Limit cycles from a four-dimensional centre in ℝ m in resonance p : q

Given positive coprime integers p and q, we consider the linear differential centre in ℝ m with eigenvalues ±pi, ±qi and 0 with multiplicity m − 4. We perturb this linear centre in the class of all polynomial differential systems of the form linear plus a homogeneous nonlinearity of degree p + q − 1, i.e. , where every component of F(x) is a linear polynomial plus a homogeneous polynomial of degree p + q − 1. When the displacement function of order ϵ of the perturbed system is not identically zero, we study the maximal number of limit cycles that can bifurcate from the periodic orbits of the linear differential centre.


Introduction
In the qualitative theory of polynomial differential systems, the study of their limit cycles and mainly the obtention of information on their number for a given polynomial differential system is one of the main topics. We recall that for a differential system a limit cycle is a periodic orbit isolated in the set of all its periodic orbits.
In dimension two, i.e. in the plane, the two main problems related with limit cycles are: first, the study of the number of limit cycles depending on the degree of the polynomial differential system. This is an old problem proposed by D. Hilbert in 1900, known as the 16th Hilbert problem (see the surveys [1,2] for details), and second the study of how many limit cycles emerge from the periodic orbits of a given centre when we perturb it inside a certain class of differential systems [3].
Since the study of limit cycles and mainly the obtention of information on their number for a given polynomial differential system is in general a very difficult problem (almost impossible), there are hundreds of papers trying to solve these questions in the plane for many particular families of polynomial systems, see the references quoted in the book [3] and in the surveys [1,2]. These problems have been studied intensively in dimension two, and unfortunately the results are far from being satisfactory. In fact, the Riemann conjecture and the 16th Hilbert problem are the two unique problems of the famous list of Hilbert which are not solved.
Our main aim is to extend these studies from dimension two to higher dimension, and to observe the differences which appear due to the increase of the dimension of the polynomial differential systems. Thus, we take a linear resonant centre p : q of dimension four living inside dimension m ! 5 and study how many of the periodic orbits of the centre persist as limit cycles once this centre is perturbed inside a class of polynomial differential systems of degree p þ q À 1. The interesting result obtained for this class of polynomial differential systems is that the number of limit cycles obtained are powers related to the dimension m having bases related to the degree of the perturbation p þ q À 1 (for the precise result, see Theorem 1).
Here we study how many limit cycles emerge from the periodic orbits of a centre when we perturb it inside a given class of differential equations in dimension higher than four. More precisely, given m ! 5, we consider the linear differential centre for some positive coprime integers p and q. We perturb system (1) in the form where " is a small parameter, and where F : R m ! R m is a polynomial of the form and F k N arbitrary homogeneous polynomials, respectively, of degrees 1 and N ¼ p þ q À 1 in the variables x ¼ (x 1 , . . . , x m ) for k ¼ 1, . . . , m, with the exception that F k 1 ¼ k x k for k ¼ 5, . . . , m. We note that the polynomial perturbations F(x) of this form cover all polynomial perturbations of system (2) of degrees 2 and 3 (this follows from the theory of normal forms, see [4] for details).
For " ¼ 0 the differential system (2) has a four-dimensional centre in resonance p : q. Without loss of generality we can assume that q 4 p. We want to study how many limit cycles can bifurcate from the periodic orbits of this centre when we perturb it inside the class of polynomial vector fields of the linear form plus a homogeneous nonlinearity of degree p þ q À 1. Our main result is the following theorem.
Theorem 1: Assume that p, q ! 1 are coprime integers with q 4 p and that m ! 5. If " 6 ¼ 0 is sufficiently small and the displacement function of order " (see (5)) is not identically zero, then the maximum number of limit cycles of the differential system (2) bifurcating from the periodic orbits of the four-dimensional linear differential centre (1) is at most We refer to Section 2 for the definition of the displacement function of order ". Theorem 1 is proved in Section 4 using the averaging theory described in Section 2. Indeed, Theorem 1 depends heavily on the computation of the averaged system associated to the differential system (2), because its singular points with Jacobian nonzero provide the limit cycles of the differential system (2) when the displacement function of order " is not identically zero. Theorem 1 improves and extends previous results for system (2) restricted to R 4 (see [4,5]) and in R m for p ¼ 1 (see [6]).
When p, q and m are relatively small the averaged system can be computed explicitly, thus allowing one to improve the upper bound for the number of limit cycles given by Theorem 1. In particular, we have established the following result in [6].
Theorem 2: If " 6 ¼ 0 is sufficiently small and the displacement function of order " is not identically zero, then the maximum number of limit cycles of the differential system (2) bifurcating from the periodic orbits of the four-dimensional linear differential centre (1) is at most We note that the corresponding upper bounds given by Theorem 1 are, respectively, 256 and 1044.

First-order averaging theory
The aim of this section is to present the first-order averaging method obtained in [7]. We first briefly recall the basic elements of averaging theory. Roughly speaking, the method gives a quantitative relation between the solutions of a nonautonomous periodic system and the solutions of its averaged system, which is autonomous. The following theorem provides a first-order approximation for periodic solutions of the original system. We consider the differential system where H : R Â D ! R n and R : R Â D Â (À" 0 , " 0 ) ! R n are continuous functions, T-periodic in the first variable and where D is an open subset of R n . We define h : D ! R n by and we denote by d B (h, V, a) the Brouwer degree of h at a (see [8] for the definition).
Theorem 3: We assume that: (i) H and R are locally Lipschitz with respect to x; (ii) for a 2 D with h(a) ¼ 0, there exists a neighbourhood V of a such that h(z) 6 ¼ 0 for all z 2 Vn{a} and d B (h, V, a) 6 ¼ 0.Then for " 6 ¼ 0 sufficiently small there exists an isolated T-periodic solution (Á, ") of system (3) such that (a, 0) ¼ a.
The system _ x ¼ "hðxÞ is called the averaged system associated to system (3).
Hypothesis (i) ensures the existence and uniqueness of the solution of each initial value problem on the interval [0, T]. Hence, for each z 2 D, it is possible to denote by x(Á, z, ") the solution of system (3) with the initial value x(0, z, ") ¼ z. We also consider the function : D Â (À" 0 , " 0 ) ! R n defined by This is called the displacement function of order ". It follows from the proof of Theorem 3 that for every z 2 D the following relations hold: xðT, z, "Þ À xð0, z, "Þ ¼ ðz, "Þ, and ðz, " where h is given by (4) and where the symbol O(" 2 ) denotes a function bounded on every compact subset of D Â (À" 0 , " 0 ) multiplied by " 2 . We note that in order to see that d B (h, V, a) 6 ¼ 0, it is sufficient to check that the Jacobian of D z h(z) at z ¼ a is not zero [8].

Averaged system
Writing Lemma 4: Doing the change of variables from (x 1 , x 2 , x 3 , x 4 , x 5 , . . . , x m ) to the new variables (, r,, s, y 5 , . . . , y m ) given by . . , m, and taking as the new independent variable, system (6) is transformed into the system where Proof: In the variables (, r, , s, y 5 , . . . , y m ) system (6) becomes For " sufficiently small, _ ðtÞ 4 0 for each (t, (, r, , s, y 5 , . . . , y m )) 2 R Â D. Now we eliminate the variable t in the above system by considering as the new independent variable. It is clear that the right-hand side of the new system is well-defined and continuous in R Â D Â (À" 0 , " 0 ), 2-periodic with respect to the independent variable and locally Lipschitz with respect to (r, , s, y 5 , . . . , y m ). From (8), Equation (7) is obtained after an expansion with respect to the small parameter ". h We recall a technical result from [4] which we shall use later on.
Lemma 5: Let and be real numbers. Given nonnegative integers i, j, k, l, there exist constants c uv and d uv such that cos i sin j cos k sin l is equal to if j þ l is even, and is equal to Now we compute the corresponding averaged functions h j (r, , s, y 5 , . . . , y m ) for j ¼ 1, . . . , m of system (7) given in (4). We write We also write h j ðr, , s, y 5 , . . . , y m Þ ¼ Proposition 6: We have for some constants a 1 , b 1 , c 1 and d 1 li 5 ÁÁÁi m depending on the coefficients of the perturbation. Proof: We write the function H 1 as and h N 1 ðr, s, , y 5 , . . . , y m Þ ¼ where for some constants c i 1 ÁÁÁi m uv and d i 1 ÁÁÁi m uv . Therefore all the integrals with respect to are zero except possibly when Without loss of generality, we continue to assume that p 5 q. Since p and q are coprime, there exists a nonnegative integer n such that we have that nq p þ q, and thus n (p þ q)/q 5 2. So either n ¼ 1 or n ¼ 0, i.e. either and it follows from (10) that This yields the term If i 3 þ i 4 À 2v ¼ 0, then 2v þ i 5 þ ÁÁÁ þ i m ¼ N À i 1 À i 2 , and 2v þ i 5 þ ÁÁÁ þ i m runs from 0 to N ¼ p þ q À 1. This yields the terms The proposition follows adding the terms from (9), (11) and (12). h for some constants a 2 , b 2 , c 2 and d 2 vi 5 ÁÁÁi m depending on the coefficients of the perturbation. Proof: As in Proposition 6, we write the function H 2 as Then h 1 2 ðr, s, , y 5 , . . . , y m Þ ¼ and using Lemma 5 we obtain h N 2 ðr, s, , y 5 , . . . , y m Þ ¼ for some constants c i 1 ÁÁÁi m uv and d i 1 ÁÁÁi m uv . All the integrals with respect to are zero except possibly when Since p and q are coprime, there exists a nonnegative integer u such that i 1 þ i 2 À 2u ¼ nq we have that np p þ q, and thus n (p þ q)/q 5 2. So either n ¼ 1 or n ¼ 0, i.e. either and hence If Thus 2v þ i 5 þ ÁÁÁ þ i m runs from 1 to p þ q, yielding the terms The proposition follows adding the terms of (13), (15) and (16).
Proceeding in a similar manner to the proofs of Propositions 6 and 7, we get h 1 3 ðr, , s, y 5 , . . . , y m Þ ¼ where and The terms whose integrals need not be zero satisfy in Equation (19), and in Equation (20). The arguments in the proof of Proposition 7 show that in (18) the terms that may remain in the first sum are and the arguments in the proof of Proposition 6 show that the terms that may remain in the second sum are The proposition follows adding the terms in (17) for some constants d 5 vi 5 ÁÁÁi m depending on the coefficients of the perturbation. Proof: As in the former proofs, we write All the integrals with respect to are zero except possibly when Proceeding as in the proof of Proposition 6, we find that either i 3 þ i 4 À 2v ¼ p or which yields a contradiction. Therefore, this case does not occur.
Hence 2v þ i 5 þ ÁÁÁ þ i m runs from 0 to p þ q À 1, and we obtain the terms This yields the desired statement. h

Proof of Theorem 1
We recall a technical result proved in [5].
Lemma 10: If p, q, and are nonnegative integers with þ ¼ q À 1 and þ ¼ p, then We will use the following proposition.
Proposition 11: The function h 3 (r, , s, y 5 , . . . , y m ) is given by Proof: Using the notation of Proposition 8, we shall prove that b 3 ¼ Àc 1 /p, c 3 ¼ b 1 /p, d 3 ¼ Àc 2 /q and e 3 ¼ b 2 /q. To simplify the proof, let a 1 By Lemma 10 the term in (24) is equal to For i 2 odd the coefficient of the monomial appears in a sum determining the coefficient of r qÀ1 p cos(pqs) in h 1 , and also appears in a sum determining the coefficient of r qÀ2 p sin(pqs) in h 3 with the opposite sign. In a similar way for i 2 even the coefficient of the monomial appears in a sum determining the coefficient of r qÀ1 p sin(pqs) in h 1 , and appears in a sum determining the coefficient of r qÀ2 p cos(pqs) in h 3 with the same sign. We can do the same for all monomials in F 2 N , F 3 N and F 4 N , and thus we conclude that Now we have all the ingredients to prove Theorem 1.
Proof of Theorem 1: It follows from Propositions 6, 7, 9 and 11 that where h j ¼ h j (r, , s, y 5 , . . . , y m ). According to the results of Section 2, we must study the real solutions of the system h k ðr, , s, y 5 , . . . , y m Þ ¼ 0 for k ¼ 1, 2, 3, 5, . . . , m that have nonzero Jacobian. In order that these solutions can provide limit cycles of system (2), we must look for those such that r 2 þ 2 6 ¼ 0. We distinguish three cases.
Substituting z and w in the equationh 4 ¼ 0, we obtain a quotient of a polynomial of degree 2(p þ q þ 1) by a polynomial of degree 4þp in the variables (A 2 , C 5 , . . . , C m ).