A Chebyshev criterion with applications

We show that a family of certain deﬁnite integrals forms a Chebyshev system if two families of associated functions appearing in their integrands are Chebyshev systems as well. We apply this criterion to several examples which appear in the context of perturbations of periodic non-autonomous ODEs to determine bounds on the number of isolated periodic solutions, as well as to persistence problems of periodic solutions for perturbed Hamiltonian systems.


Main results
Chebyshev systems (T -systems), complete Chebyshev systems (CT -systems) and extended complete Chebyshev systems (ECT -systems) are the natural extensions of polynomials of a given degree m to more general functions. Notice that degree m polynomials can be seen as elements of the vector space 1, x, . . . , x m of dimension m + 1, for which each element has at most m roots, counting multiplicities, such that this bound is attained. In the next section we give the precise definition of T , CT and ECT -systems, which essentially introduce them as vector spaces of functions satisfying these properties.
When studying perturbations of Hamiltonian systems with a continuum of periodic orbits, the level sets of the periodic orbits that persist as limit cycles are given by the zeroes of a line integral, or Abelian integral, see for instance [8]. A commonly used method to control the number of zeroes of such integrals when the perturbation depends on parameters is to prove that they form a basis which is a Chebyshev system. With this aim, a criterion was developed in [6] which shows that if some functions constructed from the integrands of the Abelian integrals form a Chebyshev system, then the Abelian integrals that generate the complete Abelian integral itself are a Chebyshev system as well. This is a powerful result since proving the Chebyshev property for functions is usually easier than to do so for functions defined as line integrals.
In the same spirit, the goal of the present paper is to prove that some definite integrals form a Chebyshev system if families of functions given by the integrands of these integrals are also Chebyshev systems. Our main result is:

. Consider a family of integrals of the form
with a, b ∈ R and h ∈ L for L ⊂ R open, satisfying the following hypotheses: • f 1 , . . . , f n are continuous functions on the open interval (a, b) such that (f 1 , . . . , f n ) is a CT-system on (a, b). • g is an analytic function on L × (a, b) such that for any fixed set of n elements {x 1 , . . . , x n } ∈ [a, b] such that x i = x j when i = j , the functions form an ECT-system on L.
Then (I 1 , . . . , I n ) is an ECT-system on L.
By relaxing the above hypotheses, in Theorem 2.4 below we give a similar result but proving that (I 1 , . . . , I n ) is a CT-system on L. The following is a corollary of Theorem 1.1, proved in Section 3, which we will use in all our applications.
Theorem 1.2 complements the main result of [4] where the family of functions is studied and it is characterized under which conditions it forms a Chebyshev family. In that paper, nothing is assumed on the monotonicity of the function q but, the functions f i appearing in (1) are simply powers of q. Moreover, the proof given in [4] differs from our present approach and is based on the fact that the Wronskians introduced in Lemma 2.3 to prove that the functions form an ECT -system can be related to Gram determinants.
In Section 3 we apply Theorem 1.2 to study the number of zeroes of two families of functions. The first one is a Melnikov type function that controls the periodic solutions that bifurcate from a one-dimensional non-autonomous periodic differential equation of Abel type. The second one controls the periodic orbits that persist in the rotary regions by perturbing several Hamiltonian potential planar systems. Recall that these persistent periodic orbits are given by zeroes of Abelian integrals, see [8]. In particular, we apply this result to study some periodic perturbations of the pendulum and of the whirling pendulum when the constant rotation rate is smaller that a given value.

Preliminary results and proof of the main results
We start by recalling the definitions of T , CT and ECT systems, the notions of continuous and discrete Wronskian and a useful characterization of CT and ECT -systems. This result and much more information on the subject can be found in the monographs [7,10].
has at most k − 1 isolated zeros on I counting multiplicity.

Lemma 2.3.
The following equivalences hold: The proofs of the following two results are inspired by the proof of Proposition 3.3 in [6].

Theorem 2.4. Consider a family of integrals of the form
open, satisfying the following hypotheses: form a CT-system on (a, b).
. . , n}, let S k be the symmetric group of k elements and denote by k the k-simplex defined by {x k ∈ [a, b] n : x 1 < · · · < x k }. Taking into account the definition of the determinant we have that and note that where R ⊂ R k is a set of Lebesgue measure zero. Therefore we can write The next step is to change coordinates in each integral of the above sum according to In view of the fact that the absolute value of the determinant of the Jacobian of ψ ρ is one, we find that Since both (f 1 , . . . , f n ) and (g 1 , . . . , g n ) are CT-systems on (a, b) the integrand in the last integral is different from zero. Since k is connected it follows that det(I i (h j )) 1≤i,j ≤n = 0 and the proof is complete.
Proof of Theorem 1.1. Now we need to prove that for any k ≤ n and for any h ∈ L, W [I k ](h) = 0. As in the previous proof fix k ∈ {1, . . . , n}, let S k be the symmetric group of k elements and denote by k the k-simplex defined by {x k ∈ [a, b] n : x 1 < · · · < x k }. Taking into account the definition of the determinant we have that We consider again for each permutation ρ ∈ S k the mapping ψ ρ : R k → R k , ψ ρ (x k ) = x ρ(k) to write the last integral as The change of coordinates x k = ψ ρ (u k ) in each integral of the above sum and the fact that the absolute value of the determinant of the Jacobian of ψ ρ is one, yields Since (f 1 , . . . , f n ) is a CT-system on (a, b) and (g(h, u 1 ), . . . , g(h, u n )) is an ECT-system on L for any ordered set {u 1 , . . . , u n } ∈ (a, b), the integrand in the last integral is different from zero. Therefore, det(I i (h j )) 1≤i,j ≤n = 0 and the proof is complete. By Theorem 1.1, the proof will follow if we prove that (g i (h)) n i=0 form an ECT-system on L. To prove this fact, note first that For k ∈ N, we introduce the notation [α] 0 = 1, [α] k := α(α −1)(α −2) · · · (α −k +1). Therefore, the Wronskian determinant is the determinant of a Vandermonde matrix. Since all a i are distinct, it is nonzero, and hence by Lemma 2.3, (g i (h)) n i=0 form an ECT-system on L as we wanted to prove.
Notice that in the above proof we show in particular that the family of functions (a i + h) α , with i = 0, 1, . . . , n and α / ∈ N and all a j distinct, form an ECT-system in a suitable interval. It is curious to observe that any permutation of this set of n + 1 functions form also an ECT-system. This is an unusual property, which is not true for the ECT-system (1, h, h 2 , . . . , h n ) for instance.

Applications
This section collects several applications of our main results.

Perturbation of periodic non-autonomous ODEs
Theorem 1.2 can be applied to determine upper bounds on the number of isolated periodic solutions obtained from the first order analysis for perturbations of certain 1-dimensional nonautonomous differential equations. Consider the initial value problem where f is real analytic and T -periodic with respect to the first variable. Let ϕ(x, h) be the solution of this problem and assume that it is T -periodic for all h ∈ (h 1 , h 2 ). Now for g(x, y) also real analytic and T -periodic with respect to the first variable we consider the perturbed problem We will look for T -periodic orbits that persist after perturbation. Denote by ψ(x, h, ε) the solution of the perturbed problem (3). Notice that ψ(x, h, 0) = ϕ(x, h). By similarity with the notation used when studying the perturbations of Hamiltonian systems, see [8], we will say that the periodic solution corresponding to h = h * persists if for ε small enough there exists h ε such that ψ (x, h ε , ε) is T -periodic and lim ε→0 h ε = h * . From the theorems on dependence of solutions on parameters we get that where D 2 f (x, y) denotes the derivative of f (x, y) with respect to y. So we have that As a more concrete example, consider the problem which is of the form (3) with k ∈ N, λ and F are T -periodic and (x) = x 0 λ(s)ds such that (T ) = 0. Direct computations give that Notice that the condition (T ) = 0 implies that a neighborhood of y = 0 is full of T -periodic solutions. Moreover, and from (4) we get that Example 3.1. In our first example we revisit some results of [4], in which the number of isolated periodic solutions obtained from first order perturbations of some generalized Abel equations was obtained. The equations under consideration, ⎧ ⎨ ⎩ dy dx = cos x k − 1 y k + εP n (cos x, sin x)y p , are of the form (5) with T = 2π , 1 < k ∈ N, k < p ∈ N and P n a degree n polynomial. Using (6) and (7), simple computations show that for some a i,j ∈ R. Since where This is exactly the type of analytic functions studied in [4], see equations (1) and (5) of that paper. Notice that Thus, we are under the hypotheses of Theorem 1.2 with [a, b] = [π/2, 3π/2], f i (x) = sin i x and p(x) = − sin(x). Therefore, when p−k 1−k / ∈ N the family (I 0 (h), I 1 (h), . . . , I n (h)) is an ECTfamily on (0, 1).

Example 3.2.
Rigid planar systems are frequently studied because they encompass all of the difficulties of Hilbert's XVIth problem in a more tangible context: the system has a unique critical point at the origin and can be globally reduced to a one-dimensional non-autonomous differential equation. Moreover, the solution of the center-focus problem is equivalent to determining the isochronous centers, see [5,12] or [2], where these systems are called uniformly isochronous centers. Polynomial rigid systems with a center or focus at the origin are of the form x = −y + xP (x, y), where P is an arbitrary polynomial. In polar coordinates they can be written as because θ = 1, the property which gives rise to their name. In the particular case P = P k−1 + εP p−1 , where P m are homogeneous polynomials of degree m and k, p ∈ N, equation (8) reads dr dθ = r k P k−1 (cos θ, sin θ) + εr p P p−1 (cos θ, sin θ), which is of the form (5). Hence, adding some additional hypotheses on P k−1 and P p−1 we have found new families of non-autonomous differential equations for which Theorem 1.2 can be applied to obtain upper bounds on the number of isolated periodic solutions by studying the zeroes of the corresponding M(h) given in (7). We skip the details.  This energy level separates the oscillatory region R 0 from the rotary regions R ± : for h <h there exist periodic orbits inside the region enclosed by the heteroclinic connections between the saddle points at (T , 0), while for h >h the periodic orbits encircle the cylinder with period 2T above and below the heteroclinic connections, see Fig. 1. In what follows we will focus on periodic orbits in the rotary regions R ± . We consider the following perturbation of the double potential x = y 2s−1 ,