CUBIC HOMOGENEOUS POLYNOMIAL CENTERS

First, doing a combination of analytical and algebraic computations, we determine by first time an explicit normal form depending only on three parameters for all cubic homogeneous polynomial differential systems having a center. After using the averaging method of first order we show that we can obtain at most one limit cycle bifurcating from the periodic orbits of the mentioned centers when they are perturbed inside the class of all cubic polynomial differential systems. Moreover, there are examples with one limit cycles. 2010 Mathematics Subject Classification: Primary: 34C07, 34C08, 37G15.


Introduction and statement of the main results
We consider polynomial differential systems in R 2 of the form (1) ẋ = P (x, y), ẏ = Q(x, y), where P and Q are real polynomials in the variables x and y.We say that this system has degree m, if m is the maximum of the degrees of P and Q.
In his address to the International Congress of Mathematicians in Paris in 1900, Hilbert [7] asked for the maximum number of limit cycles which real polynomial differential systems of degree m could have, this problem is known as the 16th Hilbert problem, for more details about it see the surveys [8,9].
Since the 16th Hilbert problem is too much difficult Arnold, in [1], stated the weakened 16th Hilbert problem, one version of this problem is to determine an upper bound for the number of limit cycles which can bifurcate from the periodic orbits of a polynomial Hamiltonian center when it is perturbed inside a class of polynomial differential systems, see for instante [3,4,6].
We recall that a singular point p ∈ R 2 of a differential system (1) is a center if there is a neighborhood U of p such that U \ {p} is filled of periodic orbits of (1).The period annulus of a center is the region fulfilled by all the periodic orbits surrounding the center.We say that a center located at the origin is global if its period annulus is R 2 \ {0}.
In this paper we shall study the weakened 16th Hilbert's problem but perturbing mainly non-Hamiltonian centers using the technique of the averaging method of first order.More precisely, our first objective will be to characterize all planar cubic homogeneous polynomial differential system having a center, and after to analyze how many limit cycles can bifurcate from the periodic orbits of these cubic homogeneous polynomial centers when we perturb them inside the class of all cubic polynomial differential systems.
The next result characterize all planar cubic homogeneous polynomial differential system having a center.
Theorem 1.Any planar cubic homogeneous polynomial differential system having a center after a rescaling of its independent variable and a linear change of variables can be written into the form where α = ±1, a, b, µ ∈ R and µ > −1/3.
Theorem 1 will be proved in section 2.
Our objective now is to study how many limit cycles can bifurcate from the periodic orbits of the cubic homogeneous polynomial centers (2) when we perturb them inside the class of all cubic polynomial differential systems.We consider for ε sufficiently small the following cubic polynomial differential systems where p and q are arbitrary polynomials of degree 3, i.e.
p(x, y) = a 0 + a 1 x + a 2 y + a 3 x 2 + a 4 xy + a 5 y 2 + a 6 x 3 +a 7 x 2 y + a 8 xy 2 + a 9 y 3 , q(x, y) Clearly from (2) it is easy to check that the cubic homogeneous polynomial centers in general are non-Hamiltonian, this justifies what we have said that mainly in (3) we are perturbing non-Hamiltonian centers.
In [10] the authors provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of any homogeneous and quasi-homogeneous center, which can be obtained using the Abelian integral method of first order.They also show that these bounds are the best possible using the Abelian integral method of first order.In particular, applying statement (c) of Theorem A of [10] with p = q = 1 and n = 3 it follows that with the Abelian integral method of first order it can be obtained for systems (3) at most 1 limit cycle.For more details on the Abelian integral method see the second part of the book [4].
What it is not provide in [10] is an explicit formula for computing the limit cycle which can bifurcate from the periodic orbits of any homogeneous and quasi-homogeneous center.We shall provide here such a formula for the cubic homogeneous polynomial centers (2).For doing that we shall use the averaging method of first order described in section 3.
We must mention that for planar polynomial differential systems since the Abelian integral method of first order as the averaging method of first order correspond to the coefficient of first order in the small parameter of the perturbation of the displacement function, both methods provide the same information.
We need to define the following three functions: (5) ) .
Let (r, θ) be the polar coordinates defined by x = r cos θ and y = r sin θ.The following result is well known, but since its proof is almost immediate we shall give it in section 4.
Proposition 2. In polar coordinates the periodic solution r(θ, z) of the cubic homogeneous polynomial center (2) For studying the limit cycle that can bifurcate from the periodic orbits of a cubic homogeneous polynomial center perturbed inside the class of all cubic polynomial differential equations we need to define the following two functions and two numbers: Theorem 3. The following two statements hold.
(a) Consider the polynomial h(z) = c 0 + c 2 z 2 .Clearly it can have at most one real positive root.Assume that z 0 is a positive root of h(z).Then, for ε ̸ = 0 sufficiently small the perturbed cubic polynomial differential system (3) has a limit cycle such that in polar coordinates tends to the periodic orbit k(θ)z 0 of the cubic homogeneous polynomial center (3) for ε = 0. (b) There are cubic homogeneous polynomial centers for which the polynomial h(z) has a real positive root.
Theorem 3 will be proved in section 4.

Proof of Theorem 1
From Proposition 4.2 of [5] and its proof we have the following result.
In Corollary 3.3 of [5] there are ten normal forms for all the cubic homogeneous polynomial differential systems.Only the normal formals ( 8) and ( 9) of that corollary satisfy that the polynomial xQ(x, y) − yP (x, y) has no real linear factors, but these two normal forms can be joined in the following unique normal form In short, in order that the cubic homogeneous polynomial differential system (7) has a center at the origin only remains that its corresponding integral I be zero.For system (7) we have ( 8) Note that when µ > −1/3 we have By using the symmetry of g 1 with respect to cos θ and sin θ, and using the change of variable θ = π 2 − ϕ, we find On the other hand, splitting the integration interval [0, π] to [0, π/2] and [π/2, π] and making the change θ = π − ϕ in [π/2, π], we have Similarly, Hence, from ( 8) we obtain ( 9) where Consequently, from ( 9) we obtain that I = 0 if and only if c = −a, and Theorem 1 follows from ( 7) with c = −a.

Averaging theory of first order
One of the averaging theories considers the problem of the bifurcation of T -periodic solutions from differential systems of the form with ε = 0 to ε ̸ = 0 sufficiently small.Here the functions F 0 , F 1 : R×Ω → R n and F 2 : R × Ω × (−ε 0 , ε 0 ) → R n are C 2 functions, T -periodic in the first variable, and Ω is an open subset of R n .The main assumption is that the unperturbed system has a submanifold of periodic solutions.Let x(t, z, ε) be the solution of system (11) such that x(0, z, ε) = z.We write the linearization of the unperturbed system along a periodic solution x(t, z, 0) as In what follows we denote by M z (t) some fundamental matrix of the linear differential system (12).We assume that there exists an open set V with Cl(V ) ⊂ Ω such that for each z ∈ Cl(V ), x(t, z, 0) is T -periodic, where x(t, z, 0) denotes the solution of the unperturbed system (11) with x(0, z, 0) = z.The set Cl(V ) is isochronous for the system (10); i.e. it is a set formed only by periodic orbits, all of them having the same period.Then, an answer to the bifurcation problem of T -periodic solutions from the periodic solutions x(t, z, 0) contained in Cl(V ) is given in the following result.

Theorem 5. [Perturbations of an isochronous set]
We assume that there exists an open and bounded set V with Cl(V ) ⊂ Ω such that for each z ∈ Cl(V ), the solution x(t, z) is T -periodic, then we consider the function Then the following statements holds: (a) If there exists a ∈ V with F(a) = 0 and det ((∂F/∂z) (a)) ̸ = 0, then there exists a T -periodic solution x(t, ε) of system (10) such that x(0, ε) → a as ε → 0. (b) The type of stability of the periodic solution x(t, ε) is given by the eigenvalues of the Jacobian matrix ((∂F/∂z) (a)).
For an easy proof of Theorem 5(a) see Corollary 1 of [2].In fact the result of Theorem 5 is a classical result due to Malkin [11] and Roseau [12].
For additional information on averaging theory see the book [13].

Proof of Theorem 3
We write the polynomial differential system (3) in polar coordinates (r, θ) through x = r cos θ and y = r sin θ.In the new coordinates the system becomes ) .