Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations

J. Differential Equations 255 (2013) 4012–4051

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Journal of Differential Equations www.elsevier.com/locate/jde

Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations R. Huzak ∗ , P. De Maesschalck, F. Dumortier Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, 3590 Diepenbeek, Belgium

a r t i c l e

i n f o

Article history: Received 20 December 2012 Revised 23 June 2013 Available online 19 August 2013 MSC: 34E15 37G15 34E20 34C23 34C26 Keywords: Slow-fast system Singular perturbations Slow divergence integral Limit cycle Blow-up

a b s t r a c t This paper deals with local bifurcations occurring near singular points of planar slow-fast systems. In particular, it is concerned with the study of the slow-fast variant of the unfolding of a codimension 3 nilpotent singularity. The slow-fast variant of a codimension 1 Hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the Van der Pol system. Similarly, codimension 2 slow-fast Bogdanov– Takens bifurcations have been characterized. Here, the singularity is of codimension 3 and we distinguish slow-fast elliptic and slowfast saddle bifurcations. We focus our study on the appearance on small-amplitude limit cycles, and rely on techniques from geometric singular perturbation theory and blow-up. © 2013 Elsevier Inc. All rights reserved.

1. Introduction We study planar singularly perturbed vector fields, concentrating more precisely on the presence of small limit cycles of a specific nilpotent singular point. The framework this paper is written in follows general ideas from geometric singular perturbation theory (e.g. [16,24]), together with the technique of family blow-up (e.g. [8,28]). In principle, slow-fast systems are systems where dynamics can be observed on two distinct time scales. The time scale separation is traditionally denoted by 0 <   1, and quite often slow-fast

*

Corresponding author. E-mail addresses: [email protected] (R. Huzak), [email protected] (P. De Maesschalck), [email protected] (F. Dumortier). 0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2013.07.057

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Fig. 1. The fast dynamics of (2) along parabolic fast fibers. At the origin, the curve of singular points has a contact with the fast fibers.

systems are written in a so-called standard form where the slow and fast variables can be readily identified: for example in the dynamical system

⎧ dx ⎪ ⎨  = f (x, y ,  ), dt

⎪ ⎩ dy = g (x, y ,  ),

(1)

dt

the slow variable is clearly y and the fast one x. In this paper we adopt a geometric point of view and prefer to consider equations associated to smooth vector fields. We prefer to define notions in a coordinate free way and consider system (1) (multiplied by  to make dx smooth) as local represendt tation of a vector field in one of its charts. Of course in other charts, the trivial foliation might change into a nontrivial fast flow. In that sense, singularly perturbed vector fields are defined as families of vector fields X  , that have for  = 0 a curve of singular points. As a relevant example, we consider

 X :

x˙ = y ,   y˙ = −xy +  f (x) + y 2 G (x, y ) .

(2)

The fast dynamics, i.e. the dynamics of X 0 , shows a line of singular points { y = 0}, and the fast orbits dy are easily seen to lie on fibers determined by the equation dx = −x. The fast fibers are parabola, intersecting the line of singular points transversally except at the origin, see Fig. 1. Given a point P = (0, 0) on the singular curve, the linear part of the vector field has one eigendirection tangent to the curve (with eigenvalue 0), and has another eigendirection transverse to the curve, corresponding to a nonzero eigenvalue. It implies that the dynamics of X 0 , i.e. the fast dynamics, near the curve is clear: orbits are attracted towards or repelled away from the curve, in the direction of the parabola. Points P at which X 0 has a nonzero eigenvalue are called normally hyperbolic points of X 0 . Near such points, there exist invariant manifolds asymptotic to the curve of singular points (in fact if we augment X  with the trivial equation ˙ = 0, these invariant manifolds would be center manifolds at (x, y ,  ) = (x0 , y 0 , 0)). Using traditional asymptotic methods we find

y=

f (x)



x

  + O 2 .

This expression of the invariant manifold near normally hyperbolic points leads to a coordinate free way of establishing the slow dynamics: we reduce the dynamics of X  to the invariant manifold, divide out  and take the limit as  → 0:

x =

f (x) x

,

x = 0.

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Remark 1. The reader can readily check that for systems in the standard form (1), this way of computing the slow dynamics agrees with setting  = 0 in (1), implicitly solving f (x, y , 0) = 0 as x = x( y ) and considering y  = g (x( y ), y , 0). For system (2), one can revert to the standard form by writing y = y˜ − 12 x2 . At the origin, normal hyperbolicity is lost. The origin is a so-called contact point (because the curve of singular points makes a contact with the fibers of the fast foliation). We observe that it is a singularity of nilpotent type. Remark 2. Though (2) was introduced as a relevant example, it is more than that: it is actually a smooth local normal form for equivalence for slow-fast vector fields having a quadratic contact between the curve of singular points and the fast foliation (under the generic assumption that the point at which the contact is made is a singularity with nilpotent linear part). We remark that when G ≡ 0, the normal form is of generalized Liénard type. The presence of such contact points makes the dynamics nontrivial in the sense that in families perturbing from X 0 , it is notoriously hard to determine the presence of one or more limit cycles, or even to merely give a bound on the number of such limit cycles. Limit cycles for 0 <   1 are close to canard limit periodic sets; these are invariant sets for  = 0 that are homeomorphic to a circle. In (2), we can observe top parts of the parabola together with a part of the x-axis as potential canard limit periodic sets. In a number of papers [4–6], the characterization of limit cycles near such canard limit periodic sets, i.e. the so-called detectable cycles, is made fairly complete. It is clear that the direction of the slow dynamics, determined by f (x)/x, needs to point from right to left to make detectable cycles possible. But also the origin as a singleton is a limit periodic set out of which limit cycles may perturb, these are called small-amplitude cycles, in contrast to the detectable ones. In this paper, we aim at studying small-amplitude limit cycles. Below, we will classify the type of contact point based on the order of zero of the function f in (2). We will consider unfoldings of the different types of contact points, leading to different bifurcations: from now on we consider

 X  ,b,λ :

x˙ = y ,   y˙ = −xy +  f (x, b) + y 2 G (x, y , λ) ,

(3)

where (b, λ) are multidimensional parameters. The parameter λ will not be essential in our study, but we will keep it to stress the generality of the results. When f (x, b) = b0 + O (x), for some b0 = 0, it is not so hard to see that there are no limit cycles; we treat this case for the sake of completeness. (In fact, the reader familiar with singular perturbations might notice that the case b0 = 0 corresponds to a jump point at the origin, well studied in for example [32], [8], [28], . . . . No bifurcation is seen, since there is no singular point for  > 0.) When f (x, b) = b0 + b1 x + O (x2 ), for some b1 = 0 and b0 ∼ 0, one or more limit cycles may appear, as is shown in [9]. As b0 is varied through the origin, a slow-fast Hopf (or singular Hopf) bifurcation takes place, see [8], [28], [9]. When G = 0, the maximum number of small-amplitude limit cycles is shown in [9] to be finite in analytic families or in families of finite codimension (we refer to this paper for more details). In [7], a study was made of the more degenerate case f (x, b) = b0 + b1 x + b2 x2 + O (x3 ), where b2 = 0 fixed and (b0 , b1 ) ∼ (0, 0). Here, (b0 , b1 ) are to be considered as unfolding parameters in a codimension 2 slow-fast Bogdanov–Takens bifurcation. To some extent, this case was easier to treat because of the presence of the symmetry-breaking term b2 x2 . It is shown that from the origin, at most one limit cycle may perturb. Interested in solving the general question, we continue the study towards slow-fast codimension 3 bifurcations, in the hope that, just like in the case of detectable cycles, a mechanism is exposed that allows afterwards to treat codimension n bifurcations for all n  3 in a recursive approach. At present however, it seems that the small-amplitude limit cycles near a codimension 3 point are far more difficult to deal with than the detectable cycles that pass near such a point.

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The goal of this paper is to start the codimension 3 study where

 

f (x, b) = b0 + b1 x + b2 x2 + b3 x3 + O x4 ,

(4)

with (b0 , b1 , b2 ) ∼ (0, 0, 0) and b3 = 0, hoping that it will yield new insights that can lead to the treatment of the general case. Like in [5], the case n = 3 is treated independently from n > 3, because the appropriate desingularization is essentially different between these two cases; see [6]. In Section 2, we subdivide the parameter space in specific regions, some of which are easy to treat, some harder. In this paper, we will give a complete analysis of the small-amplitude limit cycles in most regions, and we isolate a parameter region for which the study is more involved; this region will be dealt with in a separate publication. It is therefore necessary to state the main result of the paper after introducing the different parameter regions. We refer to Section 2 for detailed statements of results. In Section 3 we give an overview of the blow-up construction needed to desingularize the system. We recall that blow-up is a technique used in singular perturbation theory that allows to treat contact points, such as the origin in the given system. It was first introduced in a slow-fast context in [8], for a slow-fast Hopf point. There, essentially blow-up suffices to desingularize the system completely. In our codimension 3 setting, we need several blow-ups: one to resolve the degeneracy of the contact point, and another one to resolve the degeneracy introduced by the slow-fast nature. A detailed proof of all statements formulated in Section 2 are also given in Section 3. The focus on the determination of limit cycles in planar systems has arisen principle in Hilbert’s 16th problem [22]: though phase plane analysis is typically straightforward using tools like topological index of singular points, invariant separatrices, Poincaré–Bendixson theorem, and the absence of chaos, the determination of limit cycles is far from straightforward, as can be observed by the fact that Hilbert 16th problem remains unresolved even in the most basic setting of degree 2. In 2000, Hilbert 16th problem was revitalized as Smale’s 13th problem [39], this time focusing on classical Liénard systems. Reductions of Smale’s 13th problem to slow-fast Liénard systems (see [12] and [37]) show the interest in systems like (2) (with G = 0). We refer for background reading to the survey [30] on Liénard systems and the survey [23] on Hilbert 16th problem. Besides this purely mathematical question, limit cycles of slow-fast type, i.e. so-called relaxation oscillations appear quite often in applications and have therefore enjoyed a lot of attention during the last decades. Descriptions of relaxation oscillations were made in systems where the contact points are all of jump type (i.e. (2) with f (0, 0) = 0), see [34], [25], [27], [41], [32], . . . . These cycles come close to parts of the curve of singular points that are attracting, not repelling. Later, so-called canard type oscillations were discovered that contain both attracting and repelling branches of the curve of singular points. These oscillations were treated with many different techniques (complex analysis [36], matched asymptotics [15], nonstandard analysis [2], Gevrey asymptotics [18], geometric singular perturbation theory [8,35]). Small-amplitude cycles of slow-fast type were only discussed briefly in [8,28], and extensively in [9] in case of the codimension 1 Hopf bifurcations near contact points, and in [7] in case of codimension 2 bifurcations. In applications, planar canards have been used to model (bio)chemical reactions ([33], [20], [29], . . . ), electrical circuits, predator–prey models ([1], [31], . . . ), and are also appear as ode-model in the study of traveling wave solutions of partial differential equations ([40], . . . ). Phenomena besides limit cycles in slow-fast systems in two variables can be found in [38] (dynamics on tori), [11,26] (homoclinics), [17] (enhanced delay). Furthermore, it is of course clear that the results on two-dimensional slow-fast systems appear as subproblems in the study of 2 slow + 1 fast systems, or 1 slow + 2 fast systems. For those systems, there is a large amount of results, like before using many different techniques among which also geometric singular perturbation theory and blow-up, like the theory used in this paper. 2. Blow-up in the parameter space and statement of the results Without loss of generality, we may suppose b3 = ±1 in the expression (4) for f . We rewrite (2) as

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X ±,b,λ :



x˙ = y ,   y˙ = −xy +  b0 + b1 x + b2 x2 ± x3 + x4 H (x, λ) +  y 2 G (x, y , λ)

(5)

where G and H are smooth,   0 is the singular parameter kept small, b = (b0 , b1 , b2 ) is regular perturbation parameter close to 0 and λ ∈ Λ, with Λ a compact subset of some euclidean space. Definition 2.1. The family X +,b,λ represents slow-fast codimension 3 saddle bifurcations and X −,b,λ represents slow-fast codimension 3 elliptic bifurcations, in analogy with the terminology introduced in [10]. In the rest of the paper we will refer to X +,b,λ as the saddle case, and to X −,b,λ as the elliptic case. In studying limit cycles near the origin (x, y ) = (0, 0) of (5) it is important to get results on a neighborhood of (x, y ) = (0, 0) that does not shrink to the origin when  → 0 and b → 0. More precisely, we say that the cyclicity of the origin of (5) is bounded by N if there exists a neighborhood V of (0, 0) in (x, y )-space, a neighborhood W of (0, 0, 0) in b-space and an 0 > 0, such that for each ( , b, λ) ∈ [0, 0 ] × W × Λ, the system (5) has at most N-limit cycles inside V . (The minimum of such N is the cyclicity of the origin.) Main question: We investigate the number of limit cycles of (5) that can appear in a fixed neighborhood of the origin (x, y ) = (0, 0), i.e. we study the cyclicity of the origin. 2.1. Reparametrization of the parameter space This section is devoted to presenting detailed statements in different regions in (b0 , b1 , b2 )-space, near (b0 , b1 , b2 ) = (0, 0, 0). Depending on how ( , b0 , b1 , b2 ) is approaching (0, 0, 0, 0), different behavior can be observed, and there are subregions in parameter space that behave exactly like in the lower codimension cases described in Section 1. For example, after a simple rescaling like

  (x, y , b0 , b1 , b2 ) = R X , R 2 Y , ± R 3 , R 2 B 1 , R B 2 , the reader can readily verify that one finds a similar slow-fast vector field that in an O (1)-neighborhood of ( X , Y ) = (0, 0) behaves like a jump point in ( X , Y )-coordinates. Part of the study presented in this paper will relate the lower codimension results in rescaled coordinates to results in a uniform neighborhood in (x, y )-coordinates. For the moment, it suffices to conclude that one has to expect different results on small-amplitude limit cycles depending on the region in parameter space. Motivated by the above discussion, we reparametrize the b-parameters, by introducing weighted spherical parameters:

  (b0 , b1 , b2 ) = r 3 B 0 , r 2 B 1 , r B 2 ,

r  0, B = ( B 0 , B 1 , B 2 ) ∈ S2 .

This is in fact a blow-up of the origin in (b0 , b1 , b2 )-space: the study of the part of the parameter space near the origin is replaced by a study near r = 0, keeping ( B 0 , B 1 , B 2 ) ∈ S2 . Instead of using coordinates on the sphere, it is common practice to use the 6 charts of the sphere as shown in Fig. 2. We conveniently give names to the different regions covering the sphere: a) Jump region (C 0+ , C 0− ). (b0 , b1 , b2 ) = (±r 3 , r 2 B 1 , r B 2 ), ( B 1 , B 2 ) in an arbitrary compact subset of the plane. b) Slow-fast Hopf region (C 1− ). (b0 , b1 , b2 ) = (r 3 B 0 , −r 2 , r B 2 ), B 0 close to 0 and B 2 in an arbitrary compact interval. The size of the neighborhood of B 0 = 0 here, has to be taken into account when choosing the sizes of the compact regions in a). c) B 1 = 1 (C 1+ ). (b0 , b1 , b2 ) = (r 3 B 0 , r 2 , r B 2 ), B 0 close to 0 and B 2 in an arbitrary compact interval. The size of the neighborhood of B 0 = 0 here, has to be taken into account when choosing the sizes of the compact regions in a).

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Fig. 2. Six charts covering the sphere in ( B 0 , B 1 , B 2 )-space. The sixth chart C 0− is not shown: it is the symmetric counterpart of C 0+ on the backside of the spherical surface. Together with the radial parameter r, ( B 0 , B 1 , B 2 ) parametrize the (b0 , b1 , b2 )-space.

d) Slow-fast Bogdanov–Takens region (C 2+ , C 2− ). In this region we consider (b0 , b1 , b2 ) = (r 3 B 0 , r 2 B 1 , ±r ), B 0 and B 1 close to 0. Of course, in order to cover the entire sphere by the chosen charts, the size of the neighborhood for ( B 0 , B 1 ) here, has to be taken into account when choosing the sizes of the compact regions in a), b) and c). If we introduce this change in the parameter space in the family of vector fields X ±,b,λ , we obtain

an ( , B , r , λ)-family of vector fields in R2 :

X ±, B ,r ,λ :



x˙ = y ,   y˙ = −xy +  r 3 B 0 + r 2 B 1 x + r B 2 x2 ± x3 + x4 H (x, λ) +  y 2 G (x, y , λ).

(6)

Let us now state the results. The distinction between the saddle and the elliptic case (see Definition 2.1) is only relevant in the proofs of the statements, not in formulation the results themselves. Theorem 2.1 (The jump region). Let B 0 = +1 or B 0 = −1. Suppose that B 11 > 0 and B 12 > 0 are arbitrarily large and K is any compact set in the (x, y )-plane. There exist 0 > 0, r0 > 0 such that X ±, B ,r ,λ has no periodic orbits in K for ( , B 1 , B 2 , r , λ) ∈ [0, 0 ] × [− B 11 , B 11 ] × [− B 12 , B 12 ] × [0, r0 ] × Λ.

Theorem 2.1 will be proved in Section 3.4. The jump region resembles the codimension 0 situation described in Section 1, where b0 = 0. The contact point in our system is not exactly a contact point of jump type, but in the analysis of the system in this jump region, a jump-type contact point will appear after blow-up of the origin. Theorem 2.2 (C 1+ ). Let B 1 = +1. Given any B 12 > 0 and any compact set K in the (x, y )-plane. There exist 0 > 0, r0 > 0 and B 10 > 0 such that X ±, B ,r ,λ has no periodic orbits in K for ( , B 0 , B 2 , r , λ) ∈ [0, 0 ] ×

[− B 10 , B 10 ] × [− B 12 , B 12 ] × [0, r0 ] × Λ.

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Theorem 2.2 will be proved in Section 3.5. The region covered in Theorem 2.2 could be called the saddle region, because in the analysis of the system in this region, a contact point of saddle type will appear after blow-up of the origin. The saddle type contact points have rarely been looked at in the literature precisely because no periodic orbits can be nearby (the slow dynamics points in the wrong way). Hence, the result in this theorem is not surprising. Theorem 2.3 (The slow-fast Bogdanov–Takens region). Let B 2 = +1 or B 2 = −1. Suppose that K is any compact set in the (x, y )-plane. There exist 0 > 0, r0 > 0 and a ( B 0 , B 1 )-neighborhood W of the origin such that X ±, B ,r ,λ has at most one (hyperbolic) limit cycle in K for ( , B 0 , B 1 , r , λ) ∈ [0, 0 ] × W × [0, r0 ] × Λ. Theorem 2.3 will be proved in Section 3.6. Like before, the contact point of our system in the slow-fast BT-region is not of the slow-fast BT kind studied in [7], but after blow-up we do find a slow-fast BT-bifurcation to which we apply the results of [7]. In the next theorem, we would like to announce results for the parameter region C 1− . However, some case will be excluded, for which different techniques are necessary to bound the number of limit cycles. We will first choose B 1 = −1, B 0 ∼ 0, and keep B 2 in a large compact set avoiding the origin. The following theorem will be proved in Section 3.7. Theorem 2.4 (The slow-fast Hopf region for B 2 = 0). Let B 1 = −1. Suppose that K ⊂ (−∞, +∞) \ {0} is an arbitrary compact set. There exist 0 > 0, r0 > 0, B 10 > 0 and there exists a neighborhood V of (x, y ) = (0, 0) such that the following statements are true. (i) System X ±, B ,r ,λ has at most one (hyperbolic) limit cycle in V , for each ( , B 0 , B 2 , r , λ) ∈ [0, 0 ] ×

[− B 10 , B 10 ] × K × [0, r0 ] × Λ. (ii) Fixing ( , B 2 , r , λ) ∈ ]0, 0 ] × K × ]0, r0 ] × Λ, the B 0 -family X ±, B ,r ,λ undergoes, in V and at B 0 = 0, a Hopf bifurcation of codimension 1. Suppose that B 2 ∈ K ∩(0, +∞). When B 0 increases, there is (in V ) an attracting hyperbolic focus and no limit cycle; when B 0 decreases, there is (in V ) a repelling hyperbolic focus and an attracting limit cycle of which the size monotonically grows as B 0 decreases. Suppose that B 2 ∈ K ∩ (−∞, 0). When B 0 decreases, there is (in V ) a repelling hyperbolic focus and no limit cycle; when B 0 increases, there is (in V ) an attracting hyperbolic focus and a repelling limit cycle of which the size monotonically grows as B 0 increases. Let us denote by P the point in the B-space given by ( B 0 , B 1 , B 2 ) = (0, −1, 0). Define now





Q δ  = q ∈ S2 d(q, P ) < δ  ⊆ S2 , where δ  > 0 and d(q, P ) =



q21 + (q2 + 1)2 + q23 with q = (q1 , q2 , q3 ). Taking into account Theo-

rem 2.1, Theorem 2.2, Theorem 2.3 and Theorem 2.4 we find an upper bound on the number of limit cycles that my occur in (6) near the origin (x, y ) = (0, 0), but away from P in the B-parameter space: Theorem 2.5 (Away from the point P ). Given δ  > 0 arbitrary, there exist 0 > 0, r0 > 0 and a neighborhood V of (x, y ) = (0, 0) such that for each ( , r , B , λ) ∈ [0, 0 ] × [0, r0 ] × (S2 \ Q δ  ) × Λ the system (6) (i.e. X ±, B ,r ,λ ) restricted to V has at most one hyperbolic limit cycle. In the next theorem, we give partial results near P , and, from the announcement, it is immediately clear that near P the situation is more intricate. On one hand, we find more than one limit cycle, and on the other hand the cyclicity seems to depend on the higher order perturbation terms in H (x, λ), in analogy with the results in [9]. Theorem 2.6 (Near the point P ). Consider system (6) with H (0, λ) = 0 for all λ ∈ Λ. For sufficiently small, system (6) contains a saddle-node bifurcation of limit cycles near P .

 > 0 and r > 0

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Fig. 3. Different charts near the blow-up sphere in (x, y , r )-space. The chart { y¯ = +1} is not shown: it is on the back side of the sphere and is the symmetric counterpart of { y¯ = −1}.

Remark 3. Theorem 2.6 will be proven in Section 3.8. In Section 3.8, we also detect all closed curves from which limit cycles may bifurcate, for  > 0, r > 0 and for parameters B near P , and find an upper bound for the cyclicity of a good number of them. The cyclicity of some delicate closed curves will be subject to further investigation. Throughout Section 3.8, we suppose that H (0, λ) = 0 for all λ ∈ Λ. 3. Proofs of Theorem 2.1–Theorem 2.6 System X ±, B ,r ,λ in (6) is a singular perturbation problem with singular parameter  and will be studied using general principles from geometric singular perturbation theory (GSPT). For  = 0, the only difficult point in the analysis of (6) is the contact point at the origin, at which the linearization admits the nilpotent matrix



0 1 0 0

.

Standard extension of GSPT would lead us to include  in the phase space and blow up the origin in (x, y ,  )-space. Like in [5], it is better however to first desingularize the system in another way prior to doing the classical blow-up. Let us explain why, by looking at the family (5). Suppose that b0 = 0 in (5). Then a blow-up would require weights (1, 2, 3) for (x, y ,  ). When b0 = 0, b1 = 0, different weights are required, and a different set of weights is needed to study the case where b0 = b1 = 0, b2 = 0. In family (6), all these different situations come together, and we need a way to separate the different blow-up regimes. In a first step to achieve this, we have rewritten the parameter space in Section 2.1. In the next section we will make a blow-up in (x, y , r )-space. 3.1. Primary blow-up Leaving

 as a parameter, we blow up the origin using   (x, y , r ) = u x¯ , u 2 y¯ , ur¯ ,

u  0, r¯  0, (¯x, y¯ , r¯ ) ∈ S2 .

(7)

The study of the dynamics in the blown-up coordinates will be done in different charts (Fig. 3). The family chart r¯ = 1 is the traditional rescaling chart, and it amounts to making a standard rescaling of the phase variables (x, y ). There are also the phase-directional rescaling charts “ x¯ = +1, x¯ = −1, y¯ = +1, y¯ = −1” that link the family chart to the original phase plane. The study of these intermediate charts is quite standard in Theorems 2.1, 2.2 and 2.3: in those cases, all limit cycles under

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Fig. 4. Blow-up phase space seen from the top of the blow-up sphere (the  -direction is perpendicular to the picture). The dynamics of the vector field X ±, B ,r ,λ + 0 ∂∂r in the phase directional charts, for  > 0, outside the region D. (a) the saddle case, (b) the elliptic case. Both P  and Q  are hyperbolic, though two eigenvalues of P  → 0 as  → 0: the separatrix that leaves the sphere at P  tends to the line of singular points as  → 0. Similarly for the singularities at the left side of the sphere.

consideration spend little time in these charts, and the majority of the computations is done in the traditional chart. From Theorem 2.4 onwards, the situation is more delicate as there are some limit periodic sets (out of which limit cycles perturb) that have large parts on the “equator” of the blow-up locus, and for which the essential parts of the study are precisely done in the phase directional charts. Let us now discuss the individual charts. a) The family chart for the primary blow-up. This chart is obtained by taking r¯ = 1 in (7) and keeping (¯x, y¯ ) in some arbitrarily large disk in R2 . In this family chart, the vector field X ±, B ,r ,λ + 0 ∂∂r yields, after division by the positive factor u,

X F,±B ,u ,λ :

⎧ ⎨ x˙¯ = y¯ ,   y˙¯ = −¯x y¯ +  B 0 + B 1 x¯ + B 2 x¯ 2 ± x¯ 3 + u x¯ 4 H (u x¯ , λ)   ⎩ + u y¯ 2  G u x¯ , u 2 y¯ , λ ,

(8)

where we treat u = r as a parameter. Expression (8) represents again a singular perturbation problem with singular parameter  and with regular parameters ( B , u , λ). At a first glance, it appears that, when comparing this form to the original form (6), not much is gained. The main difference lies in the fact that the coefficients in front of 1, x¯ and x¯ 2 in (8) can no longer take small values simultaneously (( B 0 , B 1 , B 2 ) ∈ S2 ). b) The phase-directional charts for the primary blow-up. To get information on a fixed (x, y )-neighborhood of the origin (x, y ) = (0, 0) (independent of r and  ) we explicitly have to look near the “equator”, i.e. in the charts {¯x = +1}, {¯x = −1}, { y¯ = +1}, { y¯ = −1}. The following analysis will show that the dynamics near the equator is like in Fig. 4. b1) The phase-directional chart {¯x = +1}. The part of the sphere where x¯ ∼ 1 can be studied in rectified coordinates, with the directional blow-up

  (x, y , r ) = U , U 2 Y¯ , U R¯ ,

U  0.

(9)

In this phase directional chart, the vector field X ±, B ,r ,λ + 0 ∂∂r yields, after division by the positive factor U ,

R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

P ±

X  ,1B ,λ :

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⎧ ⎨ U˙ = U Y¯ , R˙¯ = − R¯ Y¯ ,   ⎩ ˙¯ Y = −Y¯ − 2Y¯ 2 +  R¯ 3 B 0 + R¯ 2 B 1 + R¯ B 2 ± 1 + O (U ) .

(10)

P ±

It is not difficult to see the following facts. On {U = 0, R¯ = 0} X  ,1B ,λ has singularities at Y¯ ± = ± + O ( 2 ) and Y¯ ± = − 12 ∓  + O ( 2 ), which we denote by P  and Q  respectively. There are no other singularities for  > 0 by taking U , R¯ sufficiently small. For  = 0 there is P ± the plane {Y = 0} of singularities of X  ,1B ,λ . An analysis of the linear part shows the following: We have at Q  a hyperbolic (resonant) saddle and at P  a hyperbolic (resonant) saddle with the exception for  = 0 where we get a semi-hyperbolic singularity with the Y¯ -axis as stable manifold and a two-dimensional center direction, transverse to the Y¯ -axis. Moreover, the P ± above expression also implies that the vector field X 0,1B ,λ is semi-hyperbolic along the plane Y¯ = 0. P ± As the point (U , R¯ , Y¯ ) = (0, 0, 0) is a semi-hyperbolic singularity of X 0,1B ,λ , there exists a P ±

C k ( B , λ)-family of center manifolds of the extended family of vector fields X  ,1B ,λ + 0 ∂∂ at (U , R¯ , Y¯ ,  ) = (0, 0, 0, 0). In fact the ( B , λ)-family of the center manifolds of the extended P ± family forms an ( , B , λ)-family of invariant manifolds of X  ,1B ,λ . It can be checked that all invariant manifolds are graphs of the form





Y¯ =  ±1 + O (U , R¯ ,  ) . It follows that the invariant behavior on U = 0 is given by





R˙¯ = − R¯ ±1 + O ( R¯ ,  ) , and the invariant behavior on R¯ = 0 is given by





U˙ =  U ±1 + O (U ,  ) . Putting all the information together we find that, for  > 0, in the saddle case the singularity P  is an attracting and hyperbolic node on U = 0 and a hyperbolic saddle on R¯ = 0. In the elliptic case the singularity P  is an attracting and hyperbolic node on R¯ = 0 and a hyperbolic saddle on U = 0. b2) The phase-directional chart {¯x = −1}. This part of the sphere can be studied in rectified coordinates, with the directional blow-up formula

  (x, y , r ) = −U , U 2 Y¯ , U R¯ ,

U  0.

There is no need to make calculations in the {¯x = −1}-chart, we merely apply the coordinate change (U , R¯ , t ) → (−U , − R¯ , −t ) to (10) and its analysis in section b1). b3) The phase-directional charts { y¯ = +1} and { y¯ = −1}. The part of the sphere where y¯ ∼ ±1 can be studied in rectified coordinates, with the directional blow-up formula

  (x, y , r ) = U X¯ , ±U 2 , U R¯ ,

U  0.

In the resulting system, it is not hard to see that (U , R¯ , X¯ ) = (0, 0, 0) is not a singular point; near X¯ = 0, the dynamics on the equator show a regular movement from left to right in the chart { y¯ = 1} and from right to left in { y¯ = −1}. All singularities on the equator can hence be studied with the charts {¯x = +1} and {¯x = −1}.

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Putting all the information together, we represent in Fig. 4 the dynamics of the blown-up vector field near the equator of the blow-up locus {u = 0}, for  > 0. In other words, we represent the dynamics of the vector field X ±, B ,r ,λ + 0 ∂∂r in the phase directional charts of the primary blow-up, for  > 0. c) Combining all charts of the primary blow-up. Our aim is first to study limit cycles of singular perturbation problem (8) in an (arbitrarily) large compact set D in the (¯x, y¯ )-plane (see Fig. 4). Bearing in mind the blow-up formula (7) we see that D blows down to a domain in the (x, y )-plane that shrinks to the origin when r → 0. Besides limit cycles confined in D the vector field (6) might have other small-amplitude limit cycles close to the blow-up sphere. We will show however that such cycles can only appear in the slow-fast Hopf region C 1− in the elliptic case. In fact, from Fig. 4(a) it is immediately clear that all small-amplitude limit cycles are confined to D in the saddle case. 3.2. Slow-fast analysis in the family chart of the primary blow-up Because the primary blow-up did not include  , it is not surprising that the blown-up vector field X F,±B ,u ,λ is still singularly perturbed in  . For  = 0 the critical curve (the curve of singularities) of

X 0F,±B ,u ,λ is given by { y¯ = 0}. We are interested in the cyclicity of so-called canard limit periodic sets,

i.e. we are interested in the number of isolated periodic orbits of the vector field X F,±B ,u ,λ that can bifurcate from a limit periodic set having a branch of singular points (a section of the critical curve). We are also interested in so-called small-amplitude limit cycles, i.e. we also want to study the number of isolated periodic orbits of X F,±B ,u ,λ that can bifurcate from (¯x, y¯ ) = (0, 0). Let us get better insight in the problem above. Families of vector fields like (8) are analyzed by considering two different limiting systems. On one hand, one studies the fast subsystem X 0F,±B ,u ,λ , which is obtained by setting  = 0 in (8). The critical curve consists of partially hyperbolic singularities, except at the origin (¯x, y¯ ) = (0, 0), where we deal with a nilpotent singularity. We see that the curve is normally attracting when x¯ > 0 and normally repelling when x¯ < 0. All movement of X 0F,±B ,u ,λ

happens along fibers y¯ = − 12 x¯ 2 + C . The dynamics of X F,±B ,u ,λ for small values of  is reflected more or less by the dynamics of the fast subsystem X 0F,±B ,u ,λ , especially away from its singularities. Close to the critical curve a second subsystem becomes important: the slow subsystem. That system is given by the slow dynamics along the critical curve but outside the contact point:

x¯  =

B 0 + B 1 x¯ + B 2 x¯ 2 ± x¯ 3 + u x¯ 4 H (u x¯ , λ) x¯

.

(11)

Orbits of X F,±B ,u ,λ are typically perturbations of combinations of trajectories of the fast subsystem and trajectories of the slow subsystem. We are interested in those combinations from which limit cycles of the vector field X F,±B ,u ,λ can bifurcate. We have two different kinds of such combinations: a) Canard limit periodic sets Γ y¯ . For each y¯ > 0 we can consider the combination of the orbit of the fast subsystem through the point (0, y¯ ) and the piece of the critical curve between the α -limit set and the ω -limit set of that fast orbit. b) The nilpotent (contact) point (¯x, y¯ ) = (0, 0). It is clear that we will have limit cycles near Γ y¯ only for those parameters ( B , u , λ) for which the slow dynamics allows the passage from the attracting part of the critical curve to the repelling part of the critical curve. Coupled with the slow dynamics, we also need to look at the nilpotent contact point. We intend to study the nilpotent point by means of a blow-up. We add the equation ˙ = 0 to (8) and, depending on the chosen chart in the parameter space, we make a so-called secondary blow-up at the origin (¯x, y¯ ,  ) = (0, 0, 0).

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Fig. 5. A (schematic) bird’s eye view of the primary and secondary blow-up in the saddle and elliptic case, for r =  = 0. The primary blow-up is done in (x, y , r )-space while the secondary blow-up is done in (¯x, y¯ ,  )-space.

As mentioned before, a different set of weights is necessary in the parameter regions. In the charts C 0+ and C 0− , we will use

  (¯x, y¯ ,  ) = v x˜ , v 2 y˜ , v 3 ˜ . In the charts C 1+ and C 1− , there are 2 relevant small parameter ( , B 0 ) and it will be necessary to rewrite these parameters as ( , B 0 ) = (δ 2 E , δ B¯ 0 ), with δ > 0, ( E , B¯ 0 ) ∈ S1 ; this has the benefit of including only 1 parameter δ in the blow-up construction. We will then use the weights

  (¯x, y¯ , δ) = v x˜ , v 2 y˜ , v δ˜ . In the charts C 2+ and C 2− , there are 3 relevant small parameters ( , B 0 , B 1 ) that need to be included in the blow-up construction. Fortunately, this case has already been completely treated in [7] and we do not repeat the blow-up construction here. 3.3. Combining the primary blow-up and the secondary blow-up In Fig. 5 we represent the phase portrait of X 0F,±B ,0,λ , inside the primary blow-up locus, on which we represent the information at infinity in a (1, 2) quasi-homogeneous compactification. The dynamics away from the origin (x, y ) = (0, 0) (the primary blow-up locus) is obtained by studying the vector field X 0±, B ,r ,λ in the phase-directional charts of the primary blow-up.

Recall that the dynamics of X F,±B ,0,λ in a family chart of the secondary blow-up depends on the chosen charts C k+ and C k− in the parameter space. It means that the dynamics on the secondary blow-up locus depends on the chosen chart in the parameter space. In later sections we will see that the dynamics of X F,±B ,0,λ in phase-directional charts of the secondary blow-up will look like the one depicted in Fig. 5. Combining now Fig. 4(a) and Fig. 5 we see once more that all limit cycles in the saddle case can be obtained by considering the vector field X F,+B ,u ,λ in a sufficiently large neighborhood of the origin in the (¯x, y¯ )-space. 3.4. Proof of Theorem 2.1 ± We consider the singular perturbation system X F,(± 1, B 1 , B 2 ),u ,λ . The parameters ( B 1 , B 2 ) are kept in an arbitrary compact set,  ∼ 0, u ∼ 0 and λ ∈ Λ. The origin (¯x, y¯ ) = (0, 0) is now a jump point for

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± X F,(± x, y¯ ,  ) = (0, 0, 0) using the blow-up 1, B 1 , B 2 ),u ,λ . As mentioned before, we blow up the origin (¯ transformation (secondary blow-up):

  (¯x, y¯ ,  ) = v x˜ , v 2 y˜ , v 3 ˜ ,

v  0, (˜x, y˜ , ˜ ) ∈ S2+ ,

(12)

where S2+ is the half-sphere where ˜  0. The family chart is obtained by taking ˜ = 1 in expression (12):

  (¯x, y¯ ,  ) = v x˜ , v 2 y˜ , v 3 ,

(13)

± and keeping in mind that (˜x, y˜ ) is in a compact set. The system X F,(± 1, B 1 , B 2 ),u ,λ changes, after dividing by v, into:

⎧ ⎪ ⎨ x˙˜ = y˜ , y˙˜ = −˜x y˜ ± 1 + B 1 v x˜ + B 2 v 2 x˜ 2 ± v 3 x˜ 3 + uv 4 x˜ 4 H (uv x˜ , λ)   ⎪ ⎩ + uv 4 y˜ 2 G vu x˜ , v 2 u 2 y˜ , λ .

(14)

For v = 0 expression (14) turns to



x˙˜ = y˜ , y˙˜ = −˜x y˜ ± 1.

(15)

System (15) has no singularities. The part of the sphere where x˜ ∼ +1 can be studied in rectified coordinates, with the directional blow-up formula

  (¯x, y¯ ,  ) = V , V 2 Y˜ , V 3 E˜ ,

V  0.

± ∂ In this phase directional chart, the vector field X F,(± 1, B 1 , B 2 ),u ,λ + 0 ∂  yields, after division by the positive factor V ,

⎧ V˙ = V Y˜ , ⎪ ⎪ ⎪ ˙˜ ⎨ E = −3 E˜ Y˜ ,   ⎪ Y˙˜ = −Y˜ − 2Y˜ 2 + E˜ ±1 + B 1 V + B 2 V 2 ± V 3 + uV 4 H (uV , λ) ⎪ ⎪   ⎩ + uV 4 E˜ Y˜ 2 G uV , u 2 V 2 Y˜ , λ .

(16)

On { V = 0, E˜ = 0} (16) has singularities at Y˜ = 0 and Y˜ = − 12 . A linear analysis shows that we have

at Y˜ = − 12 a resonant saddle and at y˜ = 0 a semi-hyperbolic singularity with the Y˜ -axis as stable manifold and a two-dimensional center manifold. Center manifolds are expressed by









y˜ = E˜ ±1 + O ( V ) + O E˜ 2 . Each center manifold contains the line of singularities { E˜ = Y˜ = 0}. The center behavior on V = 0, in ˜ is given by: terms of E,

˙





E˜ = ∓3 E˜ 2 + O E˜ 3 .

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Fig. 6. Dynamics near the secondary blow-up locus. (a) The case B 0 = 1. (b) The case B 0 = −1. In both cases, no orbit can connect attracting branch of singular points to the repelling branch of singular points, so there is no possibility of having canards.

Remark 4. Notice that ± in the formulas above means: + for the case B 0 = 1 and − for the case B 0 = −1. Similarly, ∓ means: − for the case B 0 = 1 and + for B 0 = −1. This sign convention should not cause confusion with the ± sign, distinguishing the saddle and elliptic cases. The part of the sphere where x˜ ∼ −1 can be studied by applying the coordinate change ( V , E˜ , t ) → (− V , − E˜ , −t ) to (16). Besides the singularities we already found in the charts {˜x = ±1}, there are no extra singularities in the charts { y˜ = ±1}. Remark 5. It can be easily seen that there exist E˜ 0 > 0 and V 0 > 0 such that the system (16) has no singularities for ( V , E˜ ) ∈ [0, V 0 ]× ]0, E˜ 0 ], with λ ∈ Λ, u ∼ 0 and ( B 1 , B 2 ) in the supposed compact set. Using a similar reasoning in the other phase-directional charts and using the structural stability ± of the system (15) we see that there are no singularities of the system X F,(± 1, B 1 , B 2 ),u ,λ in a fixed ¯ neighborhood of the origin (¯x, y ) = (0, 0) for  > 0. It follows that there are no small-amplitude ± periodic orbits of the vector field X F,(± 1, B , B ),u ,λ under the given conditions on the parameters. 1

2

± In order to see that we have no limit cycles of X F,(± 1, B 1 , B 2 ),u ,λ close to canard limit periodic sets Γ y¯ , we refer to Fig. 6. In Fig. 6 we represent the phase portrait of (15), inside the circle, on which we represent the information at infinity in a ( 13 , 23 ) quasi-homogeneous compactification. The information at infinity is included in the phase-directional charts of the secondary blow-up (12). As a connection ± between the two branches of the critical curve of X F,(± 1, B , B ),u ,λ is absent, no canard limit cycles of 1

2

± the vector field X F,(± 1, B 1 , B 2 ),u ,λ are present. On account of the absence of the connection we see that X ±,(±1, B , B ),r ,λ has no limit cycles in any 1 2 compact set in the (x, y )-plane, for  , r small enough. This completes the proof of Theorem 2.1.

3.5. Proof of Theorem 2.2 We consider the singular perturbation system X F,(±B ,1, B ),u ,λ where B 2 is in an arbitrary compact 0 2 set, B 0 ∼ 0,  ∼ 0, u ∼ 0 and λ ∈ Λ. In order to desingularize the system X F,(±B ,1, B ),u ,λ we introduce 0 2 rescaling in the parameter space:

  ( , B 0 ) = δ 2 E , δ B¯ 0 ,

( E , B¯ 0 ) ∈ S1 ,

The calculations will be performed, as usual, in charts.

δ ∼ 0, E  0.

(17)

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3.5.1. ( , B 0 ) = (δ 2 , δ B¯ 0 ), B¯ 0 in a large compact set . We blow up the origin (¯x, y¯ , δ) = We study the singular perturbation system X F2± ¯ δ ,(δ B 0 ,1, B 2 ),u ,λ (0, 0, 0) using the blow-up transformation (secondary blow-up):

  (¯x, y¯ , δ) = v x˜ , v 2 y˜ , v δ˜ ,

(18)

˜ ∈ S2+ and S2+ is the half-sphere with δ˜  0. The family chart is obtained by where v  0, (˜x, y˜ , δ) taking δ˜ = 1 in expression (18):

  (¯x, y¯ , δ) = v x˜ , v 2 y˜ , v , and keeping in mind that (˜x, y˜ ) is in a compact set. The system X F2± ¯ changes, after dividing δ ,(δ B 0 ,1, B 2 ),u ,λ by v, into:

⎧ ⎪ ⎨ x˙˜ = y˜ , y˙˜ = −˜x y˜ + B¯ 0 + x˜ + B 2 v x˜ 2 ± v 2 x˜ 3 + uv 3 x˜ 4 H (uv x˜ , λ) ⎪   ⎩ + uv 3 y˜ 2 G vu x˜ , v 2 u 2 y˜ , λ .

(19)

For v = 0 expression (19) turns to



x˙˜ = y˜ , y˙˜ = −˜x y˜ + B¯ 0 + x˜ .

(20) B¯ 0 ±( B¯ 2 +4)1/2

0 System (20) has one singularity (˜x, y˜ ) = (− B¯ 0 , 0) with eigenvalues . Hence system (20) 2 has one hyperbolic saddle. The part of the sphere where x˜ ∼ +1 can be studied in rectified coordinates, with the directional blow-up formula

  ˜ , (¯x, y¯ , δ) = V , V 2 Y˜ , V

V  0.

(21)

∂ In this phase directional chart, the vector field X F2± ¯ + 0 ∂δ yields, after division by the δ ,(δ B 0 ,1, B 2 ),u ,λ positive factor V ,

⎧ V˙ = V Y˜ , ⎪ ⎪ ⎪ ⎪ ˜˙ ⎨ ˜ Y˜ ,

= −

  ˙ ˜2

˜ B¯ 0 + 1 + B 2 V ± V 2 + uV 3 H (uV , λ) ⎪ Y˜ = −Y˜ − 2Y˜ 2 +

⎪ ⎪ ⎪   ⎩ ˜ 2 Y˜ 2 G uV , u 2 V 2 Y˜ , λ . + uV 3

(22)

˜ = 0} (22) has singularities at Y˜ = 0 and Y˜ = − 1 . The eigenvalues of the linear part at On { V = 0,

2 Y˜ = 0 are given by (0, 0, −1) and at Y˜ = − 12 by (− 12 , 12 , 1). So we have at Y˜ = − 12 a resonant saddle

and at Y˜ = 0 a semi-hyperbolic singularity with the Y˜ -axis as stable manifold and a two-dimensional center manifold. Center manifolds are expressed by





˜ 2 1 + O ( V ) + O ( ) ˜ . Y˜ =

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Fig. 7. Dynamics on the secondary blow-up locus for B 1 = 1 and B¯ 0 = 0. The dynamics shows that only false canard orbits can appear (i.e. orbits near the repelling branch that connect to the attracting branch but fail to return and form cycles).

˜ = Y˜ = 0}. The center behavior on V = 0, in Each center manifold contains the line of singularities {

˜ , is given by: terms of

  ˙˜ = −

˜ . ˜ 3 1 + O ( )

˜ t) → The part of the sphere where x˜ ∼ −1 can be studied by applying the coordinate change ( V , , ˜ −t ) to (22). Besides the singularities we already found in the charts {˜x = +1} and {˜x = −1}, (− V , − , there are no extra singularities in the charts { y˜ = +1} and { y˜ = −1}. look After putting all the information together, we find how the phase portraits of X F2± ¯ δ ,(δ B 0 ,1, B 2 ),u ,λ like near the origin (¯x, y¯ ) = (0, 0), for δ ∼ 0. In Fig. 7 we represent the phase portraits of (20) for B¯ 0 = 0, inside the circle, on which we represent the information at infinity in a (1, 2) quasi-homogeneous compactification. We deal with similar phase portraits if B¯ 0 = 0 and B¯ 0 in the supposed compact set.

˜ 0 > 0 and V 0 > 0 such that the system (22) has no Remark 6. It can be easily seen that there exist

˜ ∈ [0, V 0 ]×]0,

˜ 0 ], with λ ∈ Λ, u ∼ 0 and ( B¯ 0 , B 2 ) in the supposed compact singularities for ( V , ) set. Using a similar reasoning in the other phase-directional charts of the secondary blow-up and using the structural stability of the system (20) in compact sets we see that there is one hyperbolic saddle of the system X F2± ¯ in a fixed neighborhood of the origin (¯x, y¯ ) = (0, 0) for δ > 0, δ ,(δ B 0 ,1, B 2 ),u ,λ

δ ∼ 0, λ ∈ Λ, u ∼ 0 and ( B¯ 0 , B 2 ) in the compact set above. It follows that there are no small-amplitude under the given conditions on the parameters. periodic orbits of the vector field X F2± ¯ δ ,(δ B 0 ,1, B 2 ),u ,λ

In order to see that we have no limit cycles of X F2± ¯ close to canard limit periodic sets δ ,(δ B 0 ,1, B 2 ),u ,λ 2 Γ y¯ , we need to study the slow dynamics (11). Since B 1 = 1,  = δ and B 0 = δ B¯ 0 , the expression (11) changes to

x¯  = 1 + B 2 x¯ ± x¯ 2 + u x¯ 3 H (u x¯ , λ). This expression implies that x¯  |x¯ =0 > 0. Consequently, the slow dynamics points from the left to the right near x¯ = 0 and we have no canard limit cycles in the (¯x, y¯ )-plane (see also Fig. 7). In the elliptic case, on account of that specific slow dynamics we also see that there can be no limit cycles near the equator of the primary blow-up locus and there can be no (detectable) canard limit cycles in the (x, y )-plane, in any compact set. Of course, there is no need to look for such limit cycles in the saddle case. Putting all facts together we obtain: Given any B¯ 10 > 0 and B 12 > 0 and any compact set K in the (x, y )-plane. There exist δ0 > 0, r0 > 0 such that X ±2 ¯ has no limit cycles in K for (δ, B¯ 0 , B 2 , r , λ) ∈ [0, δ0 ] × [− B¯ 10 , B¯ 10 ] × [− B 12 , B 12 ] × δ ,(δ B 0 ,1, B 2 ),r ,λ

[0, r0 ] × Λ.

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3.5.2. ( , B 0 ) = (δ 2 E , ±δ), E ∼ 0 We consider the singular perturbation system X δF2±E ,(±δ,1, B ),u ,λ with δ as singular perturbation 2 parameter. We perform a blow-up (a secondary blow-up), similar to the secondary blow-up (18):

  (¯x, y¯ , δ) = v x˜ , v 2 y˜ , v δ˜ ,

(23)

˜ ∈ S2+ and S2+ is the half-sphere with δ˜  0. The family chart is obtained by where v  0, (˜x, y˜ , δ) taking δ˜ = 1 in expression (23):

  (¯x, y¯ , δ) = v x˜ , v 2 y˜ , v , and keeping in mind that (˜x, y˜ ) is in a compact set. The system X δF2±E ,(±δ,1, B ),u ,λ changes, after divid2 ing by v, into:

⎧ ˙ ⎪ ⎨ x˜ = y˜ ,   y˙˜ = −˜x y˜ + E ±1 + x˜ + B 2 v x˜ 2 ± v 2 x˜ 3 + uv 3 x˜ 4 H (uv x˜ , λ) ⎪   ⎩ + uv 3 y˜ 2 E G vu x˜ , v 2 u 2 y˜ , λ ,

(24)

with E as small parameter. The origin (˜x, y˜ ) = (0, 0) is now a jump point for (24) (see Section 3.4). In order to desingularize system (24), we use a so-called tertiary blow-up at the origin (˜x, y˜ , E ) = (0, 0, 0), identical to the secondary blow-up (12). The study of (24) in the different charts of the tertiary blow-up is completely analogous to the study of the jump case in Section 3.4. In fact, we can expect limit cycles of (24) near the nilpotent contact point (˜x, y˜ ) = (0, 0) or near the canard limit periodic sets attached to y˜ > 0 (see Section 3.2). As a consequence of the fact that a connection, on the corresponding tertiary blow-up locus, between the two branches of the critical curve { y˜ = 0} is absent, there can be no canard limit cycles of (24). Bearing in mind Remark 5 there can be no small-amplitude limit cycles of (24), near (˜x, y˜ ) = (0, 0). In order to be able to say something about limit cycles of X δF2±E ,(±δ,1, B ),u ,λ in a fixed neighbor2

∂ hood of the origin (¯x, y¯ ) = (0, 0), we need to study the vector field X δF2±E ,(±δ,1, B ),u ,λ + 0 ∂δ in phase 2 directional charts of the secondary blow-up (23). The part of the sphere where x˜ ∼ +1 can be studied in rectified coordinates, with the directional blow-up formula

  ˜ , (¯x, y¯ , δ) = V , V 2 Y˜ , V

V  0.

∂ In this phase directional chart, the vector field X δF2±E ,(±δ,1, B ),u ,λ + 0 ∂δ yields, after division by the 2 positive factor V ,

⎧ V˙ = V Y˜ , ⎪ ⎪ ⎪˙ ⎪ ⎨

˜ = −

˜ Y˜   ˙ ˜ + 1 + B 2 V ± V 2 + uV 3 H (uV , λ) ˜ 2 E ±

⎪ Y˜ = −Y˜ − 2Y˜ 2 +

⎪ ⎪ ⎪   ⎩ ˜ 2 E G uV , u 2 V 2 Y˜ , λ . + uV 3 Y˜ 2

(25)

˜ = 0} (25) has singularities at Y˜ = 0 and Y˜ = − 1 . The eigenvalues of the linear part at On { V = 0,

2 Y˜ = 0 are given by (0, 0, −1) and at Y˜ = − 12 by (− 12 , 12 , 1). So we have at Y˜ = − 12 a resonant saddle

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and at Y˜ = 0 a semi-hyperbolic singularity with the Y˜ -axis as stable manifold and a two-dimensional center manifold. Center manifolds are expressed by





˜ 2 E 1 + O ( V ) + O ( ) ˜ . Y˜ =

˜ = Y˜ = 0}. If E = 0, then each center maniEach center manifold contains the line of singularities {

˜ , is given fold contains the plane of singularities {Y˜ = 0}. The center behavior on V = 0, in terms of

by:

  ˙˜ = −

˜ . ˜ 3 E 1 + O ( )

˜ t) → The part of the sphere where x˜ ∼ −1 can be studied by applying the coordinate change ( V , , ˜ −t ) to (25). Besides the singularities we already found in the charts {˜x = +1} and {˜x = −1}, (− V , − , there are no extra singularities in the charts { y˜ = +1} and { y˜ = −1}.

Putting the information from the different charts together, we see that dynamics of X δF2±E ,(±δ,1, B ),u ,λ 2 in the phase directional charts of the secondary blow-up (23) are identical to the dynamics shown ˜ 0 > 0, V 0 > 0 in Fig. 7, for E > 0 and E ∼ 0. Furthermore, it can be easily seen that there exist

˜ E ) ∈ [0, V 0 ]×]0,

˜ 0 ]× ]0, E 0 ], and E 0 > 0 such that the system (25) has no singularities for ( V , , with λ ∈ Λ, u ∼ 0 and B 2 in the supposed compact set. Using a similar reasoning in the other phase-directional charts and using the fact that system (24) has one hyperbolic saddle in an arbitrary compact set in the (˜x, y˜ )-plane for E > 0 and v ∼ 0, we find an ( E , δ, B 2 , u , λ)-uniform neighborhood of (¯x, y¯ ) = (0, 0) in which X δF2±E ,(±δ,1, B ),u ,λ has one hyperbolic saddle. It follows that we have no 2

small-amplitude periodic orbits. Using the fact that the point (˜x, y˜ ) = (0, 0) is the jump point for (24) once more we see that there can be no canard limit cycles of X δF2±E ,(±δ,1, B ),u ,λ , close to canard limit periodic sets Γ y¯ in the 2

(¯x, y¯ )-plane, no limit cycles at infinity (see Fig. 4) and no canard limit cycles of X δ±2 E ,(±δ,1, B ),r ,λ in an 2 arbitrary compact set in the (x, y )-plane.

Putting all facts together we obtain: Given any B 12 > 0 and any compact set K in the (x, y )-plane. There exist δ0 > 0, r0 > 0 and E 0 > 0 such that X δ±2 E ,(±δ,1, B ),r ,λ has no limit cycles in K for ( E , δ, B 2 , r , λ) ∈ [0, E 0 ] × [0, δ0 ] × [− B 12 , B 12 ] × [0, r0 ] × Λ. 2

We finish the proof of the theorem by combining this result and the result in Section 3.5.1. 3.6. Proof of Theorem 2.3 In the slow-fast Bogdanov–Takens region, we consider the singular perturbation system X F,(±B , B ,+1),u ,λ in the primary blow-up: 0 1



x˙¯ = y¯ ,      y˙¯ = −¯x y¯ +  B 0 + B 1 x¯ + x¯ 2 ± x¯ 3 + O u x¯ 4 + O u y¯ 2 ,

(26)

where ( B 0 , B 1 ) ∼ (0, 0),  ∼ 0, u ∼ 0 and λ ∈ Λ. The O (u x¯ 4 ) and O (u y¯ 2 ) terms come from contributions of G and H , and are not relevant in this subsection. System (26) deals with parameter chart C 2+ ; the study of the parameter chart C 2− can be reduced to this study upon applying the coordinate change (¯x, B 0 , u , t ) → (−¯x, − B 0 , −u , −t ) to (26), so in view of proving Theorem 2.3, it suffices to consider (26). We refer to singular perturbation problem (26) as a standard slow-fast Bogdanov–Takens bifurcation that has been studied in detail in [7]. From [7] it follows that in a standard slow-fast Bogdanov– Takens bifurcation we encounter canard-type relaxation oscillations, of a quite small amplitude, i.e. the size of which tends to zero for ( B 0 , B 1 ) → (0, 0). In other words, we encounter small-amplitude limit cycles near the nilpotent contact point (¯x, y¯ ) = (0, 0).

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Fig. 8. A bird’s eye view of the primary blow-up with the limiting boundary and indication of the slow dynamics. (a) The elliptic case. (b) The saddle case. The boundary consists, like in a flow-box, of two orbits, an inset and an outset.

More precisely, there is a uniform neighborhood V of (¯x, y¯ ) = (0, 0) and a neighborhood for ( , B 0 , B 1 , u ) = (0, 0, 0, 0) such that X F,(±B 0 , B 1 ,+1),u ,λ has at most one limit cycle in V , and when it

appears, it is hyperbolic. The shape of such a neighborhood V is very specific, as can be seen in Fig. 8. For more detail on V , we refer to [7]. Here, it suffices to keep in mind that V does not shrink to the origin as ( , B 0 , B 1 , u ) → (0, 0, 0, 0). We recall from Section 3.2 that limit cycles can occur near (¯x, y¯ ) = (0, 0) (as is the case here), but they could a priory also occur near canard limit periodic sets Γ y¯ . Such limit cycles would however be “detectable canard cycles” for the blown-up system (26). In [6], it is shown that the presence of such limit cycles would contradict the orientation of the slow dynamics of (26), see also Fig. 8. As a consequence, the original family (6) of vector fields, considered in the slow-fast BT parameter region, and for  > 0 small enough, can have no other limit cycles besides the ones in the blow-down of V . Since there is at most one, we have proved Theorem 2.3. 3.7. Proof of Theorem 2.4

We consider the singular perturbation system X F,(±B ,−1, B ),u ,λ where  ∼ 0, B 0 ∼ 0, u ∼ 0, λ ∈ Λ 0 2 and B 2 = 0. Without loss of generality we can suppose that B 2 ∈ K for any compact set K ⊂]0, +∞[. F± It is sufficient to see that X  ,( B ,−1, B ),u ,λ is invariant under the symmetry 0

2

S : (¯x, B 0 , B 2 , u , t ) → (−¯x, − B 0 , − B 2 , −u , −t ). Remark 7. The system X F,(±B ,−1, B ),u ,λ can be obtained by replacing the plus sign in front of x¯ in 0 2 X F,(±B ,+1, B ),u ,λ by a minus sign. The study of the former will be more or less analogous to the study 0

2

of the latter as presented in Section 3.5. Hence, in order to desingularize the system X F,(±B ,−1, B ),u ,λ 0 2 near the origin (¯x, y¯ ) = (0, 0), we will use the blow-up formulas (17), (18), (21) and (23), keeping the same notations as in Section 3.5.

3.7.1. ( , B 0 ) = (δ 2 , δ B¯ 0 ), B¯ 0 in a large compact set We study the singular perturbation system X F2± ¯ . We blow up the origin (¯x, y¯ , δ) = δ ,(δ B 0 ,−1, B 2 ),u ,λ (0, 0, 0) using the blow-up transformation defined in (18). The family chart is obtained by taking δ˜ = 1 and keeping (˜x, y˜ ) in a compact set in expression (18). The vector field X F2± ¯ changes, after dividing by v, into: δ ,(δ B 0 ,−1, B 2 ),u ,λ

R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

⎧ ⎪ ⎨ x˙˜ = y˜ , y˙˜ = −˜x y˜ + B¯ 0 − x˜ + B 2 v x˜ 2 ± v 2 x˜ 3 + uv 3 x˜ 4 H (uv x˜ , λ), ⎪   ⎩ + uv 3 y˜ 2 G vu x˜ , v 2 u 2 y˜ , λ .

4031

(27)

For v = 0 expression (27) turns to



x˙˜ = y˜

(28)

y˙˜ = −˜x y˜ + B¯ 0 − x˜ .

System (28) has one singularity (˜x, y˜ ) = ( B¯ 0 , 0). A linear analysis shows that it is a strong focus when B¯ 20 − 4 < 0 and B¯ 0 = 0, a center when B¯ 0 = 0, or a hyperbolic node when B¯ 20 − 4  0. Bearing in mind the directional blow-up formula (21), in the phase directional chart {˜x = 1} the ∂ vector field X F2± ¯ + 0 ∂δ yields, after division by the positive factor V , δ ,(δ B 0 ,−1, B 2 ),u ,λ

⎧ ˙˜ ⎪ ˜ ˜ ⎪ ⎨ = − Y ,   ˙Y˜ = −Y˜ − 2Y˜ 2 +

˜2

˜ B¯ 0 − 1 + B 2 V ± V 2 + uV 3 H (uV , λ) ⎪   ⎪ ⎩ ˜ 2 Y˜ 2 G uV , u 2 V 2 Y˜ , λ . + uV 3

(29)

˜ = 0} (29) has singularities at Y˜ = 0 and Y˜ = − 1 . The eigenvalues of the linear part at On { V = 0,

2 Y˜ = 0 are given by (0, 0, −1) and at Y˜ = − 12 by (− 12 , 12 , 1). So we have at Y˜ = − 12 a resonant saddle

and at Y˜ = 0 a semi-hyperbolic singularity with the Y˜ -axis as stable manifold and a two-dimensional center manifold. Center manifolds are expressed by





˜ 2 −1 + O ( V ) + O ( ) ˜ . Y˜ =

˜ = Y˜ = 0}. The center behavior on V = 0, in Each center manifold contains the line of singularities {

˜ , is given by: terms of

  ˙˜ = −

˜ . ˜ 3 −1 + O ( )

As usual, the part of the sphere where x˜ ∼ −1 can be studied by applying the coordinate change ˜ t ) → (− V , − , ˜ −t ) to (29). Besides the singularities we already found in the charts {˜x = +1} ( V , , and {˜x = −1}, there are no extra singularities in the { y˜ = +1}- and { y˜ = −1}-charts. Putting the results together, we find how the phase portraits of the vector field X F2± ¯ , δ ,(δ B 0 ,−1, B 2 ),u ,λ

for δ ∼ 0, look like near the origin (¯x, y¯ ) = (0, 0). In Fig. 9 are represented the different phase portraits of (28) inside the circle, on which we represent the information at infinity in a (1, 2) quasihomogeneous compactification. Besides the center at B¯ 0 = 0, all singularities of (28) are hyperbolic. Consequently, the system (28) is stable in any compact set in the (˜x, y˜ )-plane, for B¯ 0 = 0. It implies that there can be no small-amplitude limit cycles of X F2± ¯ for δ ∼ 0 and B¯ 0 = 0 (see Fig. 9). Since a connection δ ,(δ B 0 ,−1, B 2 ),u ,λ

between the two branches of the critical curve { y¯ = 0} is absent (see Fig. 9), there can be no limit cycles of X ±2 ¯ near the origin in the (x, y )-plane for B¯ 0 = 0, δ ∼ 0, r ∼ 0, B 2 ∈ K and

λ ∈ Λ.

δ ,(δ B 0 ,−1, B 2 ),r ,λ

Remark 8. Of course, the system X ±2

δ ,(δ B¯ 0 ,−1, B 2 ),r ,λ

has no limit cycles for B¯ 0 = 0, δ ∼ 0, r ∼ 0, B 2 ∼ 0

and λ ∈ Λ. Hence K ⊂ R can be any compact set (including zero) in the case B¯ 0 = 0. This is not true in the case B¯ 0 = 0 and here lies the reason why we need to study separately the ( B 2 = 0)-case (see Theorem 2.6) and the ( B 2 = 0)-case (see Theorem 2.4).

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Fig. 9. Phase portraits of (28): phase portraits in the slow-fast Hopf region: at B¯ 0 = 0, a Hopf bifurcation takes place. At the same time, the attracting and repelling branches of the line of singular points are connected, hence giving the possibility for the existence of canards nearby.

Bearing in mind that B 2 ∈ K ⊂ ]0, +∞[ we now treat the case B¯ 0 = 0. Let us recall the system X F2± ¯ : δ ,(δ B 0 ,−1, B 2 ),u ,λ

˙ x¯ = y¯ ,     y˙¯ = −¯x y¯ + δ 2 δ B¯ 0 − x¯ + B 2 x¯ 2 ± x¯ 3 + u x¯ 4 H (u x¯ , λ) + u y¯ 2 δ 2 G u x¯ , u 2 y¯ , λ .

(30)

Fig. 9 suggests that the system (30) may have limit cycles near the origin in the (¯x, y¯ )-plane for B¯ 0 ∼ 0. Before we continue considering (30) near the origin (¯x, y¯ ) = (0, 0), we study canard limit cycles of (30) near canard limit periodic sets Γ y¯ defined in Section 3.2. Remark 9. In Section 3.2 we defined a canard limit periodic set Γ y¯ for each y¯ > 0 as the combination

of the orbit of the fast subsystem X 0F,±B ,u ,λ through the point (0, y¯ ) and the piece of the critical curve { y¯ = 0} between the α -limit set and the ω -limit set of that fast orbit. We can also parameterize limit periodic sets by the regular parameter x¯ > 0. In fact, for each x¯ > 0 we consider the combination of the orbit of the fast subsystem X 0F,±B ,u ,λ through the point (0, 12 x¯ 2 ) and the piece of the critical curve { y¯ = 0} between the α -limit set and the ω -limit set of that fast orbit. Thus x¯ represents the x¯ -component of the ω -limit set of that fast orbit. We can denote the limit periodic set attached to x¯ > 0 by Γx¯ . Remark 10. It can be easily seen that the

α -limit set of the fast orbit in definition of Γx¯ 0 is {(−¯x0 , 0)}.

The slow dynamics of (30) are given by (11), which we rewrite here for B 0 = 0 and B 1 = −1:

x¯  = −1 + B 2 x¯ ± x¯ 2 + u x¯ 3 H (u x¯ , λ). We distinguish two possibilities:

(31)

R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

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S a) The saddle case (+¯x2 in (31)): (31) has two simple singularities, at x¯ = x¯ − ( B 2 , u , λ) < 0 and x¯ =

S S x¯ + ( B 2 , u , λ) > 0. Systems X δF2+,(δ B¯ ,−1, B ),u ,λ have a hyperbolic saddle near (¯x, y¯ ) = (¯x− ( B 2 , u , λ), 0) 0 2 S and (¯x, y¯ ) = (¯x+ ( B 2 , u , λ), 0) for δ ∼ 0, δ > 0. To prove this we use the center manifold theorem near these points providing C ∞ center manifolds given as graphs of functions y¯ c (¯x) with

y¯ c (¯x) = δ 2

  δ B¯ 0 − x¯ + B 2 x¯ 2 + x¯ 3 + u x¯ 4 H (u x¯ , λ) + O δ4 . x¯

S On such a center manifold, near (¯x± ( B 2 , u , λ), 0), system (30) is described by

x˙¯ = δ 2

¯   δ B 0 − x¯ + B 2 x¯ 2 + x¯ 3 + u x¯ 4 H (u x¯ , λ) + O δ2 . x¯

Applying the implicit function theorem to the right hand side of this expression we find that

S x¯ ± ( B 2 , u , λ) =

−B2 ±



B 22 + 4

2

+ O (u )

and





x¯ ± S ,0 =

−B ± 2

 2

B 22 + 4

+ O (u , δ), 0 ,

δ > 0,

F+ where (¯x, y¯ ) = (¯x± S , 0) are hyperbolic saddles of X δ 2 ,(δ B¯ 0 ,−1, B 2 ),u ,λ . Bearing in mind Remark 10 we can now detect canard limit periodic sets Γx¯ that can generate limit cycles by perturbation in the saddle case:

Γx¯ ,









S S x¯ ∈ 0, min x¯ − ( B 2 , u , λ) , x¯ + ( B 2 , u , λ) .

(32)

Since B 2 ∈ K , we have that

S x¯ ( B 2 , u , λ) > x¯ S ( B 2 , u , λ). −

+

(33)

Using (33) we see that we deal with those canard limit periodic sets Γx¯ the fast orbit of which may end up in a simple zero of the slow dynamics in at most one side of the critical curve (see Fig. 10(a)). “Big” limit periodic sets Γx¯ (i.e. with x¯  1) do not have nearby limit cycles; such cycles are not permitted by the slow dynamics, see Fig. 10(a). b) The elliptic case (−¯x2 in (31)): we have 3 possibilities: b1) 0 < B 2 < 2, u ∼ 0: (31) has no singularities in an arbitrary compact set for u small enough. It follows that the slow dynamics (31) allows the passage from the normally hyperbolic and attracting part of the critical curve { y¯ = 0} to the normally hyperbolic and repelling part of that critical curve. In this case we detect canard limit periodic sets Γx¯ that can generate limit cycles by perturbation (see Fig. 10(b1)):

Γx¯ ,

x¯ ∈ ]0, +∞[.

(34)

Unlike in case (a) there is no bound on the size of the limit periodic set Γx¯ , and we have to consider Γx¯ as x¯ → +∞ (such limit periodic set tends partially to the equator of the primary blow-up locus, see Fig. 5).

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Fig. 10. Indication of slow dynamics and canard limit periodic sets Γx¯ that can generate limit periodic sets by perturbation. (a) The saddle case. (b1), (b2), (b3) The elliptic case, in different stages of the saddle-node bifurcation of the equilibrium in the slow dynamics.

b2) B 2 ∼ 2, u ∼ 0: the slow dynamics (31) contains a saddle-node singularity x¯ = 1 for B 2 = 2 and u = 0. Passage near this saddle-node for parameter values ( B¯ 0 , B 2 , δ, u ) close to (0, 2, 0, 0) might hence be possible. We again detect canard limit periodic sets Γx¯ that can generate limit cycles by perturbation (see Fig. 10(b2)):

Γx¯ ,

x¯ ∈ ]0, +∞[.

(35)

Like in the case (b1) we also have to study Γx¯ , x¯ ∼ +∞. b3) B 2 > 2, u ∼ 0: (31) has two simple singularities, given by

E x¯ ± ( B 2 , u , λ) =

B2 ±



B 22 − 4

2

+ O ( u ) > 0.

It can be easily seen (see a) the saddle case) that systems X F2− ¯ have a hyperbolic saddle δ ,(δ B 0 ,−1, B 2 ),u ,λ E E , 0) and a hyperbolic and attracting node near (¯x, y¯ ) = (¯x+ , 0), for δ ∼ 0, δ > 0. We near (¯x, y¯ ) = (¯x− detect canard limit periodic sets Γx¯ that can generate limit cycles by perturbation (see Fig. 10(b3)):

Γx¯ ,





E x¯ ∈ 0, x¯ − ( B 2 , u , λ) .

(36)

Thus, we have to deal with canard limit periodic sets Γx¯ the fast orbit of which may end up in a hyperbolic zero of the slow dynamics in at most one side of the critical curve. To find the cyclicity of the limit periodic sets Γx¯ prescribed in (32) (the saddle case) and (34), (35), (36) (the elliptic case), we use the results given in [4]. Based on the calculations made in this section we see that X F2± ¯ , B¯ 0 ∼ 0, are C ∞ -families of vector fields satisfying the assumptions δ ,(δ B 0 ,−1, B 2 ),u ,λ

T0–T5 given in [4]. Thus, we are in a perfect position to use the results given in [4]. We can see in [4] that important information in order to study the cyclicity of the graphic Γx¯ is given by the slow dynamics (31) along [−¯x, x¯ ] and the slow divergence integral along [−¯x, x¯ ]. The slow divergence integral is defined as

R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

 x¯

±

I (¯x, B 2 , u , λ) =

w dw

−1 + B 2 w ± w 2 + u w 3 H (u w , λ)

−¯x

4035

(37)

,

where I + is the slow divergence integral in the saddle case and I − is the slow divergence integral in the elliptic case. Indeed the divergence of (30) on y¯ = 0 is given by −¯x when δ = 0, while dt = dw . The slow divergence integral I ± is well defined and smooth for x¯ ∼ 0. −1+ B w ± w 2 +u w 3 H (u w ,λ) 2

Denote the right hand side of (31) by N ± ( w , B 2 , u , λ), where w = x¯ . Since B 2 is strictly positive and contained in the compact set K , we can prove that (37) is strictly negative, both for the saddle and for the elliptic case:

Lemma 3.1. Let ρ > 0 be arbitrarily small. There is an ν0 > 0 and u 0 > 0 so that for any (¯x, B 2 , u , λ) ∈ [ρ , ρ1 ] × K × [−u 0 , u 0 ] × Λ, with the property that N + ( w , B 2 , u , λ) < 0 (resp. N − ( w , B 2 , u , λ) < 0) for w ∈ [−¯x, x¯ ], we have that I + (¯x, B 2 , u , λ) < −ν0 (resp. I − (¯x, B 2 , u , λ) < −ν0 ). Proof. We only consider the case ( I + , N + ), the other one leading to similar calculations. We have

 x¯

+

I (¯x, B 2 , u , λ) = −¯x

w dw N + ( w , B 2 , u , λ)

 x¯ =

1 N + ( w , B 2 , u , λ)

0

 x¯ =−

−

N + (− w , B 2 , u , λ)



w2 N +(w , B

0

2

, u , λ). N + (− w , B

Since B 2 ∈ K , u ∼ 0 and w in a compact set, there exists Due to (38) we obtain

I + (¯x, B 2 , u , λ)  −ν1

 x¯

ρ  −ν1 0

2 , u , λ)

w dw



2B 2 + u O w 2



dw .

(38)

ν1 > 0 such that 2B 2 + u O ( w 2 )  ν1 .

w 2 dw N +(w , B

0



1

2 , u , λ). N

+ (− w , B

2 , u , λ)

w 2 dw N + ( w , B 2 , u , λ). N + (− w , B 2 , u , λ)

 −ν0 ,

for some ν0 > 0. In the last step, we used the fact that the last integral is strictly positive, uniformly in B 2 ∈ K , u ∼ 0 and λ ∈ Λ. 2 Cyclicity of Γx¯ in the saddle case. Because of (32) and (33) we find that it is sufficient to study S the cyclicity of Γx¯ for x¯ ∈ ]0, x¯ + ( B 2 , u , λ)]. The slow divergence integral I + is well defined for x¯ ∈ S ]0, x¯ + ( B 2 , u , λ)[. Taking into account Lemma 3.1 we find that for any ρ > 0 small there exist u 0 > 0 and ν0 > 0 S sufficiently small such that I + (¯x, B 2 , u , λ)  −ν0 for x¯ ∈ [ρ , x¯ + ( B 2 , u , λ)[, B 2 ∈ K , u ∈ [−u 0 , u 0 ] and λ ∈ Λ. S Applying the results of [4] we know that the cyclicity of Γx¯ with x¯ ∈ [ρ , x¯ + ( B 2 , u , λ)] is one (I + = 0). In fact, each limit cycle generated by such Γx¯ has to be hyperbolically attracting (I + is

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R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

strictly negative), so that necessarily the global cyclicity of

( B 2 , u , λ) ∈ K × [−u 0 , u 0 ] × Λ.



S x¯ ∈[ρ ,¯x+ ( B 2 ,u ,λ)] Γx¯

is one for each

Cyclicity of Γx¯ in the elliptic case. As we mentioned above, we have the three possibilities: b1) B 2 ∈ C ⊂ (0, 2), u ∼ 0, C is a compact set in R. From Lemma 3.1 we know that for any ρ > 0 small there exist u 0 > 0 and ν0 > 0 sufficiently small such that I − (¯x, B 2 , u , λ)  −ν0 for x¯ ∈ [ρ , ρ1 ], B 2 ∈ C , u ∈ [−u 0 , u 0 ] and λ ∈ Λ. Here we used the fact that the slow dynamics is regular on any compact set in the x¯ -space, taking u small enough. Because of [4] (or [3]) we know that the cyclicity of Γx¯ with x¯ ∈ [ρ , ρ1 ] is one. In fact, each limit − cycle generated by such Γx¯ has to  be hyperbolically attracting (I is strictly negative), so that necessarily the global cyclicity of x¯ ∈[ρ , 1 ] Γx¯ is one for each ( B 2 , u , λ) ∈ C × [−u 0 , u 0 ] × Λ. ρ

b2) B 2 ∼ 2, u ∼ 0. Like in the case  b1) we again use Lemma 3.1 and [4] and find that for any ρ > 0 small the global cyclicity of x¯ ∈[ρ , 1 ] Γx¯ is one, and when a limit cycle appears, it is hyperbolic ρ

and attracting. b3) B 2 ∈ C ⊂ (2, +∞), u ∼ 0 C is a compact set in R. Taking into account (36) we need to study the E ∈ ]0, x¯ − ( B 2 , u , λ)]. Like in the saddle case we find that for any ρ > 0 small cyclicity of Γx¯ for all x¯  the global cyclicity of x¯ ∈[ρ ,¯x E ( B 2 ,u ,λ)] Γx¯ is one for each B 2 ∈ C , u ∼ 0 and λ ∈ Λ. We deal again −

with a hyperbolically attracting limit cycle. Cyclicity of the origin (x¯ , y¯ ) = (0, 0) in the saddle and elliptic cases. In terms of [9], Eq. (30) (in both the saddle and elliptic case) represents a slow-fast Hopf bifurcation of codimension one. To see that we refer to [7], more precisely to the subsection “Cyclicity of the origin (¯x, y¯ ) = (0, 0)”. The codimension is one because the coefficient B 2 in front of x¯ 2 ∂∂y¯ in (30) is strictly positive (B 2 ∈ K ). In Section 3.1.4 of [7], more precisely in the subsections “Cyclicity of the origin (¯x, y¯ ) = (0, 0)” and “Unicity of the limit cycle near Γ0 ”, it has been shown, based on [9], that one can in fact have at most one limit cycle in a slow-fast Hopf bifurcation of codimension 1, and it has to be simple (hyperbolic). Remark 11. Besides the proof given in [7] there exists another proof of the fact that we have at most one limit cycle near (¯x, y¯ ) = (0, 0). We remark that the results in [9] were subject to a conjecture, but this conjecture has recently been solved for cases up to codimension 2, see [19]. Consequently [9] implies that we have at most one limit cycle near (¯x, y¯ ) = (0, 0) in the codimension one case. It follows that systems (30) have at most one limit cycle in a ( B¯ 0 , B 2 , u , λ)-uniform neighborhood of the origin in (¯x, y¯ , δ)-space; it is hyperbolic and attracting. For δ > 0, [7] implies that the systems (30) undergo a Hopf bifurcation of codimension 1 at B¯ 0 = 0. When B¯ 0 increases, there is an attracting hyperbolic focus (Fig. 9) and no limit cycle; when B¯ 0 decreases, there is repelling hyperbolic focus and a hyperbolic and attracting limit cycle of which the size monotonically grows as B¯ 0 decreases. Combining (x¯ , y¯ ) = (0, 0) and limit periodic sets Γx¯ in the saddle and elliptic cases. Taking into account the subsections “Cyclicity of Γx¯ in the saddle case”, “Cyclicity of Γx¯ in the elliptic case” and “Cyclicity of the origin (¯x, y¯ ) = (0, 0) in the saddle and elliptic cases” we find that systems , defined in (30), have at most one hyperbolic and attracting limit cycle in an arX F2± ¯ δ ,(δ B 0 ,−1, B 2 ),u ,λ

bitrarily large but fixed compact set in the (¯x, y¯ )-space, for B 2 ∈ K , λ ∈ Λ, (δ, B¯ 0 , u ) ∼ (0, 0, 0). The undergo a Hopf bifurcation of codimension 1 at B¯ 0 = 0, for δ ∼ 0, δ > 0, systems X F2± ¯ δ ,(δ B 0 ,−1, B 2 ),u ,λ u ∼ 0, B 2 ∈ K and λ ∈ Λ.

will be Remark 12. The unique hyperbolic and attracting limit cycle of the system X F2+ ¯ δ ,(δ B 0 ,−1, B 2 ),u ,λ ¯ located in the region in the (¯x, y¯ )-space between the two saddles (¯x, y¯ ) = (¯x− , 0 ) and (¯ x , y ) = (¯ x+ S S , 0). We suppose that D in Fig. 4(a) is chosen in such a way that the boundary of D is close to the equator

R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

4037

of the primary blow-up and that the saddles are located in interior of D for any B 2 ∈ K , λ ∈ Λ, (δ, B¯ 0 , u ) ∼ (0, 0, 0) and δ > 0. Hence the unique limit cycle is located in the interior of D. Cyclicity of (x, y ) = (0, 0) in the saddle case. Considering the dynamics of the vector field + 0 ∂∂r , defined in (6), in the phase directional charts of the primary blow-up (7) X +2 ¯ δ ,(δ B 0 ,−1, B 2 ),r ,λ

(Fig. 4(a)) and bearing in mind Remark 12 we see that X +2

δ ,(δ B¯ 0 ,−1, B 2 ),r ,λ

has at most one limit cycle

in a fixed neighborhood of the origin (x, y ) = (0, 0), for B 2 ∈ K , λ ∈ Λ, (δ, B¯ 0 , r ) ∼ (0, 0, 0); the limit cycle is hyperbolically attracting. Although the systems X F2+ ¯ may have a limit cycle for u = 0, systems X +2 ¯ δ ,(δ B 0 ,−1, B 2 ),u ,λ

δ ,(δ B 0 ,−1, B 2 ),r ,λ

have no limit cycles near the origin for r = 0 because the primary blow-up (7) is singular for u = 0. undergo a Hopf bifurcaFrom what has been said above, it follows that the systems X +2 ¯ δ ,(δ B 0 ,−1, B 2 ),r ,λ

tion of codimension 1 at B¯ 0 = 0, for δ ∼ 0, δ > 0, r ∼ 0, r > 0, B 2 ∈ K and λ ∈ Λ.

Cyclicity of (x, y ) = (0, 0) in the elliptic case. Taking into account the specific dynamics of the vector + 0 ∂∂r , defined in (6), in the phase directional charts of the primary blow-up (7) field X −2 ¯ δ ,(δ B 0 ,−1, B 2 ),r ,λ

(see Fig. 4(b)) we have to study the transition from limit cycles near (large) canard limit periodic sets Γx¯ to limit cycles near small (but detectable) canard limit periodic sets in the (x, y )-space. There is no need to study the transition mentioned above in the b3)-case because we deal with E ( B 2 , u , λ) in the slow dynamics (31) in the elliptic case. Thus the cyclicity the hyperbolic zero x¯ = x¯ − of (x, y ) = (0, 0) is one in the b3)-case. To find the cyclicity of (x, y ) = (0, 0) in the other cases (the b1)-case and the b2)-case) we refer to [21]. The transition mentioned above has been studied in detail in [21]. We finish Section 3.7.1 by providing a few more details on what is shown in [21] and how it can be applied. To use the results in [21] we apply the coordinate change x = x∗ , y = y ∗ − 12 x∗ 2 to X −2 ¯ . δ ,(δ B 0 ,−1, B 2 ),r ,λ

We obtain

X ∗−¯

δ, B 0 , B 2 ,r ,λ

:

⎧ 1 2 ⎪ ⎪ x˙∗ = y ∗ − x∗ , ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎨  2 3 4  2 ∗ ˙ y = δ δr 3 B¯ 0 − r 2 x∗ + B 2 rx∗ − x∗ + x∗ H x∗ , λ ⎪ ⎪ ⎪

2

⎪ ⎪ 1 1 ⎪ ⎪ + y ∗ − x∗ 2 G x∗ , y ∗ − x∗ 2 , λ . ⎩ 2

(39)

2

In [21] a number of results has been proven for the system X ∗−¯ , depending on the parameter δ, B 0 , B 2 ,r ,λ B 2 . To understand it better we desingularize (39), in a way compatible to the blow-ups in this paper. In Fig. 4(b) and Fig. 5 we represented a bird’s eye view of the primary blow-up in the elliptic case. We get a similar picture in blowing up (39) at (x∗ , y ∗ , r ) = (0, 0, 0), by means of so-called primary blow-up (x∗ , y ∗ , r ) = (u ∗ x¯ ∗ , u ∗ 2 y¯ ∗ , u ∗ r¯ ∗ ), except that the curve of singularities { y = 0} is now the parabola { y ∗ = 12 x∗ 2 }. The family chart is obtained by taking r¯ ∗ = 1 in the blow-up formula above. The vector field (39) changes, after dividing by u ∗ , into:

X ∗ F¯−

δ, B 0 , B 2 ,u ∗ ,λ

:

⎧ 1 ⎪ ⎪ x˙∗ = y¯ ∗ − x¯ ∗ 2 , ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎨   2 ∗ ˙ y = δ δ B¯ 0 − x¯ ∗ + B 2 x¯ ∗ 2 − x¯ ∗3 + u ∗ x¯ ∗4 H u ∗ x¯ ∗ , λ ⎪ ⎪ ⎪

2



⎪ ⎪ 1 ∗2 1 ∗2 ⎪ ∗ ¯∗ ∗ ∗ ∗ 2 ∗ ⎪ ,λ . G u x¯ , u y¯ − x¯ + u y − x¯ ⎩ 2

2

(40)

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Fig. 11. A bird’s eye view of the primary and secondary blow-up with limit periodic sets in the elliptic case, for B¯ 0 = 0.

Remark 13. The systems X ∗ F¯− can also be obtained by applying the coordinate change x¯ = δ, B 0 , B 2 ,u ∗ ,λ F− 1 ∗2 ∗ ∗ x¯ , y¯ = y¯ − 2 x¯ to X 2 ¯ ∗ . δ ,(δ B 0 ,−1, B 2 ),u ,λ

The study of the vector field X ∗−¯

of X −2 ¯ δ ,(δ B 0 ,−1, B 2 ),r ,λ

+ 0 ∂∂r in phase directional charts is analogous to the study δ, B 0 , B 2 ,r ,λ ∂ + 0 ∂ r in the phase directional charts of (7). For a detailed analysis of the latter,

we refer to Section 3.1 and [21]. To desingularize the vector field X ∗ F¯− at the origin (¯x∗ , y¯ ∗ ) = (0, 0) we blow up (¯x∗ , y¯ ∗ , δ) = δ, B 0 , B 2 ,u ∗ ,λ (0, 0, 0) by use of so-called secondary blow-up









x¯ ∗ , y¯ ∗ , δ = v ∗ x˜ ∗ , v ∗ 2 y˜ ∗ , v ∗ δ˜ ∗ ,

see (18). Taking into account Remark 13 the study of X ∗ F¯− in different charts of the secondary δ, B 0 , B 2 ,u ∗ ,λ F− blow-up is analogous to the study of X 2 ¯ in different charts of the blow-up (18). ∗ δ ,(δ B 0 ,−1, B 2 ),u ,λ

Bearing in mind all the information obtained above, we represent in Fig. 11 a bird’s eye view of the primary blow-up at (x∗ , y ∗ , r ) = (0, 0, 0) w.r.t. (39) and the secondary blow-up at (¯x∗ , y¯ ∗ , δ) = (0, 0, 0) w.r.t. (40), for B¯ 0 = 0. We see closed curves (so-called limit periodic sets) that can generate limit cycles by perturbation in the elliptic case. Notice that Fig. 11 covers only the b1)-case. In the b2)-case we deal with a saddle-node singularity in the slow dynamics and we consider the same limit periodic sets as in the b1)-case. The paper [21] is devoted to the study of the cyclicity of the singular cycle L 00 consisting of semi-hyperbolic singularities S 1 , S 2 , R 1 , R 2 and the regular and singular orbits that are connected (heteroclinic) to them. It has been proven that the cyclicity of L 00 is one in the b1)-case and in the b2)-case, and a limit cycle generated by such L 00 has to be hyperbolically attracting (see [21], Theorem 3.2). Taking into account the subsection “Combining (¯x, y¯ ) = (0, 0) and limit periodic sets Γx¯ in the defined in (40) have saddle and elliptic cases” and Remark 13 we find that systems X ∗ F¯− ∗ δ, B 0 , B 2 ,u ,λ

at most one hyperbolic and attracting limit cycle in an arbitrarily large but fixed compact set in the (¯x∗ , y¯ ∗ )-space, for B 2 ∈ K , λ ∈ Λ, (δ, B¯ 0 , u ∗ ) ∼ (0, 0, 0). Since it is not possible to have two

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hyperbolically attracting limit cycles, we find that X ∗−¯ defined in (39) have at most one hyδ, B 0 , B 2 ,r ,λ perbolically attracting limit cycle in a fixed neighborhood of the origin (x∗ , y ∗ ) = (0, 0), for B 2 ∈ K , λ ∈ Λ, (δ, B¯ 0 , r ) ∼ (0, 0, 0). In other words, X −2 ¯ has at most one hyperbolically attracting δ ,(δ B 0 ,−1, B 2 ),r ,λ

limit cycle in a fixed neighborhood of (x, y ) = (0, 0) under the given conditions on the parameters.

3.7.2. ( , B 0 ) = (δ 2 E , ±δ), E ∼ 0 The study of X δ±2 E ,(±δ,−1, B ),r ,λ is analogous to the study of X δ±2 E ,(±δ,1, B ),r ,λ in Section 3.5.2. For 2 2 δ > 0, E > 0 and r > 0, systems X δ+2 E ,(±δ,−1, B ),r ,λ have two hyperbolic saddles, a hyperbolic node be2

tween them and, using “secondary and tertiary blow-up”, it is clear that no limit cycles are present. Similarly, systems X δ−2 E ,(±δ,−1, B ),r ,λ have one hyperbolic node (and extra singularities after a saddle2

node bifurcation) and no limit cycles. Remark 14. This is true even if B 2 ∼ 0. 3.8. Analysis near the point P in parameter space The parameter point P has been introduced in Section 2. This section deals with limit cycles near

(x, y ) = (0, 0), for ( B 0 , B 1 , B 2 ) ≈ P = (0, −1, 0) and  , r small enough. Besides proving Theorem 2.6, we will do some further investigation, see also Remark 3. Throughout this section, we suppose that H (0, λ) = 0 for all λ ∈ Λ. Taking into account Remark 8 near the origin and Remark 14 it remains to study the singular perturbation system X ±2 ¯ δ ,(δ B 0 ,−1, B 2 ),r ,λ

(x, y ) = (0, 0) for δ ∼ 0, B¯ 0 ∼ 0, B 2 ∼ 0, r ∼ 0 and λ ∈ Λ. For some reasons that will become clear

later in this section it is better to work with the vector field X ∗−¯ , defined in (39), in the elliptic δ, B 0 , B 2 ,r ,λ case. In the saddle case we will study the vector field X ∗+¯ obtained by putting a + sign in the δ, B 0 , B 2 ,r ,λ

front of x∗3 ∂ ∂y ∗ in (39):

X ∗+¯

δ, B 0 , B 2 ,r ,λ

:

⎧ 1 2 ⎪ ⎪ x˙∗ = y ∗ − x∗ , ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎨  2 3 4  2 ∗ ˙ y = δ δr 3 B¯ 0 − r 2 x∗ + B 2 rx∗ + x∗ + x∗ H x∗ , λ ⎪ ⎪ ⎪

2

⎪ ⎪ 1 1 ⎪ ⎪ + y ∗ − x∗ 2 G x∗ , y ∗ − x∗ 2 , λ . ⎩ 2

(41)

2

From now on, we study X ∗±¯ near the origin (x∗ , y ∗ ) = (0, 0) for δ ∼ 0, B¯ 0 ∼ 0, B 2 ∼ 0, r ∼ 0 δ, B 0 , B 2 ,r ,λ and λ ∈ Λ. 3.8.1. Limit periodic sets for parameters near P First we want to detect closed curves that can generate limit cycles by perturbation in the saddle and elliptic cases. We use the same primary and secondary blow-up in the saddle and elliptic cases as we did in the subsection “Cyclicity of (x, y ) = (0, 0) in the elliptic case”. In the family chart of the primary blow-up the vector field X ∗±¯ can be written, after dividing by u ∗ , as: δ, B 0 , B 2 ,r ,λ

X ∗ F¯±

δ, B 0 , B 2 ,u ∗ ,λ

:

⎧ 1 ⎪ ⎪ x˙∗ = y¯ ∗ − x¯ ∗ 2 , ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎨   2 ∗ ˙ y = δ δ B¯ 0 − x¯ ∗ + B 2 x¯ ∗ 2 ± x¯ ∗3 + u ∗ x¯ ∗4 H u ∗ x¯ ∗ , λ ⎪ ⎪ ⎪

2



⎪ ⎪ 1 1 ⎪ ⎪ + u ∗ y¯ ∗ − x¯ ∗ 2 G u ∗ x¯ ∗ , u ∗ 2 y¯ ∗ − x¯ ∗ 2 , λ . ⎩ 2

2

(42)

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Fig. 12. A bird’s eye view of the primary and secondary blow-up with limit periodic sets in the saddle case for (δ, B¯ 0 , B 2 , u ∗ ) = (0, 0, 0, 0), with indication of (primary) slow dynamics. There are five kinds of limit periodic sets: L h inside the secondary blow-up locus, the singular polycycle L 0 , the canard cycle L y ∗ and the 2-saddle-cycle L 1/2 .

For δ = 0 the critical curve of X ∗ F¯± is given by { y¯ ∗ = δ, B 0 , B 2 ,u ∗ ,λ critical curve is given by:



1 ∗2 x¯ } 2



x¯ ∗ = −1 + B 2 x¯ ∗ ± x¯ ∗ 2 + u ∗ x¯ ∗3 H u ∗ x¯ ∗ , λ ,

and the slow dynamics along the

x¯ ∗ = 0.

(43)

Two possibilities exist: a) The saddle case (+¯x∗ 2 in (43)): for B 2 = u ∗ = 0, (43) has two simple singularities, at x¯ ∗ = −1 and x¯ ∗ = 1. In Fig. 12 we see closed curves that can generate limit cycles by perturbation in the saddle case for (δ, B¯ 0 , B 2 , u ∗ ) ∼ (0, 0, 0, 0). There are five different kinds of these limit periodic sets in the saddle case: (i) and (ii) The singular point in the middle and the closed orbits L h on the secondary blow-up locus; (iii) The singular cycle L 0 consisting of singularities S 1 , S 2 and the regular orbits that are heteroclinic to them; (iv) The canard limit periodic sets L y¯ ∗ , y¯ ∗ ∈ ]0, 12 [, in the (¯x∗ , y¯ ∗ )-space; (v) The slow-fast two-saddle-limit periodic set L 1 . 2

b) The elliptic case (−¯x∗ 2 in (43)): for B 2 = u ∗ = 0, (43) has no zeros. In Fig. 11 we see closed curves that can generate limit cycles by perturbation in the elliptic case for (δ, B¯ 0 , B 2 , u ∗ ) ∼ (0, 0, 0, 0). There are five different kinds of such curves in the elliptic case: (i), (ii) and (iii) The singular point in the middle, the closed orbits L h and the singular cycle L 0 (see the saddle case); (iv) The canard limit periodic sets L y¯ ∗ , y¯ ∗ ∈ ]0, +∞[, in the (¯x∗ , y¯ ∗ )-plane; (v) The singular cycle L 00 consisting of singularities S 1 , S 2 , R 1 , R 2 and the regular and singular sections that are connected (heteroclinic) to them. We remark that the canard limit periodic sets L y ∗ in the (x∗ , y ∗ )-plane are not relevant for the study of the cyclicity of (x∗ , y ∗ ) = (0, 0). For a detailed analysis we refer to [5].

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In this paper, we will give a complete study of the limit periodic sets of type (iv) in both the saddle and elliptic cases (see Section 3.8.4), and the limit periodic set (v) in the saddle case (see Section 3.8.5). Theorem 2.6 will be shown in Section 3.8.4, and additionally the limit periodic sets in bothsections are shown to produce at most 2 limit cycles. More precisely, it will be shown that the ¯ ∗1 < y¯ ∗2 < 12 in the saddle case and set y¯ ∗ ∈[ y¯ ∗ , y¯ ∗ ] L y¯ ∗ can produce at most 2 limit cycles for any 0 < y 1

2

for any 0 < y¯ ∗1 < y¯ ∗2 < +∞ in the elliptic case. Similarly, the limit periodic sets of type (v) in the elliptic case are shown to produce at most 2 limit cycles, based on the work in [21]. The calculations in Sections 3.8.2 and 3.8.3 isolate two open problems, one of which has to do with abelian integrals and the other with the limit periodic set L 0 . These two problems will be studied in a separate paper. Here we point out that techniques needed to solve the problems are similar to the ones introduced in [9]; for more details we refer to Sections 3.8.2 and 3.8.3. To find the cyclicity of the origin (x, y ) = (0, 0) of (5) in the case where H (0, λ) = 0 for all λ ∈ Λ, we will need to study the relation between informations obtained near each limit periodic set. This is also a topic of further study. 3.8.2. Cyclicity of the singular point in the middle and L h in the saddle and elliptic cases It can be easily seen that we deal with the following system on the secondary blow-up locus:

⎧ ⎨ x˜˙∗ = y˜ ∗ − 1 x˜ ∗ 2 , 2 ⎩ ˙∗ y˜ = −˜x∗ .

(44)

The vector field (44) is of center type with the center at the origin (˜x∗ , y˜ ∗ ) = (0, 0). We also see that the vector field (44) is the dual of the differential 1-form:



1



ω0 = x˜ ∗ dx˜ ∗ + y˜ ∗ − x˜ ∗ 2 d y˜ ∗ 2

∗

which admits the function −e − y˜ as integrating factor and the function



˜∗

˜∗

H x ,y



=e

− y˜ ∗



1 y − x˜ ∗ 2 + 1 2

˜∗



as first integral. This means that ∗

−e − y˜ w 0 = dH. ∗

Of course the integrating factor is not unique, but −e − y˜ has the advantage that the related Hamil∗ tonian H is zero on { y˜ ∗ = 12 x˜ ∗ 2 − 1} and also at infinity (e − y˜ is flat for y˜ ∗ = +∞). Hence {H(˜x∗ , y˜ ∗ ) = 0} contains the limit periodic sets L 0 . Notice that {H(˜x∗ , y˜ ∗ ) = 1} represents the center (˜x∗ , y˜ ∗ ) = (0, 0) (the singular point in the middle in Fig. 11 and Fig. 12 is denoted by L 1 ) and that {H(˜x∗ , y˜ ∗ ) = h} = L h where h ∈ ]0, 1[. We study limit cycles of X ∗ F¯± ∗ , defined in (42), near L h where h ∈ [h 0 , 1] with h 0 > 0 arbiδ, B 0 , B 2 ,u ,λ

trarily small  and fixed. In the Liénard setting, i.e. where G ≡ 0, it can be easily seen, based on [9] and [19], that h∈[h0 ,1] L h can produce at most 2 limit cycles. This will be studied in the above-mentioned separate paper, together with the case G ≡ 0.

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3.8.3. Cyclicity of L 0 in the saddle and elliptic cases We study limit cycles of X ∗ F¯± ∗ , defined in (42), near L 0 . In fact we study the transition from δ, B 0 , B 2 ,u ,λ

limit cycles near (large) limit periodic sets L h to limit cycles near small but detectable canard limit periodic sets L y¯ ∗ in the (¯x∗ , y¯ ∗ )-space. A similar problem has been treated in [9]. The paper [9] deals with Eqs. (42) of Liénard type, i.e. with G ≡ 0. To see this connection we refer to [7]. The paper [9] implies that L 0 produces at most 3 limit cycles. Presence of the function G in (42) is important in studying the limit cycles near L 0 because we ∗F ± deal with perturbations of the vector fields X δ, 0,0,0,λ of center type (δ > 0); in this case L 0 is not shown to have a finite cyclicity. It implies that extra calculations are required, taking into account that H (0, λ) = 0 for all λ ∈ Λ (the cyclicity of L 0 depends on the higher order perturbation terms in H ). This case will be studied in the above-mentioned separate paper. When we deal with smooth systems of finite codimension and G ≡ 0 (for example (42) with B 2 = 0), then [9] implies that L 0 has a finite cyclicity. 3.8.4. Cyclicity of L y¯ ∗ in the saddle and elliptic cases We study the cyclicity of the closed curves as in (iv), i.e. the limit periodic sets L y¯ ∗ consisting of



with {(− 2 y¯ ∗ , y¯ ∗ )} as a fast orbit of X ∗ F¯± 0, B 0 , B 2 ,u ∗ ,λ



α -limit set and {( 2 y¯ ∗ , y¯ ∗ )} as ω-limit set and 



between x¯ ∗ = − 2 y¯ ∗ and x¯ ∗ = 2 y¯ ∗ . containing the piece of the critical curve of X ∗ F¯± 0, B 0 , B 2 ,u ∗ ,λ Let C + and C − be arbitrary compact sets such that C + ⊂ (0, 12 ) and C − ⊂ (0, +∞). We suppose that the limit periodic sets L y¯ ∗ are parametrized by y¯ ∗ ∈ C + in the saddle case and by y¯ ∗ ∈ C − in the elliptic  case.  It is important to note that in the saddle case the slow dynamics (43) is regular along [− 2 y¯ ∗ , 2 y¯ ∗ ]for y¯ ∗ ∈ C + , B 2 ∼ 0 and u ∗ ∼ 0. In the elliptic case the slow dynamics (43) is regular along [− 2 y¯ ∗ , 2 y¯ ∗ ] for y¯ ∗ ∈ C − , B 2 ∼ 0 and u ∗ ∼ 0. Since the slow dynamics is regular, we are in a position to use the results in [3]. We first define the slow divergence integral as 

I

 ∗±

2 y¯



y¯ ∗ , B 2 , u ∗ , λ =



−

∗

w dw

2 y¯ ∗

−1 + B 2 w ± w 2 + u ∗ w 3 H (u ∗ w , λ)

.

(45)

I ∗± ( y¯ ∗ , B 2 , u ∗ , λ) is well defined and smooth for y¯ ∗ ∈ C ± , B 2 ∼ 0, u ∗ ∼ 0, λ ∈ Λ. Note that





I ∗± y¯ ∗ , 0, 0, λ = 0.

(46)

Like in [3] we try to study the limit cycles of X ∗ F¯± near L y¯ ∗ as zeros of a difference map. δ, B 0 , B 2 ,u ∗ ,λ We first consider sections



S ± = x¯ ∗ = 0, y¯ ∗ ∈ C ± , and a second section



T = x˜ ∗ = 0 , that we define along the (secondary) blow-up locus of the origin (¯x∗ , y¯ ∗ ) = (0, 0). More precisely, T is defined in the traditional rescaling chart {δ˜ ∗ = 1} and parametrized by the (secondary) blow-up in forward and backward time, we can define coordinate y˜ ∗ . By following the orbits of X ∗ F¯± ∗ δ, B 0 , B 2 ,u ,λ

smooth transition maps from S ± to T , denoted by respectively F ± and B ± .

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Closed orbits of X ∗ F¯± are given by zeros of the difference map δ, B 0 , B 2 ,u ∗ ,λ

± = F ± − B ± , where ± is smooth in ( y¯ ∗ , δ, B¯ 0 , B 2 , u ∗ , λ), for δ ∼ 0, δ  0, ( B¯ 0 , B 2 , u ∗ ) ∼ (0, 0, 0) and λ ∈ Λ. ∗ ¯∗ From [3] follows that there exist smooth functions B¯ ± 0 ( y , δ, B 2 , u , λ) such that periodic or-

bits of systems X ∗ F¯± , laying near limit periodic sets L y¯ ∗ , with y¯ ∗ ∈ C + in the saddle δ, B 0 , B 2 ,u ∗ ,λ ∗ ¯∗ case and y¯ ∗ ∈ C − in the elliptic case, can only occur for B¯ 0 = B¯ ± 0 ( y , δ, B 2 , u , λ). In fact, since ∂ ± ¯ ∗ ( y , 0, 0, B 2 , u ∗ , λ) ∂ B¯ 0

= 0 ( B¯ 0 is a “breaking parameter”), the im∗ ¯∗ plicit function theorem implies existence of unique smooth functions B¯ ± 0 ( y , δ, B 2 , u , λ) such that ∗ , δ, B , u ∗ , λ). ¯ solution of ± = 0, for δ ∼ 0, B¯ 0 ∼ 0, can only occur for B¯ 0 = B¯ ± ( y 2 0 Moreover the system X ∗ F¯± represents a center for B¯ 0 = B 2 = u ∗ = 0 and δ > 0, and we have δ, B , B ,u ∗ ,λ

± ( y¯ ∗ , 0, 0, B 2 , u ∗ , λ) = 0 and

0

2

 

± y¯ ∗ , δ, 0, 0, 0, λ = 0, as well as





¯∗ B¯ ± 0 y , δ, 0, 0, λ = 0. Since our aim is to study the linear part of the slow divergence integral I ∗± , we prefer to work with

      ¯ c± y¯ ∗ , δ, B 2 , u ∗ , λ = ± y¯ ∗ , δ, B¯ ± c , δ, B 2 , u ∗ , λ , B 2 , u ∗ , λ ,

0

(47)

¯ c± is a smooth c-family of functions where c ∈ C + in the saddle case and c ∈ C − in the elliptic case.

∗ ∗ ± ∗ ¯ in ( y¯ , δ, B 2 , u , λ), c ( y¯ , δ, 0, 0, λ) = 0 and we keep in mind that all expressions that depend on y¯ ∗ , except for I ∗± and its derivatives, also depend on c. In [14] this procedure has been called “cloning a variable”. In what follows, we avoid writing the parameter c. ¯± ∂

Again from [3], we know that ∂ y¯ c∗ is on {δ > 0} C ∞ -contact equivalent to













I ∗± y¯ ∗ , B 2 , u ∗ , λ + ψ1± y¯ ∗ , δ, B 2 , u ∗ , λ + ψ2± δ, B 2 , u ∗ , λ δ 2 ln δ,

(48)

where ψ1± and ψ2± are smooth, including at δ = 0, and ψ1± = O (δ). Taking into account (46) we know

¯± ∂

that I ∗± as in (45) and ∂ y¯ c∗ are identically zero for B 2 = u ∗ = 0 and I ∗± is a smooth function, so that we can write (48) as

     B 2 I 1∗± y¯ ∗ , B 2 , u ∗ , λ + Ψ1± y¯ ∗ , δ, B 2 , u ∗ , λ

     + u ∗ I 2∗± y¯ ∗ , B 2 , u ∗ , λ + Ψ2± y¯ ∗ , δ, B 2 , u ∗ , λ ,

(49)

where Ψ1± and Ψ2± are O (δ) and δ -regularly smooth in ( y¯ ∗ , B 2 , u ∗ , λ), i.e. Ψ1± and Ψ2± and all their derivatives w.r.t. ( y¯ ∗ , B 2 , u ∗ , λ) are continuous in δ  0. After rescaling in the parameter space ( B 2 , u ∗ ) = κ ( B¯ 2 , u¯ ∗ ), where ( B¯ 2 , u¯ ∗ ) ∈ S1 , u¯ ∗  0 and κ  0, expression (49) can be written as



κ B¯ 2

  ∂ I ∗±  ∗ ∂ I ∗±  ∗ ¯ y¯ , 0, 0, λ + u¯ ∗ y , 0 , 0 , λ + O (δ) + O ( κ ) , ∂ B2 ∂ u∗

(50)

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where O (κ ) represents smooth functions and O (δ) δ -regularly smooth functions. Instead of working with (50) one may consider its derivative with respect to y¯ ∗ :





∗±    ∂ I ∗±  ∗ ∗ ∂I ¯ ¯ κ B¯ 2 y , 0 , 0 , λ + u y¯ ∗ , 0, 0, λ + O (δ) + O (κ ) , ∗ ∗ ∗

∂ B 2 ∂ y¯

∂ u ∂ y¯

which is a similar type of equation as (50). For κ = 0, we deal with a center. In what follows, we suppose that

(51)

κ = 0.

Lemma 3.2. The slow divergence integral (45) has the following properties: 

 2 y¯ ∗ − w 2 dw ∗± i) ∂∂IB ( y¯ ∗ , 0, 0, λ) =  ∗ (− ; 1± w 2 )2 2 − 2 y¯  −2 2 y¯ ∗

∗±

ii) ∂ B∂ I ∂ y¯ ∗ ( y¯ ∗ , 0, 0, λ) = (−1±2 y¯ ∗ )2 ; 2 

 2 y¯ ∗ − w 4 dw ∗± iii) ∂∂Iu ∗ ( y¯ ∗ , 0, 0, λ) = H (0, λ)  ∗ (− ; 1± w 2 )2 − 2 y¯  −2 2 y¯ ∗ 2 y¯ ∗

∗±

iv) ∂ u∂ ∗I ∂ y¯ ∗ ( y¯ ∗ , 0, 0, λ) = H (0, λ) (−1±2 y¯ ∗ )2 . Proof. It follows directly from (45).

2

If we bear in mind Lemma 3.2, then the dominant part in (51) can be written as ∗±    ∂ I ∗±  ∗ ∗ ∂I ¯ ¯ y , 0 , 0 , λ + u y¯ ∗ , 0, 0, λ ∂ B 2 ∂ y¯ ∗ ∂ u ∗ ∂ y¯ ∗   −2 2 y¯ ∗  ¯ B 2 + 2H (0, λ)u¯ ∗ y¯ ∗ . = ∗ 2 (−1 ± 2 y¯ )

B¯ 2

(52)

Let us recall that y¯ ∗ ∈ C + in the saddle case and y¯ ∗ ∈ C − in the elliptic case. We know that H (0, λ) = 0 for all λ ∈ Λ. Since ( B¯ 2 , u¯ ∗ ) ∈ S1 and u¯ ∗  0, (52) has at most one zero (counting multiplicity). Using Rolle’s theorem we find that

B¯ 2

  ∂ I ∗±  ∗ ∂ I ∗±  ∗ y¯ , 0, 0, λ + u¯ ∗ y¯ , 0, 0, λ ∗ ∂ B2 ∂u

(53)

∗±

∗±

has at most 2 zeros (counting multiplicity). Since ∂∂IB (0, 0, 0, λ) = 0 and ∂∂Iu ∗ (0, 0, 0, λ) = 0 (see 2 Lemma 3.2), the expression (53) has at most one zero counting multiplicity. It implies that expression ¯± ∂

(50) has at most one zero counting multiplicity for (δ, κ ) ∼ (0, 0). In other words, ∂ y¯ c∗ has at most one simple zero for δ = 0. ¯ c± has at most two zeros (counting multiUsing Rolle’s theorem once more we finally get that

∗ ± ¯ plicity) w.r.t y ∈ C . Of course, in that statement we suppose that (δ, B¯ 0 , B 2 , u ∗ ) ∼ (0, 0, 0, 0), δ = 0 have at most two limit cycles, multiplicity and B 22 + (u ∗ )2 > 0. It follows that the systems X ∗ F¯± ∗ taken into account, which are close to the set

δ, B 0 , B 2 ,u y¯ ∗ ∈C ±

,λ

L y¯ ∗ .

Remark 15 (Proof of Theorem 2.6). Using Lemma 3.2 it can be directly seen that (53) (and consequently ∗± ∗± I ∗± ) has a simple zero. We merely write M 1± = ∂∂IB ( y¯ ∗ , 0, 0, λ), M 2± = ∂∂Iu ∗ ( y¯ ∗ , 0, 0, λ) and put 2





¯∗ B¯ ± 2 y ,λ =  in (53).

±

M2

M 1±2 + M 2±2

,





u¯ ∗± y¯ ∗ , λ = − 

M 1± M 1±2 + M 2±2

R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

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Supposing that I ∗± ( y¯ ∗0 , B¯ 02 , u ∗0 , λ0 ) = 0, where y¯ ∗ = y¯ ∗0 is a simple zero, it follows from [13] that

the systems X ∗ F¯± with δ > 0, and δ ∼ 0, contain a saddle-node bifurcation of limit cycles near δ, B 0 , B 2 ,u ∗ ,λ 0 ∗ ∗ L y¯ ∗ for ( B 2 , u ) ∼ ( B¯ 2 , u 0 ) and ( B¯ 0 , λ) ∼ (0, λ0 ). 0

3.8.5. Cyclicity of L 1 (in the saddle case) 2

We study the cyclicity of the slow-fast two-saddle-limit periodic set as in (v) in the saddle case, with {(−1, 12 )} i.e. the cyclicity of the limit periodic set L 1 consisting of a fast orbit of X ∗ F¯+ ∗ 0, B 0 , B 2 ,u ,λ

2

as

α -limit set and {(1, 12 )} as ω-limit set and containing the piece of the critical curve of X 0∗,FB¯+, B 0

2 ,u

∗ ,λ

between x¯ ∗ = −1 and x¯ ∗ = 1. Hence the fast orbit ends up (on both sides) in the hyperbolic zeros of the slow dynamics for B 2 = u ∗ = 0. We know that near (¯x∗ , y¯ ∗ ) = (1, 12 ) the family X ∗ F¯+ has a persistent hyperbolic saddle ∗ δ, B 0 , B 2 ,u ,λ

∗F + μ1 > 0, and that near (¯x∗ , y¯ ∗ ) = (−1, 12 ) the family X δ, has B¯ 0 , B 2 ,u ∗ ,λ 2 a persistent hyperbolic saddle with ratio of eigenvalues −δ μ2 , μ2 > 0. On account of symmetries, it is clear that μ1 = μ2 for B¯ 0 = B 2 = u ∗ = 0.

with ratio of eigenvalues −δ 2 μ1 ,

Limit cycles near canard limit periodic sets for which the fast orbit connects two saddles on the critical curve have been studied in [4], under the non-degeneracy condition that the quantities μ1 and μ2 are not equal. Of course, this condition is violated in our case. We will present an analysis in the limited framework of this paper, but it should be clear that a similar treatment is possible in the more general framework of [4]. Inspired by [4] we define transition maps near L 1 . Consider an open section S 0 = {¯x∗ = 0, 2

y¯ ∗ > τ } ⊂ R2 such that 0 < τ < 12 and τ ∼ 12 . We parametrize S 0 by coordinate y¯ ∗ and define S ∗ = S 0 × [0, δ0 ] where δ0 > 0 is small. Consider also the section T as prescribed in Section 3.8.4, and define T ∗ = T × [0, v ∗0 ] (with δ0 = v ∗0 ). We parametrize T ∗ by the (secondary) blow-up coordinates ( y˜ ∗ , v ∗ ). Suppose that W ∗ is a ball of small radius at the origin ζ := ( B¯ 0 , B 2 , u ∗ ) = (0, 0, 0). In what fol∗F + the system X ∗ F¯+ lows, we suppose that ζ ∈ W ∗ and we denote by X δ,ζ,λ ∗ . We know that near δ, B 0 , B 2 ,u ,λ

(¯x∗ , y¯ ∗ , δ) = (±1, 12 , 0) there is a smooth curve S ± in the (¯x∗ , y¯ ∗ , δ)-space, consisting of hyperbolic ∗F + ∂ + 0 ∂δ . Let M unst (resp. M st ) be the union of unstable (resp. stable) manifolds at saddles of X δ,ζ,λ points of S − (resp. S + ). The smooth (ζ, λ)-families of manifolds M unst and M st subdivide the section

S ∗ in a “left” and a “right” open region. By definition the “right” open region is closer to the contact point (¯x∗ , y¯ ∗ ) = (0, 0) (see [4]). An open set U ⊂ S 0 × ]0, δ0 ] × W ∗ × Λ is admissible when for each ( y¯ ∗ , δ, ζ, λ) ∈ U ( y¯ ∗ , δ) is contained within the “right” open region (w.r.t. the subdivision of S ∗ by both M unst and M st ). The following theorem has been proven in [4] (Theorem 2.1.): Theorem 3.3. Let U ⊂ S 0 × ]0, δ0 ] × W ∗ × Λ be an admissible set. For all ( y¯ ∗ , δ, ζ, λ) ∈ U the point ∗F + ∗F + ∂ ∂ ( y¯ ∗ , δ, ζ, λ) reaches the plane T ∗ in finite time when following the orbits of X δ,ζ,λ + 0 ∂δ and − X δ,ζ,λ − 0 ∂δ . Furthermore, for all k > 0 there exists a 0 < δk  δ0 so that the mappings

        F : U ∩ {δ  δk } → T ∗ : y¯ ∗ , δ, ζ, λ → y˜ ∗ , v ∗ = F y¯ ∗ , δ, ζ, λ , δ

(54)

        B : U ∩ {δ  δk } → T ∗ : y¯ ∗ , δ, ζ, λ → y˜ ∗ , v ∗ = B y¯ ∗ , δ, ζ, λ , δ

(55)

and

∗F + ∂ (defined by following the orbits of X δ,ζ,λ + 0 ∂δ in respectively forward and backward time) are C ∞ and have

a C k -extension to the closure of U ∩ {δ  δk } (U ∩ {δ  δk }).

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R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

Especially, Theorem 3.3 is valid if we take the following admissible set:

U1 =





y¯ ∗ , δ, ζ, λ :





τ < y¯ ∗ < min y¯ ∗unst (δ, ζ, λ), y¯ ∗st (δ, ζ, λ) , (δ, ζ, λ) ∈ ]0, δ0 ] × W ∗ × Λ ,

where y¯ ∗unst (δ, ζ, λ) (resp. y¯ ∗st (δ, ζ, λ)) represents the smooth (including δ = 0) intersection of the M unst (resp. M st ), at the δ -level, and S 0 . ∗F + Closed orbits of X δ,ζ,λ near L 1 are given now by zeros of the difference map 2

     

y¯ ∗ , δ, ζ, λ = F y¯ ∗ , δ, ζ, λ − B y¯ ∗ , δ, ζ, λ , where ( y¯ ∗ , δ, ζ, λ) ∈ U 1 ∩ {δ  δk } =: U 1k and where F (resp. B) is defined in (54) (resp. (55)). Since ∗F + X δ,ζ,λ is invariant under the symmetry (¯x∗ , ζ, t ) → (−¯x∗ , −ζ, −t ), we can write as

     

y¯ ∗ , δ, ζ, λ = F y¯ ∗ , δ, ζ, λ − F y¯ ∗ , δ, −ζ, λ .

(56)

where ( y¯ ∗ , δ, ζ, λ) ∈ U 1k . Taking into account (56) and [4] it can be proven, based on Rolle’s theorem, that for fixed ∗F + (δ, ζ, λ) ∈ ]0, δk ] × W ∗ × Λ the number of periodic orbits of X δ,ζ,λ in the region U 1k , at the (δ, ζ, λ)-level, is bounded by 1 + the number of solutions (multiplicity taken into account) of

    δ 2 I y¯ ∗ , δ, ζ, λ − δ 2 I y¯ ∗ , δ, −ζ, λ + δ 2 O (ζ ) = 0,

(57)

where O (ζ ) is C ∞ in U 1k and has a C k -extension to the closure U 1k and where I ( y¯ ∗ , δ, ζ, λ) is the

divergence integral taken along the orbit through the point ( y¯ ∗ , δ, ζ, λ) ∈ U 1k in positive time until it

hits T ∗ . In what follows, we prove that (57) has at most 1 solution (counting multiplicity) near y¯ ∗ = 12 for each fixed δ > 0, ζ = (0, 0, 0), (δ, ζ ) ∼ (0, 0, 0, 0) and λ ∈ Λ. This will imply that the cyclicity of L 1 is 2

bounded by 2. When δ > 0 and ζ = (0, 0, 0), then we deal with a center. In order to study the divergence integral I ( y¯ ∗ , δ, ζ, λ) we use the following lemma (see [4], Lemma 4.3.): Lemma 3.4. Let Ψ : V ⊂ Rn → V  ⊂ Rn : y → x = Ψ ( y ) be a diffeomorphic transformation between two local charts of an n-dimensional manifold. Let X be a vector field defined on V  and let Y = Ψ ∗ ( X ) be the pull back of this vector field on V . Then



 div

Rn

X dt =

divRn Y dt + log O

Ψ (O )

J ( y2 ) J ( y1 )

where O is an orbit of Y from one point y 1 of V to another point y 2 and where J ( y ) is the Jacobian determinant of the transformation Ψ . Let h.Y be an equivalent vector field on V for some strictly positive function h. Then



 divRn (Y ) dt =

O

where dt  = dt /h.

O

divRn (hY ) dt  − log

h( y 2 ) h( y 1 )

,

R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

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Lemma 3.4 allows to compute the divergence integral in normal form coordinates. First we split up I ( y¯ ∗ , δ, ζ, λ) in three parts: 1) The passage along the fast fiber does not have a significant effect on the divergence inte∗F + gral: this contribution in δ 2 I ( y¯ ∗ , δ, ζ, λ) is O (δ 2 ) and C ∞ . In fact the vector field X δ,ζ,λ is (locally) ∞ ∗ C -conjugate to a divergence free flow box, so the contribution in I ( y¯ , δ, ζ, λ) only consists of extra (C ∞ -) log-terms that appear as in Lemma 3.4. 2) To study contribution in δ 2 I ( y¯ ∗ , δ, ζ, λ) of the passage near the curve S + of hyperbolic saddles we use a C k -normal form (see [4]):

⎧ 2 ⎨ v˙1 = δ μ1 v 1 , v˙ = − v 2 , ⎩ 2 δ˙ = 0, where −δ 2 μ1 = −δ 2 μ1 (δ, ζ, λ) is the above mentioned ratio of eigenvalues. We consider the planes { v 1 = 1} and { v 2 = 1} in these normal form coordinates and we assume that the orbit through ( y¯ ∗ , δ, ζ, λ) intersects the plane { v 2 = 1} in a point with v 1 = ς ( y¯ ∗ , δ, ζ, λ), where ς is C k . Taking into account Lemma 3.4 passage from { v 2 = 1} to { v 1 = 1} leads to the following contribution in the divergence integral I ( y¯ ∗ , δ, ζ, λ) multiplied by δ 2 :

1 δ

2

ς ( y¯ ∗ ,δ,ζ,λ)

  1 − μ1 δ 2     −1 + μ1 δ 2 dv 1 + O δ 2 = ln ς y¯ ∗ , δ, ζ, λ + O δ 2 , 2 μ1 μ1 δ v 1

where O (δ 2 ) is C k . (To see that O (δ 2 ) is C k we refer to Theorem 3.1 in [4].) 3) The passage from { v 1 = 1} to T ∗ leads to a contribution in δ 2 I ( y¯ ∗ , δ, ζ, λ). More specifically, it is an O (1)-expression that is δ -regularly C k in ( y¯ ∗ , ζ, λ), i.e. O (1) and all its derivatives up to order k w.r.t. ( y¯ ∗ , ζ, λ) are continuous including at δ = 0. We refer to [3] for more details. (There is no need for further specification of this O (1)-term, as it will not be the leading order part in the analysis.) Putting the contributions 1), 2) and 3) together (57) can be written as

1 − μ1 (δ, ζ, λ)δ 2

μ1 (δ, ζ, λ) where





ln ς y¯ ∗ , δ, ζ, λ −

1 − μ1 (δ, −ζ, λ)δ 2

μ1 (δ, −ζ, λ)





ln ς y¯ ∗ , δ, −ζ, λ + O (ζ ) = 0,

(58)

ς is C k and where O (ζ ) is δ -regularly C k . In order to simplify (58) we first calculate μ1 : 











μ1 (δ, ζ, λ) = 2 + B 2 1 + O B 2 , u ∗ + u ∗ 3H (0, λ) + O B 2 , u ∗ + O (δ),

(59)

where O ( B 2 , u ∗ ), O ( B 2 , u ∗ ), O (δ) are C ∞ . Taking into account (59) Eq. (58) is equivalent to











ln ς y¯ ∗ , δ, ζ, λ − 1 + B 2 1 + O B 2 , u ∗ , δ



      + u ∗ 3H (0, λ) + O B 2 , u ∗ , δ + O ( B¯ 0 δ) ln ς y¯ ∗ , δ, −ζ, λ + O (ζ ) = 0,

where O ( B 2 , u ∗ , δ), O ( B 2 , u ∗ , δ), O ( B¯ 0 δ) are C ∞ and where O (ζ ) is δ -regularly C k . Next, we calculate the difference

     

ς y¯ ∗ , δ, ζ, λ := ς y¯ ∗ , δ, ζ, λ − ς y¯ ∗ , δ, −ζ, λ . It can be easily seen that

(60)

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R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

y¯ ∗st (δ, ζ, λ) =



    1 H (0, λ) + B 2 − + O B 2 , u∗ + u∗ − + O B 2 , u ∗ + O (δ),

1 2

2

2

(61)

where O ( B 2 , u ∗ ), O ( B 2 , u ∗ ), O (δ) are C ∞ , and that





 





ς y¯ ∗ , δ, ζ, λ = y¯ ∗ − y¯ ∗st (δ, ζ, λ) .ς¯ y¯ ∗ , δ, ζ, λ .

(62)

Because ς is a diffeomorphism between the section S ∗ and the plane { v 2 = 1}, we have ς¯ < 0. Without loss of generality we suppose that ς¯ ( 12 , 0, 0, 0, 0, λ) ≡ −1. Based on (61) and (62) we have now



1 ∗

ς y , δ, ζ, λ = B 2 −1 + O y¯ − , ζ, δ 

¯∗



2







1 1 + B¯ 0 O y¯ ∗ − , ζ, δ , + u ∗ − H (0, λ) + O y¯ ∗ − , ζ, δ 2

2

(63)

where O ( y¯ ∗ − 12 , ζ, δ), O ( y¯ ∗ − 12 , ζ, δ), O ( y¯ ∗ − 12 , ζ, δ) are C k .

∗F + There exists a C k -function b˜ 0 ( y¯ ∗ , δ, B 2 , u ∗ , λ) such that periodic orbits of systems X δ,ζ,λ , near

the limit periodic set L 1 , can only occur for parameter B¯ 0 = b˜ 0 ( y¯ ∗ , δ, B 2 , u ∗ , λ). Let us prove this 2

statement. The forward transition map F is defined in the “right” open region (w.r.t. the division of S ∗ by M st ) and has a C k -extension to the closure of the “right” open region (see Theorem 3.3). Note that ( y¯ ∗ , δ, ζ, λ) = ( 12 , 0, 0, 0, 0, λ) is in that closure. We claim that the function F has, near

( 12 , 0, 0, 0, 0, λ), a C k -extension to a neighborhood of ( 12 , 0, 0, 0, 0, λ), for each λ ∈ Λ. Indeed, in nor-

mal form coordinates we can treat the closed “right” open region as a (convex) rectangle and we then use the Whitney extension theorem (see [42]). We denote by F˜ the extension of F . Taking into account (56) we consider now the C k -extension of the difference map :

      ˜ y¯ ∗ , δ, ζ, λ := F˜ y¯ ∗ , δ, ζ, λ − F˜ y¯ ∗ , δ, −ζ, λ .

(64)

˜ ≡ in U k . Since ( 1 , 0, B¯ 0 , 0, 0, λ) ∈ U k (see (61)), we have that ( ˜ 1 , 0, 0, 0, 0, λ) = Clearly, we have

1 1 2 2 ˜ 0 and ∂ ¯ ( 12 , 0, 0, 0, 0, λ) = 0 ( B¯ 0 is a “breaking parameter”). The implicit function theorem implies ∂ B0

now existence of a unique C k -function b˜ 0 ( y¯ ∗ , δ, B 2 , u ∗ , λ), with ( y¯ ∗ , δ, B 2 , u ∗ ) ∼ ( 12 , 0, 0, 0), such that

˜ = 0, for ( y¯ ∗ , δ, ζ ) ∼ ( 1 , 0, 0, 0, 0), can only occur for B¯ 0 = b˜ 0 ( y¯ ∗ , δ, B 2 , u ∗ , λ). Hence, it solution of

2 is sufficient to study zeros of the following c-family:

     

c y¯ ∗ , δ, B 2 , u ∗ , λ := y¯ ∗ , δ, b˜ 0 c , δ, B 2 , u ∗ , λ , B 2 , u ∗ , λ for each c ∼ 12 . On account of (64) we have

b˜ 0 (c , δ, 0, 0, λ) = 0

(65)

for c ∼ 12 . We make a rescaling ( B 2 , u ∗ ) = κ ( B¯ 2 , u¯ ∗ ) in the parameter space, where κ > 0, u¯ ∗  0 and ¯ ( B 2 , u¯ ∗ ) ∈ S1 . If we substitute the control curve b˜ 0 for the parameter B¯ 0 in (60), and if we use (65), then our attention goes to the study of solutions of







ln ς+ − 1 + κ B¯ 2 + 3H (0, λ)u¯ ∗ + O (κ , δ)

ln ς− + O (κ ) = 0,

(66)

R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

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where O (κ , δ) is C k and does not depend on y¯ ∗ , O (κ ) is δ -regularly C k and where







ς+ = ς y¯ ∗ , δ, b˜ 0 c , δ, B 2 , u ∗ , λ , B 2 , u ∗ , λ 





and





ς− = ς y¯ ∗ , δ, −b˜ 0 c , δ, B 2 , u ∗ , λ , − B 2 , −u ∗ , λ .

(67)

From (63) we find that





1

ς+ − ς− = κ − B¯ 2 − H (0, λ)u¯ ∗ + O y¯ ∗ − , κ , δ



2

(68)

where O ( y¯ ∗ − 12 , κ , δ) is C k . We recall that H (0, λ) = 0 for each λ ∈ Λ. First, we claim that Eq. (66) has at most one solution (counting multiplicity) when B¯ 2 + H (0, λ)u¯ ∗ = 0. It is sufficient to prove that the derivative of the left hand side of (66) w.r.t. y¯ ∗ is never zero (for κ > 0). We get ∂ ς+ ∂ y¯ ∗

ς+ =





 − 1 + κ B¯ 2 + 3H (0, λ)u¯ ∗ + O (κ , δ)

∂ ς− ∂ y¯ ∗

ς−

+ O (κ )

ς− ∂∂ςy¯+∗ − ς+ ∂∂ςy¯−∗ − κ ( B¯ 2 + 3H (0, λ)u¯ ∗ + O (κ , δ))ς+ ∂∂ςy¯−∗ + O (κ ) ς+ ς−

=κ

− B¯ 2 − H (0, λ)u¯ ∗ + O ( y¯ ∗ − 12 , κ , δ)

ς+ ς−

+ O (1)

κ (±∞) for ς+ ς− → 0. Here we have used ς+ = O ( y¯ ∗ − 12 , κ , δ), ∂∂ςy¯±∗ = −1 + κ , δ) and ς− ∂∂ςy¯+∗ − ς+ ∂∂ςy¯−∗ = κ (− B¯ 2 − H (0, λ)u¯ ∗ + O ( y¯ ∗ − 12 , κ , δ)). More delicate is the

which clearly tends to O ( y¯ ∗

−

1 , 2

case B¯ 2 + H (0, λ)u¯ ∗ ∼ 0. In that case, we use the fact that B¯ 2 + 3H (0, λ)u¯ ∗ = ( B¯ 2 + H (0, λ)u¯ ∗ ) + 2H (0, λ)u¯ ∗ is strictly positive or strictly negative depending on the sign of H (0, λ). Let us recall that ( B¯ 2 , u¯ ∗ ) ∈ S1 and u¯ ∗  0. Without loss of generality we suppose that H (0, λ) is strictly positive and B¯ 2 + 3H (0, λ)u¯ ∗ + O (κ , δ) in (66) is equal to 1, and we write (66) as

ln ς+ − (1 + κ ) ln ς− + O (κ ) = 0.

(69)

Remark 16. When H (0, λ) < 0 for some λ ∈ Λ, then we replace (69) by ln ς+ − (1 − κ ) ln ς− + O (κ ) = 0. The study of this case will be analogous to the study of the case where H (0, λ) > 0 for some λ ∈ Λ. For the sake of simplicity, we write (68) as ς+ − ς− = κ  where  ∼ 0. We then show that (69) has no solutions for   0. Indeed, in that case, we have

ln ς+ − (1 + κ ) ln ς− + O (κ )  ln ς− − (1 + κ ) ln ς− + O (κ )

  = κ − ln ς− + O (1) → κ (+∞)

as ς− → 0. Suppose now that  < 0. First we take − = ς+ ¯ where ¯ ∈ ]0, ¯ 0 ] for ¯ 0 arbitrarily large and fixed. We then show that (69) has no solutions. Indeed, we have

ln ς+ − (1 + κ ) ln ς− + O (κ ) = ln ς+ − (1 + κ ) ln(ς+ + κς+ ) ¯ + O (κ )

= −κ ln ς+ − (1 + κ ) ln(1 + κ ) ¯ + O (κ )   = κ − ln ς+ + O (1)

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R. Huzak et al. / J. Differential Equations 255 (2013) 4012–4051

which tends to κ (+∞) as ς+ → 0. Suppose now that ς+ = − ς¯+ where ς¯+ > 0 is close to zero. We then show that (69) has at most one solution counting multiplicity. We consider the derivative of the left hand side in (69): ∂ς

∂ς

+ −  ∂  ∂ y¯ ∗ ∂ y¯ ∗ ln ς − ( 1 + κ ) ln ς + O ( κ ) = − ( 1 + κ ) + O (κ ) + − ∂ y¯ ∗ ς+ ς−

=

∂ ς+ ∂ y¯ ∗

− ς¯+

=−

− (1 + κ )

∂ ς− ∂ y¯ ∗

−(ς¯+ + κ )

+ O (κ )

κ ( ∂∂ςy¯+∗ − ς¯+ ∂∂ςy¯−∗ ) + ς¯+ ( ∂∂ςy¯+∗ − ∂∂ςy¯−∗ ) + O (κ )  ς¯+ (ς¯+ + κ ) 1



∂ ς+ 1 ∂ y¯ ∗ + O (ς¯+ ) =κ − + O (1) .  ς¯+ (ς¯+ + κ ) ∂ς

Since ∂ y¯+∗ is strictly negative, this expression tends to κ (−∞) as − ς¯+ → 0. Putting all informations together we find that the set L 1 produces at most 2 limit cycles (see also 2

Remark 17).

Remark 17. When B¯ 2 + H (0, λ)u¯ ∗ = 0, then applying Rolle’s theorem we find at most 2 limit cycles near L 1 . In the case B¯ 2 + H (0, λ)u¯ ∗ ∼ 0, we have applied a blow-up to (−, ς+ ) ∼ (0, 0) and then 2

we have considered (69) in two different charts of that blow-up, in one of which we have proved that (69) is nonzero and in the other that the derivative of (69) w.r.t. y¯ ∗ is nonzero. Since  and ς+ depend on y¯ ∗ , it is possible that (−, ς+ ) “goes” through the different charts as y¯ ∗ ∼ 12 varies. Since

≈ −1 and ς¯+ ∼ 0, it is clear that, once we have left one of these two charts by varying y¯ ∗ , we do not enter back into that chart. This fact allows us to use Rolle’s theorem and to see that L 1 produces ∂ ς+ ∂ y¯ ∗

at most 2 limit cycles, for B¯ 2 + H (0, λ)u¯ ∗ ∼ 0. 3.8.6. Cyclicity of L 00 (in the elliptic case) We study limit cycles of X ∗−¯

δ, B 0 , B 2 ,r ,λ

2

, defined in (39), near L 00 but this time for B 2 ∼ 0 and

H (0, λ) = 0 for each λ ∈ Λ. As we mentioned above [21] is devoted to the study of the limit cycles of (39) near L 00 . From Theorem 3.3 in [21], we infer that the cyclicity of L 00 is bounded by 2. In other words, no more than 2 limit cycles of (39) may bifurcate from L 00 for (δ, B¯ 0 , B 2 , r ) ∼ (0, 0, 0, 0) and λ ∈ Λ. References [1] H.W. Broer, V. Naudot, R. Roussarie, K. Saleh, A predator–prey model with non-monotonic response function, Regul. Chaotic Dyn. 11 (2) (2006) 155–165. [2] J.-L. Callot, F. Diener, M. Diener, Le problème de la “chasse au canard”, C. R. Acad. Sci. Paris Sér. A–B 286 (22) (1978) A1059–A1061. [3] P. De Maesschalck, F. Dumortier, Time analysis and entry-exit relation near planar turning points, J. Differential Equations 215 (2) (2005) 225–267. [4] P. De Maesschalck, F. Dumortier, Canard cycles in the presence of slow dynamics with singularities, Proc. Roy. Soc. Edinburgh Sect. A 138 (2) (2008) 265–299. [5] P. De Maesschalck, F. Dumortier, Singular perturbations and vanishing passage through a turning point, J. Differential Equations 248 (9) (2010) 2294–2328. [6] P. De Maesschalck, F. Dumortier, Detectable canard cycles with singular slow dynamics of any order at the turning point, Discrete Contin. Dyn. Syst. 29 (1) (2011) 109–140. [7] P. De Maesschalck, F. Dumortier, Slow-fast Bogdanov–Takens bifurcations, J. Differential Equations 250 (2) (2011) 1000–1025. [8] F. Dumortier, R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc. 121 (577) (1996), x+100, with an appendix by Cheng Zhi Li.

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