the proof of Theorem 1.3.

Disclosure statement
No potential conflict of interest was reported by the authors.