An example of rapid evolution of complex limit cycles

In the current article we study complex cycles of higher multiplicity in a specific polynomial family of holomorphic foliations in the complex plane. The family in question is a perturbation of an exact polynomial one-form giving rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the fundamental group of a leaf from the foliation. In addition to that, we introduce the notion of a multi-fold cycle and show that in our example there exists a limit cycle of any multiplicity. Furthermore, such a cycle gives rise to a one-parameter family of cycles continuously depending on the perturbation parameter. As the parameter decreases in absolute value, the cycles from the continuous family escape from a very large subdomain of the complex plane.


Introduction
Limit cycles of planar polynomial vector fields have long been a focus of extensive research. For instance, one of the major problems in this area of dynamical systems is the famous Hilbert's 16th problem [12] asking about the number and the location of the limit cycles of a polynomial vector field of degree n in the plane. Since the original Hilbert's problem continues to be very persistent, some simplifications have been considered as well. Among them is the so called infinitesimal Hilbert's 16 problem [12], [13] concerned with the number of limit cycles that can bifurcate from periodic solutions of a polynomial Hamiltonian planar system by a small polynomial perturbation. Recently, an answer to this question has been given in an article by Binyamini, Novikov and Yakovenko [2].
When studying a planar polynomial vector field, an extension to the complex domain proves to be helpful, an idea that can be attributed to Petrovskii and Landis [15], [16]. In this way a polynomial complex vector field is obtained and the holomorphic curves tangent to it form a partition of the complex plane by Riemann surfaces, called a polynomial complex foliation with singularities, or in short polynomial complex foliation [12], [13].
Following the idea of complexification, polynomial deformations of planar Hamiltonian vector fields could be extend to C 2 . More precisely one could consider the complex line field ker(dH + εω) with a one-form ω = Adx + Bdy, where A, B and H ∈ C[x, y] are polynomials with complex coefficients and ε is a small complex parameter. For the purposes of the current study, we focus our attention on a specific example. Let H be the simple polynomial Choose polynomial one-forms ω 1 and ω 2 as follows: ω 1 = (H − 1)(ydx − xdy) and ω 2 = y dH.
Consider the two parameter family of complex line fields F a,ε = ker dH + ε(ω 1 + aω 2 ) , where ε and a are the parameters. Notice that the family is of the form (1). As mentioned earlier, the holomorphic curves tangent to F a,ε form a foliation of Riemann surfaces in C 2 further denoted by F a,ε (C 2 ). For example, consider the Riemann surface S 1 = {(x, y) ∈ C 2 | x 2 + y 2 = 1}. As we are going to see in the next section 2, the surface S 1 is tangent to the complex line field F a,ε for any value of the parameters a and ε so it is a leaf of F a,ε (C 2 ). Fix the unit circle δ 0 = S 1 ∩R 2 . Notice, that in the case of real a and ε the phase curves of (2) restricted to R 2 are topologically either lines or circles, i.e. curves with either a trivial or a non-trivial (isomorphic to Z) fundamental group. For example, δ 0 is such a circular phase curve. This simple observation leads us to the definition of a marked complex cycle. Definition 1. A marked complex cycle of a complex foliation is a nontrivial element of the fundamental group of a leaf from the foliation with a marked base point.
We denote a marked complex cycle by (∆, q) where ∆ is the homotopy class of loops on the leaf, all passing through the same base point q. Each loop from ∆ will be called a representative of the cycle. In general, a real phase curve of a polynomial vector field in R 2 extends to a Riemann surface tangent to the vector field's complexification in C 2 . Thus, a closed phase curve in R 2 defines a loop on the corresponding complex leaf, giving rise to a nontrivial element from the fundamental group of that leaf [12]. In other words, a real closed phase curve is a marked complex cycle on its complexification. As an illustration, the leaf S 1 is the complexification of the real trajectory δ 0 . The surface S 1 is topologically a cylinder and δ 0 is a nontrivial loop on it. Denoting by q 0 the point (1, 0) ∈ S 1 and by ∆ 0 the homotopy class of δ 0 relative to q 0 we obtain a marked complex cycle (∆ 0 , q 0 ) of F a,ε (C 2 ).
When ε = 0 the line field (2) will be denoted by F 0 and its corresponding foliation by F 0 (C 2 ). From now on, we are going to refer to F 0 (C 2 ) as the integrable foliation and to F a,ε (C 2 ) as the perturbed foliation. Notice that F 0 (C 2 ) consists of algebraic leaves of the form S c = {(x, y) ∈ C 2 | H(x, y) = c} embedded in C 2 , where c ∈ C. All leaves with c = 0 are topological cylinders. Our basic approach will be to study the complex cycles of the more complicated F a,ε (C 2 ) by taking advantage of the simplicity of F 0 (C 2 ).
One of the very useful tools for converting some of the topological properties of the foliation into dynamical properties of a holomorphic map of complex dimension one is the so called Poincaré displacement map [12], [13]. Next, we present a construction of it in the case of example (2). Let T ′ be a complex segment (a small disc on a complex line in C 2 ) passing through q 0 and transverse to the surface S 1 . Consider an annular neighborhood A(δ 0 ) of δ 0 on the surface S 1 . Next, take a tubular neighborhood N (δ 0 ) of A(δ 0 ) in C 2 . It is diffeomorphic to a direct product A(δ 0 )×D, where D ⊂ C is the unit disc. Let ̺ be the projection of N (δ 0 ) onto A(δ 0 ) with respect to that direct product structure. Without loss of generality, we can think that T ′ = ̺ −1 (q 0 ). Let T ⊂ T ′ be a small enough open neighborhood of q 0 in T ′ . Take any point q ∈ T and consider the leaf L a,ε (q) from the foliation F a,ε (C 2 ) that passes through q. Starting from q ∈ L a,ε (q), lift the loop δ 0 to the unique path on L a,ε (q) that covers δ 0 under the projection ̺. The second end-point of this lift is again on T ′ and we denote it by P a,ε (q). As a result, we obtain a one-to-one correspondence P a,ε : T → T ′ which, by the analytic dependence of the leaves of F a,ε (C 2 ) on initial conditions [13], is a holomorphic map. In addition, notice that P a,ε (q 0 ) = q 0 for all a and ε.
Observe that by construction, if we consider another loop δ ′ 0 ⊂ S 1 passing thorough q 0 and homotopic to δ 0 on S 1 then the Poincaré map with respect to δ ′ 0 will be identical to P a,ε , possibly on a smaller cross section T . This is because the homotopy between δ 0 and δ ′ 0 can be lifted to a homotopy on any leaf L a,ε (q) passing close enough to S 1 . Therefore, the endpoints on T ′ of the lifts of δ 0 and δ ′ 0 on L a,ε (q) will be the same. Similarly, P a,ε does not depend on the choice of a product structure on N (δ 0 ). In fact, by the tubular neighborhood theorem [10] any two product structures on N (δ 0 ) are isotopic via an isotopy of N (δ 0 ) that fixes A(δ 0 ) point-wise. Therefore, as a point-set, the lift of δ 0 on any near-by leaf with respect to a projection from another product structure will be the same as the lift obtained via ̺. The difference will be only in the parametrization of the lift.
The Poincaré map P a,ε has the property that if two points from the cross-section T are in the same orbit of the map then they belong to the same leaf of the foliation. Moreover, a marked complex cycle of F a,ε (C 2 ), with a base point on T and a representative in N (δ 0 ) that covers m times the loop δ 0 gives rise to an m-periodic orbit of P a,ε . The converse is also true [15], [16]. An m-periodic orbit corresponds to a marked complex cycles of F a,ε (C 2 ) with a base point on T and a representative in N (δ 0 ) that covers δ 0 a number of m times. Definition 2. A marked cycle of (2) that corresponds to an m-periodic orbit of P a,ε is called an m-fold cycle. Whenever m > 1 and we do not want to specify the number m, we call the m-fold cycle a multi-fold cycle.
Notice that whenever an m−fold cycle of F a,ε (C 2 ) corresponds to an m-periodic orbit of P a,ε , the cycle also gives rise to m fixed points of the iterated map P m a,ε .
Definition 3. An m-fold limit cycle of F a,ε (C 2 ) is an m-fold cycle that corresponds to an isolated fixed point of P m a,ε . The case m = 1 has been extensively studied. In fact, the real cycles of a planar polynomial line field of the form (1) extend to 1−fold cycles of its complexificaion. The aforementioned infinitesimal Hilbert's 16th problem [2], [12] treats exactly the special case m = 1. The following classical result, known as Pontryagin's criterium [17] can be stated in the following form.
Pontryagin's Theorem. Let δ c be an analytic family of simple closed curves on the corresponding leaves S c of foliation (1) when ε = 0. Consider the analytic function I(c) = δc ω. If there exists a value c 0 such that I(c 0 ) = 0 and I ′ (c 0 ) = 0 then there exists a continuous family δ ε of loops, each representing a 1-fold complex limit cycle of (1). Moreover, for ε close to 0, the loops δ ε always stay close to δ c0 and δ ε → δ c0 as ε → 0.
In contrast to 1-fold cycles, little is known about multi-fold ones. That is why, the goal of this article is to shed some light on the case m > 1. During a series of informal discussions, Y. Ilyashenko proposed the following questions in the spirit of Petrovskii and Landis' works [15] and [16]: Are there polynomial families of form (1) with Poncaré maps that have isolated periodic orbits of arbitrary period m > 1?
Q2. If m > 1, what may happen to an m-fold limit cycle when ε approaches 0?
Q3. Does a multi-fold limit cycle settle on a leaf of F 0 (C 2 ) as ε → 0?
For the rest of this article we try to give some answers to Ilyashenko's questions posed above for the particular family (2). Loosely stated, the main statement of the current paper is the following: Main Result. Multi-fold limit cycles of all possible periods appear in the family (2) when the complex parameters a and ε are chosen appropriately. Moreover, each of these cycles extends to a continuous family with respect to ε. Finally, when ε tends to zero, the multi-fold limit cycles from the family escape from a very large open subdomain of C 2 that contains the surface S 1 .
The precise formulation of the claim above will be stated in the next section as theorem 1.
The proof of the main result starts with a fairly explicit construction of the Poincaré map P a,ε . After that, it is established that periodic orbits of all periods of P a,ε bifurcate from the fixed point q 0 . This immediately yields that m-fold limit cycles of all possible m ∈ N bifurcate from the cycle (∆ 0 , q 0 ) of the family (2). For the second part of the statement, we exploit more thoroughly the connection between the topological properties of F a,ε (C 2 ) and the dynamics of the Poincaré map P a,ε . First, we construct a very large smooth surface transverse to the foliation F a,ε (C 2 ). Then, we extend the Poincaré map P a,ε on this cross-section. We call it a non-local Poincaré map. Each multi-fold cycle of F a,ε (C 2 ) generated in the first part of our main result corresponds to a periodic orbit of P a,ε and together with that, determines a well-defined free homotopy class of loops in an open fibred subdomain of C 2 . The topology and the fiber structure of this subdomain comes from F 0 (C 2 ). Moreover, as the cross-section surface is transverse to F a,ε (C 2 ), we can induce a complex structure on it so that P a,ε is holomorphic. Finally, the construction of the non-local complex analytic Poincaré map allows us to establish that the behavior of a multi-fold limit cycle is quite different from the behavior of a 1-fold limit cycle as ε tends to zero. By Pontryagin's theorem, the latter always stays close to some cycle from F 0 (C 2 ) and converges to it as ε converges to zero. In contrast to the behavior of a 1−fold limit cycle, a multi-fold one tends to escape from a very large domain in C 2 when ε approaches 0. We call this phenomenon a rapid evolution of the multi-fold limit cycle.
The occurrence of quick escape of cycles is not that surprising if one recalls the dynamics of holomorphic maps with parabolic fixed points and their perturbations [3], [14]. In our case, P a,ε is a two parameter perturbation of the identity map. It is well known that as P a,ε approaches the identity its m−periodic orbits, for m > 1, leave the map's domain. Since a periodic point of P a,ε represents a multi-fold cycle of F a,ε (C 2 ), the multi-fold cycles should escape too. The challenge in our study is to establish the existence of periodic orbits of period m > 1 for the Poincaré map P a,ε of the foliation F a,ε (C 2 ) and to extend the map's domain as much as possible. For that reason we need to analyze the link between the foliation and the map as well as to explore the topology of F 0 (C 2 ) globally which makes the problem quite more interesting.
So far, the third question from the list above stays unanswered. The information we have on rapid evolution reveals an interesting insight. If the answer to that question is positive, then before a multi-fold limit cycle can reach an algebraic leaf as ε → 0, its representatives should change their topological properties somewhere along the way. This means that there is a possibility that the cycle settles on a critical leaf of F 0 (C 2 ) or goes through one or several critical leaves of F 0 (C 2 ), settling on a regular leaf. Since (1) is polynomial, it extends to a foliation on CP 2 . Thus, another possibility is an interaction with the line at infinity.
We finish this section with a discussion about another interesting and important issue, related to question 1 above. A central problem in the study of multi-fold limit cycles is their existence in families of polynomial foliations of the form (1). Ideally, one would like to establish existence of multi-fold limit cycles in general families of type (1). Heuristically, we can follow the following steps. Using Pontryagin's theorem, we could find a family of 1-fold cycles which gives a family of isolated fixed points for the corresponding Poincaré map P ε . For infinitely many values of ε in any neighborhood of 0, the derivative of P ε evaluated at the fixed point will be an m-th root of unity. Thus, for such ε a local continuous family of m-periodic isolated orbits could bifurcate from the fixed point. This will happen as long as some of the resonant terms of the map's normal form do not vanish, i.e. the map is not analytically equivalent to a rotation. Since having all zero resonant terms is an extremely special property for maps with root-of-unity multiplier, we can expect that the Poincaré transformations for most foliations of the form (1) will have a lot of isolated periodic orbits and thus, the foliations themselves will have many multi-fold limit cycles. The only obstacle in this strategy is the verification that some of the resonant term coefficients of the map's normal form are nonzero. This fact imposes a challenge since the connection between the polynomial foliation and its Poincaré transformation is implicit and indirect.

The main theorem
Before giving a precise statement of the main result of the paper, we are going to fix some notations and give some definitions.
Let us verify that both S 0 = {(x, y) ∈ C 2 | H(x, y) = 0} and S 1 = {(x, y) ∈ C 2 | H(x, y) = 1} are leaves of the foliation F a,ε for any value of the parameters a and ε. Having in mind that dH ∧ dH = 0, consider the wedge product Since H = 0 on S 0 and H = 1 on S 1 , the wedge product (3) becomes zero when restricted to either S 0 or S 1 , hence both of them are tangent to the complex linefield F a,ε , which implies that both of them are leaves of F a,ε (C 2 ) for any a and ε. Look at the polynomial H = x 2 + y 2 as a map H : C 2 → C. Consider the punctured plane of regular values B = C − {0} and its preimage E = H −1 (B). Clearly, E is just C 2 with the critical level set S 0 of H removed. Recall 1 ∈ B and hence S 1 ⊂ E, which is a topological cylinder (or a twice punctured sphere if you prefer).
Observe that every leaf of F a,ε (C 2 ) different from S 0 is entirely contained in the domain E. Denote by F a,ε the foliation F a,ε (C 2 ) with the leaf S 0 removed. Then F a,ε is a foliation without singularities in E. In particular, when ε = 0 the restricted foliation F 0 consists of all fibers of H with the exception of the critical one S 0 = H −1 (0).
Let 0 < ρ 0 < R 0 , thinking of ρ 0 as very small and R 0 as large. Define the annulus A 0 = {c ∈ C | ρ 0 < |c| < R 0 }. Consider the preimage E 0 = H −1 (A 0 ). Then, the set E 0 is the large open subdomain of C 2 from which the multi-fold cycles are going to escape, according to our main statement.
Next, we look at multi-fold cycles from a topological point of view rather than dynamically.
Definition 4. A loop contained in E is called m-fold vertical provided that it is free homotopic to δ m 0 inside the domain E. A marked complex cycle of F a,ε is called m-fold vertical provided that it has an m-fold vertical representative contained in E.
As one would expect, if a marked complex cycle of F a,ε has at least one mfold vertical representative in E, then all of its representatives are m-fold vertical. Indeed, let δ and δ ′ be two loops from the same marked cycle of F a,ε and let δ be free-homotopic in E to δ m 0 . Then both δ and δ ′ are homotopic on the same leaf of F a,ε which, in its own turn, is contained entirely in E. Therefore, both representatives are homotopic to each other inside E and since δ is free-homotopic in E to δ m 0 , so is δ ′ . More interesting is the question whether the number m is a topological invariant of an m-fold vertical cycle. Assume δ ⊂ E is a loop representing some m-fold vertical cycle of F a,ε . Also, assume that δ is free homotopic in E to another loop δ ′ 0 ⊂ S 1 . As both δ m 0 and δ ′ 0 belong to the cylinder S 1 , whose fundamental group is Z, the closed curve δ ′ 0 should be free homotopic on S 1 to δ k , for some k ∈ Z. Therefore, the representative δ is simultaneously m-fold and k-fold vertical. Later, in proposition 1, we are going to verify that δ 0 is not null-homotopic in E and k = m always.
The leaves of foliation F a,ε , given by the line field (2), depend analytically on the two parameters a and ε. In order to study the phenomenon of rapid evolution, we need to define continuous dependance of marked limit cycles on parameters.
Definition 5. A family {(∆ ε , q ε )} ε of marked limit cycles of F a,ε is called continuous with respect to ε provided that there exists a continuous family of loops {δ ε } ε such that: a) for each ε, the closed curve δ ε belongs to the class ∆ ε ; b) the base point q ε varies continuously with respect to ε.
Let D r (0) = {ε ∈ C : |ε| < r} for r > 0. We claim that as long as r > 0 is chosen small enough, rapid evolution of marked complex cycles occurs in the following form: Theorem 1. For the two-parameter family of foliations F a,ε given by (2) the following statements hold: 1. For any m ∈ N large enough there exists a complex parameter ε m near 1 m and a parameter a m such that for all ε in a neighborhood of ε m , the polynomial foliation (2) has an m-fold vertical limit cycle with a representative inside the domain E 0 ∈ C 2 . 2. Furthermore, there is a parameter disc D r (m) (0) containing ε m such that for any simple curve η ⊂ D r (m) (0) connecting ε m to 0 there exists a relatively open subset σ of η, such that the m-fold cycle from point 1 extends on σ to a continuous family {(∆ ε , q ε )} ε∈σ of marked cycles of F a,ε . 3. Finally, as ε moves along σ towards 0, it reaches a value ε * ∈ σ such that for any ε ∈ σ past ε * no m−fold vertical representative of (∆ ε , q ε ) will be contained in E 0 anymore.

The local Poincaré map
We begin our investigations with the construction of the Poincaré transformation locally and the computation of some of its terms.
Define A(δ 0 ) as a tubular neighborhood of δ 0 on the surface S 1 and N (δ 0 ) as a tubular neighborhood of A(δ 0 ) in C 2 . Let be a an infinite horizontal band in C of width r 0 and let be the disc of radius r 0 centered at 1. Consider the map Without loss of generality, we can think that f 1 (B r0 × D r0 (1)) = N (δ 0 ). In other words, f 1 can be thought of as the universal covering map of N (δ 0 ). Notice, that we also have . For 0 < r 1 < 1, define the map Composing the maps f 1 and f 2 we obtain (1−w) 2 is well defined and nonzero for w ∈ D r1 (0), we can cancel it out and the line field becomes The holomorphic function µ ε (z) = e −εz is nonzero everywhere, so Our next step is to define the Poincaré transformation for the foliation F a,ε , using the local chart f on the tubular neighborhood N (δ 0 ) of the loop δ 0 . Denote the desired map by P a,ε : D r1 (0) −→ C.
We are going to explain how it is constructed. Define the pathδ  can be lifted to a path δ (m) a,ε (u) on the leaf of F a,ε passing through the point (0, u) ∈ {0} × D r1 (0), so that if pr 1 : (z, w) → z then pr 1 (δ (m) a,ε (u)) =δ (m) 0 . Again, as before, whenever m = 1 we omit the superscript (m) and we write δ a,ε (u) = δ (1) a,ε (u). The lift δ a,ε (u) has two endpoints. The first one is (0, u) and the second one we denote by (2π, P a,ε (u)). When a=0, the map P 0,ε (u) comes from the foliation F 0,ε which in our tubular neighborhood is given by ker(d(we −εz )). Then, δ 0,ε (u) = {(t, ue εt ) : t ∈ [0, 2π]} and so P 0,ε (u) = e 2πε u. Since δ a,ε (0) =δ 0 , the equality P a,ε (0) = 0 holds for all (a, ε). As a result, the Poincaré transformation can be written down as P a,ε (u) = e 2πε u + aI(u, ε)u + a 2 G(u, a, ε)u and its k-th iteration can be expressed as In this case, denote the lift ofδ In order to study the periodic orbits of P a,ε (u), we are going to look at the difference P m The one-form dJ (i/m) is exact and yields Notice that when a = 0 the paths take the special explicit form with endpoints (0, u) and (2πm, u). Divide equation (4) by a. When a → 0 the limit of the left hand side of (4) is 0 (u) as a → 0. As a result, we can conclude that Since both sides of the equation are divisible by u, To compute the integral in (6), notice that 1/ √ 1 − w is well defined and holomorphic in the disc D r1 (0) ∋ 1 so it expands as uniformly convergent series The value of the integral depends on the coefficients of (7) which, in their own turn, depend on the integral When k = m the primitive of the function e i k+m m t − e i k−m m t under the integral is again 2πm-periodic, leading to the conclusion that the integral is zero. When The computations above lead to Finally, setting c m = πb m = 0, we can conclude that for ε = i m , the Poincaré map takes the form

Existence of multi-fold cycles
In this section we show how multi-fold limit cycles bifurcate from the cycle (∆ 0 , q 0 ), located on the leaf S 1 = H −1 (1). Remember that δ 0 ⊂ S 1 is the unit circle in the real plane R 2 ⊂ C 2 centered at the origin. The set ∆ 0 is the element of the fundamental group of S 1 determined by the loop δ 0 with a base point q 0 = (1, 0) ∈ S 1 .
From the discussion in the introduction, the existence of a multi-fold limit cycle of F a,ε follows from the existence of an isolated m-periodic orbit of the Poincaré transformation P a,ε . We can see from the construction of the map that a representative of the cycle will be contained in the tubular neighborhood N (δ 0 ) ⊂ E and therefore free homotopic to δ m 0 in it. This fact immediately implies that the limit cycle will be m-fold vertical. Therefore, all we need to show is that P a,ε has an isolated m-periodic orbit.
We fix the radii r 1 > 0, r 2 > 0 andr 3 > 0 so that for any (a, ε) ∈ D r2 (0)× Dr 3 (0) the map P a,ε : D r1 (0) −→ C is well defined. Let m > 0 be such that i/m ∈ Dr 3 (0). Lemma 4.1. There exists ε m near i m and a parameter a m such that for all ε in a neighborhood of ε m , the map P a,ε has an isolated periodic orbit of period m.
Proof. The verification of the claim depends on four facts. Putting them together will help us determine the values of the parameters a and ε. In order to find a periodic orbit for the map P a,ε (u), we are going to look at the equation Whenever a = 0 we can rewrite (9) in the form Furthermore, having in mind that u = 0 is always a solution of (9), we can divide by u and obtain
Hence, by the implicit function theorem, it follows that for a possibly decreased r 2 > 0 there exists a holomorphic function χ : i m such that χ(0) = i m and g 0 (a, χ(a)) = 0 for all a ∈ D r2 (0). From here, we can see that the zero locus of g 0 inside the product domain D r2 (0)×D r3 Apply the results from fact 4 to obtain Since (a m , ε m ) ∈ W , it follows from fact 3 that whenever |u| = r 1 the following inequality holds Therefore, by Rouche's theorem, the equation has as many solutions as By fact 1, equation (14) has m roots u 1 (a m ), ..., u m (a m ) contained in D r1 (0). For that reason, equation (13) has m solutions contained in D r1 (0). Let us denote them by u 1 (a m , ε m ), ...,u m (a m , ε m ). As it was established earlier, none of them is zero.
Having in mind that u j ∈ D r0 (0) and each of them is nonzero for j = 1, .., m, we estimate Hence, the orbit u 1 ,...,u m consists of different points and therefore is periodic of period m in D r1 (0).

Topology of the fiber bundle
Some of the constructions we would need in order to complete the proof of theorem 1 depend on the topology of the domain E ⊂ C 2 . That is why our goal is to understand it well.
We begin with the introduction of some useful notations. By [δ] M we denote the set of all loops homotopic to a loop δ on a manifold M , all passing through a base point x 0 ∈ M . As usual, the homotopy class [δ] M is an element of the fundamental group π 1 (M, x 0 ) of M . Our first step is to compute the fundamental group of the domain E.
To make our arguments more standard, we introduce new coordinates in C 2 . Let . This is a unitary linear transformation of C 2 and it preserves the standard Hermitian dot product z 1 z 2 + w 1 w 2 . Therefore, the change of variables is an isometry and preserves all the metric properties of C 2 . With respect to these coordinates Proof. Let S 3 = {(z, w) ∈ C 2 | |z| 2 + |w| 2 = 1} be the unit three-sphere in C 2 and K = S 3 ∩ {(z, w) ∈ C 2 | z = w = 0}. Then T is embedded in S 3 . The intersection of a complex line with S 3 is always a great circle, and more precisely, a fiber of the Hopf bundle [18]. Therefore, K is the classical Hopf link in the three-sphere, consisting of two great circles linked once. Let M K = S 3 \ K be its complement. As C 2 \{(0, 0)} deformation retracts onto S 3 , we have that E deformation retracts onto M K . In its own turn, M K deformation retracts onto the torus T, so we conclude that E deformation retracts onto T.
For the second part of the lemma, notice that From here, immediately follows that δ 0 lies on the surface of the torus. If we take an s 1 -circle and an s 2 -circle on T as a homology basis for H 1 (T, Z) then δ 0 has homology coordinates (1, −1) and therefore is not null-homologous. Hence, it is not null-homotopic either.
Proposition 1. Let δ ⊂ E be an m-fold vertical representative of an m-fold vertical cycle of F a,ε . Assume that δ is also free homotopic in E to another loop δ ′ 0 ⊂ S 1 . Then δ ′ 0 is free homotopic on S 1 to δ m 0 . Proof. Both δ m 0 and δ ′ 0 belong to the cylinder S 1 , whose fundamental group is Z. That is why, the closed curve δ ′ 0 should be free homotopic on S 1 to δ k 0 , for some k ∈ Z. All we need to show is that m = k. Indeed, consider the corresponding . Both loops δ k 0 and δ m 0 are free-homotopic in E to the same loop δ. Hence, they are free-homotopic to each other in E. Therefore, the elements [δ 0 ] m E and [δ 0 ] k E are conjugate in π 1 (E, q 0 ). Since, according to corollary 5.1, the fundamental group = 1 (here we use multiplicative notation). By the same corollary, we can see that [δ 0 ] E = 1 and π 1 (E, q 0 ) = Z ⊕ Z is torsion-free. Therefore m = k The map H : E → B defined by the polynomial H = x 2 + y 2 is a smooth, locally trivial fiber bundle. For any regular value c ∈ B, the fibers S c = {p ∈ C 2 | H(p) = c} are topological cylinders, diffeomorphic to each other [1], [13]. We use S 1 as a model fiber. The map ν : C → S 1 given by ν(ζ) = (cos ζ, sin ζ), for ζ ∈ C, is the universal covering map of S 1 .
Here are some facts about the topology of the fiber bundle H : E → B. The unit circle γ 0 = {c ∈ C | |c| = 1} is a simple closed loop in B starting from 1, going around 0 counterclockwise and coming back to 1. The homotopy class of γ 0 with a base point 1 is the generator of the fundamental group π 1 (B, 1) ∼ = Z. For c ∈ γ 0 consider the fiber S c . Then, if the parameter c starts from 1 and moves along the loop γ 0 until it comes back to 1 then the corresponding fiber S c will also make one turn around the critical value 0 starting and ending up at S 1 . This procedure gives rise to an isotopy class of diffeomorphisms with a representativeD 0 : S 1 → S 1 which is a Dehn twist. The mapD 0 can be chosen so that it twists the cylinder C 0 and is the identity on S 1 \ C 0 [1].
Let π(z) = e 2πiz . Then π : C → B is the universal covering map of the punctured plane B. Denote its group of deck transformations by Γ = {γ m 0 | m ∈ Z and γ 0 (z) = z + 1}. As usual, it is isomorphic to the fundamental group of B. The vertical strip B = {z ∈ C | 0 ≤ Re(z) ≤ 1} is a closed fundamental domain. Also, whenever we have a cartesian product M 1 × M 2 of two sets, by pr Mj we are going to denote the projection pr Mj : The next lemma shows that we can "unfold" the bundle H : E → B into a trivial covering bundle pr C : C × S 1 → C. Moreover, we can do so by making sure the deck groupΓ acts in a very special manner. It not only takes vertical fibers {z} × S 1 to vertical fibers, but what really important is that it also takes horizontal fibers C × {p} to horizontal fibers.
There is a smooth covering map Π : C × S 1 → E with the following properties: In other words, apart from being a covering map, Π is also a bundle map.

The deck group of
where γ 0 (z) = z + 1 is the earlier described generator of Γ and the map D 0 =D −1 0 is the Dehn twist of the cylinder c 0 ⊂ S 1 and identity everywhere else on the surface S 1 . Thus, the factor bundle (C×S 1 )/Γ is diffeomorphically isomorphic to the bundle E.
Proof. Consider the pullback of the bundle H : E → B over the plane C under the covering map π. To carry out this construction, first define the total space π * E = {(z, q) ∈ C × E | π(z) = H(q)}. Then, the restricted projection κ = (pr C ) | π * E : π * E → C gives us the desired pullback bundle. Also, there is a map Π ′ = (pr E ) | π * E : π * E → E that satisfies the condition H •Π ′ = κ • π and so it is a bundle map over the map π. Together with that,Π ′ : π * E → E is a covering map. Because C is contractible, the pullback bundle κ : π * E → C is trivializible, i.e. there is a smooth bundle isomorphism ς : C × S 1 → π * E so that κ • ς = pr C • id C where id C is the identity map on C. Then, the compositionΠ =Π ′ • ς : C × S 1 → E satisfies the condition H •Π = π • pr C and therefore is a bundle map and a covering map at the same time. Without loss of generality we can think thatΠ(0, p) = p, that is we identify the fiber {0} × S 1 with the surface S 1 , where π(0) = 1.
We are going to look at the deck groupΓ of the covering mapΠ. For anyγ ∈Γ we have the relation pr C •γ = γ m 0 • pr C for some m ∈ Z. That is why, just like Γ, the groupΓ is free Abelian with one generatorγ 0 (z, p) = (γ 0 (z), ψ(z, p)), where (z, p) ∈ C × S 1 . The map ψ : C × S 1 → S 1 is smooth and if we use the notation ψ z (p) = ψ(z, p), then for any fixed z ∈ C the resulting map ψ z : S 1 → S 1 is a diffeomorphism on the standard fiber S 1 . If we factor C × S 1 by the action of the deck groupΓ we obtain the manifold (C × S 1 )/Γ which is isomorphic to E as a fiber bundle over B.
The strip N ǫ deformation retracts onto the point 0 ∈ N ǫ . Therefore, there exists a smooth deformation retraction ϑ : N ǫ × [0, 1 2 ] → N ǫ . Then, for t = 0 the map ϑ t : N ǫ → N ǫ is the identity on N ǫ , for t = 1 it is the constant map ϑ 1 ≡ 0 and for all t ∈ [0, 1 2 ] we have that ϑ t (0) = 0. With the help of ϑ t , we define the isotopy φ : As a result, when t = 0 we have the earlier defined map φ 0 . Moreover, when t = 0.5 we obtain the map φ 0.5 (z, p) = (γ 0 (z), ψ(0, p)) for (z, p) ∈ N ǫ × S 1 . Notice that the second component of φ 0.5 does not depend on the variable z but only on p. Whenever z = 0, the map ψ 0 (p) = ψ(0, p) is isotopic to the Dehn twist D 0 =D −1 0 . This follows from Picard-Lefchetz's theory as discussed previously in the current section and in [1]. Thus, we can extend smoothly the isotopy φ t for t ∈ [0, 1] so that for t = 0 the gluing map is φ 0 , for t = 0.5 the map becomes φ 0.5 from above and finally when t = 1 we obtain φ 1 (z, p) = (γ 0 (z), D 0 (p)) for all (z, p) ∈ N ǫ × S 1 .
Notice that φ t respects the vertical fibers {z} × S 1 , that is the isotopy takes place only with respect to the second coordinate, along the surface S 1 , while the first coordinate is kept the same. Therefore, (B × S 1 )/φ 0 and (B × S 1 )/φ 1 are smoothly isomorphic as fiber bundles over B. As we already saw, (B × S 1 )/φ 0 and E are isomorphic as well. Hence, (B × S 1 )/φ 1 and E are isomorphic as bundles over B. Since by construction (B × S 1 )/φ 1 and (C × S 1 )/Γ are also isomorphic as bundles over B, we can conclude that there exists a smooth bundle isomorphism Φ : (C × S 1 )/Γ → E. If υ : C × S 1 → (C × S 1 )/Γ is the quotient map, then it is a bundle map over the covering map π. When we compose it with Φ we obtain the desired bundle covering map Π = Φ • υ : C × S 1 → E satisfying the condition H • Π = π • pr C and havingΓ as its group of deck transformations. This completes the proof of the lemma.
The results from lemma 5.2 are a main tool in the proof of theorem 1. As it was mentioned already, a deck transformationγ 0 m (z, p) = (γ m 0 (z), D m 0 (p)) fromΓ maps not only vertical fibers {z} × S to vertical fibers {γ m 0 (z)} × S 1 but also horizontal fibers C × {p} to horizontal fibers C × {D m 0 (p)}. In particular, since D m 0 acts on S 1 − c 0 as the identity map, whenever p ∈ S 1 \ C 0 , the horizontal plane C × {p} is invariant under the action ofΓ. These facts lead us to the following conclusion.
Corollary 5.2. For p ∈ S 1 \ C 0 , the projection Π(C × {p}) = B p is a smoothly embedded surface in E, diffeomorphic to the punctured plane B. Moreover, B p intersects each leaf from the integrable foliation F 0 transversely at a single point.
In particular, this corollary applies to the point q 0 = (1, 0). Thus, we have obtained a global cross-section B q0 .

The non-local Poincaré map
For the rest of the article, we are going fix some m ∈ N and take the parameter a = a m as in lemma 4.1. Since a = a m is fixed, from now on we are going to drop a from all notations that contain it either as a subscript or a superscript. For example, when a = a m we will write F ε instead of F a,ε , the folaition leaves will be denoted by L ε instead of L a,ε and the notation P ε will replace the previously accepted notation P a,ε for the Poincaré map.
Apart from the annulus A 0 from section 2, define two more domains in B. Remember that for a pair of numbers 0 < ρ < R we defined the annulus A(ρ, R) = {c ∈ C | ρ < |c| < R}. Fix the small positive numbers ρ 0 > ρ ′ 0 > ρ ′ 1 > 0 and the large ones 0

As a result we obtain three nested open sets
. Next, we lift on C all annuli form the previous paragraph to obtain the three horizontal strips π −1 (A 0 ), π −1 (A ′ 0 ) and π −1 (A ′ 1 ). We fix the following cross-sections in C × S 1 : Projecting by Π 0 , let From now on, we also use the shorter notationsẑ = (z, q 0 ) ∈ C × {q 0 }. Consider the pulled-back foliationF ε = Π * F ε on the covering space C × S 1 . It is invariant with respect to the action ofΓ. In other words, ifγ ∈Γ andL ε (z, p) is a leaf ofF ε passing through the point (z, p) ∈ C × S 1 , thenγ(L ε (z, p)) =L ε (γ(z, p)).
Proof. The statement follows directly from corollary 6.1.
On a side note, but still worth mentioning is a fact that follows from the constructions in the proof of lemma 6.1. It is not difficult to see that the Poincaré map is not sensitive to (local) homotopies of the base loop δ 0 . In other words, if δ 0 is homotopic on S 1 to another loop δ ′ 0 passing through q 0 , then the two maps obtained by the lifting of δ 0 and δ ′ 0 onto the leaves of the foliationF ε under the projection pr S1 will be equal, as long as δ ′ 0 is close enough to δ 0 on S 1 or the radius r is kept small enough. Thus, if we slightly wiggle δ 0 on S 1 but keep the base point q 0 fixed, the resulting Poincaré map will stay the same. Consequently, the same is true for P ε .

Complex structure on the cross-section
Apart from the smooth structure of a fiber bundle, the space E, being a subset of C 2 , has a complex structure with respect to which the foliation F ε is holomorphic and depends analytically on the parameter ε. This fact provides the foliation with very specific properties. On the other hand, the Poincaré map P ε : associated with F ε captures some topological properties of the foliation. Since some of those properties are strongly related to the holomorphic nature of the foliation, we would like our Poincaré map to reflect the complex analyticity of F ε . So far P ε is defined as a smooth map on a subdomain of the smooth surface A ′ 1 (q 0 ) and therefore our next step is to induce a complex structure on A ′ 1 (q 0 ) in which the Poincaré transformation is holomorphic.
Since the closure of . Take a point q ′ ∈ A ′ 1 (q 0 ) and a complex cross-section T q ′ through q ′ , transverse to F ε . More precisely, T q ′ is a complex segment, i.e. it lies on a complex line through q ′ and is a real two dimensional disc.
The fact that the foliation F ε is holomorphic and A ′ 1 (q 0 ) is smoothly embedded surface transverse to F a,ε provides us with convenient holomorphic flow-box charts of C 2 . A chart of this kind consists of an open neighborhood F B(q ′ ) ⊂ E of q ′ and a biholomorphic map β q ′ ,ε : D × D −→ F B(q ′ ) with the following properties: ε (ζ, 0)) through the point β q ′ ,ε (ζ, 0) for any ζ ∈ D; 4. The portion of A ′ 1 (q 0 ) passing through F B(q ′ ) looks like the graph of a smooth map α q ′ ,ε : D → D in the chart D × D. In other words

Consider the family of pairs
) is a holomorphic atlas for the surface A ′ 1 (q 0 ) with charts depending complex-analytically on ε. Proof. The proof is a direct verification that the transition functions φ q1,ε • φ −1 q2,ε of two intersecting charts are holomorphic in both the coordinate variable and the parameter ε. It comes from the fact that the charts are basically projections of the open patches U q ′ ⊂ A ′ 1 (q 0 ) onto the holomorphic cross-sections T q ′ along the leaves of the foliation F ε . Therefor the transition transformations are going to be maps from one holomorphic cross-section to another following locally the leaves of F ε . As the leaves depend holomorphically on the initial condition and the parameter ε, we obtain the desired result. For more details, one can look at [6] or [7].
The choice of complex structure on the surface A ′ 1 (q 0 ) is justified by the next lemma. As it turns out, the map P ε is holomorphic in the complex structure A ε (A ′ 1 (q 0 )).
Lemma 7.2. The Poincaré map P ε : A ′ 0 (q 0 ) → A ′ 1 (q 0 ) from corollary 6.2 associated to the foliation F ε is holomorphic in the complex structure defined by the atlas A ε (A ′ 1 (q 0 )) and depends complex-analytically on the parameter ε. Proof. The idea is based on the heuristic arguments of the previous lemma 7.1.
Restricted to an open patch U q1 ⊂ A ′ 0 (q 0 ), the map P ε sends U q1 inside another open patch U q2 . If we look at P ε in the two corresponding coordinate charts, we obtain a map form the holomorphic cross-section T q1 to the holomorphic crosssection T q2 by following the "lifts" of the loop δ 0 on the leaves of the foliation F ε . The leaves depend holomorphically on the initial condition and the parameter ε, as well as both T q1 and T q2 are complex segments, so we conclude that the map is as desired. More details can be found in [6] or [7].
is holomorphic with respect to the complex atlas A ε (A ′ 1 (q 0 )). The new atlas makes the lifted Poincaré mapP ε holomorphic, depending complex-analytically on ε.
Proof. Since as a smooth covering map Π 0 is a local diffeomorphism, simply pull back the complex structure given by A ε (A ′ 1 (q 0 )) to the surfaceÂ ′ 1 .
For each j = 1, ..., m denote byδ j the closed curveδ ε but assuming that it starts form the pointẑ j . As point-sets all loopsδ j are the sameδ ε . The only difference between them is that the parametrization for eachδ j starts from a different point from the periodic orbit ofP ε .
Let P r S1 : C × S 1 → {0} × S 1 be the map P r S1 (z, p) = (0, p). Having in mind that C is contractible, P r S1 is a deformation retraction, so clearly eachδ j is free homotopic in C × S 1 to {0} × δ m 0 . As Π(C × S 1 ) = E, the map Π sends this free homotopy to a free homotopy in E between δ j = Π(δ j ) and δ m 0 = Π(0, δ m 0 ). Hence all loops δ j are m−fold vertical.
The third point of the current theorem follows directly from definition 4 and the discussion in the paragraph right after it. Lemma 8.2. Let ε ′ belong to the parameter disc D r (0), where the radius r > 0 is chosen as in lemma 6.1.
The third point of the current statement follows directly from lemma 8.1 part two. For each ε ∈ η ′ and j = 1, ..., m the constructed representative δ j (ε) ⊂ E ′ 1 depends in fact continuously on the parameter ε ∈ η ′ because the leaves of all foliations we work with depend continuously on the initial points and the parameter ε. Thus the loops δ j (ε) and the base points q j (ε), continuously depending on ε, give rise to continuous families {(∆ j (ε), q j (ε))} ε∈η ′ 9. Proof of theorem 1 From now on we assume that a = a m as provided by lemma 4.1. Let η be an embedded in D r (0) curve, connecting ε 0 to 0. For convenience, define a natural linear order on it so that 0 ≺ ε 0 . We begin this section with a summary of the proof.
First, with the help of lemma 4.1, we show that the Poincaré map P a,ε of the foliation F a,ε has an isolated periodic orbit {q j } m j=1 for the fixed parameter a = a m and ε 0 very close to ε m ∈ D r (0). Then, {q j } m j=1 can be extended to a continuous family of periodic orbits {q j (ε)} m j=1 on A ′ 0 (q 0 ) for values of the parameter ε defined on an relatively open subset η max of the path η. We also find out that there exists ε * ∈ η max such that if ε ∈ η max and ε ≺ ε * then some q j0 (ε) ∈ A ′ 0 (q 0 ) \ A 0 (q 0 ). We denote q j0 (ε) = q(ε). By point 3 of lemma 8.2, there exists a continuous family of marked cycles {(∆(ε), q(ε))} ε∈ηmax defined on η max . For the value ε = ε 0 the cycle (∆(ε 0 ), q(ε 0 )) is limit m−fold vertical and has a representative δ(ε 0 ) contained in the domain E 0 = H −1 (A 0 ). As ε moves on η max towards 0, it passes through the point ε * and as a result of this the point q(ε) leaves A 0 (q 0 ) and therefore it leaves E 0 as well. Consequently, for any ε ∈ η max with ε ≺ ε * no representative of (∆(ε), q(ε)) is contained in E 0 because all of them pass through the same base point q(ε) and q(ε) is not in E 0 anymore. This last fact concludes the proof of theorem 1 with σ = η max .
Next, we continue with the detailed proof of the main result of this article. The Poincaré map constructed in section 3 is in fact the local Poincaré map from the introduction of the article, defined on the complex cross-section T ′ q0 = {(x, 0) ∈ C 2 | |x − 1| < r ′ 0 } where r ′ 0 is very small. We denote this local map by P loc ε : T q0 → T ′ q0 . Notice that P loc ε is constructed using the tubular neighborhood N (δ 0 ) together with a projection, we call ̺ : N (δ 0 ) → A(δ 0 ), coming from the direct product structure on N (δ 0 ) selected in section 3. Since the covering map Π : C × S 1 → E is a local diffeomorphism, the tubular neighborhood N (δ 0 ) inherits via Π the product structure of C×S 1 together with a projection ̺ ′ coming from the natural projection pr S1 which maps C × S 1 onto S 1 . As already mentioned in the introduction, there exists an isotopy on N (δ 0 ) that keeps A(δ 0 ) fixed point-wise and sends one product structure to the other [10]. Therefore, the lifts of δ 0 via ̺ and ̺ ′ on any nearby leaf of F ε will coincide as point-sets. In conclusion, the map P loc ε is in fact a representation of the non-local Poincaré map P ε in a complex chart from the atlas A ε (A ′ 1 (q 0 )) defined in section 7. Together with this, by corollary 7.1, the local map P loc ε on the cross-section T ′ q0 is a representation of the lifted Poincaré transformation P ε in a complex chart from the atlas A ε (Â ′ 1 ). Combining this fact with lemma 4.1, we conclude that for a choice of ε 0 near ε m ∈ D r (0) the transformationP ε has an isolated m− periodic orbit {ẑ j } m j=1 . By lemma 8.1, for j = 1, ..., m there are m−fold vertical limit cycles (∆ j , q j ) each having a representative δ j contained in E ′ 1 . Comparing the construction of P loc ε with that ofP ε and P ε as well as having in mind the fact that the lifts are independent on the choice of a product structure on N (δ 0 ), one concludes that δ j is contained in N (δ 0 ) which is a thin neighborhood of δ 0 and in its own turn is a subset of E 0 . Thus, the representative δ j of the limit cycle (∆ j , q j ) is an m−fold vertical loop contained in the domain E 0 for j = 1, .., m.
We are interested in the zero locus ofF in the domain D ′ ×D r * (ε * * ). If we assume for a moment thatF (ζ, ε) ≡ 0 on D ′ then we would have the identity P (m) ε (ζ) ≡ ζ on D ′ and therefore P m ε (q) ≡ q on the open subset U ′ q * 1 ⊂ A ′ 0 (q 0 ). Because of the analyticity of P m ε (q) with respect to both q and ε, the identity P m ε (q) ≡ q will hold on all of A ′ 0 (q 0 ) and for all ε ∈ D r (0). In particular, it will be true for ε = ε 0 . But for that value the map P m ε has an isolated fixed point q 1 (ε 0 ) ∈ A 0 (q 0 ) ⊂ A ′ 0 (q 0 ) which leads to a contradiction. ThereforeF is not identically zero.

Concluding remarks
The choice of the family (2) comes into play mostly in the proof of the existence of multi-fold vertical cycles for line fields of type (1). It is specifically designed to facilitate the computation in the first part of the article, where we study the bifurcation of periodic orbits from a resonant parabolic fixed point of the Poincaré map. Establishing the link between a foliation of type (1) and the resonant terms in the normal form of its corresponding Poincaré map seems hard. We can see that in our simple example (2) we have quite involved computations in order to show the non-triviality of the resonant normal form of P a,ε . For the second part of the article, in which we construct a non-local Poincaré transformation and we the study the topological properties and rapid evolution of the multi-fold limit cycles of (2), we do not seem to need that much the explicit form of the foliation of type (1). In fact the central role is played by the polynomial H and its geometric-topological properties. Since H is quite simple, so is its geometry and consequently the topology of the fiber bundle H : E → B. With some additional modifications one could choose a more complicated polynomial H and carry out similar constructions and prove rapid evolution for more general families of type (1), provided that the existence of a periodic orbit for the Poincaré map is assumed. In [6], or alternatively in [7], the reader could see (1) studied in a more general form. In these works the approach and the general philosophy of the current article are preserved, but the interplay between the topology of the foliation and the dynamics of the Poincaré map are quite more interesting. The map branches and its branching is inherently related to the homotopy class of the loop δ 0 on the surface of a fixed fiber S c0 of H.