CANARD CYCLES WITH TWO BREAKING PARAMETERS

CANARD CYCLES WITH TWO BREAKING PARAMETERS Freddy DUMORTIER Universiteit Hasselt, Campus Diepenbeek Agoralaan - Gebouw D B-3590 Diepenbeek, Belgium E-mail: freddy.dumortier@uhasselt.be and Robert ROUSSARIE Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S. Université de Bourgogne, B.P. 47 870 21078 Dijon Cedex, France E-mail: roussari@u-bourgogne.fr Abstract We consider two-dimensional slow-fast systems with a layer equation exhibiting canard cycles. The canard cycles under consideration contain both a turning point and a fast orbit connecting two jump points. At both the turning point and the connecting fast orbit we suppose the presence of a parameter permitting generic breaking. Such canard cycles depend on two parameters, that we call phase parameters. We study the relaxation oscillations near the canard cycles by means of a map from the plane of phase parameters to the plane of breaking parameters. 2000 Mathematics Subject Classification : 34C05,34C07,34C23,34C26,34E15. Key words and phrases : Slow-fast system, bifurcation, canard cycle, Liénard equation. 1 Introduction In the papers [DR2] and [DR3] we have studied how multiple relaxation oscillations of canard type can bifurcate from canard cycles. The results essentially deal with families of planar differential systems in which two different times scales play a role. In the so-called fast time such 1

systems take the form X λ, : ẋ = f(x, y, λ) ẏ = g(x, y, λ). (1) where 0 is a small 1-parameter, λ is a multi-dimensional parameter and f and g are smooth functions. The limit for = 0 in equation (1) is called the layer equation, representing for each λ a y-family of 1-dimensional subsystems and containing a curve, called critical curve (or set), defined by {f(x, y, λ) =0}, and consisting of singularities. We are especially interested in periodic orbits that are close, in the Hausdorff sense, to the limit periodic sets of the layer equation X λ,0. These limit periodic sets consist of curves of the critical set together with regular orbits of the layer equation; the latter are called fast orbits. We call such a degenerate limit periodic set a slow-fast cycle, and the curves of singularities are also called slow curves. In general, at the different points of a slow curve, the layer equation has a non-zero eigenvalue and the slow curve is normally hyperbolic at such a point. It is attracting (respectively repelling) in case this eigenvalue is negative (respectively positive). In the other case we speak about contact point. Wealways restrict to slow-fast cycles having a finite number of contact points. Near the most generic contact points, called jump points, aperiodicorbitofx λ,, for > 0, staying close to an attracting slow curve for a while, can only continue close to a fast orbit (see Figure 3). For a precise definition of jump point and proof of the claim, we can e.g. refer to [DR3]. A more degenerate contact point can permit in its neighborhood periodic orbits of X λ, that, after staying a while near an attracting slow curve, continue by following a piece of a repelling curve. Such contact points are called turning points (see e.g. [DR1] or [DR2] and also Figure 2). Such orbits of X λ, that, near a turning point, are close to non-trivial pieces of both attracting and repelling slow curves are called canard solutions. In this spirit we call a slow-fast cycle a canard cycle if it contains both attracting and repelling slow curves. The other slow-fast cycles are said to be common ones. Periodic orbits of X λ, that are in the Hausdorff sense, close to slow-fast cycles are called relaxation oscillations. We called them common (respectively canard) if they are close to a common (respectively canard) slowfast cycle. The papers [DR2] and [DR3] deal with multiple relaxation oscillations and their bifurcations near well-chosen canard cycles. 2

S T P 1 P 2 (a) Figure 1: Canard cycles S (b) In [DR2] the canard cycles are like in Figure 1(a), consisting of a single fast orbit and a single slow curve, containing one turning point. The slow curve is attracting on one side of the turning point, and repelling on the other side. Such a canard cycle belongs to a 1-parameter family of canard cycles, that can be described by any regular parameter used on a segment S transverse to the fast orbits of the canard cycles. Such a parameter plays a different role than the parameter (λ, ) in equation (1). We would like to call it a phase parameter. To study the relaxation oscillations near this 1-parameter family of canard cycles, we use a (family) blow up at the turning point (see [DR2]), leading to a picture like in Figure 2. S T C C 1 C 2 Figure 2: Blow up of turning point The picture represents a three dimensional situation in (x, y, )-variables. The inner part of the circle C has to be considered as a blister on the (x, y)- plane with the circle as edge. The planes { = Const.} in this picture have to be seen as sticking plasters above the blister. The two slow curves in Figure 2 are normally hyperbolic, even at the intersection points with the circle C. We denote them by C 1,C 2. At each C i the sphere obtained by blow-up contains a unique center manifold W i. One can choose along the sphere (the blister) a section T, as in Figure 2, transversally cutting the orbits of the blown up field; also W i will cut T at a point h i (for some regular parameter h on T ). In fact 3

the point h i depends on λ, defining a smooth function h i (λ). In [DR2] it is supposed that λ h 1 (λ) h 2 (λ) is a submersion at some parameter value λ 0, with the property that h 1 (λ 0 )= h 2 (λ 0 ). This permits to choose the parameter λ λ 0 in a way that a = h 1 (λ) h 2 (λ) can be used as an independent parameter; we call it a breaking parameter. In former papers it has sometimes be called a rotational parameter. Changing a leads to quick changes in shape of the canard relaxation oscillations. This breaking mechanism is often linked to a Hopf-bifurcation (see [DR1] and [DR2]) so that we call it the Hopf breaking mechanism. In [DR3] different types of canard cycles have been studied, among which the most interesting ones are represented in Figure 1(b). They consist of two slow curves and two fast curves of which one connects two jump points. They occur e.g. for some parameter value λ = λ 0. Again they occur in a 1-parameter family, that can be described by some regular phase parameter on a section S that is chosen transversally to the fast orbits of the canard cycles. To define an appropriate section T and a related breaking parameter is now easier. We choose any section T, transverse to the fast orbit connecting the two jump points for λ = λ 0. For λ = λ 0, we can expect that there is no longer a fast orbit connecting the two jump points. There is an orbit O 1 having z 1 as α-limit and a fast orbit O 2 having z 2 as ω-limit. Both orbits O i cut T at a single point h i (λ). We again suppose that λ h 1 (λ) h 2 (λ) is a submersion near λ = λ 0, so that we can again consider a = h 1 (λ) h 2 (λ) to be a breaking parameter near λ 0. We call this breaking mechanism the jump breaking mechanism. To study the relaxation oscillations near these canard cycles (see [DR3]), we use (family) blow up at the two jump points z i. It leads to a picture like in Figure 3, representing again a 3-dimensional situation in (x, y, )-space. 4

s 2 s 4 T S Figure 3: Blow up of jump points So, up to now, all canard cycles that we have studied were linked to a single breaking mechanism, requiring a single breaking parameter. The whole study could be performed with a single phase parameter, since the canard cycles under consideration only pass through one layer of fast orbits. It leads to a study of a specific kind of functions depending on a single variable. The nature of this function will be recalled in section 2. In this paper we want to study canard cycles depending on two phase variables and that are broken by two breaking mechanisms. We could also call them two-layer canard cycles, since they pass trough two layers of fast orbits, each one characterized by a phase parameter, the two phase parameters being independent of each other. Instead of having to check the occurrence and bifurcations of critical points for a family of functions in one variable, we will have to consider singularities of a plane to plane map, from the two phase variables into the two breaking parameters. This map will be stable under generic assumptions. Like in the papers [DR2] and [DR3] we will again present our results and give ours proofs for Liénard equations. The setting permits an easy way of describing a precise context and presenting the results. Another advantage is that the formulas we get are ready for applications to specific Liénard systems. Our methods of proof however have a broader range of validity. Similar results as the ones that we will describe for Liénardsystemscouldbestatedandproven for a much larger class of slow-fast systems, not only on the plane, but even on an arbitrary surface, orientable or not. Let us present the results in a more detailed way. We suppose that λ = (a, b) IR 2 and that X λ, is the smooth slow-fast Liénard system ẋ = y F a (x) X λ, (2) ẏ = g b (x). On the interval of definition ] x 0,x 0 [ the function F 0 has two Morse maxima p 1,p 2 with the same value, giving two jump points P 1,P 2. It also has a 5

Morse minimum at 0, situated between the two minima and giving a turning point at Q =(0, 0). The parameter a is a breaking parameter for the jump mechanism and the parameter b can be rescaled to give a parameter b that is a breaking parameter for the Hopf breaking mechanism at the turning point Q (see [DR1],[DR2]). There exists an interval ]α, β[ such that for any (u, v) ]α, β[ 2, we have a canard cycle Γ uv containing the turning point Q, the fast orbit between P 1,P 2 and two other fast orbits, respectively on the left and on the right of Q, at the levels {y = u} and {y = v} (seethefigure4). b P 1 P 2 I Sl a Sr J a {y = u} b {y = v} K Q L y -x 0 p 1 0 p 2 x 0 x Figure 4: Canard cycles with two breaking and phase parameters We define the slow divergence integral of the slow curve between x 1 and x 2 as follows : x2 1 df0 2dx Int(x 1,x 2 )= x 1 g 0 (x) dx (x) (3) Then, at the canard cycle Γ uv are associated four functions : I(u),K(u),J(v) and L(v) that are the slow divergence integrals of the four slow curves contained in Γ uv. Two of these curves are located on the left of Q and are function of u and two of them are on the right of Q and are function of v (see Section 2 for their precise definition). The theory of canard cycle bifurcation as developed in [DR1],[DR2],[DR3] permits to defined four functions Ĩ(u, ), K(u, ), J(v, ) and L(v, ) whichare -regularly smooth in u, v (i.e. continuous in, as well as all their partial derivatives with respect to u, v :seedefinition 2.1), which are equal to I,J,K,L for =0andamap(u, v) Φ (u, v) which controls the relaxation oscillations (i.e. the bifurcating limit cycles) in the following sense : 6

Let Σ l and Σ r be two segments transverse to the fast orbits, parametrized respectively by u and v. Then for > 0 small enough, a limit cycle of X (a,b), cuts Σ l in u and Σ r in v if and only if (a, b) =Φ (u, v). Let D(u, v) = I(u) J(v) + L(v) K(u) be the (total) slow divergence of Γ uv. In Sections 4, 5 we will prove the following result, as a consequence of the Propositions 4.1 and 4.2 : Theorem 1.1. Consider a value (u 0,v 0 ) ]α, β[ 2. 1. If D(u 0,v 0 ) = 0, then (u 0,v 0 ) is a regular point of Φ, for > 0, small enough. A hyperbolic relaxation oscillation bifurcates from Γ uv. 2. If D(u 0,v 0 )=0and I(u 0 ) J(v 0 ) = 0, then (u 0,v 0 ) is a generic fold singularity of Φ, for > 0, small enough. A codimension 1 semi-stable relaxation oscillation bifurcates from Γ uv. This semi-stable limit cycle is generically unfolded by the parameter (a, b), for > 0 small enough, producing a pair of hyperbolic limit cycles. 3. If D(u 0,v 0 )=0and I(u 0 ) J(v 0 )=0and I (u 0 )L (v 0 ) K (u 0 )J (v 0 ) = 0, then (u 0,v 0 ) is a generic cusp singularity of Φ, for > 0, small enough. Acodimension2 relaxation oscillation bifurcates from Γ uv. This degenerate limit cycle is generically unfolded by the parameter (a, b), for > 0 small enough, producing systems having three hyperbolic limit cycles in the vicinity of Γ uv. In Section 7 we will apply Theorem 1.1 to Liénard systems of the form : ẍ +(x x 3 + a)ẋ + (b x(1 + cx + dx 3 )) = 0, with (a, b, c, d) (0, 0, 0, 0), and apply to it Theorem 1.1 to show the existence of such systems (with > 0 arbitrarily small) having 3 limit cycles. 2 Precise setting and definition of the map Φ We consider a smooth slow-fast Liénard system with an expression (2) : ẋ = y F a (x) X λ, ẏ = g b (x). with λ =(a, b) near(0, 0). The smooth functions F a (x) andg b (x) fulfill the following conditions : 1. On some interval ] x 0,x 0 [thefunctionf a has just 3 singular points: 2 maxima at p 1,p 2 and one minimum that we position at 0, with x 0 < p 1 < 0 <p 2 <x 0. The values F a (±x 0 ) are below the minimum value. The parameter a is just the difference a = F a (p 2 ) F a (p 1 ). Let F 0 (p 1 )= F 0 (p 2 )=β and F 0 (0) = α. 7

g 2. Write g b (x) =g(x, b). One supposes that g(0, 0) = 0, x (0, 0) = 0and g b (0, 0) = 0. One supposes also that g 0(x) > 0forx<0andthatg 0 (x) < 0forx>0. For a = b = 0 one has canard cycles containing 3 horizontal segments : one between the two maxima σ h, one below the left maximum σ l (u), at the height y = u between x 1 (u) andx 2 (u) ( x 0 <x 1 (u) < 1 <x 2 (u) < 0), and one below the right maximum : σ r (v), at the height y = v between x 1 (v) and x 2 (v) (0 < x 1 (v) < 1 < x 2 (v) <x 0 ). One has (u, v) [α, β] 2 and we call Γ uv the corresponding canard cycle. For any x 1,x 2 [ x 0,x 0 ] (not ordered), with 0 ]x 1,x 2 [, one considers the slow divergence integral introduced in (3) : x2 1 df0 2dx Int(x 1,x 2 )= x 1 g 0 (x) dx (x) Let us consider the four following integrals I(u) =Int(x 1 (u), 1) J(v) =Int( x 2 (v), 1) K(u) =Int(x 2 (u), 0) L(v) =Int( x 1 (v), 0) These choices are made such that each of these functions is < 0foru or v ]α, β[. Another trivial property is: I,J > 0 and K,L < 0 (4) We choose one vertical section C 1 at x =0, cutting the segment σ h and, as explained in the introduction, one section C 2 transversal to the blow-up halfsphere of 0 cutting transversally the separatrix at infinity on the half-sphere (see [DR1] and [DR2]). The parameter a is the breaking parameter for the section C 1 and a rescaling b of b ( b = 1/2 b for some δ > 0) is the breaking parameter at C 2. Write λ =(a, b). We also consider the vertical sections Σ l = {p 1 } [α, β] and Σ r = {p 2 } [α, β] which cut the two families of horizontal segments σ l (u) and σ r (v) respectively. Definition 2.1. We say that a function f(z,), with z IR p for some p, is - regularly smooth in z (or -regularly C in z) if f is continuous and all partial derivatives of f with respect to z exist and are continuous in (z, ). Remark 2.2. In this paper we will constantly work with functions that are -regularly smooth in z with z equal to (u, v, λ) or to a subset of it. As we will explain later on, the -regularly smooth functions that we will encounter are in fact smooth functions in (u, v, δ, δ log δ) with δ = 1/6. 8

We blow-up the family X λ, at the three singular points (see [DR1] or [DR2] and [DR3] for instance). Let X λ, be the blown up family. Now, it follows from previous papers ([DR1],[DR2],[DD1],[DD2],[DR3]), that there exist functions Ĩ(u, λ, ), K(u, λ, ), J(v, λ, ), L(v, λ, ), which are -regularly C in (u, v, λ), such that Ĩ(u, 0, 0) = I(u), K(u, 0, 0) = K(u), J(v, 0, 0) = J(v), L(u, 0, 0) = L(v) and the transitions have the following expressions : (1) From Σ l to C 1 : u exp( Ĩ(u, λ,) )+f l ( λ, ) (2) From Σ l to C 2 : u exp( K(u, λ,) )+g l ( λ, ) (3) From Σ r to C 1 : v exp( J(v, λ,) )+f r ( λ, ) (4) From Σ r to C 2 : v exp( L(v, λ,) )+g r ( λ, ) with f l,g l,f r,g r functions that are -regularly smooth in λ. This implies that the system of equations for the limit cycles is exp( Ĩ(u, λ,) exp( K(u, λ,) ) exp( J(v, λ,) )=f r ( λ, ) f l ( λ, ) =a F ( λ, ) ) exp( L(v, λ,) )=g r ( λ, ) g l ( λ, ) = b G( λ, ). with F ( λ 0, 0) = 0andG( λ 0, 0) = 0. This can be written as : exp( Ĩ(u, λ,) ) exp( J(v, λ,) )=a exp( K(u, λ,) ) exp( L(v, λ,) )= b. (5) (6) with new functions Ĩ, J, K and L, which differ from the previous ones by terms of order O(), which are -regularly C in (u, v, λ). A problem is that the parameters (a, b) appear in the left hand terms of the equations. But one can solve this system in a and b because the partial derivatives of the right hand terms with respect to a, b, are flat in. One can hence solve (6) to obtain a = a(u, v, ) and b = b(u, v, ) such that the functions a(u, v, ) and b(u, v, ) haveexactlythesameformasthelefthandtermsof(6) except that the functions Ĩ, J, K, L are replaced by functions not depending on λ and which are -flat perturbations of the previous ones. We shall continue to call them Ĩ(u, ),... LetusremarkthatwealwayshaveĨ(u, 0) = I(u),... Then we can rewrite our system Φ : a = exp( Ĩ(u,) b = exp( K(u,) ) exp( J(v,) ) ) exp( L(v,) ). We can see Φ as a family of maps from a plane IR 2, with coordinates (u, v) [α, β] 2, to another plane IR 2, with coordinates (a, b) near0 IR 2 (7) 9

depending on a parameter. For each (a, b) in the image of Φ, with > 0 small enough, each counter-images (u, v) corresponds to a limit cycle Γ uv for the value cutting Σ l in u and Σ r in v. 3 Study of the map Φ 3.1 The curve of singular points We fix small enough and we look at Φ as a plane-to-plane map. To simplify the expressions, we do not indicate everywhere the dependence of the functions Ĩ,... on u, v,. First, let us compute the Jacobian matrix of Φ dφ = 1 Ĩ exp Ĩ 1 K exp K 1 J exp J 1 L exp L where Ĩ, K are the partial derivatives in u and J, L are the partial derivatives in v. We now can compute the singular points of Φ. To find them, let us compute (v, u, ) =detdφ (u, v) = 1 Ĩ L exp 2 exp We introduce the function Ĩ L +exp J exp (8) K K J (9) D = Ĩ J + L K (10) D is a function of (u, v, ) andd(u, v) = D(u, v, 0) is precisely the complete slow divergence integral computed along the canard cycle Γ uv. Using the function D, we can write = 1 J 2 Ĩ L exp exp K K J D exp Ĩ L (11) TheequationforthesingularpointsofthemapΦ which is (u, v, ) =0is equivalent to the equation S(u, v, ) = D(u, v, ) log K J (u, v, ) = 0 (12) Ĩ L Let us check the coherence of this equation : the term in the logarithm is strictly positive as it follows from (4) and moreover S(u, v, 0) = D(u, v) asit can be expected. The function S and the functions Ĩ,... are -regularly smooth in (u, v). Then, any C -stable property of D will be preserved for small values of. Let us consider D(u, v). We can write D(u, v) =(I K)(u) (J K)(v). as (I K) (u) > 0 for all u and also (J K) (v) > 0, the function D has no 10

singular points. Moreover, {D(u, v) = 0} can be solved as the graph of strictly increasing function : v = ψ(u) (i.e. suchthatψ (u) > 0 for every u). On the other side the domain of definition is not known. What you can say, using that D(α, β) < 0andD(β, α) > 0, is that S = {D(u, v) =0} = is a curve joining a point on {α} [α, β] [α, β] {β} to a point on {β} [α, β] [α, β] {α}. Several cases are a priori possible. Using the Implicit Function theorem we find a family of functions ψ (u), for small enough, solutions of {S(u, ψ (u), ) =0}, which are -regularly smooth in u. The domain of definition of ψ depends on and ψ (u) > 0forany(u, ). The set of singular points {S(u, v, ) =0} is a surface S, graphof(u, ) v = ψ (u). For any fixed 0, let us write S for the graph of the curve u v = ψ (u). In particular S 0 = {D(u, v) =0}. 4 Generic singularities 4.1 The fold case Let Φ(u, v) be a smooth plane-to-plane map. The conditions to have a generic fold at a point (u 0,v 0 ) are the following : 1. (u 0,v 0 ) belongs to the critical set of Φ given by S = { (u, v) =detdφ(u, v) = 0}, 2. d (u 0,v 0 ) = 0, i.e. (u 0,v 0 ) is a regular point of S, which is locally a regular curve at (u 0,v 0 ), 3. The eigenvalues of the Jacobian matrix dφ(u 0,v 0 )aredifferent, i.e. this Jacobian has a non-zero eigenvalue, 4. At (u 0,v 0 ), the direction of the eigen-axis of the eigenvalue 0 is transverse to S. Then, as it was proved by Whitney ([W],one can also see [M]), the map Φ can be reduced to the fold model (U, V ) (U, V 2 ). (13) by a local right-left equivalence (i.e by composition by local diffeomorphisms at (u 0,v 0 )andφ(u 0,v 0 )). Let us translate these generic conditions to our context : Proposition 4.1. We consider (u 0,v 0 ) S 0 such that I(u 0 ) J(v 0 ) = 0. Then there is a neighborhood W of (u 0,v 0, 0) in S such that any (u 0,v 0, ) W with > 0, is a generic fold singular point of the map Φ : IR 2 IR 2. There is right-left equivalence of Φ with the fold model (13) given by a couple of families of smooth diffeomorphisms (R (u, v),l (a, b)), defined for > 0 small enough. These diffeomorphisms and their inverses are -regularly smooth 11

in (u, v), respectively in (a, b). The first one R (u, v) is defined on a fixed neighborhood W of (u 0,v 0 ) and the second one on a neighborhood of (0, 0) of size exp( A ) for some positive constant A. Proof We already know that the first two conditions for a generic fold are fulfilled. Let us consider the last two conditions. The eigenvalue equation for the Jacobian matrix (8) at (u, v, ) S is L(v, ) λ 2 L Ĩ(u, ) Ĩ + exp (v, ) exp (u, ) λ = 0 (14) L Ĩ The two eigenvalues are 0 and exp L(v,) (v, ) +exp Ĩ(u,) (u, ). As L (v, ) andĩ (u, ) have opposite signs, the second eigenvalue is different from zero. The slope of the eigenaxis P 0 (u, v, ) for the value 0 is equal to exp p 0 (u, v, ) = exp J(v,) Ĩ(u,) J (v, ) = J (v,) Ĩ (u, ) I (u, ) exp J(v, ) Ĩ(u, ) (15) Let us choose a neighborhood W such that inside W the function J(v, ) Ĩ(u, ) is bounded away from 0 and so has a constant sign. Then if 0, the slope p 0 (u, v, ) + or depending on whether J(v 0 ) I(u 0 ) is positive or negative. Then, if a neighborhood W of (u 0,v 0, 0) in S is chosen small enough, the direction of P 0 (u, v, ) will be almost horizontal or vertical and hence will be transverse to the curve S for (u, v) S. As such at these points, the conditions for a generic fold are fulfilled. Let us look now at a right-left equivalence with the fold model. The conditions for a generic fold for Φ are fulfilledinawholeneighborhoodw of (u 0,v 0 ) independent of > 0when 0, although these conditions degenerate at =0. It follows that the family of the right coordinate changes can be defined on a fixed neighborhood of (u 0,v 0 ). The family of the left coordinate changes is then defined on the image of W by Φ, which is of a size : exp( A )for some positive constant A. The continuous dependence on for the right-left equivalence comes from the fact that Φ verifies similar properties. 4.2 The cusp case Let again Φ(u, v) be a smooth plane-to-plane map. The conditions to have a generic cusp singular point (point de fronce in French) at a point (u 0,v 0 )are the following : 1. (u 0,v 0 ) belongs to the critical set of Φ given by S = { (u, v) =detdφ(u, v) = 0}, 12

2. d (u 0,v 0 ) = 0, i.e. (u 0,v 0 ) is a regular point of S, which is locally a regular curve at (u 0,v 0 ), 3. The eigenvalues of the Jacobian matrix dφ(u 0,v 0 )aredifferent, i.e. this Jacobian has a non-zero eigenvalue, 4. At (u 0,v 0 ), the direction of the eigenaxis for the eigenvalue 0 is tangent to S. This can be expressed in the following way. We suppose that locally S is a graph {v = ψ(u)} and that the slope (ratio of the v-component on the u-component) of the eigen-space of the eigenvalue 0 at (u, ψ(u)) S is p(u), then : p(u 0 ) ψ (u 0 )=0 5. This tangency at (u 0,v 0 ) is generic. then the genericity condition is that : d du (p ψ )(u 0 )=p (u 0 ) ψ (u 0 ) = 0. Then, as it was also proved by Whitney ([W],one can also see [M]), the map Φ can be reduced to the cusp model (U, V ) (U, V 3 + UV). (16) by a local right-left equivalence (i.e by composition by local diffeomorphisms at (u 0,v 0 )andφ(u 0,v 0 )). Let us translate these generic conditions to our context : Proposition 4.2. We consider (u 0,v 0 ) S 0 (v 0 = ψ(u 0 )) such that I(u 0 ) J(v 0 )=0. (17) Let us suppose that I (u 0 )L (v 0 ) K (u 0 )J (v 0 ) = 0. (18) Then there is a function u() defined for 0 small enough, smooth for > 0 with u(0) = u 0 and such that the plane-to-plane map Φ has a generic cusp point at (u(), ψ (u()). There is a right-left equivalence of Φ with the cusp model (16) given by a couple of families of smooth diffeomorphisms (R (u, v),l (a, b)), defined for > 0 small enough. These diffeomorphisms and their inverses are -regularly smooth in their respective variables. The first map R (u, v) is defined on a fixed neighborhood W of (u 0,v 0 ) and the second one on a neighborhood of (0, 0) of size exp( A ), for some positive constant A. Proof 13

Suppose that conditions (17), (18) are fulfilled. Recall that the critical curve S is the graph of a function v = ψ (u) which is implicitly given by the equation J(v, ) L(v, )+ log J (v, ) L (v,) = Ĩ(u, ) K(u, )+ log Ĩ (u, ) K (u, ) Putting v = ψ (u), it follows that the derivative of ψ is equal to ψ Ĩ (u, ) K Ĩ (u, )+ (u,) K (u,) (u) = Ĩ (u,) K (u,) J (v,) L (v, )+ J (v,) J L (v,) (v,) L (v,) Now, as it was shown in the proof of the Proposition 4.1, the slope of the 0-eigenspace at the point (u, ψ (u)) is equal to p (u) = J (ψ (u)) Ĩ (u) exp J(ψ (u)) Ĩ(u) (a) First we look for a function u() such that the condition (4) of tangency : p (u) = ψ (u) isverified at u = u(), for any > 0 small enough. This condition is given by J(ψ (u)) Ĩ(u) = log Ĩ (u) J (ψ (u)) Ĩ (u) K (u) J (ψ (u)) L (ψ (u)) that we consider as an equation for an implicit function u(). This equation reduces to J(ψ(u)) I(u)+o (1) = 0 (19) The symbol o (1) stands for a function of (u, ), tendingto0for 0, which is -regularly smooth in u. One has that (J(ψ(u)) I(u)) = J (ψ(u))ψ (u) I (u) or, putting v for ψ(u) to simplify the expression : (J(ψ(u)) I(u)) = J (v) I (u) K (u) J (v) L (v) I (u) = I (u)l (v) K (u)j (v) J (v) L (v) Then the condition (18) implies that (J ψ I)(u 0) = 0. We can apply the Implicit Function Theorem to (19) to find a function u() with initial condition u(0) = u 0, solution of the tangency equation p (u()) = dψ du (u()). By definition one has that v 0 = ψ(u 0 ). (b) Wewantnowtoverifyatu() the generic condition (p ψ )(u()) = p (u()) ψ (u()) = 0 14

Putting u for u(), this is equivalent to : J (ψ (u)) Ĩ (u) exp J(ψ (u)) Ĩ(u) ψ (u) = 0 The left hand term is equal to : 1 χ (u) = ( J(ψ (u)) Ĩ(u))+ J (ψ (u)) Ĩ (u) But, under the tangency condition at u = u(), we have exp J(ψ (u)) Ĩ(u) exp J(ψ (u)) Ĩ(u) = Ĩ (Ĩ K + O()) J ( J L + O()) ψ (u) where the derivatives in the right hand term are evaluated at u() orψ (u()). Putting this in the above expression of χ (u) weobtain J ( J L + O()) Ĩ (Ĩ K + O()) χ (u()) = J ψ I (u 0 )+o (1) where the remainder o (1) which tends to 0 with, is an -regularly smooth function in u. As we have seen above, the principal part J ψ I (u 0 )of this expression is equal to J (v 0 ) I (u 0 ) K (u 0 ) J (v 0 ) L (v 0 ) I (u 0 )= I (u 0 )L (v 0 ) J (v 0 )K (u 0 ) J (v 0 ) L (v 0 ) which is non-zero by the hypothesis (18). Then χ (u()) = 0for > 0small enough. This is equivalent to the genericity condition for the tangency and finishes the part (b) of the proof. The remarks about the conjugacy with the model have the same proof as is the fold case. Remark 4.3. The generic condition (18) can be interpreted as the independence of the two functions D(u, v) and I(u) J(v) at the point (u 0,v 0 ). As D =(I J) (K L) it the same as the independence of D and K L or of I J and K L. For this reason, we will say that a canard cycle Γ u0 v 0 which fulfills the condition I(u 0 ) J(v 0 )=K(u 0 ) L(u 0 )=0and the generic condition (18), is a (generic) canard cycle of codimension 2. A (generic) canard cycle Γ u0 v 0 such that D(u 0,v 0 )=0 and I(u 0) J(v 0) = 0is said of codimension 1. 5 Return map and plane-to-plane map It could look rather strange to study the bifurcation of a limit cycle by considering a plane-to-plane map. In fact, one can always decompose a return map 15

along a limit cycle as a composition of different maps, implying the study of a map in higher dimension. This idea has already extensively been used (see Khovanskii :[Kh], Moussu :[Mo], Kaloshin : [K]). In the previous paragraphs, we looked at a limit cycle Γ uv cutting the two transverse sections Σ l and Σ r at the points u and v respectively. Up to the notation, we have considered transition maps i(u),j(v),k(u),l(v) from these sections to the sections C 1 and C 2. For two parameters a, b the return map we want to consider is essentially the composition of the four maps u P a b(u) =(k b) 1 l j 1 (i a)(u). This is a one-to one function of u depending on the parameter (a, b). Of course, it is completely equivalent to consider either the equation for the fixed points {P a b(u) =u} or the difference equation { a b(u) =0} where a b(u) =j 1 (i a)(u) l 1 (k b)(u) (20) Putting v = j 1 (i a)(u) it follows that solving the equation a b(u) =0is equivalent to finding the counter-images of the map a = i(u) j(v) Φ(u, v) : (21) b = k(u) l(v). Proposition 5.1. The function a b(u) has a generic saddle-node bifurcation point at u 0 for the parameter (a 0, b 0 ) (this means that a0 b0 (u 0 )= a b (u 0)=0 and 2 a b (u 2 0 ) = 0)if and only if the map Φ has a generic fold singularity at (u 0,v 0 ) (where v 0 = j 1 (i a 0 )(u 0 )), with value Φ(u 0,v 0 )=(a 0, b 0 ). The function a b(u) has a generic bifurcation point of codimension 2 at u 0 for the parameter (a 0, b 0 ) (this means that a0 b0 (u 0 )= a b (u 0)= 2 a b (u 2 0 )= 0 and 3 a b (u 3 0 ) = 0)if and only if the map Φ has a generic cusp singularity at (u 0,v 0 ) (where v 0 = j 1 (i a 0 )(u 0 )), with value Φ(u 0,v 0 )=(a 0, b 0 ). Proof We have already noticed that the conditions a0 b0 (u 0 )=0 and v 0 = j 1 (i a 0 )(u 0 ) are equivalent to the condition Φ(u 0,v 0 )=(a 0, b 0 ). Let us introduce the three following functions which are used in order to write the conditions of having a generic fold or a generic cusp for the map Φ : 1. D(u, v) =DetdΦ(u, v) =k (u)j (v) i (u)l (v). 2. T (u, v) =dd(u, v)[j (v),i (u)] 3. G(u, v) defined by dd(u, v) dt (u, v) =G(u, v)du dv 16

From now on we suppose fulfilled the condition a b(u) = 0. We have a generic fold at the point (u 0,v 0 ) if and only if D(u 0,v 0 )=0andT (u 0,v 0 ) = 0. We have a generic cusp if and only if D(u 0,v 0 )=T (u 0,v 0 )=0andG(u 0,v 0 ) = 0. Now the result is a consequence of the following claims, where v = v(u, a) = j 1 (i(u) a) : 1. 2. a b (u) = D(u,v) j (v)l (v) : a b (u) =(j 1 ) (i(u) a) i (u) (l 1 ) (k(u) b) k (u 0 ). As v = j 1 (i(u) a) =l 1 (k(u) b), we have that (j 1 ) (i(u) a) = 1 j (v) and also (l 1 ) (k(u) a) = 1 l (v). From this it follows that a b (u) = i (u) j (v) k (u) D(u, v) l = (v) j (v)l (v) 2 a b 2 (u) = T (u,v) j 2 (v)l (v) modulo a b : From the first claim it follows that : As v 2 a b 2 v v)[1, (u, a)] (u) = dd(u, j (v)l D(u, v) d (v) dv (u, a) = i (u) j (v), one has that 1 j (v)l (v) v dd(u, v)[1, (u, a)] j (v)l = dd(u, v)[j (v),i (u)] T (u, v) (v) j 2 (v)l = (v) j 2 (v)l (v), and the result follows, using the first claim.. 3. 3 a b G(u,v) j 2 (v)l (v) D (u) = modulo { 3 a b v (u,v), 2 a b }. Taking into account 2 the two first claims, we can also prove the equality modulo {D, T}. Let us notice that, as D j + D v i = T and i = 0,j = 0, we have that D = 0 modulo T. This gives meaning to the above expression for 3 a b 3 v (u). To simplify the notation, we no longer write the variables u, v. Using the two above claims we have that 3 a b 3 (u) = 1 T j 2 (v)l (v) + T i v j modulo {D, T} But, taking in account the expression of T we have : T + T i v j = which gives the claim. 1 D D T v v D T = v 1 D G modulo T v 17

Remark 5.2. In the expression of the functions D, T, G one finds derivatives of the functions i, j, k, l of order 1, 2 and 3 respectively. In the present paper the transitions i, j, k, l are defined for > 0, small enough. They have, not mentioning the parameter λ, the following form Ĩ(u, ) J(v, ) i (u) =f l ()+exp K(u, ) k (u) =g l ()+exp,j (v) =f r ()+exp,l (v) =g r ()+exp, L(v, ) They have hence the form f(s, ) =α()+exp F (s,), where s = u or v and F (s, ) is -regularly smooth in s. Let F (s) = F (s, 0). Then, any partial derivative in s of f(s, ), of order at least one,has a principal part in which depends only on the first derivative F s. More precisely, we have for any k IN, k 1 : F (s, ) k k f (s, ) = sk k F s (s) + o (1) exp It is the reason why the conditions on the functions I,J,K,L that we have obtained in the Proposition 4.1 and in the Theorem 1.1, are just on the first derivatives of these functions in u or v. In fact, the limit expressions for the fold curve : D(u, v) =0and for the cusp : I(u) J(v) =0are of order 0., 6 Rescaling codimension 2 canard cycles Let us consider a codimension 2 canard cycle Γ u0 v 0, i.e. a canard cycle that fulfills the conditions I(u 0 ) J(v 0 )=K(u 0 ) L(v 0 )=0 and I (u 0 )L (v 0 ) K (u 0 )J (v 0 ) = 0. As we have already noticed, the second condition means that the map (u, v) (I(u) J(v),K(u) L(v)) has a maximal rank at the point (u 0,v 0 ). The functions Ĩ(u, ) J(v,) and K(u, ) L(v,) being-regularly smooth in (u, v), we can use the Implicit Function Theorem to find a continuous fonction (u(),v()), which verifies u(0) = u 0,v(0) = v 0 and : Ĩ(u(), ) J(v(), ) = K(u(), ) L(v(), ) =0 In order to get a better understanding of the bifurcation diagram near such a codimension 2 canard cycle, we introduce rescaled local phase variables U, V, transverse to the canard cycle Γ u0 v 0, by putting u = u()+u, v = v()+v (22) We will take (U, V ) in an arbitrarily large disk D 1 in IR 2. If we write, for small enough : Ĩ(u(), ) = J(v(), ) =I 0 (), K(u(), ) = L(v(), ) =K0 () 18

and I 1 () =Ĩ (u(), ), L 1 () = L (v(), ), K 1 () = K (u(), ), J 1 () = J (v(), ) we have Ĩ(u()+U, ) = I 0 ()+I 1 ()U + O( 2 ) K(u()+U, ) =K 0 ()+K 1 ()U + O( 2 ) J(v()+V,) = I 0 ()+J 1 ()V + O( 2 ) L(v()+V,) = K 0 ()+L 1 ()V + O( 2 ) (23) The remainder terms O( 2 ) and any remainder term that we will encounter below are -regularly smooth in U, V, λ. We will not repeat this. In particular, the coefficients I 0 (),K 0 () andi 1 (),J 1 (), K 1 (),L 1 () are continuous in. Ofcourse,for,for =0wehave: and I 0 (0) = J 0 (0) = I(u 0 )=J(v 0 ), K 0 (0) = L 0 (0) = K(u 0 )=L(v 0 ) I 1 (0) = I (u 0 ),K 1 (0) = K (u 0 ),J 1 (0) = J (v 0 ),L 1 (0) = L (v 0 ), that we will write : I 1 (0) = I 1, K 1 (0) = K 1,J 1 (0) = J 1,L 1 (0) = L 1 The map Φ, as defined in (7), has the following form in the rescaled variables : ã =exp(i 1 ()U + O()) exp(j 1 ()V + O()) Φ (U, V ) : (24) b =exp(k1 ()U + O()) exp(l 1 ()V + O()) where ã, b are the rescaled parameters : ã = a exp( I 0() ), b = b exp( K 0() ). This parameter (ã, b) is chosen in an arbitrarily large disk D 2 in IR 2 while the initial parameter (a, b) isconfined to an -exponentially flat domain. Let us remark that Φ (U, V ) in (24) has now a smooth non trivial limit for =0, while the limit of (7) was equal to (0, 0). The determinant of the differential dφ (U, V )is (U, V )= I 1 L 1 exp(i 1 U + L 1 V )+J 1 K 1 exp(k 1 U + J 1 V )+O() The curve F of fold points given by { (U, V )=0} can be written : (I 1 K 1 )U (J 1 L 1 )V =log J 1K 1 I 1 L 1 + O() (25) Its limit for = 0 is a line F 0, not passing through (0, 0) (let us notice that I 1 K 1 > 0,J 1 L 1 > 0andthat0< J 1K 1 I 1L 1 =1). Along F, the 0-eigenvalue 19

direction can be represented by a vector (J 1 exp J 1 V +O(),I 1 exp I 1 U +O()) T. This vector is tangent to F under the condition : which reduces to (I 1 K 1 )J 1 exp J 1 V (J 1 L 1 )I 1 exp I 1 U + O() =0 J 1 V I 1 U =log (J 1 L 1 )I 1 (I 1 K 1 )J 1 + O(). (26) Let G be the curve defined by the equation (26). Its limit for =0isthe equation of a line G 0 which is transverse to the line F 0. Then, for > 0small enough, F and G are two regular curves on the disk D 1, which intersect transversally at a unique point C = (U,V ). This point is a generic cusp singularity (the genericity is a direct consequence of the transversality). For =0wehavealimitC 0 =(U 0,V 0 ) = (0, 0) : U0 = J1 L1 J C 0 : 1 K 1 I 1 L 1 log (J 1 L 1 )I 1 (I 1 K 1 )J 1 + J 1 log J1K1 I 1 L 1 V 0 = I1 K1 J 1 K 1 I 1 L 1 log (J 1 L 1 )I 1 (27) (I 1 K 1 )J 1 + I 1 log J1K1 I 1 L 1 To finish we notice that this study for (U, V ) D 1 describes the situation for (u, v) D 1 in the initial variables. The size of this domain is of order O(). On the other hand, we have proved above in the Paragraph 3 that the cusp singularity is in fact the unique codimension 2 singularity in a whole neighborhood of (u 0,v 0 ) in the initial variables (u, v), for any > 0 small enough. What we have more in the rescaled variables is a smooth non trivial limit for =0. This property will be exploited in a forthcoming paper. 7 Application to polynomial Liénard equations of type (4, 3) As announced in the last paragraph of the introduction we will now apply Theorem 1.1 to the polynomial Liénard equations X cd (a,b), ẋ = y (ax + x2 2 x4 4 ) ẏ = (b x(1 + cx + dx 3 )), (28) keeping (a, b, c, d) A B C D a small neighborhood of (0, 0, 0, 0) and checking that the conditions of Theorem 1.1 and more precisely of Proposition 4.2 can be fulfilled. It will be in function of the 3-parameter (a, b, ) at certain canard cycles of system (28), when we take (c, d) = (0, 0) sufficiently small, along a well chosen curve in the (c, d)-plane. So, the parameter (c, d) plays acompletelydifferent role that the parameter (a, b, ). This is reflectedinthe notation X(a,b), cd. As in Section 2 we first consider the layer equation for a = b =0. This layer equation is independent of the parameters. We remark that p 1 = 1,p 2 =1. 20

The common value at these maxima is equal to 1 4 and the value at the minimum 0isequalto0, as in Section 2. For any y ]0, 1 4 [, the equation in x : {F 0(x) =y} has4simpleroots: ± 1 ± 1 4y. In order to simplify the notations, it is convenient to use the variable Y = 1 4y ]0, 1[ ( the maxima are at the height Y = 0 and the minimum is at the height Y =1). We will keep the notations u, v to denote heights of horizontal segment respectively on the left and on the right of 0, defined now in terms of Y and not y. These coordinates differ from the previous ones by the diffeomorphism y 1 4y from ]0, 1 4 [to]1, 0[, and now we have that (u, v) [0, 1]2. The ends of the horizontal segment σ l (u) are x 1 (u) = 1+u and x 2 (u) = 1 u. The ends of the horizontal segment σ r (v) are x 1 (v) = 1 v and x 2 (v) = 1+v. We will denote by Γ uv the canard cycle containing the segments σ l (u)andσ r (v). Let us introduce the 3 following polynomials : P (x) = 1 2 x2 1 2 x4 + 1 6 x6, Q 1 (x) = 1 3 x3 2 5 x5 + 1 7 x7, (29) Q 2 (x) = 1 5 x5 2 7 x7 + 1 9 x9. Letusnoticethat:1+cx + dx 3 =(1+cx + dx 3 ) 1 + O( c 2 + d 2 )onany fixed interval [ x 0,x 0 ]. As a consequence, we obtain by a direct computation : I(u, c, d) =P (1) P ( 1+u)+c( Q 1 (1) + Q 1 ( 1+u)) + d( Q 2 (1) + Q 2 ( 1+u)) + O( c 2 + d 2 ) J(v, c, d) =P (1) P ( 1+v)+c(Q 1 (1) Q 1 ( (30) 1+v)) + d(q 2 (1) Q 2 ( 1+v)) + O( c 2 + d 2 ) and K(u, c, d) =P (0) P ( 1 u)+c(q 1 (0) + Q 1 ( 1 u) + d(q 2 (0) + Q 2 ( 1 u)) + O( c 2 + d 2 ) L(v,c, d) =P (0) P ( 1 v)+c(q 1 (0) + Q 1 ( 1 v) + d(q 2 (0) + Q 2 ( 1 v)) + O( c 2 + d 2 ) (31) For c = d =0, it is easy to verify that each function I J and K L has 0 as regular value, but these two functions are divisible by u v. By the Implicit Function Theorem, there exist two functions v 1,v 2, with the following expansions : v 1 (u, c, d) =u + cv 1 (u)+dw 1 (u)+o(( c + d ) 2 ) v 2 (u, c, d) =u + cv 2 (u)+dw 2 (u)+o(( c + d ) 2 (32) ) 21

smooth for (u, c, d) ]0, 1[ C D, such that J(v 1 (u, c, d),c,d) I(u, c, d) =0 L(v 2 (u, c, d),c,d) K(u, c, d) =0 Putting (32) in the expressions (30) and (31), we obtain : V 1 (u) =4 1+u P ( 1+u) 1 Q1 (1) Q 1 ( 1+u) V 2 (u) =4 1 u P ( 1 u) 1 Q1 ( 1 u) (33) (34) and W 1 (u) =4 1+u P ( 1+u) 1 Q2 (1) Q 2 ( 1+u) W 2 (u) =4 1 u P ( 1 u) 1 Q2 ( 1 u) Taking into account that P (x) =x(1 x 2 ) 2, we have that (35) P ( 1+u) =u 2 1+u and P ( 1 u) =u 2 1 u. Then we can simplify the above expressions : V 1 (u) =4u 2 Q 1 (1) Q 1 ( 1+u) V 2 (u) =4u 2 Q 1 ( 1 u) (36) and W 1 (u) =4u 2 Q 2 (1) Q 2 ( 1+u) W 2 (u) =4u 2 Q 2 ( (37) 1 u) Consider a value (c, d) = (0, 0) and a point u 0 ]0, 1[. The genericity condition (18) in Proposition 4.2 for the canard cycle Γ u0 u 0 is clearly equivalent to the condition : (v 1 v 2 )(u 0,c,d) = 0 (38) In order to transform this condition, we blow up the (c, d)-parameter space at (0, 0). This means that we introduce a new parameter e E =[ E 0,E 0 ], where E 0 > 0 can be chosen arbitrarily large, and the parameter change : C E C D is given by d = ce. In this way we just cover a sector at (0, 0) in C D; this will be sufficient to obtain the expected result (although a study in a complementary sector could be completely similar). In the parameter (c, e), we have (keeping the same notations v 1,v 2, ) : v 1 (u, c, e) v 2 (u, c, e) =cv(u, c, e) with V(u, c, e) =(V 1 (u) V 2 (u)) + e(w 1 (u) W 2 (u)) + O(c). (39) As we will suppose that c = 0, it suffices to verify the condition (38) for the function V. To begin, let us consider the functions V 1 (u) V 2 (u) andw 1 (u) W 2 (u). We can write V 1 (u) V 2 (u) =4u 2 ξ 1 (u) and W 1 (u) W 2 (u) =4u 2 ξ 2 (u) 22

with : ξ 1 (u) =Q 1 (1) Q 1 ( 1+u) Q 1 ( 1 u) ξ 2 (u) =Q 2 (1) Q 2 ( 1+u) Q 2 ( 1 u) Lemma 7.1. We have that (40) 2u 2 ξ 1(u) =2u 4 ξ 2(u) = 1 u 1+u (41) Proof For i =1, 2wehaveQ i (x) =x2i (1 x 2 ) 2. From this, the result follows directly. From (41) it follows that the functions ξ 1 and ξ 2 are strictly decreasing. Moreover we have that ξ 1 (0) = Q 1 (1) = 8 105 < 0andξ 2(0) = Q 2 (1) = 10 315 < 0. So, these functions have no zero on the interval [0, 1]. It is precisely because the function ξ 1 has no zero on ]0, 1[ that we cannot just consider a Liénard system of type (2, 3), corresponding here to d =0. We will take advantage of the fact that ξ 2 has no zero by replacing the function V by the function V(u) V(u, c, e) = W 1 (u) W 2 (u) = u2 V(u) 4 ξ 2 (u) = e + ξ 1(u) + O(c) (42) ξ 2 (u) Let p(u) = ξ 1(u) ξ 2 (u). (43) From (41) it follows that ξ 2 (u) ξ1 (u) = u2 and as a consequence we obtain that ū ]0, 1[ is a singular point of p(u) if and only if ū is a root of the equation u 2 p(u) =1. (44) The function u u 2 p(u) being analytic on [0, 1] in 1 u, and non constant, equation (44) has a finite set of roots : Σ, on [0, 1]. Then,ifwetakeany value u 0 ]0, 1[ Σ, we can apply the Implicit Function Theorem to the function V(u, c, e) atthepointu = u 0,c =0,e = e 0 = p(u 0 ). There exists a smooth function u(c, e) definedinaneighborhoodw =] c 1,c 1 [ ]e 0 e 1,e 0 + e 1 [of c =0,e= e 0, such that V(u(c, e),c,e) 0. In terms of the initial slow divergence integral, it is precisely the conditions (17) of Proposition 4.2 : I(u, c, e) J(v,c, e) =K(u, c, e) L(v, c, e) =0, where u = u(c, e) and v = v 1 (u(c, e),c,e)=v 2 (u(c, e),c,e). Moreover, we choose W such that V (u(c, e),c,e) = 0. For c = 0, this inequality is equivalent to (38), itself equivalent to the genericity condition (18) of Proposition 4.2. Let us summarize what we have obtained : 23

Theorem 7.2. Consider X(a,b), cd as defined in (28) Then there exists a finite set Σ ]0, 1[ with the following property. Let us take any u 0 ]0, 1[ Σ, and let us consider e 0 = p(u 0 )), where p(u) is defined by (43). Then, for c = 0small enough, the layer equation X(0,0),0 cd, with d = ce 0, has a canard cycle near Γ u0 u 0, that satisfies the conditions (17),(18) of Proposition 4.2. As a consequence, this canard cycle bifurcates in function of (a, b) and, producing 3 hyperbolic limit cycles. Remark 7.3. Numerically, we find that the equation u 2 p(u) =1has no root at all, indicating that the set Σ in Theorem 7.2 is most probably empty. Acknowledgments The authors would like to thank the Royal Flemish Academy of Belgium for Science and the Arts, for the financial support and the hospitality during the preparation of this paper. 24

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