Abelian Integrals and Non-generic Turning Points

In this paper we initiate the study of the Chebyshev property of Abelian integrals generated by a non-generic turning point in planar slow-fast systems. Such Abelian integrals generalize the Abelian integrals produced by a slow-fast Hopf point (or generic turning point), introduced in Dumortier et al. (Discrete Contin Dyn Syst Ser S 2(4):723–781, 2009), and play an important role in studying the number of limit cycles born from the non-generic turning point.

We stress that the goal of this paper is not to prove finite cyclicity of non-generic turning points, which is a result that needs further research to approach and it is beyond the scope of this manuscript. We mostly focus on the ECT-property of the ordered set (Ī −n ,Ī 0 ,Ī 1 , . . . ,Ī m−1 ).
We prove that a result similar to [7,Corollary 3.5] and [14,Theorem A] is true for each fixed n > 1 (see Theorems 1 and 2 in Sect. 2). The main difference between the statement of [7,Corollary 3.5] (n = 1) and Theorem 1 (n > 1) is that for n > 1 the boundary point h = 1 is not included in the interval on which the ECT-property holds. The functionĪ −n goes to infinity as h → 1 − (see Lemma 2).
Applying methods from [13] to the case where n > 1 it can be seen that, for any fixed integer n > 1, (Ī −n ,Ī 0 ) is an ECT-system on [ , 1 − ]. The proof is analogous to the proof for n = 1 and we therefore omit it for the sake of brevity. We did not succeed in using [13] to prove the same for the ordered set (Ī −n ,Ī 0 ,Ī 1 ) when n > 1. This is a topic of further study.
Further, we prove the monotonicity property of the quotient I 0 /I −n on the interval ]0, 1[ for each n > 1 (see Theorem 3). Theorem 3 naturally generalizes [4,Theorem 18] which covers the case of n = 1. Theorem 3 can be used to prove existence and uniqueness of limit cycles of planar systems obtained after desingularization of nongeneric turning points. For more details see Theorem 4 in Sect. 3.
In Sect. 2 we recall the definition of ECT-systems and state the main results of this paper. As already mentioned above, in Sect. 3 we motivate our study of the Abelian integrals. We prove the main results in Sect. 4.

Definitions and Statement of Results
Definition 1 Let f 0 , f 1 , . . . , f n−1 be analytic functions on a real interval with nonempty interior I . The ordered set of functions ( f 0 , f 1 , . . . , f n−1 ) is an extended complete Chebyshev system (in short, ECT-system) on I if, for all k = 1, 2, . . . n, any nontrivial linear combination has at most k − 1 isolated zeros on I counted with multiplicity.
(Notice that in this abbreviation "T" stands for Tchebycheff, which in some sources is the transcription of the Russian name Chebyshev.) One can prove (see [7,Lemma 3.7]) that ( f 0 , f 1 , . . . , f n−1 ) is an ECT-system on I if and only if the sequence F 1 , . . . , . . , f k n−1 }, can be constructed such that: , for k = 1, . . . , n − 2 and i = k + 1, . . . , n − 1, are analytic on I and 2. f 0 and ( f k k ) , k = 1, . . . , n − 1, are nowhere zero on I . This equivalent definition of ECT-system has been used in [5] with I being a closed interval [a, b]. If I = [a, b], then we have the following stability property of ECTsystem (see [5,Proposition 7.6] and if g i is an analytic function sufficiently close to f i in the C n−1 -topology, for i = 0, . . . , n − 1, then (g 0 , g 1 , . . . , g n−1 ) is also an ECT-system on [a, b]. We will use this stability property in Sect. 3.
The following results show that the number of zeros of any nontrivial linear combination of the Abelian integrals are bounded locally near the endpoints of the interval ]0, 1[. We note that in the following statements, when m = 0 the set of functions is only formed byĪ −n .
For n = 1, Theorem 3 has been proved by Chengzhi Li in [4] using the method from [1]. We use the same technique to prove it in the case where n > 1 (see Sect. 4.3). (1)

Lemma 1 The formula in
where we use the notationx forx ± . This gives (1).

Motivation
Consider slow-fast polynomial Liénard equations where m, n ≥ 1, a := (a 1 , . . . , a m ) is kept in a compact set K ⊂ R m , ≥ 0 is a small singular perturbation parameter and α ∼ 0 is a small regular parameter. To study the number and configurations of limit cycles of (4), we can use geometric singular perturbation theory. The theory is essentially composed of two parts, one of which, called Fenichel theory [6], describes the dynamics of X ,α,a near normally hyperbolic manifolds. The other part is family blow-up [4,11]; it is used to desingularize X ,α,a for example near the origin (x, y) = (0, 0) where the normal hyperbolicity is lost. More precisely, the fast subsystem X 0,α,a has the curve of singularities S = {(x, y) ∈ R 2 |y = x 2n + m k=1 a k x 2n+k } and horizontal fast movements. The critical curve S contains near the origin (x, y) = (0, 0) a normally repelling part x < 0, a normally attracting part x > 0 and a nilpotent contact point x = 0 which separates them (see Fig. 2). The contact point is generic (resp. non-generic) when n = 1 (resp. n > 1). For > 0 and ∼ 0, the dynamics of X ,α,a , uniformly away from S, can be described using regular horizontal orbits of the fast subsystem X 0,α,a . Near the normally hyperbolic parts of S, the dynamics of X ,α,a is given by the slow flow (often called slow dynamics) where F(x, a) := x 2n + m k=1 a k x 2n+k . When x ∼ 0 and x = 0, the slow dynamics points from the attracting part of S to the repelling part of S (note that To see how the Abelian integrals defined in Sect. 1 come into play, we blow up the origin (x, y, ) = (0, 0, 0) in X ,α,a + 0 ∂ ∂ using the following "singular" coordinate change (see [2,10]) We work with different charts in (5).
The phase directional charts {x = ±1,ȳ = ±1}. The most interesting phase directional chart is the chart {ȳ = +1}. In this chart we find two semi-hyperbolic singularities p ± located on the equator of the blow up locus (they are the end points of the invariant curve {ȳ =x 2n − 1 2n }). For a detailed study of X ,α,a in the {ȳ = +1}direction see e.g. [10]. The other phase directional charts ({x = ±1,ȳ = −1}) are not relevant when we study limit cycles of X ,α,a near the origin in the (x, y)-space.
Define a section = {x = 0,ȳ ≥ 0} parametrized by h ∈]0, 1] by means of the relation H (0,ȳ) = h and a section 0 from 0 ⊂ to . Notice that we focus on the return map P with ( , α) ∼ (0, 0), ≥ 0 and a ∈ K , defined uniformly away from γ 0 and γ 1 . We have P(h, , α, a) of X F ,α,a can be written as Proof We first study the Poincaré map P(h, α, A) of

Remark 1 Theorem 4 gives the uniqueness of limit cycles of X F
,α,a in the compact set U, for |a 1 | ≥ κ. Using (11) and The Implicit Function Theorem we see that X F ,α,a has a limit cycle near γ h , with h ∈ [ρ, 1 − ρ], for α = − a 1 2n where o(1) → 0 as → 0.

Remark 2
To find an optimal upper bound for the number of limit cycles of (4) in a fixed neighborhood of (x, y) = (0, 0), independent of → 0, it is more suitable to study the Chebyshev property of (Ī −n ,Ī 0 ,Ī 1 , . . . ,Ī l ). These integrals appear in the expression for the derivative of P given in (8). The reason for this comes from [5] where the same has been done for n = 1. In fact, the Chebyshev property of the derivatives is relevant in a gluing process with the polycycle γ 0 (see for example [5,Proposition 7.17] or [8]). As already observed in [14] for n = 1, we recall that the Chebyshev property of (Ī −n ,Ī 0 ,Ī 1 , . . . ,Ī l ) in the limit h → 0, obtained in Theorem 2, does not say anything about the number of limit cycles produced by γ 0 for n > 1. One has to use different techniques to study the cyclicity of γ 0 (see [5]). This topic is therefore not a subject of the present paper. (a j + g j (h)) where a j = 0 and g j is an analytic function at h = 1 with g j (1) = 0.
Proof When n = 1, this has been proved in [7,Lemma 3.4]. When n > 1, Lemma 2 can be proved in similar fashion. However, for the sake of completeness, we will give a sketch of the proof of this lemma. We know that γ h can be described by 2n ds, with j = −n, 0, 1, . . . . In the last step we use the change of coordinates g(ȳ) = √ 1 − hs and C(ȳ) = g(ȳ) 2 . Note that the function is analytic at z = 0, and thus can be written as k≥0 b k z k . We obtain now 2n ds.
where is the Gamma function.

Proof of Theorem 1
For each α ∈ Q we say that a function f belongs to the set R α if there exists > 0 such that is an analytic function at h = 1 satisfying F(1) = 0. We notice that if f ∈ R α and g ∈ R β with α > β then ( f g ) ∈ R α−β−1 . Let us fix 0 ≤ k ≤ m − 1 and consider any function in the linear span of I −n ,Ī 0 , . . . ,Ī k , that is where η −n , η 0 , . . . , η k ∈ R. We can consider η k = 0 since otherwise (h) belongs to the linear span with lesser index k. For the sake of compactness we rewrite the previous equality as From Lemma 2 we know thatĪ 0 −n ∈ R 1−n 2n . In particular, there exists 0 > 0 such that Therefore we can ensure that we have thatĪ 2 i ∈ R i−n n for i = 1, 2, . . . , k. In particular,Ī 2 1 ∈ R 1−n n and we can perform the next step in the division-derivation algorithm. Following this procedure, the ( j + 1)-step in the algorithm is Since the sequence of 0 , 1 , . . . , k+1 does not depend on but only on the Abelian integralsĪ −n ,Ī 0 , . . . ,Ī k , this shows that, by taking = k+1 , any function in the linear span ofĪ −n ,Ī 0 , . . . ,Ī k has at most k + 1 zeros, counted with multiplicity, on the interval [1 − , 1[.

Lemma 3 For each j
Proof From the fact thatȳ + (h) is the positive solution of f (ȳ) = h we have that y + ( f (u)) = u. Then, performing the change of variableȳ = 1−t 2ns we get for some a = 0. We claim at this point that if f ∈ S α and g ∈ S β with α = β then ( f /g) ∈ S α−β−1 . Indeed, deriving the quotient we obtain A simple computation shows that and so Let us show that lim s→0 + s k Q Since lim s→0 + s F (s) = lim s→0 + sG (s) = 0 and lim s→0 + G(s) is a non-zero constant, lim s→0 + s Q 0 (s) = 0. Assuming the property to be true up to k − 1 we use the following recursive expression for the derivative of the quotient of two functions proposed in [15], Then multiplying by the term s k the property follows by lim s→0 + s k F (k) = 0, lim s→0 + s k+1− j G (k+1− j) (s) = 0 and lim s→0 + s j−1 Q ( j−1) 0 (s) = 0 for j = 1, . . . , k.
This shows lim s→0 + s k Q (k) 0 (s) = 0 for all k > 0 and so lim s→0 + s k Q (k) (s) = 0 for all k > 0. Moreover, for some a = 0. This proves the claim.
SinceĪ − j (h) can be extended analytically at h = 0 and the map s → f ( 1 2ns ) is flat at s = 0 the functionÎ − j (s) can be written asÎ − j (s) = a j + g j (s), where a j is a constant and g j is a flat function at s = 0. In consequence, by Lemma 3, and by Lemma 4, whereã j is a constant andg j is a flat function at s = 0. The last equality together with Lemma 5 allows to writeÎ j (s) = 2(2ns) − 2n+2 j+1 for each j ∈ {−n, 0, 1, 2, . . . }. Let us fix 0 ≤ k ≤ m − 1 and consider any function in the linear span ofÎ −n ,Î 0 ,Î 1 , . . . ,Î k , where η −n , η 0 , . . . , η k ∈ R. We can consider η k = 0 since otherwise (s) belongs to the linear span with lesser index k. For the sake of compactness we rewrite From the previous discussion, we haveÎ 0 −n ∈ S −1/2n . In particular, there exists 0 > 0 such thatÎ 0 −n (s) = 0 for all s ∈]0, 0 ] and the first step of the division-derivation algorithm can be performed, producing whereh ∈]h 0 , h[. In the last step we differentiate q and use the definition of P. If we write Q(h) := I 0 (h) − P(h 0 )I −n (h), then, using (13) and the fact that I −n < 0, it suffices to prove that Q(h) < 0 for h > h 0 and h ∼ h 0 . We have Q(h 0 ) = 0 because P (h 0 ) = 0. Since I −n (h) = − 1 4n 2Ī−n (h) and I 0 (h) = − 2n+1 4n 2Ī0 (h), withĪ −n andĪ 0 defined in (1), we get Consider two vertical linesx = ± 2n P(h 0 ) 2n+1 . These two vertical lines intersect the oval γ h 0 through the interior (Fig. 3) since Q(h 0 ) = 0. See also (14). Let's denote by the region between γ h and γ h 0 , with h > h 0 and h ∼ h 0 . We can write = 1 ∪ 2 ∪ 3 ∪ 4 (see Fig. 3). Then we have In the last step we used Green's Theorem and using d H(x,ȳ) = 0 and the fact that the numerator in the above integrals vanishes along the vertical linesx = ± 2n P(h 0 ) 2n+1 . It is clear that the first integral in (15) is positive. Since the curve {x 2n −ȳ = 0} does not intersect 2 and 4 (more precisely, the curve {x 2n −ȳ = 0} intersects γ h 0 in its right-most point and its left-most point), we have that the second integral in (15) is also positive. We conclude that Q(h) < 0 for h > h 0 and h ∼ h 0 . This completes the proof of Theorem 3.