PRIMARY BIRTH OF CANARD CYCLES IN SLOW-FAST CODIMENSION 3 ELLIPTIC BIFURCATIONS

. In this paper we continue the study of “large” small-amplitude limit cycles in slow-fast codimension 3 elliptic bifurcations which is initiated in [8]. Our treatment is based on blow-up and good normal forms.


1.
Introduction. This paper deals with the canard phenomenon and corresponding limit cycles of canard type, as they appear in slow-fast families of vector fields in the (x, y)-plane. The prototypical example where the canard phenomenon appears is the Van der Pol system, where the unique singular point of a specific slow-fast structured vector field undergoes a Hopf bifurcation upon variation of a control parameter. Letting denote the singular parameter separating the two time scales in the Van der Pol system, a weighted rescaling of (x, y, ) exposes a rescaled system of differential equations in rescaled variables (X, Y ), where the slow-fast structure has disappeared and where a traditional Hopf bifurcation is observed. The periodic orbit(s) seen in the (X, Y )-space are considered small-amplitude limit cycles in the traditional (x, y)-plane, since they typically are contained in an O( )-neighbourhood of the Hopf point. Traditional techniques from dynamical systems and bifurcation theory can be used to deal with those small-amplitude limit cycles. Besides these cycles, the (x, y)-plane may also contain so-called slow-fast cycles of canard type, with properties typically associated to slow-fast systems. It is precisely at the interface between small-amplitude cycles and canard cycles that a delicate analysis is needed. Those families of cycles are unbounded in (X, Y )-space and yet close to the origin in (x, y)-coordinates. Pioneering work has been done in [12] (in a codimension 1 scenario) and in [2] for systems that are locally similar to the Van der Pol system in higher codimension.
In [2], the way the interface between small-amplitude cycles and canard cycles is examined is by blowing up the origin. The traditional rescaling from above is seen in one of the charts of the blow-up construction. In the blow-up construction, one can also examine other charts, the so-called phase directional rescaling charts. It is precisely those charts that become important in a study of the birth of canard cycles. We remind the reader that in most papers on slow-fast systems using blowup, the phase directional rescaling charts are only studied minimally, just enabling the contributing authors to trace orbits passing through these charts and to focus attention to the traditional rescaling charts. Indeed, only a few properties are needed of these phase directional rescaling charts to examine large amplitude canard cycles, the so-called detectable cycles, see also [10].
Here, we also study the birth of canard cycles, i.e. cycles at the interface between small-amplitude cycles and canard cycles, but in a more degenerate context: while the underlying bifurcation mechanism in the Van der Pol equation is a codimension 1 Hopf bifurcation, we now consider a codimension 3 singularity in a slow-fast context. The essential extra difficulty is that besides the singular parameter , we now have 3 extra parameters unfolding the codimension 3 singularity, and even without a slowfast structure, a blow-up is needed to give a complete study. As can be expected, we present a study where two blow-up constructions are combined: one to unfold the codimension 3 singularity (a "primary blow-up"), and one to dissolve the slowfast structure (a "secondary blow-up"). Cycles that are bounded in coordinates after the secondary blow-up could be called small-amplitude cycles. As they grow in size under influence of some perturbation parameter, they meet the boundary of the secondary blow-up and give birth to intermediary-sized cycles: those cycles are of size O(1) in coordinates after the primary blow-up. This birth of canard cycles is largely similar to the one treated in [2]. In primary blow-up coordinates, the intermediary-sized cycles are already of canard type, but may continue growing until they meet the boundary of the primary blow-up and give birth to canard cycles of size O(1) in original coordinates. This birth process actually differs more than a bit from the situation discussed in [2] and [12], and the study of limit cycles in this situation is the topic of this paper.
One of the main properties of the primary blow-up is that it is not that is included in the blow-up construction, since the primary blow-up does not involve dissolving the slow-fast structure; instead, the primary blow-up is involved with desingularizing a codimension 3 singularity. As a consequence, the family of vector fields being blown up has a slow-fast structure both before and after blow-up.
Instead of presenting a general technique for treating a birth of canard situation in case of a blow-up preserving the slow-fast structure, we choose to demonstrate the (quite intricate) techniques in a situation for which a lot of results already have been obtained: • We study a slow-fast codimension 3 singularity that is the slow-fast variant of a well-studied codimension 3 singularity (see [5]). • The desingularization of the singularity using a primary and secondary blowup has been worked out before (see [8]). Both the detectable canard cycles ( [3]) and the small-amplitude limit cycles ( [8]) have been characterized before. On top of that, the birth problem associated to the secondary blow-up has been dealt with. (See [8].) We claim that the results are general enough to be useful in other situations, and furthermore the results contribute to a complete understanding of the slow-fast codimension 3 singularities and nearby limit cycles.
In Section 2, we present a more detailed setting and introduce the setting using a precise system of equations in mind. In Section 3, we carry out the desingularization via blowing-ups. Section 4 contains precise statements of the results we aim to prove. Section 5 finally contains detailed proofs of the results.
The technique of blow-up is crucial in this paper, not only in proving the results, but already in stating the results. We therefore refer the interested reader to [13] for more informations about desingularizations of nilpotent singularities in families of planar vector fields, and continue under the assumption that reader has sufficient background.
2. Slow-fast codimension 3 elliptic bifurcations. We deal with the slow-fast family of planar systems X¯ ,b,λ : where G andH are smooth,¯ ≥ 0 is the singular parameter that is kept small, b = (b 0 , b 1 , b 2 ) are regular perturbation parameters close to 0 and λ ∈ Λ, with Λ a compact subset of some euclidean space. The family X¯ ,b,λ represents slow-fast codimension 3 elliptic bifurcations, in analogy with the terminology introduced in [5] and [8].
Let us recall that [5] is devoted to the study of generic bifurcations of threeparameter families of planar vector fields around singular points whose linear parts are nilpotent. The authors dealt with three categories: the saddle case, the elliptic case and the focus case. Dealing with slow-fast systems (¯ ∼ 0,¯ > 0), in [8] we distinguish only between the elliptic case (1) and the saddle case, obtained by putting the +-sign in front of x 3 in (1). The focus case is possible in the family (1) if the parameter¯ > 0 is sufficiently large, but then (1) is not of slow-fast type. For more details we refer to [8].
The family X¯ ,b,λ contains for¯ = 0 a curve of singular points given by {y = 0}. All points on the curve except for the origin are normally hyperbolic. Of course, the dynamics for¯ = 0 can be studied by canceling the common factor y and seeing that orbits of X 0,b,λ take the form y = y(x), with dy dx = −x. In other words, orbits lie on parabolas that intersect the curve of singular points transversally, except at the origin, where the parabola has a second-order contact with the curve (see Figure  1). The origin (x, y) = (0, 0) is a so-called contact point, and we observe that it is of nilpotent type. The¯ -perturbation may cause the curve y = 0 to perturb into some invariant curve with a dynamics on it, but with a speed that is O(¯ ). This way, so-called detectable canard limit cycles may appear: a fast movement along the top of a parabola above {y = 0} is followed by a slow movement connecting the two ends of the parabola along y = 0.
As mentioned above the papers [8] and [3] are devoted to the study of the limit cycles that may appear in the family X¯ ,b,λ perturbing from X 0,(0,0,0),λ . The paper [3] deals with systems (1) but emphasizes passage near the generic turning point (x, y) = (0, 0) in order to study the detectable canard limit cycles, whereas in the paper [8] the focus is on small amplitude limit cycles near (x, y) = (0, 0).
Let us first explain what is our goal of the present paper. We reparametrize the b-parameters, by introducing weighted spherical parameters as used in [8]: We obtain an (¯ , B, r, λ)-family of vector fields in R 2 : For r = 0, system (2) has no limit cycles near (x, y) = (0, 0). For r = 0, r ∼ 0, system (2) has been studied in an (¯ , B, λ)-uniform neighborhood of the origin in (x, y)-space (see [2], at least in the Liénard setting, i.e. G ≡ 0, and [1]). This local study is valid in a small neighborhood of (x, y) = (0, 0) and the domain on which the arguments of [2] and [1] can be applied shrinks to (x, y) = (0, 0) when r → 0. The idea in [8] was to start the study of limit cycles of (2) in a neighborhood of the origin (x, y) = (0, 0) that does not shrink to the origin when¯ → 0 and r → 0. This means that the transition from small amplitude limit cycles to small (but detectable) canard limit cycles has to be considered. The goal of the present paper is to study this "birth of canards" in the (x, y) − plane, i.e. so-called "large" small amplitude limit cycles.
As mentioned above, in [8] slow-fast codimension 3 saddle bifurcations have been studied. In slow-fast codimension 3 saddle bifurcations, one finds no birth of canards because detectable canard limit cycles are not present (see [8] and [3]). Now we want to provide few details on what is shown in [8] and how the above mentioned birth of canards comes into play. In [8], the best way we saw to study the small limit cycles problem of (2) was by applying the so-called "primary" blow-up: withx 2 +ȳ 2 +r 2 = 1 andr ≥ 0, u ≥ 0, u ∼ 0. We also divided the system we got by u. The precise elaboration can be found in [8] or in Section 3. Roughly speaking this blow-up transforms the r-family of quite "degenerate" two-dimensional problems (2) into a less degenerate, but still¯ -singular, three-dimensional problem. Instead of having to work in a r-uniform neighborhood of the origin, we now have to consider a neighborhood inside {u ≥ 0} of the primary blow-up locus {(u,x,ȳ,r); u = 0,x 2 +ȳ 2 +r 2 = 1,r ≥ 0}.
As it is usual in working along a sphere it is preferable to work in different charts. He have the family chart "r = 1" and the phase-directional charts "x = ±1,ȳ = ±1". The family chart is the traditional rescaling chart, and it amounts to making a standard rescaling of the phase variables (x, y). The phase-directional charts link the family chart to the original phase plane and we use them to study dynamics of (2) near the "equator" {(u,x,ȳ,r); u = 0,r = 0,x 2 +ȳ 2 = 1} of the primary blow-up locus.
In [8] we detected, depending on region in the B-space, all possible closed curves (so-called limit periodic sets) on the (primary) blow-up locus which can produce limit cycles of the blown-up vector field for ur > 0 and¯ > 0, and we saw that a birth of canards, near a limit periodic set with large parts on the "equator" of the blow-up locus, is possible only for those values of B which are in the slow-fast Hopf region:B 0 ∼ 0, B 1 = −1 and B 2 in an arbitrarily large compact interval. The subject of this paper is a study of cyclicity of the limit periodic set near which the birth of canards occurs. In the slow-fast Hopf region, some of the results from [8] have been proved by using the cyclicity results that we will prove in this paper (see Theorem 2.4, Theorem 2.5 and Section 3.8 in [8]).
Let us recall that, instead of using coordinates on S 2 (B ∈ S 2 ), we used different charts of the sphere (jump chart, slow-fast Bogdanov-Takens chart, slow-fast Hopf chart, etc.) and we proved that, outside the slow-fast Hopf region (i.e. chart), system (2) has at most one hyperbolic limit cycle, in an (¯ , r, λ)-uniform neighborhood of (x, y) = (0, 0). The size of this limit cycle goes to zero when r → 0 because the essential parts of the study of this case have been done in the family chart "r = 1"; for more details we refer to [8].
Here we point out that the proofs of all these results from [8] were based on performing an extra blow-up at the origin (x,ȳ,¯ ) = (0, 0, 0) which depends on region in the B-space in which one looks. We called it "secondary" blow-up.
Being interested in the birth of canards, we consider (2) where B is in the slowfast Hopf region. We introduce the following rescaling: The calculations will be performed, as usual, in charts. When E is in any compact interval and B 0 = ±1, then the system (2) has no small-amplitude limit cycles (hence no birth of canards occurs); for details see [8]. When E = 1 and B 0 ∼ 0, then (2) changes into where G andH are smooth, > 0 is the singular parameter that is kept small, r > 0 is a regular parameter that is kept small, B 0 is a regular parameter close to 0, B 2 is a regular parameter in an arbitrary compact subset of R and λ ∈ Λ, with Λ a compact subset of some euclidean space. If we introduce a new variable Y = y + 1 2 x 2 , then (4) changes into X ,r,B0,B2,λ : where we denote Y again by y. We prefer to work in the so-called Liénard plane. We recall that when G = 0 andH is polynomial, the given family (1) of vector fields is of (generalized) polynomial Liénard type of degree (1, n), where n = deg¯ (...). The 1 in (1, n) comes from the degree of the polynomial in front of y inẏ. Determining the maximum number of limit cycles of a Liénard type vector field of degree (m, n) is one of the major open problems in the field of planar dynamics (see [14]), and [8], [3], [9] and this paper contribute to the extensive research in this area, in the case n ≥ 4 and m = 1. We point out that there is a strong link between results on slow-fast type Liénard equations, as treated in this paper, and general Liénard equations, see [6] an [7].
As mentioned in the introduction, in Section 3 we study system (5) in the family chart and in the phase-directional charts of the primary blow-up (3), and we detect a limit periodic set on the blow-up locus near which a birth of canards occurs (see Figure 2). In [8], this limit periodic set was denoted by L 00 . In this paper, we denote it by Γ. The fact that Γ is on the primary blow-up locus, with a part on the "equator" of the primary blow-up locus, explains the title "Primary birth of canard cycles ..." that we have chosen for this paper.
In Section 4 we define a difference map near Γ which enables us to introduce the notion of cyclicity of Γ. Then we state the results about the cyclicity of Γ, depending on the parameter B 2 . When B 2 = 0, then at most one limit cycle may appear near Γ. When B 2 ∼ 0, then the cyclicity of Γ depends on the higher order terms inH, in analogy with the results in [2]. If we suppose thatH(0, λ) = 0 for all λ ∈ Λ, then at most two limit cycles may come near Γ.
In Section 5, we give a detailed proof of all statements formulated in Section 4. We use similar methods as in [2] to study the difference map near Γ; we exploit symmetries that are present both in the system (5) and in the blow-up construction (3), we use C k -normal forms (see [11]) near semi-hyperbolic singularities P + 0 and P − 0 on Γ (see Figure 2), etc.
3. Blow-up construction and detection of Γ. If we add the equationṙ = 0 to (5), we obtain a τ = ( , B 0 , B 2 , λ)-family of vector fields in R 3 : As mentioned in the introduction, we consider the so-called primary blow-up map (defining a singular change of coordinates) The blown-up vector field is defined as the pullback of the original vector field X τ divided by u:X In order to detect Γ for = u = B 0 = 0, we have to study the blown-up vector fieldX τ near the blow-up locus {0} × S 2 + , in the family chart "r = 1" (see Section 3.1) and in the phase-directional chart "ȳ = 1" (see Section 3.2). (We denote by S 2 + the half-sphere in (x,ȳ,r)-space withr ≥ 0.) The phase-directional charts "x = ±1,ȳ = −1" are not relevant when studying the limit periodic set Γ.
3.1. Family chart "r = 1". We use the following family rescaling of (5): with (x,ȳ) in an arbitrarily large disk in R 2 and u = r ≥ 0. The blown-up field is a τ -family of 3-dimensional vector fields (after division by the positive factor r): The vector field X (1) τ represents a singular perturbation problem with as singular parameter. If we treat X (1) τ as a (τ, r)-family of 2-dimensional vector fields, it can be easily seen that X (1) τ for = 0 has a curve of singularities (a critical curve) given by {ȳ − 1 2x 2 = 0}. The critical curve consists of partially hyperbolic singularities, except at the origin (x,ȳ) = (0, 0), where we deal with a nilpotent singularity (so-called turning point). We see that the curve is normally attracting whenx > 0 and normally repelling whenx < 0.
The -perturbation may cause the curve {ȳ − 1 2x 2 = 0} to perturb to some invariant curve which follows the attracting part of the critical curve until it reaches the section {x = 0} and then follows the repelling part of the critical curve. To see that this connection between the attracting part and the repelling part of the critical curve is possible forx ∼ 0, we desingularize X (1) τ near the origin (x,ȳ, ) = (0, 0, 0) using the so-called secondary blow-up (x,ȳ, ) = (v x, v 2 y, v ) where v ≥ 0, ( x, y, ) ∈ S 2 and ≥ 0.
In the family chart " = 1" one obtains, after dividing by v, the family of vector fields For B 0 = v = 0, dynamics of the above system are of center type with a regular orbit γ = { y = 1 2 x 2 − 1} connecting the end point of the attracting part of the critical curve and the end point of the repelling part of the critical curve, and pointing from the attracting part to the repelling part. The end points are located on the "equator" of the secondary blow-up locus {v = 0} and can be studied in phase-directional charts " x = ±1, y = ±1"(see also [8]). Forx = 0, we can consider the slow dynamics along {ȳ − 1 2x 2 = 0}: When parameter B 2 is kept in any compact set K ⊂] − 2, 2[, then (9) is strictly negative in an arbitrary compact set in thex-space by taking r small enough. For (B 2 , r) ∼ (±2, 0), a saddle-node singularity appears in (9), located nearx = ±1. When B 2 is kept in any compact set K ⊂ R \ [−2, 2], then one has two simple singularities in the slow dynamics, for r ∼ 0. Combining the casex ∼ 0 and the casex = 0 we find that a passage from x = +∞ tox = −∞, along the critical curve, is possible when the slow dynamics has no simple singularities. As a consequence, the critical curve, for r = 0, will be a part of the limit periodic set Γ (see Figure 2).

3.2.
Phase-directional chart "ȳ = 1". As we are interested in the points of intersection of the critical curve {ȳ − 1 2x 2 = 0} with the "equator" of the primary blow-up locus, we consider the phase-directional chart "ȳ = 1" where the blow-up map is where U ∼ 0, R ≥ 0 and (X, R) is in an arbitrarily large disk in R 2 . We obtain a blown-up field which, after dividing by U , can be written as has singularities atX = ± √ 2 + O( 2 ) which represent the above mentioned intersection points for = 0. The eigenvalues of the linear part at ). Hence, we find that P ± are hyperbolic (resonant) saddles for > 0 and semi-hyperbolic singularities for = 0.
For > 0, dynamics of X (2) τ near P ± , restricted to {U = 0} (the blow-up locus), are of saddle type. In P + (resp. P − ) we have theX-axis as stable manifold (resp. unstable manifold). For U = R = 0 and ≥ 0, dynamics of X (2) τ points from P − to P + . As a consequence, Γ will contain the part of the "equator" of the (primary) blow-up locus between P − 0 and P + 0 . 3.3. Combining the family chart "r = 1" and the phase-directional chart "ȳ = 1". The primary blow-up locus {0}×S 2 + can be considered as a 2-dimensional closed disc which we denote byD. Let Γ denote the limit periodic set onD defined as the union of the critical curve {ȳ − 1 2x 2 = 0} and the regular arc A of ∂D between P − 0 and P + 0 (see Figure 2). Hence the limit periodic set Γ represents a limiting situation, for = u = 0.
As mentioned in the introduction, we are interested in an upper bound on the number of limit cycles ofX τ that can bifurcate from Γ, for > 0 and u > 0.

Statement of the results. We suppose that
0 > 0 are small and fixed, and B 0 2 > 2 is arbitrarily large and fixed. We denote by B the compact set As it is usual in studying the cyclicity of a limit periodic set with two "corners" (e.g. Γ with P − 0 and P + 0 ), it is preferable to link limit cycles to zeros of a difference map rather than to fixed points of a return map. We choose two sections and parametrized by (ȳ, r) whereȳ ∼ 0 and r ∼ 0 (see Figure 3).
We are interested in examining orbits ofX τ and −X τ , defined in (6), that start at the section Σ 1 0 and meet the section Σ 2 0 in finite time.
To be more precise, we denote by o + U,R,τ (resp. o − U,R,τ ) the forward orbit (resp. the backward orbit) ofX τ starting at the point (U, R) ∈ Σ 1 0 , for τ ∈ [0, 0 ] × B × Λ. Now we can define the following set in (U, R, τ )-space: follows the arc A until it comes close to point P + 0 (resp. P − 0 ), then follows the attracting (resp. repelling) branch of the critical curve {ȳ − 1 2x 2 = 0}, for an appropriate parameter B 2 , and meets Σ 2 0 , close to the turning point (x,ȳ) = (0, 0). Since γ defined in Section 3.1 contains ( x, y) A first result deals with the smoothness of the transition map from Σ 1 0 to Σ 2 0 along the trajectories ofX τ in positive and negative time.
There exists a small ball W around (U, R) = (0, 0) such that for any degree k ≥ 1 of smoothness there exist 0 < k ≤ 0 so that the mappings and (defined by following respectively the orbits o + U,R,τ and o − U,R,τ until they reach Σ 2 0 ) are C ∞ and have C k -extensions to D k . Moreover, functions h + and h − are strictly positive and C ∞ with C k -extensions to D k .
A proof of Theorem 4.1 is given in Section 5.1.

Remark 2.
Whereas the functions H ± are C ∞ on D k (X τ is C ∞ ), presence of hyperbolic saddles P ± , > 0, weakens the obtained smoothness: one has C ksmoothness on D k for all k, in the sense that possibly k → 0 as k → +∞ (see also [3]). The boundary of the domain D k of H ± includes the sets { = 0} and {R = 0}, and might include other parts of the parameter space, where orbits o ± U,R,τ get trapped at a saddle-node but nearby orbits can meet {x = 0}. Let us recall that such a saddle-node may appear for B 2 ∼ ±2, in the family chart "r = 1" (Section 3.1).
The functions h + and h − in respectively (12) and (13) play a central role in the search for the limit cycles near Γ. Zeros of δ := h + − h − , for U > 0, R > 0 and > 0, correspond to periodic orbits Hausdorff-close to Γ. To be more precise, for each fixed value of (r, τ ), r > 0, we can treat the function δ as 1-variable function defined on "segment" l τ r := {(U, R); (U, R, τ ) ∈ D k , U R = r, U ≥ 0}, for some k ≥ 1. We want to study the number of isolated zeros (counted with multiplicity) of the function δ on l τ r for each fixed value of (r, τ ) where r > 0. To study the isolated zeros of δ on l τ r , we will consider its Lie-derivative The reason to introduce this Lie-derivative is that the equation {L Y δ = 0} can be reduced to a simpler form than the equation {δ = 0} which contains exponential terms (see Section 5.2).
As the vector field Y has no zeros on {U R = r}, r > 0, Rolle's theorem will permit to find the maximum number of the zeros of δ from zeros of L Y δ. This trick with a Lie-derivative along a vector field is used in [2].
We say that The notion of cyclicity of δ enables us to state main results in this paper.
is bounded by N (the number of isolated zeros of a function with an empty domain is 0). b) the minimum of such N is called the cyclicity of δ at B 2 =B 2 . The cyclicity of Γ at B 2 =B 2 is the cyclicity of δ at B 2 =B 2 .
Remark 3. Since we study the system (5) for r > 0, it is sufficient to define the cyclicity of δ for U > 0 and R > 0 (r = U R). As we can see in Theorem 4.1, the functions h + and h − are defined not only for U ≥ 0, but also for U < 0. Here we point out that symmetries defined in Section 5.2 include the variable U and play an important role in the study of the cyclicity of δ: in Section 5.2 we will use Our first main result states that for B 2 = 0 at most one (hyperbolic) limit cycle may perturb from Γ. The case B 2 = 0 is easy to treat because of the presence of the term B 2 rx 2 in (5) which makes sure that for B 0 = 0 the blown-up vector field X τ is far away from center behavior on the primary blow-up locus. Theorem 4.3. a) If −2 <B 2 < 2 andB 2 = 0, then the cyclicity of Γ at B 2 =B 2 is equal to 1. WhenB 2 > 0, then we deal with a hyperbolic and attracting limit cycle. WhenB 2 < 0, then we deal with a hyperbolic and repelling limit cycle. b) IfB 2 = ±2, then the cyclicity of Γ at B 2 =B 2 is equal to 1. WhenB 2 = 2, then we deal with a hyperbolic and attracting limit cycle. WhenB 2 = −2, then we deal with a hyperbolic and repelling limit cycle. c) If 2 < |B 2 | ≤ B 0 2 , then the cyclicity of Γ at B 2 =B 2 is 0.
A proof of Theorem 4.3 is given in Section 5.3.
When B 2 ∼ 0, then one might have more than one limit cycle, as explained in [8], and it is due to the fact that one studies perturbations of a vector field of center type (symmetries introduced in Section 5.2 imply thatX τ is of center type on the blow-up locus, for B 0 = B 2 = 0). In order to make this situation less degenerate, we assume thatH(0, λ) = 0 for each λ ∈ Λ, like in [8].
Remark 5. Theorem 4.4 implies that at most two limit cycles can be found in an This result is also used in [8].
A proof of Theorem 4.4 is given in Section 5.4.

5.
Proofs of Theorem 4.1-Theorem 4.4. We start by proving Theorem 4.1. Besides proving Theorem 4.1, in Section 5.1 we get a nice structure of a forward transition map between Σ 1 0 and Σ 2 + (see Figure 3). (For a precise definition of Σ 2 + we refer to later sections.) That structure is given by (46) in Theorem 5.9, and in Section 5.2 it is used to obtain the exponential form (63) of L Y δ.
We repeat once more that the exponential form of L Y δ will be used in the proof of Theorem 4.3 and in the proof of Theorem 4.4. 5.1. Proof of Theorem 4.1. We will provide an explicit proof for H + , hence working in forward time; the treatment of H − is completely analogous.
We split H + into three parts. Choose the sections Σ 1 0 and Σ 2 0 , as explained above. We also choose a section Σ 1 + (resp. Σ 2 + ) near P + transverse to arc A (resp. transverse to the critical curve {ȳ = 1 2x 2 }) ( Figure 3). Σ 1 + (resp. Σ 2 + ) is parametrized by two regular parameters. We refer to Section 5.1.2 for precise definitions of Σ 1 + and Σ 2 + . One defines (see Figure 3): 1. the regular transition map R τ + near the arc A from Σ 1 0 to Σ 1 + , defined by the flow ofX τ , 2. the Dulac map D τ + describing the corner passage near P + from Σ 1 + to Σ 2 + defined by the flow ofX τ , 3. the singular transition map S τ + near the critical curve from Σ 2 + to Σ 2 0 , defined by the flow ofX τ , 4. the transition maps H τ ± (U, R) := H ± (U, R, τ ) near Γ from Σ 1 0 to Σ 2 0 defined by the flow of ±X τ . In particular H τ Our goal is to study the transition maps R τ + , D τ + and S τ + . We start with D τ + . Choosing appropriate normalizing coordinates near P + 0 will appear to be a helpful tool in simplifying the calculation of the transition map D τ + . Since we want to study the corner passage near P + , we will change X τ , defined in (11), near P + by the equivalent family Y τ defined as: As τ can be chosen in the compact set [0, 0 ] × B × Λ, we can suppose that Y τ is defined in a fixed neighborhood of P + 0 . We now introduce the coordinate change In the coordinates (z, U, R) the vector field Y τ can be written as Our goal is to normally linearize the vector field (16), i.e. to linearize the differential equation of the hyperbolic variable z in (16). We aim at getting a normal linearization of (16) that is as smooth as possible. The theorems that are presented in [11] provide normal linearizations that respect a lot of essential structure, combined with a maximal smoothness.

CANARD CYCLES IN SLOW-FAST CODIMENSION 3 ELLIPTIC BIFURCATIONS 2653
We consider first local invariant manifolds of (16) near the singularity Q := P + − ( √ 2, 0, 0). Section 3.2 implies that Q is a hyperbolic saddle for > 0 and a semi-hyperbolic singularity for = 0. The well-known center manifold theorem implies now that for each k ≥ 1 there exists a C k (B 0 , B 2 , λ)-family of local center manifolds of the extended family of vector fields Y τ +0 ∂ ∂ at (z, U, R, ) = (0, 0, 0, 0), where (B 0 , B 2 , λ) ∈ B × Λ. In fact the (B 0 , B 2 , λ)-family of center manifolds of the extended family forms a τ -family of invariant manifolds of Y τ at Q .
It is known that in general one cannot expect the existence of a C ∞ center manifold of a C ∞ vector field. Since we deal with a specific C ∞ vector field, we utilize Theorem 1 in [11] to see that there exists a ball V around (U, R) = (0, 0), 0 < 1 ≤ 0 and a C 1 -graph z = φ(U, R, , B 0 , B 2 , λ) on V ×[0, 1 ]×B ×Λ that forms a τ -family of invariant manifolds of (16), with φ(0, 0, 0, B 0 , B 2 , λ) = 0. Moreover, φ is C ∞ for U = 0 and there exists a decreasing sequence The main benefit of Theorem 1 in [11] is hence the fact that a single center manifold of Y τ +0 ∂ ∂ is constructed that can be made as smooth as required, provided one takes small enough. It is clear that this result improves the center manifold theorem.
Remark 6. We fix the center manifold z = φ of Y τ + 0 ∂ ∂ given in Theorem 1 in [11]. Now we have to straighten the center manifold z = φ. After applying the coordinate change to (16), the z-component of (16) can be written as Here we used the fact that the family of the invariant manifolds z = φ(U, R, τ ) of (16) is expressed by a solution to the partial differential equation Taking into account (18), one gets a family of vector fields The functions A and κ are C ∞ for U = 0 and C k for (U, R) ∈ V , |Z| small, ∈ [0, k ], (B 0 , B 2 ) ∈ B and λ ∈ Λ. Bearing in mind Remark 7, we find that where O( ) is C ∞ for U = 0 and C k for (U, R) ∈ V , ∈ [0, k ], (B 0 , B 2 ) ∈ B and λ ∈ Λ. Hence A(0, 0, 0, B 0 , B 2 , λ) = 1.

CANARD CYCLES IN SLOW-FAST CODIMENSION 3 ELLIPTIC BIFURCATIONS 2655
Hence we have that We make in the integral the change of variable: R = R exp( 2 s). This gives To end this let us use the change of variableR = R R1 . Then (24) changes into If we denote R/R 1 by V , then we have that where f (V ,R, U, τ ) = A(U V ,RR 1 , τ ). Let us recall that the domain of A(U, R, τ ) only shrinks in the -direction when degree of smoothness k increases. If we use a Taylor formula at first order inR = 0, then we have that f (V ,R, U, τ ) = f (V , 0, U, τ ) +Rf 1 (V ,R, U, τ ). As a consequence we have the following expression for the integral in (25): Let us first study the first integral in (26). Using the change of variable w = V /R we have that whereF is arbitrarily (finitely) smooth by taking small . To study the second integral in (26), we use the following lemma (see [3]): 2) Should f depend smoothly on extra parameters up to some order, then so will the resulting functions α and β be smooth w.r.t. these parameters up to the same order.
Hence Lemma 5.3, with b = 0, implies that where α k , β k are C k functions provided f 1 is C k+2 for k ≥ 1. Let us recall that f 1 is arbitrarily (finitely) smooth by taking small enough. Expressions (27) and (28), together with the fact that V = R R1 , imply that the integral in (25) can be written as If we denote the first line in (29) by α k and the expression in front of ln R by β k , then we obtain that where for each k ≥ 1 α k and β k are C k , by taking sufficiently small, and It remains to study the smoothness of the transition map d τ + (U, R). We focus here to the fact that, by choosing small enough, the exponent A(0,0,τ ) 2 can be chosen arbitrarily high, so that any degree of smoothness can locally be obtained. For the sake of completeness, let us prove it.
For R > 0 and > 0, we have nothing to prove due to the smoothness of α k and β k .
Since β k (U, 0, τ ) < 0, d τ + (U, R) → 0 when R → 0 or → 0. Choose now any k ≥ 1. In what follows, we show that, by choosing small enough, . Straightforward calculations show that any derivative ∂ I d τ + (U, R), 1 ≤ |I| ≤ k, is a finite linear combination of the following expressions: where |I j | ≥ 1 and l j=1 |I j | = |I|. On account of (31) it is sufficient to show that by choosing small enough the expression goes to 0 as R → 0 whereᾱ is continuous. We suppose that m ∈ N 1 , n 1 ∈ N 0 and n 2 ∈ N 0 are arbitrary and fixed. α in (32) is bounded and, with no loss of generality, we suppose thatᾱ = 1. The logarithm of (32) is As ln | ln R| ≤ ln 1 R for R ∼ 0, (33) is bounded above by Since β k ∼ −1, we have that −β k − n 1 2 − n 2 2 > 0 by taking small enough. Hence (34) goes to −∞ as R → 0. It also goes to −∞ as → 0, for R small.

Regular transition map R τ
+ . Let us recall that the map R τ + , from a subset of Σ 1 0 to Σ 1 + , is defined by the flow of X (2) τ . The transition map R τ + can be represented as going from a subset of Σ 1 0 to σ 1 + , i.e. with values in the normalizing coordinates ( Z, U, R). One obtains map R + (U, R, τ ). Hence R + is expressed in (U, R) with values in (U, R).
Proof. Notice first that the parameter (B 0 , B 2 , λ) takes values in the compact set B × Λ. Hence theX-component of X In other words, the orbit of (11), starting at (U, R) ∈ Σ 1 0 , (U, R) ∼ (0, 0), reaches {X =X 0 } in a finite time. Remark that this fact is clear for = 0. The fundamental results on the existence, uniqueness and continuity with respect to parameters of solutions to initial value problems imply now that we deal with a C ∞ transition map T + = (T 1 is C ∞ ). We know that T + preserves U R, i.e. T 1 + (U, R, τ )T 2 + (U, R, τ ) = U R for any τ . We also know that T + preserves {U = 0} and {R = 0}. It means that there exist C ∞ -functions t 1 + and t 2 + , such that: . Hence we find that t 1 + (0, 0, τ ) = 0 and t 2 + = 1/t 1 + , for any τ . We obtain that ).
If = 0, then T + (U, R, τ ) = (U, R). This means that there exist a C ∞ -functionT such that t 1 + (U, R, τ ) = 1 + 2T (U, R, τ ). Choose sections π + τ := ϕ({(X 0 , U, R) | U ∼ 0, R ∼ 0, R ≥ 0}). We parametrize π + τ through ϕ by (U, R). In order to finish the study of R + , we need to consider the regular transition map F + : π + τ → σ 1 + along the trajectories of (22), expressed using the chosen parametrization on π + τ , σ 1 + . Notice that ϕ leaves the (U, R)-component unchanged. This means that the regular map T + is expressed by (35) if we use for {X =X 0 } the normalizing coordinates ( Z, U, R). Then we have R + (U, R, τ ) = F + (T + (U, R, τ ), τ ). whereF is C k on Ω 2 k . Proof. The study of F + is analogous to the study of T + . Notice that the domain of A in (22) only shrinks in the -direction as the smoothness requirements increase, and that π + τ is a graph of a C k -function which has the same smoothness property as A.
Combining Lemma 5.4 and Lemma 5.5 we get: whereR is C k on Ω 3 k . Proof. We have
Proposition 5.7. There exists a ball W 4 around (U, R) = (0, 0) such that D τ + • R + is well defined and C k on Ω 4 k := ( W 4 ∩ {R > 0})×]0, k ] × B × Λ and has a C kextension to Ω 4 k , up to shrinking k if necessary. Moreover, k . The function l does not depend on k. Proof. The above-mentioned smoothness properties of D τ + • R + follow directly from Proposition 5.2 and Proposition 5.6. From Section 5.1.2 and (36), it is clear that Using expressions (23) (we write α = α k and β = β k ) and (36) we obtain that On account of Proposition 5.2, we have that Based on (39), we have Using (40) and the fact that ln(1 + 2R (U, R, τ )) = O( 2R ), the expression (38) changes to This completes the proof.
The section σ 2 + is contained in a family directional chart and we can use for it better adapted coordinates. On section σ 2 + we can consider coordinates (x, r), where (x,ȳ, r) are the coordinates of X (1) τ , defined in (8). The changes of coordinates ( Z, U ) → (x, r) are given by r = U R 1 (ϕ leaves the (U, R)-component unchanged) andx =x τ,r ( Z) (we have eliminated U = r R1 in the expression ofx). Lemma 5.8. One has where φ is fixed in Remark 6, z 0 > 0 is defined at the end of Section 5.1.2 and the domain of O( Z) only shrinks in the -direction as the smoothness requirements increase.
Proof. On account of blow-up formulas (7) and (10) we get the following relation between the coordinatesx andX, on σ 2 + : The mappings defined in (15) and (17), together with Theorem 5.1 and the dilatation defined in Section 5.1.2, imply the following relation between the coordinates X and ( Z, U ), on σ 2 + : Based on Theorem 5.1, O( Z) in (43) has the required property stated in Lemma 5.8. Putting together (42) and (43) we find the relation betweenx and ( Z, r) on σ 2 + : Parameterizing section σ 2 + by (x, r), the transition map from (a subset of) Σ 1 0 to σ 2 + along the trajectories ofX τ is equal to We denote byx(U, R, τ ) the first component in (45). Combining Proposition 5.7 and Lemma 5.8 we obtain the final form for the transition map from (a subset of) Σ 1 0 to Σ 2 + = ϕ −1 (σ 2 + ), where Σ 2 + is parametrized by (x, r): Theorem 5.9. There exists a ball W 5 around (U, R) = (0, 0) such thatx(U, R, τ ) is well defined and C k on Ω 5 k := ( W 5 ∩{R > 0})×]0, k ]×B×Λ and has a C k -extension to Ω 5 k , up to shrinking k if necessary. Moreover, are C k on Ω 5 k . The function L does not depend on k. Proof. The above-mentioned smoothness property ofx(U, R, τ ) follows directly from Proposition 5.7 and Lemma 5.8. Taking into account (41) we get Of course, we have that where ln(1 + O(d τ + (R + (U, R, τ )))) is arbitrarily (finitely) smooth by taking small enough. This follows directly from the fact that functions O( Z) in Lemma 5.8 and d τ + (R + (U, R, τ ))) are arbitrarily (finitely) smooth by taking small enough. If we use (37), then the expression (47) changes tō The rest of the proof is now trivial. 5.1.5. Transition map S τ + and conclusion. We consider the transition map S τ + from Σ 2 + to Σ 2 0 ⊂ {x = 0} along the trajectories of the vector field X (1) τ defined in (8). Remark that S τ + can be treated entirely in the family chart "r = 1". Section Σ 2 + is parametrized by (x, r) and section Σ 2 0 is parametrized by (ȳ, r). One obtains map S τ + (x, r) = (s + (x, r, τ ), r). As mentioned in Section 3.1, in the family chart we consider r as a regular parameter and observe that is a singular perturbation parameter. It is important to realize that the slow dynamics (9) are characterized by a regular flow box, with possible isolated saddle-node singularities for (B 2 , r) ∼ (±2, 0), located away from the origin (x,ȳ) = (0, 0). This case has already been treated in [4].
In order to be able to use the results in [4], we define a section along the secondary blow-up locus of the origin (x,ȳ) = (0, 0) (see Section 3.1). More precisely, T is defined in the family chart { = 1}, and parametrized by the (secondary) blow-up coordinate y. We denote by y =ŝ + (x, r, τ ) the transition map between Σ 2 + and T . By following the orbits through the curve {x =x(U, R, τ ),ȳ = 1/R 2 1 , r = U R}, (U, R, τ ) ∈ D, ( , U, R) ∼ (0, 0, 0), in forward time until they reach T , we end up with a C k -transition map y =ŝ + (x(U, R, τ ), U R, τ ) with a C kextension to the closure of its domain. Degree of smoothness k can be chosen arbitrarily high by taking small enough. We refer to Theorem 3.1 of [4] and Theorem 5.9 in this paper. Sinceȳ = 2 y, we obtain s + = 2ŝ + . For ( , B 0 ) ∼ (0, 0),ŝ + is strictly negative because γ ∩ T = (0, −1) where γ is defined in Section 3.1.

5.2.
Difference map and Lie-derivative. In order to prove Theorems 4.3 and 4.4, we need to consider the difference map where (U, R, τ ) ∈ D k and D k , H ± are defined in (12) and (13).
The r-component of ∆ is equal to 0. Theȳ-component of ∆ can be written as where (U, R, τ ) ∈ D k . The expression of (5) is invariant under the symmetry: From the family rescaling (7) and the symmetry S, defined in (50), it follows that the vector field X The directional blow-up formula (10) and the symmetry S, defined in (50), imply that X By the invariance of X To study isolated zeros of δ, we will consider its Lie-derivative L Y δ = U ∂δ ∂U −R ∂δ ∂R (see Section 4). It is clear that L Y (U R) = 0. We first apply the Lie-derivation to the expression − 2 h + (U, R, τ ): where (U, R, τ ) ∈ D k .
Remark 8. From now on, we avoid mentioning the fact that degree of smoothness of h + (U, R, τ ),x(U, R, τ ), etc., depends on . We also do not specify domains of h + (U, R, τ ),x(U, R, τ ), etc., and we avoid pointing out that α k , β k , etc., depend on degree k of smoothness. We merely say that all these functions are C k bearing in mind that any (finite) degree of smoothness can be obtained. Similarly, we say that F (U, R, τ ) is C k in (U, R, R ln R, τ ) if we can choose a sufficiently smooth function f such that F (U, R, τ ) = f (U, R, R ln R, τ ).

CANARD CYCLES IN SLOW-FAST CODIMENSION 3 ELLIPTIC BIFURCATIONS 2663
5.2.1. Study of L Y x(U, R, τ ) . On account of (23) and (46) we get where α and β are the C k -functions defined in (23), and L(U, R, τ ) is the C k -function in (U, R, R ln R, τ ) introduced in Theorem 5.9. We can write P = L Y α + β ln R + 2 L . In order to study P , we need the following easy properties of the Lie-derivation: and moreover ζ(0, 0, τ ) ≡ 0 (one can also write: ζ = o(1)).
is strictly positive and C k , and its logarithm is a C k -function. Hence, (60) can be written as where L is a C k -function.
Proof. We have:
We first simplify (64). It is clear that We aim at controlling the difference Lemma 5.13. One has that where f 1 is a C k -function, depending only on , and where f 2 is a C k -function in (U, R, R ln R, τ ).
Using (21) and Lemma 5.13, we find that wheref 2 is a C k -function in (U, R, R ln R, τ ) and identically zero for U = B 0 = B 2 = 0, and where Taking into account (71), (64) changes to where L * is a C k -function in (U, R, R ln R, τ ) and identically zero for U = B 0 = B 2 = 0. It can be easily seen that C, defined in (72), is bounded for R ∼ 0. We make this statement precise in the following lemma: Lemma 5.14.
where c 1 and c 2 are C k -functions in (U, R, R ln R, B 2 , λ) and where c 1 is strictly negative for R ∼ 0 and R ≥ 0.

CANARD CYCLES IN SLOW-FAST CODIMENSION 3 ELLIPTIC BIFURCATIONS 2667
Proof. Using (72), we can write C as where C 1 and C 2 are C ∞ -functions. Bearing in mind (70), we see that the first integral in (75) is a C k -function in (U, R, R ln R, B 2 , λ). We denote it by c 1 (U, R, B 2 , λ). It is clear from (72) that c 1 is strictly negative.
The second integral in (75) we can write as where C 3 is a C ∞ -function. The second expression in (76) can, by (70), be written as U C 4 (U, R, B 2 , λ) where C 4 is a C k -function in (U, R, R ln R, B 2 , λ).
If we use the change of variables: R R = w, then the first expression in (76) changes to The above integral can be written as U C 5 (U, R, B 2 , λ), for some C ∞ -function C 5 . We define now c 2 (U, R, B 2 , λ) = C 4 (U, R, B 2 , λ) + C 5 (U, R, B 2 , λ).
It remains to simplify the difference of divergence integrals in (73). The integral of divergence I has been studied in detail in [10], if the slow dynamics (9) has no zeros, and in [4], if the slow dynamics (9) has isolated saddle-node singularities located away from the contact point (x,ȳ) = (0, 0). As we mentioned above, a saddle-node singularity appears in (9) for (B 2 , r) ∼ (±2, 0), located nearx = ±1. Hence we may rely on [4]. When parameter B 2 is kept in any compact set K ⊂] − 2, 2[, then (9) is regular in an arbitrary compact set in thex-space by taking r small enough. As a consequence, we are allowed to refer to [10]. When 2 < |B 2 | ≤ B 0 2 , then one has simple zeros in the slow dynamics and hence limit cycles near Γ are not possible.
We are now in a good position to prove Theorems 4.3 and 4.4.

5.3.
Proof of Theorem 4.3. The symmetries S F and S P defined, respectively, by (51) and (52) imply that it suffices to prove Theorem 4.3 for B 2 strictly positive. Based on the discussion above, we distinguish three possibilities.
Our goal is to show that, by taking U ∼ 0, R ∼ 0, R ≥ 0, ∼ 0 and ≥ 0, the left hand side of the equation (73) is strictly negative for any B 0 ∼ 0, B 2 ∈ K and λ ∈ Λ. This implies that the equation {L Y δ = 0} has no solutions along {U R = r}, under the given conditions on the parameters and variables, and for (R, ) > (0, 0). Using the Rolle's Theorem one finds that the cyclicity of Γ at B 2 ∈ K is bounded by one. Consider first the expression 2 I(x(U, R, τ ), U R, τ ) − 2 I(x(−U, R, τ * ), −U R, τ * ).
We may use Theorem 5.9 and results in [10], for B 2 ∈ −K ∪K, to write the function 2 I(x(U, R, τ ), U R, τ ) as where ϕ 1 and ϕ 2 are C k -functions, including at = 0 and R = 0, ϕ 1 = O( ) and where I(B 2 , r, λ) represents the slow divergence integral, defined as: Using (77) we obtain that for some new C k -function L * in (U, R, R ln R, τ ).
Remark 10. When B 2 ∼ 0, we also deal with a regular slow dynamics and we may hence use (80), where L * andφ 2 are identically zero by taking U = B 0 = B 2 = 0 (see Section 5.4).
Note that the left hand side of the equation (80) can be treated as a continuous (including = 0) perturbation of I(B 2 , U R, λ) − I(−B 2 , −U R, λ) + C(U, R, B 2 , λ). Hence it suffices to show that the first line in (80) is strictly negative for B 2 ∈ K, λ ∈ Λ, U ∼ 0, R ∼ 0 and R ≥ 0.
To see that the first line in (80) is strictly negative for B 2 ∈ K, λ ∈ Λ, U ∼ 0, R ∼ 0 and R ≥ 0, it suffices to observe that C(U, R, B 2 , λ) is, by Lemma 5.14, strictly negative for B 2 ∈ K, λ ∈ Λ, U ∼ 0, R ∼ 0 and R ≥ 0. Hence the cyclicity of δ at B 2 ∈ K is bounded by one.
Since the left hand side of the equation (80) is strictly negative, we deal with a hyperbolically attracting limit cycle. 5.3.2. B 2 ∼ 2. We invoke Lemma 5.14 and see that the function C in (73) is bounded. Of course we have that L * introduced in (73) is bounded due to the fact that L * is C k in (U, R, R ln R, τ ). It remains to study 2 I(x(U, R, τ ), U R, τ ) − 2 I(x(−U, R, τ * ), −U R, τ * ), i.e. the first line in (73).
Hence we find that the cyclicity of Γ at B 2 = 2 is bounded by one. Since the left hand side of (73) is always negative for (U, R, B 2 , ) near (0, 0, 2, 0), we deal with a hyperbolically attracting limit cycle. We refer to Section 5.3.1 to see that the cyclicity of Γ at B 2 = 2 is one.
Lemma 5.15. We have that where f 1 and f 2 are C ∞ -functions in R and identically zero when R = 0.
Proof. We get where O(R) is a C ∞ -function in R. In the first step we used the change of variable: where O(R) is a C ∞ -function in R.