Integrable zero-Hopf singularities and three-dimensional centres

In this paper we show that the well-known Poincaré–Lyapunov non-degenerate analytic centre problem in the plane and its higher-dimensional version, expressed as the three-dimensional centre problem at the zero-Hopf singularity, have a lot of common properties. In both cases the existence of a neighbourhood of the singularity in the phase space completely foliated by periodic orbits (including equilibria) is characterized by the fact that the system is analytically completely integrable. Hence its Poincaré–Dulac normal form is analytically orbitally linearizable. There also exists an analytic Poincaré return map and, when the system is polynomial and parametrized by its coefficients, the set of systems with centres corresponds to an affine variety in the parameter space of coefficients. Some quadratic polynomial families are considered.


Introduction and statement of the main results
In this work, we consider an analytic three-dimensional systeṁ x = −y + F 1 (x, y, z), y = x + F 2 (x, y, z), z = F 3 (x, y, z), where F = (F 1 , F 2 , F 3 ): U → R 3 is a real analytic vector field on the neighbourhood of the origin U ⊂ R 3 with F(0) = 0 and whose Jacobian matrix DF(0) = 0. The origin is a zero-Hopf (also called a fold-Hopf ) singularity of system (1.1) because its associated eigenvalues are {±i, 0} with i 2 = −1.
Since the linear part of system (1.1) generates a rotation, it makes sense to extend the classical Poincaré-Lyapunov centre problem for planar analytic vector fields to the zero-Hopf singularity. The origin of system (1.1) will be called a threedimensional centre if there is a neighbourhood of it completely foliated by periodic orbits of (1.1), including continua of equilibria as trivial periodic orbits. On the other hand, we say that system (1.1) is completely analytically integrable if it admits two independent locally analytic first integrals.
Remark 1.1. It is well known that there are vector fields without singular points only on odd-dimensional spheres. A consequence of this topological result is that isolated singularities of vector fields in R n having a punctured open neighbourhood filled by non-trivial periodic orbits may only exist for even phase space dimension n. Therefore, since n = 3 for system (1.1), any three-dimensional centre at the origin of (1.1) is a non-isolated singularity and, consequently, there is an invariant curve Γ filled by equilibria passing through it tangent to the z-axis. For any analytic vector field in R 3 with an isolated singularity, it was proved in [3] that there is always a solution tending to the singularity (in the future or in the past) with a well-defined tangent.
In fact, applying formal normal-form theory to (1.1) yields the existence of a formally invariant one-dimensional manifold given by the z-axis (see § 4). The reader is also referred to [2], where the existence of the rotation axis in the C ∞ case is proved. Indeed, we will see that the formal invariant one-dimensional manifold really does exist in the analytic category too. Proposition 1.2. System (1.1) possesses a one-dimensional local analytic invariant manifold Γ tangent to the z-axis at the origin. In particular, there are local analytic coordinates tangent to the identity (x, y, z) → φ(x, y, z) = (x + · · · , y + · · · , z + · · · ) stretching the manifold Γ towards the z-axis and transforming (1.1) intoẋ Proof. In any case, be the origin a three-dimensional centre of (1.1) or not, there is always a one-dimensional local analytic invariant manifold Γ of (1.1) tangent to the z-axis at the origin. This fact is a consequence of the spectrum structure of the linear part of the analytic system (1.1): one eigenvalue is real and the others are not (see [6]). From here, the existence of the local analytic change of coordinates φ stretching Γ follows. Clearly, since the linear part of φ is the identity, the linear parts of (1.1) and (1.2) are equal. The fact thatF j (0, 0, z) = 0 for j = 1, 2 comes from the invariance of the z-axis in the new coordinates.
Remark 1.5. The need to restrict the values of w to the arbitrary but fixed compact set K containing the origin is clarified in the proof of theorem 1.4. Now, we recall the geometry associated with the polar blow-up (1.4). First, we notice the key point that (x, y, z) ∈ U \ C δ when w ∈ K. Indeed, (1.4) is a diffeomorphism in U \ C δ . Once we set the value of δ, the cone C δ is fixed and, consequently, in principle we cannot control (via the zeros of the displacement map d) the periodicity of those orbits of (1.1) in U that intersect C δ . Despite this, although the polar blow-up (1.4) does not cover the neighbourhood U of the origin, the gap C δ can be made very thin provided that we take δ large enough. In this direction, we emphasize that any periodic orbit of (1.1) in U not intersecting the z-axis is contained in U \ C δ for δ sufficiently large. Thus, in the three-dimensional centre case, most of the continuum of periodic orbits of (1.1) in U are completely contained in U \ C δ . Each one of these orbits corresponds to a 2π-periodic solution of system (1.5) with |r| sufficiently small and w(θ; r 0 , w 0 ) ∈ K for all θ. Consequently, the zeros of the displacement map d(r 0 , w 0 ), with w 0 ∈ K and |r 0 | 1, pick up all these periodic orbits. Thus, if the origin is a three-dimensional centre of (1.1), then d(r 0 , w 0 ) should have non-isolated zeros filling subsets of full Lebesgue measure of a neighbourhood of (r 0 , w 0 ) = (0, 0). At this point, we recall that a real analytic function of several variables that vanishes on a set of positive measure must be identically zero. Taking into account the analyticity of d near the origin, it follows that a necessary and sufficient condition in order that the origin of (1.1) becomes a three-dimensional centre is that d(r 0 , w 0 ) ≡ 0 for all (r 0 , w 0 ) close to (0, 0). This last assertion has been checked with several polynomial families of the form (1. In summary, the above discussion leads to the following result. From now on, we shall assume that (1.1) is a family of polynomial differential systems parametrized by its coefficients, which we collect in the real vector parameter λ ∈ Λ ⊆ R p . Hence, its associated displacement map d(r 0 , w 0 ; λ) can be expanded in a Taylor series at (r 0 , In analogy with the theory of the classical two-dimensional centres (see, for example, [9]) we call the two-dimensional vector functions i,j (λ)) Poincaré-Lyapunov constants. Clearly, d i,j (λ * ) = 0 for all admissible (i, j) is a necessary condition for family (1.1) with λ = λ * to have a three-dimensional centre at the origin.

) be a polynomial family having as parameters its coefficients λ. Then the components of the Poincaré-Lyapunov constants
Therefore, for polynomial families (1.1), the characterization of its three-dimensional centres leads to a collection of polynomials in the coefficients λ of (1.1) whose simultaneous vanishing picks out those systems for which the singularity is a threedimensional centre. This implies that each member of family (1.1) having a centre at the origin corresponds with a point λ = λ * of an affine variety V C in the parameter space of coefficients called the centre variety.
i,j (λ): (i, j) ∈ N 2 be the ideal in the polynomial ring R[λ] generated by all the components of the Poincaré-Lyapunov constants. Again by analogy with the two-dimensional centre theory, B will be termed the Bautin ideal at the origin of (1.1). Of course the variety V (B) associated with the ideal B is just V C . The ideal B is Noetherian and so it is generated by a finite number of polynomials by the Hilbert basis theorem but, unfortunately, we do not know this basis a priori.
The second aim of this work is to characterize three-dimensional centres of (1.1) via integrability and normal-form theory.  The structure of the paper is as follows. In § § 2 and 3 we give the proof of theorem 1.4 and corollary 1.7, respectively. We devote § 4 to proving theorem 1.8, and the final section focuses on examples.

Proof of corollary 1.7
We assume that (1.1) is a family of polynomial differential systems with parameters λ, that is, where all the F i depend in a polynomial way on λ. The associated Poincaré-Lyapunov quantities d i,j (λ) can be determined in a recursive way, although many computations are involved. Following the proof of theorem 1.4, we can check that after the polar blow-up (1.4), family (3.1) is transformed into the familẏ r = R(θ, r, w; λ),θ = 1 + Θ(θ, r, w; λ),ẇ = W(θ, r, w; λ), (3.2) where R, Θ and W are polynomials in the parameters λ of the family. Therefore, the corresponding system (1.5) now becomes the family and Θ(θ, 0, w; λ) = 0. Hence we can write the Taylor series where R ij (θ; λ) and W ij (θ; λ) are 2π-periodic functions in the variable θ and are polynomials in λ. Let Ψ (θ; r 0 , w 0 ; λ) = (r(θ; r 0 , w 0 ; λ), w(θ; r 0 , w 0 ; λ)) be the solution of (3.3) with initial condition Ψ (0; r 0 , w 0 ; λ) = (r 0 , w 0 ). We can expand it as for all (i, j) = (0, 0) and (p, q) = (0, 1). Notice that with this notation we have that the Poincaré-Lyapunov constants are Differentiating the former series Ψ with respect to θ and inserting into (3.3) yields so that equating coefficients of like powers of r 0 and w 0 we obtain dΨ r and in general for all admissible (i, j) we get that where p + q i + j, k + i + j and P r i,j and Q r i,j are polynomial functions of their arguments. This fact together with (3.4) proves the corollary.

Proof of theorem 1.8
Poincaré-Dulac normal forms of system (1.1) have been studied in several works. The underlying idea is to write the vector field as a sum of homogeneous polynomials and search for near-identity changes of variables that iteratively simplify the homogeneous parts of the vector field; see, for example, [1,4].
Clearly, this means that system (1.1) has two formal independent first integrals z + · · · and x 2 + y 2 + · · · . Using results of Zhang [10,11] (see also [7, theorem 9]), we know that in fact (1.1) has two independent analytic first integrals z + · · · and x 2 + y 2 + · · · , and moreover there is a convergent normalizing transformation. This proves one part of the theorem. The converse is easy to prove. Assume now that (1.1) is completely analytically integrable. This implies that (1.1) has two independent analytical first integrals H 1 (x, y, z) and H 2 (x, y, z). Since the linearizationẋ = −y,ẏ = x,ż = 0 of (1.1) has the first integrals z and x 2 + y 2 , it is clear that H 1 and H 2 can be chosen in the form H 1 (x, y, z) = x 2 + y 2 + · · · and H 2 (x, y, z) = z + · · · . Therefore (1.1) possesses a three-dimensional centre at the origin. This can be checked just by intersecting the level sets (topological cylinders and planes) of H 1 and H 2 in a neighbourhood of the origin.
Remark 4.1. Since the z-axis is a formal invariant rotation axis of the normal form (4.1), we can use cylindrical coordinates (x, y, z) → (ϕ, ρ, z) so that (4.1) can be reduced to z]]. Continuing in this direction, the proof of theorem 1.8 can be made simpler (there is no need even to introduce the small parameter ε) when we show that the normal form (4.3) of a three-dimensional centre must satisfy f = g ≡ 0. Anyway, we retain that proof since it introduces Melnikov functions (see (4.4)) and this is an appropriate way to calculate necessary three-dimensional centre parameter conditions. We will check the above in the forthcoming examples.

Examples
The computation of the necessary three-dimensional centre conditions are performed using the technique of the proof of theorem 1.8 developed in [5]. Hence we first do the rescaling (x, y, z) → (x/ε, y/ε, z/ε) with a small parameter ε and next we perform the polar blow-up (x, y, z) → (θ, r, w) defined in (1.4)

Example 1
We study the three-dimensional centre problem for the quadratic familẏ Proposition 5.1. The origin is a three-dimensional centre of family (5.2) if and only if a 1 + a 2 = 0.
Then d 1 (r 0 , w 0 ) ≡ 0 if and only if a 1 + a 2 = 0. Finally, taking a 2 = −a 1 it is easy to check that family (5.2) has two polynomial first integrals thereby finishing the proof.
Remark 5.2. Of course, instead of computing the Melnikov function, we could use reduction to normal form. If we perform the first step of such a computation on family (5.2), we geṫ and hence the condition a 1 + a 2 = 0 is obtained from corollary 1.9.
Proposition 5.5. The origin is a three-dimensional centre of family (5.7) if and only if b = e = af = cd = 0.
Hence, d 2 (r 0 , w 0 ) ≡ 0 if and only if af = cd = 0. We remark that with these parameter constraints one has d j (r 0 , w 0 ) ≡ 0 for j = 3, 4, 5, 6, making it probable that the origin becomes a three-dimensional centre of family (5.7). We are going to prove that, actually, this is the case.