Cyclicity of unbounded semi–hyperbolic 2–saddle cycles in polynomial Liénard systems

The paper deals with the cyclicity of unbounded semi–hyperbolic 2–saddle cycles in polynomial Liénard systems of type (m,n) with m < 2n + 1, m and n odd. We generalize the results in [1] (case m = 1), providing a substantially simpler and more transparant proof than the one used in [1].


Introduction
In this paper we will study families of Liénard systems (X (a,b) ) : with (m, n) ∈ N 2 , m and n odd and m < 2n + 1. We fix such (m, n). As an important ingredient of the construction, we observe that X (a,b) is invariant under with a o = (a 1 , a 3 , . . . , a n ), a e = (a 2 , a 4 , . . . , a n−1 ), b o = (b 1 , b 3 , . . . , b m−2 ) and b e = (b 2 , b 4 , . . . , b m−1 ). The motivation to study systems (1) comes from the scalar equations: x + Q(x)ẋ + P (x) = 0, with P and Q polynomials of respective strict degrees m and n and with the highest degree coefficient of P positive. In the phase plane equation (3) can Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://aimsciences.org/journals/displayArticles.jsp?paperID=4986 be written as: ẋ = y, By means of the transformation y = y + F (x), with F (x) = Q(x) and F (0) = 0, one can represent (4) in the so-called Liénard plane as: After putting a singularity at the origin (which can always be done in case m and n are both odd) and a linear rescaling in (x, y, t), expression (5) reduces to (1), at least for a good choice of the parameters (a, b).
In aiming at studying the large amplitude limit cycles of (1) (see [1] for a definition), one uses a compactification of the plane described in [2] and [4]. The best way to do this is by using, near infinity: for r ∼ 0, and multiplying the obtained expression of (1) by r n in order to desingularize. The procedure adds a circle at infinity (given by {r = 0}) near which the extended vector field (that we denote by X (a,b) ) looks like in Figure 1. In this figure double arrows stand for hyperbolic behaviour and simple arrows for semi-hyperbolic behaviour. Because of the chosen conditions on (m, n), it is possible that for some parameter value (a, b) = (a 1 , . . . , a n , b 1 , . . . , b m−1 ), X (a,b) contains a heteroclinic connection between the two semi-hyperbolic saddles at infinity, giving rise to an unbounded semi-hyperbolic 2-saddle cycle.
Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://aimsciences.org/journals/displayArticles.jsp?paperID=4986 In this paper we aim at finding an upperbound on the number of limit cycles that can perturb from such L. When we say (a, b) ∼ (a, b), we will restrict to (a, b) = (a 1 , . . . , a n , b 1 , . . . , b m−1 ) ∈ W, with W a neighbourhood of (a, b) in parameter space. This problem has been treated in [1] under the condition that m = 1. We want to generalize the results from [1] to the case m > 1. As shown in [4] unbounded semi-hyperbolic 2-saddle cycles only occur for Liénard systems of type (m, n) when m < 2n + 1 and both m and n are odd.
We are not yet able to provide a complete generalization for m > 1. We can however prove the results that follow below.
In Section 4, more precisely in (26), we define a sequence of polynomials (P i (a, b)) i=1,2,...,N −1 , with N = 2n + 1 − m, that reveal to play an important role in the subsequent calculations. More precisely we will consider the related sequence of polynomials for some K ∈ N that can be defined as follows: if P 1 , P 3 , . . . , P 2i 2 −1 belong to the ideal generated by c 1 and P 2i 2 +1 does not, . . .
We end with l = K in a way that all (P 1 , P 3 , . . . , P N −1 ) belong to the ideal generated by (c 1 , . . . , c K ). We see that K ≤ N/2.
We call (c 1 , . . . , c K ) the leading large amplitude Lyapunov quantities of the chosen family.

Remarks:
1. In the process just described we can choose for c l the polynomial P 2i l +1 itself or any other (preferably simpler) polynomial that is equal to P 2i l +1 mod (c 1 , . . . , c l−1 ).
2. In the process just described there is no need to work with a full family of Liénard systems as in (1), we can also work with a subfamily. In Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://aimsciences.org/journals/displayArticles.jsp?paperID=4986 case of a full family it will reveal that K = N/2, while for a subfamily it can be strictly smaller. We will only work with subfamilies that are obtained from (1) by restricting one or more parameters a i or b j to a constant value, while keeping the other; let us call them full subfamilies.
We can now formulate our main theorem. a,b) ) be a full subfamily of (1) and let c 1 (a, b), . . . , c K (a, b) be the related leading large amplitude Lyapunov quantities. Let (a, b) = (a, b) be a value for which Y (a,b) is defined and has an unbounded semi-hyperbolic 2-saddle cycle L (a,b) . Then Remark: Instead of saying cyclicity of L (a,b) in the statement of Theorem 1, we can also say large amplitude limit cycles of Theorem 1 does not permit to treat the cyclicity problem of unbounded semi-hyperbolic 2-saddle cycles in families (1) completely. It however induces a complete answer for a number of interesting cases, including the classical Liénard equations that have been treated in [1].
The main problem in trying to apply Theorem 1, consists in calculating the {c j (a, b) | 1 ≤ j ≤ K}. This is rather easy in case for each i we have either a 2i+1 = 0 or b 2i+1 = 0. In the other case the calculation might get quite involved. This is similar to the kind of problems that are encountered in calculating Lyapunov quantities at a non-degenerate singularity of centerfocus type.
In Section 9 we will show that following theorems can be obtained as corollaries of Theorem 1.
Theorem 2 Consider a full subfamily (Y (a,b) ) of (1) such that (where we write a i = 0 when i > n and b i = 0 when i ≥ m): Let (a, b) = (a, b) be a value, satisfying the above conditions, such that Y (a,b) has an unbounded semi-hyperbolic 2-saddle cycle L (a,b) . Then: Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://aimsciences.org/journals/displayArticles.jsp?paperID=4986 Special cases are the following of which Theorem 4 contains the theorem proven in [1]. a,b) ) be a family of polynomial Liénard equations of type (m, n), m ≤ n, with odd friction term: Theorem 4 Let (Y ( a, b)) be a family of polynomial Liénard equations of type (m, n), m ≤ n, with odd forcing term.
Remark: When m > n (⇒ 2n − m < n < m) in the 2 previous theorems, condition (ii) in Theorem 2 imposes extra conditions on the families of Liénard equations with odd forcing or odd friction term.
Of course it might be possible to apply Theorem 1 to specific cases in which some a 2i+1 b 2i+1 are not identically zero. As an example we prove: Theorem 5 Consider the following full subfamily of (1): The methods that we will use in the proof of these results are partly inspired from [1]. However, based on a recent normal linearization theorem at semi-hyperbolic singularities (see [3]) and a systematic use of the symmetry Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS.
We will now first introduce, as in [1], a difference map near the unbounded semi-hyperbolic 2-saddle cycle in a way that large amplitude limit cycles agree with small positive zeros of the difference map.

Study near infinity
As is usual, in working with a compactification as given in (6), it is preferable to work with charts. For a system X (a,b) like in (1), both semi-hyperbolic saddles at infinity lie in the chart in the positive y-direction. This is obtained by means of the tranformation and by multiplying the result with s n , leading to the family (X (a,b) ) : Because of the invariance of X (a,b) under transformation (2), we see that System (X (a,b) ) has two singularities s ± = (±1, 0) that are both semihyperbolic saddles with linear part The behaviour near s − follows easily from the behaviour near s + using Therefore the behaviour of the flow of −X Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://aimsciences.org/journals/displayArticles.jsp?paperID=4986 Combining (9) and (10), we see thatX (a,b) is also invariant under the composition: To understand the behaviour near s + = (1, 0), we study the behaviour on a center manifold W c (a,b) that locally can be written as a graphic for some smooth function (11), we also know that the (a, b)-family of center manifolds is invariant under the same transformation. We hence The behaviour on a center manifold is given bẏ Since m is odd,X (a,b) is of saddle-type near s + and the center manifold is unique.
Proposition 6 Let {u = 1 + u 0 (s, a, b)} be the center manifold ofX (a,b) at s + , then u 0 can be written as where Proof. Using the invariance of u = u 0 (s) = u 0 (s, a, b) under the flow of X (a,b) , one finds: Up to order O(s 2n+1−m ), s → 0 the above equation is given by: This already shows that the ν i do not depend on b.
We continue by substituting in (17). The result will follow by induction on i.
Remark: Since u 0 is invariant under (14) we also see that

The difference map
We will now introduce, in a different way than in [1], an appropriate difference map near an unbounded semi-hyperbolic 2-saddle cycle L as represented in Figure 2. We strongly refer to that figure for the notions and notations that we will introduce now. Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS.
The final publication is available at http://aimsciences.org/journals/displayArticles.jsp?paperID=4986 By the invariance of X (a,b) under (2) one has: a o , a e , b o , b e ) = H + (w, −a o , a e , −b o , b e ).
Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS.
The final publication is available at http://aimsciences.org/journals/displayArticles.jsp?paperID=4986 The difference map Δ (a,b) : Σ 1 0 → Σ 2 0 , expressed in the chosen parameters on the sections Σ 1 0 and Σ 2 0 , can now be defined as: for w ∼ 0 and (a, b) ∼ (a, b). Notice that Δ(w, 0, a e , 0, b e ) = 0. The large amplitude limit cycles correspond to small positive zeros of Δ (a,b) . The cyclicity Cycl (X (a,b) , L, (a, b)) is equal to the least upper bound of the number of isolated zeros of Δ (a,b) , for w ↓ 0, (a, b) → (a, b). An upperbound on this cyclicity will be found by applying a division-derivation algorithm to Δ (a,b) , based on Rolle's theorem.

Normalizing coordinates
Choosing appropriate normalizing coordinates near the semi-hyperbolic saddles at infinity will appear to be a helpful tool in simplifying the calculation of the difference map.
We can and will restrict to considering s + and we changeX (a,b) near s + by the equivalent family Y (a,b) defined as: We now introduce z = u − (1 + u 0 ) with u 0 = u 0 (s, a, b) and write and In the new coordinates (z, s), the family Y (a,b) , as defined in (20), can be written as: with The function A(s, a, b) is strictly positive and is invariant under From Theorem 1.3 of [3], we know that, on a (a, b)-uniform neighbourhood V of (a, b), there exists a smooth (a, b)-family of coordinate changes of the form (Z, s) = (z (1 + zZ(z, s, a, b)), s), conjugating (21) to  Z(z, s, a, b) is invariant under (23); we will however not have to rely on the latter. From (22), we know that A is given by: as s → 0 for some polynomials P i (a, b), 1 ≤ ∀i ≤ N , with P i (0, 0) = 0 and where N = 2n + 1 − m.
In order to calculate the difference map, we will now first study the expression of the coefficients P i (a, b).

Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS.
The final publication is available at http://aimsciences.org/journals/displayArticles.jsp?paperID=4986 5 Expression of the coefficients P i (a, b).
As in [1] it will reveal that the most interesting information will be contained in the functions P 2k+1 (a, b), with 0 ≤ 2k + 1 ≤ N − 1, and more specifically in the related functions c 1 (a, b), . . . , c K (a, b) as defined in (7). Each P 2k+1 (a, b) is an algebraic expression in (a, b) of which the complexity increases considerably as k is getting bigger.
In the case m = n = 3, we have for instance: When m = 3, n = 5, one has: and given that P 1 (a, b) = P 3 (a, b) = 0. It appears that the complexity of the expression of P 2k+1 (a, b) is caused by the mixing of terms {(a 2i+1 , b 2i+1 ) | 0 ≤ i ≤ k}, that appear in (1). The following theorem can however easily been proven. A(s, a, b) be the C ∞ function defined in (22) and let P i be the coefficients defined in (26). Then  i (a, b) only depends on (a 1 , . . . , a i , b 1 , . . . , b i ),   and when (a 1 , a 3 , . . . a 2k−1 , b 1 , b 3

Proof. (i) follows immediately from the fact that
(ii) follows from the fact that in the expression of A(s, a, b) a i and b i are always accompanied by s i (or s j , with j > i).
(iii) is a direct consequence of (i) and (ii).

Proof. Clearly
Now, because of (26):  N (a, b)) ln x s s 0 as s → 0. The result now easily follows.

Further study of the difference map
Denote by ϕ(u, s) = ϕ (a,b) (u, s) = (ψ (a,b) (u, s), s) the (a, b)-family of coordinate changes conjugating (Y (a,b) ), as defined in (20), to (25) locally near s + . ϕ is the succession of the mapping introducing z, defined below (20), and the mapping defined in (24). Let . By taking s 0 > 0 sufficiently small and after some dilatation in Z one can suppose that these sections lie in the domain of ϕ −1 .
In a similar way the regular transition maps R (a,b) can be represented as going from Σ 1 0 to σ 1 + and from σ 2 + to Σ 2 0 respectively. One obtains maps R + (s, a, b), S + (y, a, b) respectively. R + is expressed in s > 0 with values in s > 0, S + is expressed in Z > 0 with values in y. Furthermore let us denote H + (s, a, b) = S + (D + (R + (s, a, b), a, b), a, b), going from Σ 1 0 to Σ 2 0 . The difference map as defined in (19) can now be expressed as: for small positive values of s.
Let us now first provide nice expressions for the regular transition map R + using normalized sections and parametrizations.
Choose u 0 ∈]0, 1[ in (u, s)-coordinates such that ψ (a,b) (u 0 , 0) = 1. Consider the sections {u = u 0 } and {u = 0} transversally cutting the u-axis and parametrized by the s-coordinate. Let T + (s, a, b) be the regular transition map from {u = 0} to {u = u 0 }. In the following lemma we give an expression for this regular transition maps T + .
Lemma 9 Consider the family (X (a,b) ) (8). Consider the regular transition maps T + from {u = 0} to {u = u 0 }. Then: Proof. We transform (8) into the equivalent differential equation s, a, b), with Q the rational function given by: s,a,b) 1−u n+1 , Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS.
Lemma 10 Let F + (s, a, b) be the regular transition map from π + (a,b) to σ 1 + . Then Proof. To shorten notation we write F + (s) = F + (s, a, b). The map F + (s) is defined by the integral equation: Totally similar as in (29), one gets: N (a, b)) ln |x| Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS.

The reduced difference map
In this section we will introduce a so-called reduced difference map that will be used in finding an upperbound on the number of limit cycles that can perturb from L. By Rolle's theorem, the difference map Δ has at most N + 1 zeros in a neighbourhood of zero if ∂Δ ∂s has at most N zeros for s near zero, multiplicity taken into account. So, by (32), an upperbound on Cycl(X (a,b) , L, (a, b)) is found by searching the number of solutions of the equation: However from Proposition 8 and the identity in (31), it is clear that a, b) is exponentially flat. For removing this exponentially flatness, we introduce a smooth map, called a reduced difference map Δ, in such a way that its zeroes represent the roots of (35) and hence the zeroes of ∂Δ ∂s .
Theorem 12 Let (X (a,b) ) be a family of general Liénard systems like in (1) admitting an unbounded semi-hyperbolic 2-saddle cycle L for some parameter value (a, b). Let D + be the normalized expression of D (a,b) + and R + , S + the expressions of R + , S + respectively using normalized sections and parametrisations as in Section 7. Let A(s, a, b) be the map defined in (22). Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS.
The final publication is available at http://aimsciences.org/journals/displayArticles.jsp?paperID=4986 Then the cyclicity Cycl (X (a,b) , L, (a, b)) of L inside (X (a,b) ) is at most one unit higher than the number of positive zeroes of the map: −a o , a e , −b o , b e ) − H + (s, a o , a e , b o , b e ), with H + (s) = H + (s, a, b) = s N −1 log S + (D + (R + (s))) + log D + (R + (s)) + log A(R + (s)) Moreover: Proof. Large amplitude limit cycles arising from L correspond to small positive zeros of the difference map Δ. Moreover an upperbound for the number of such zeros is one unit higher than the number of solutions of equation ( = S + (D + (R + (s)))D + (R + (s)) A(R + (s)) The exponential flatness in this formula is caused by D + (R + (s)), see Proposition 8. To remove it, we introduce: leading to expression (37). From Propositions 8 and 11 and the fact that A(0) > 0, S + (0) > 0 and R + is smooth, one has:   s, a, b). The further statements now follow from (39) and Theorem 7, inducing: −a o , a e , −b o , b e ) − H + (s, a o , a e , b o , (Φ(s, −a o , a e , −b o , b e ) − Φ(s, a o , a e , b o , b e )) 9 Cyclicity in full subfamilies of (X (a,b) ) We are now ready to treat the cyclicity of unbounded semi-hyperbolic 2saddle cycles inside full subfamilies of (1) as described in the introduction. We will give a proof of Theorems 1, 2, 3, 4 and 5.
Proof of Theorem 1: From Theorem 12, we know By definition of the Lyapunov quantities (7), one now has: s, a, b).
The final publication is available at http://aimsciences.org/journals/displayArticles.jsp?paperID=4986 families are given by: where for each 2i + 1 either a 2i+1 or b 2i+1 is considered to be zero. In case m > n, one considers families of the form with the indices (k 1 , k 2 , k 2 ) defined such that 2n − m = 2k 1 − 1, n = 2k 2 + 1, m = 2k 3 + 1 and where for each 2i + 1 either a 2i+1 or b 2i+1 is considered to be zero. It is easily seen now that Theorems 3 and 4 are special cases of Theorem 2. We can only treat the full subfamilies with odd friction term or odd forcing term in full generality if m ≤ n. If m > n one has to impose a 2i+1 and b 2i+1 to be zero for 2i + 1 > 2n − m as in (44).
Proof of Theorem 2: Consider first the case where (Y (a,b) ) is a full subfamily such that a o = b o = 0. Then, by Theorem 12, Δ(s, a, b) ≡ 0 such that there are clearly no large amplitude limit cycles ∀(a, b).
Finally we treat an example where the coefficients among a o and b o occur in non-trivial pairs inside the full subfamily.
Proof of Theorem 5: We are dealing with a full subfamily of Liénard