Hilbert’s Sixteenth Problem for Polynomial Liénard Equations

This article reports on the survey talk ‘Hilbert’s Sixteenth Problem for Liénard equations,’ given by the author at the Oberwolfach Mini-Workshop ‘Algebraic and Analytic Techniques for Polynomial Vector Fields.’ It is written in a way that it is accessible to a public with heterogeneous mathematical background. The article reviews recent developments and techniques used in the study of Hilbert’s 16th problem where the main focus is put on the subclass of polynomial vector fields derived from the Liérd equations.


Introduction
Polynomial Liénard equations are planar differential equations associated to the second order scalar differential equations (1.1) where the functions f and g are polynomials of degree n and m respectively. They occur as models or at least as simplifications of models in many domains of science. Besides singularities, which are well-understood for ( 1.1), isolated periodic orbits or so-called limit cycles represent the asymptotic state of the other solutions of (1.1), see Fig. 1, which are subject of Hilbert's 16th Problem and Smale's 13th Problem. A formulation of these problems and their recent developments are recalled in Sect. 2. There were several attempts to solve Hilbert's 16th Problem and so far all of them failed. However the problem yet has a source of inspiration for significant progress in the geometric theory of planar vector fields, as well as bifurcation theory, normal forms, foliations and some topics in algebraic geometry.
This survey aims at collecting important developments and techniques in the study of Hilbert's 16th Problem where the main focus is put on the subclass of polynomial vector fields derived from the Liénard equations (1.1): x = y,ẏ = −g (x) − f (x) y, (1.2) or written in the coordinates (X, Y ) = (x, y + F (x)) of the Liénard plane: This survey paper is organized as follows. In Sect. 2 Hilbert's 16th Problem and Smale's 13th Problem (i.e. the version for classical Liénard equations) are recalled as well as known results on it. In Sect. 3 it is recalled how these global problems can be localized. In Sect. 4 we summarize the proof for Theorem 2.1 that gives an answer to Smale's 13th Problem for bounded Liénard equations. This proof includes a compactification of phase plane and parameter space as presented in Sects. 4.1 and 4.5 respectively. Next in Sect. 4.2 we reduce Hilbert's 16th Problem for this compact family to the analysis of the bifurcation phenomenon of large amplitude limit cycles. We describe this analysis in Sects. 4.3 and 4.4 for Liénard equations of odd and even degree respectively.

Polynomial Vector Fields
Hilbert's 16th Problem essentially asks for a uniform upper bound H (n) for the maximum number of limit cycles of a planar polynomial vector field uniformly in terms of the degree n. This problem is more than 100 years old and its investigation has produced many results contributing to the wide development of the theory of Dynamical Systems. It is not known whether a uniform upper bound only depending on the degree of the vector field might exist, even not when the degree is two. This part of Hilbert's 16th Problem is generally called the uniform finiteness problem. Even Dulac's theorem to prove that for individual vector fields the number of limit cycles is finite was far from trivial (see e.g. [18]); this problem is also referred to as the individual finiteness problem. In 1923 Dulac presented a proof for this theorem. Later it was found that the proof contained a gap. This lack was first solved for quadratic systems by Bamon in 1985 and for arbitrary degree by Ecalle and Ilyashenko independently in the 1990s.
Solution programmes for Hilbert's 16th Problem mostly consist in its reduction to several subproblems, based on either considering local cyclicity problems [29] or restricting the class of vector fields to a particular simpler class, see e.g. [18,26] or [29] for an overview.
Finally note that the generalization of Hilbert's 16th problem to higher order dimensions ≥3 is solved by Bobienski and Zoladek; in [1] they present a concrete example of a polynomial vector field on R 4 for which there exists infinitely many limit cycles.

Liénard Equations
In the following we denote by H L (m, n) the maximum number of limit cycles of (1.1), (1.2) or (1.3). Part of the 13th Problem that Smale put forward on his list of challenging problems for the 21st century deals with Hilbert's 16th Problem restricted to the classical Liénard equations, i.e. the case g (x) = x in (1.1), (1.2) or (1.3), see [31]. Moreover Smale suggests that the maximal number of limit cycles H L (1, n) for classical Liénard equations grows at most by an algebraic law of type n d where d is a universal constant.
The problem for classical Liénard equations when the degree of f is equal to 2 or 3 is solved; the result in [24] shows that H L (1, 2) = 1 (i.e. the so-called Van der Pol Equation has at most one limit cycle, see Fig. 1) and very recently Li and Llibre proved in their preprint [22] that H L (3, 1) = 1. Besides there was the so-called Lins, de Melo and Pugh Conjecture, stating that the maximal number of limit cycles is equal to l if g(x) = x and the degree of f is 2l or 2l + 1.
Of course there is the counter-example to that conjecture, due to Dumortier et al. [15], for limit cycles in singular perturbations, but it does not contradict the possibility for the growth of the number of limit cycles to be linear. In [15] classical Liénard equations are presented with degree of f equal to 2l and having at least l + 1 limit cycles (more precisely H L (1, 6) ≥ 4); hence one limit cycle more than conjectured by Lins, de Melo and Pugh. Recently this result is generalized in [10]: H L (1, n) ≥ n 2 + 2, ∀n ≥ 5, finding two limit cycles more than conjectured by Lins, de Melo and Pugh. In fact, in [24], they prove that, under these assumptions, there are at most l small amplitude limit cycles. Lloyd and Lynch considered the similar problem for generalized Liénard equations [21]. In most cases, they prove an upper bound for the number of small amplitude limit cycles, that can bifurcate out of a single non-degenerate singularity.
In [4,30] progress has been made towards proving the finiteness part of Hilbert's 16th Problem for classical Liénard equations. By the results from [4,30] the study of the finiteness part of Smale's 13th Problem is reduced to singular perturbation problems; more precisely the following theorem is proven.

Theorem 2.1
The global number of limit cycles of (1.1) with g (x) = x is uniformly bounded if f is restricted to some compact set of polynomials of degree exactly n (see [30] for n even and [4] for n odd).
In Sect. 4 we provide with some insight in the techniques underlying Theorem 2.1. Here below, in (2.1), we summarize the currently known results for the Hilbert numbers for Liénard equations (1.2) that we mentioned here above. (2.1)

Local and Global Finiteness Problems
Hilbert's 16th Problem is a global finiteness problem in the sense that one aims at bounding the number of limit cycles of X n (a,b) in the plane R 2 for all possible values of the parameter (a, b) . In this section we briefly explain how this global problem can be 'localized'. For this purpose we first introduce the notions of limit periodic set and cyclicity. Limit periodic sets are subsets consisting of singularities and regular orbits, that can produce limit cycles by perturbation, and the cyclicity is the maximal number of limit cycles that they can generate in a perturbation. More precisely these notions are defined as: Definition 3.1 Let P ⊂ R p be the parameter space with λ 0 ∈ P and let (X λ ) λ∈P be a family of vector fields on a regular 2-dimensional surface S. Then, 1. a compact set ⊂ S is a limit periodic set of (X λ ) λ for λ → λ 0 if and only if there exists a sequence (λ n ) n∈N in P with λ n → λ 0 for n → ∞ and such that for all n ∈ N, there exists a limit cycle γ n of X λ n with γ n → for n → ∞. 2. Let be a limit periodic set of (X λ ) λ for λ → λ 0 . Then we say that has finite cyclicity in the unfolding (X λ ) λ for λ → λ 0 if there exist N ∈ N and constants ε, δ > 0 such that for every parameter value λ with λ − λ 0 < δ the vector field X λ has at most N limit cycles γ with d H (γ, ) < ε. If has finite cyclicity in (X λ ) λ for λ → λ 0 , then the cyclicity of inside the unfolding (X λ ) λ for λ → λ 0 is defined as the positive integer The distance d H is the Hausdorff distance on the metric space of compact sets, and the limits γ n → and γ → are considered in this metric space.
A major question is the so-called local finiteness problem or local cyclicity problem that exists in finding sufficient conditions under which the cyclicity is finite, and in this case, to find an explicit estimate for the cyclicity.
Limit periodic sets are the so-called organizing centers for limit cycles. Therefore to detect limit cycles one looks first for all limit periodic sets, for which there exists, in case there are only isolated singularities, a structure theorem analogous to the Poincaré-Bendixson Theorem for αand ω-limit sets (see [29]). This theorem classifies the possible types of limit periodic sets of (X λ ) λ for λ → λ 0 as a singular point, a periodic orbit or a graphic of X λ 0 . Recall that a graphic of a vector field X λ 0 is the union of (not necessarily distinct) singular points p 0 , . . . , p m with p m = p 0 , and regular orbits γ 1 , . . . , γ m−1 of X λ 0 , connecting these singular points, in the sense that p i = α (γ i ) and p i+1 = ω (γ i ) , 1 ≤ i ≤ m − 1. In Sect. 4.3 we encounter the so-called '2-saddle cycle' which is a simple example of a graphic, see Fig. 3 By compactification of the parameter space and the phase plane of X n (a,b) , the family of polynomial vector fields extends to a compact analytic family, such that Hilbert's global finiteness problem is reduced to several local cyclicity problems (see Theorem 3.2 below and [30]).

Localization Method
Compactifying phase plane as well as parameter space one can apply the so-called localization method of Roussarie [30], which is based on the following theorem whose proof can be found in [28] or [29]. Theorem 3.2 Let K be a compact set in parameter space and let S be a compact 2-dimensional manifold, let X λ be an analytic vector field on S for each λ ∈ K . Then there exists a uniform upper bound for the number of limit cycles of X λ in S for λ ∈ K if and only if for any λ 0 ∈ K any limit periodic set of X λ 0 has finite cyclicity inside the family X λ for λ → λ 0 , i.e.
In Sect. 4, to prove Theorem 2.1, we apply the localization method of Roussarie reducing the global Smale's problem to several local cyclicity problems: small, medium and large amplitude limit cycles and cyclicity problems for slow-fast systems. This localization method requires an appropriate compactification of the phase plane as well as the chosen space of Liénard equations itself (see Sect. 4.5).

Classical Tools to Study Local Cyclicity Problems
Traditionally the study of limit cycles of planar vector fields (X λ ) λ near a limit periodic set for λ → λ 0 , as the ones we consider, is replaced by the study of isolated fixed points s of associated 1-dimensional Poincaré-maps (P λ ) λ , where s 0 corresponds to . Or equivalently, by the study of isolated zeroes s of socalled displacement maps δ λ = P λ − I or difference maps λ (e.g., see Sect. 4.3 and Fig. 3). In such a way configurations of isolated zeroes of δ λ (resp. λ ) correspond to configurations of limit cycles of X λ . In particular the cyclicity in (3.1) can be expressed in terms of zeroes of λ (or δ λ ): To control zeroes of these maps near s 0 one can rely on classical theorems as the Implicit Function Theorem, Rolle's Theorem, the Preparation Theorem, as long as the map λ 0 is of finite order at s 0 . In the other case some degeneracies first have to be removed. To that end, when is approached by a period annulus, one can analyze the division of the map in terms of Melnikov functions or the Bautin ideal (see [29]). The Bautin ideal also serves to characterize the parameter values for which the corresponding vector field is of center type. Moreover an upper bound for the cyclicity can be expressed in terms of this ideal. In [3] it is proven that this Bautin ideal corresponds to the ideal generated by Lyapunov quantities, which determine the order and stability of a weak focus; in [6] Lyapunov quantities are determined for classical and generalized Liénard equations using Cherkas transformation, and a local division of the displacement map is given in terms of the Lyapunov quantities. Furthermore in [6] a detailed study of the bifurcation phenomenon of small amplitude limit cycles is provided and generic Hopf-Takens bifurcations (degenerate or not) are precisely described.

Classical Liénard Equations of Degree n
In the Liénard plane the classical Liénard equation (1.3) is written as the polynomial vector field L n a with Besides the compactification process, due to the fact that the 'central system' is too degenerate to permit a study of its unfolding without a blow up, the method includes a desingularization. In this way the boundary of the space of Liénard equations is made by Hamiltonian and singular perturbation problems. These boundary problems both exhibit different phenomena. In this survey we only include the Hamiltonian perturbation problems, that can be solved completely. The study of the cyclicity problem for classical Liénard equations of odd degree (i.e. n is odd) that do not belong to the boundary is easy and well-known among specialists. In this case limit cycles stay at a uniform distance from infinity (see [30]). For classical Liénard equations L n a of even degree (i.e. n = 2l is even) this is no longer true and then the main problem consists in studying limit cycles that come close to infinity. These limit cycles are so-called large amplitude limit cycles of which it is shown in [4] that there are at most l + 1 (see Definition 4.1).

Bounded Liénard Equations
By 'bounded Liénard equations' of degree n we refer to a family of Liénard equations L n a a , where a belongs to a compact set K in R n−1 .
Applying the Lyapunov-Poincaré compactification, that is a quasi-homogenous version of the Poincaré compactification, the Liénard system L n a extends to an analytic vector field on the sphere S 2 (see [11]). In this compactification the non-elementary singularities at infinity are spread out over the equator of S 2 being all of elementary type.

Local Cyclicity Problems for Bounded Liénard Equations
Since the origin is the only singularity of L n a in R 2 , that is a weak focus or center, limit periodic sets that lie entirely in the finite plane correspond to either the singularity in the origin or a regular periodic orbit. Limit cycles bifurcating from these limit periodic sets are called respectively small amplitude limit cycles and medium amplitude limit cycles. These cyclicity problems are well understood and result in finite cyclicity, because these limit cycle bifurcations can be studied by isolated zeroes of analytic maps (displacement maps), see e.g. [6].
Since there are no other limit periodic sets in the finite plane the only bifurcation phenomenon of limit cycles that remains to be studied is the one of large amplitude limit cycles, defined here below. Let B R (0) denote the open ball in R 2 centered at (0, 0) and with radius R > 0 : Definition 4. 1 We say that a family (X λ ) λ of vector fields on the plane R 2 has exactly N large amplitude limit cycles for λ → λ 0 if and only if 1. there exist R > 0, a neighbourhood W of λ 0 such that for all λ ∈ W the vector field X λ has at most N limit cycles having a non-empty intersection with R 2 \ B R (0) ; 2. for all R > 0 and for every neighbourhood W of λ 0 , there exists λ ∈ W such that X λ 0 has N limit cycles having a non-empty intersection with R 2 \ B R (0).
In Sects. 4.3 and 4.4 the proofs will be sketched of the facts that no large amplitude limit cycles appear for the family of Liénard equations of odd degree, while there appear at most l + 1 for Liénard equations of even degree n = 2l.

No Large Amplitude Limit Cycles in Case n is Odd
For n is odd one can construct a uniform domain of attraction by considering the Lyapunov function V (x, y) = x 2 + y 2 /2. The time derivative of V is given bẏ Therefore given an arbitrary compact set K ⊂ R n−1 there exists R K > 0 such that for all a ∈ K all orbits of L n a enterB R K (0) after some finite time, and stay there for all later times. As a consequence there are no large amplitude limit cycles.

At Most l + 1 Large Amplitude Limit Cycles in Case n is Even
For n = 2l even, the infinity is no longer repelling and we here sketch briefly how the analysis of large amplitude limit cycles proceeds.
1. Compactification of phase space: take local coordinates (x,ȳ) near infinity given by x =x/s, y =ȳ/s 2l with s > 0 andx 2 +ȳ 2 = 1. After a time-rescaling (t = s 2l−1 t) the vector field L n a extends analytically to a vector fieldL n a on the Poincaré-Lyapunov disc of type (1, 2l). The phase portrait near the equator of this disc (which corresponds to ∞, or s = 0) is presented in Fig. 2a. There are four singularities at ∞ : two semi-hyperbolic saddle singularities that we denote by s ± in the upper half sphere and two elementary nodes (a repelling and an attracting one) in the lower half-sphere. The semi-hyperbolic singularities s ± are connected by a regular orbit 1 at ∞ for all a ∈ R n−1 . If for a 0 arbitrary but fixed there is no connection in the finite plane between s + and s − , then there will be no large amplitude limit cycles bifurcating for a → a 0 , see Fig. 2b. If there does exist a connection 2 in the finite plane between s + and s − , and thus giving rise to an unbounded 2-saddle cycle , then large amplitude limit cycles can bifurcate from for a → a 0 , see Fig. 2c. In the following we assume that such exists for a = a 0 , see Fig. 3c. 2. Let i be a transversal section with respect to i , i = 1, 2. As is convenient in the study of limit cycles near a 2-saddle cycle we define the so-called difference map as the difference between the transitions 1 and 2 from 1 to 2 , defined by the flow ofL n a in forward and backward time respectively: where w is a local regular parameter on 1 , such that w = 0 corresponds to sections ± i with respect to i near s ± the transition maps i from − 1 to + 2 , i = 1, 2 are written as the composition of a regular transition R i along i and a Dulac transition D ± near the saddle point s ± , see Fig. 3b. 3. To facilitate the calculations near 1 one can use local coordinates (u, s) in which L n a reads asL Clearly G (u, s, a) = G (−u, −s, a) ; henceL n a is invariant with respect to the symmetry (u, s, t) → (−u, −s, −t) , see Fig. 4a. 4. By this symmetry we can use the same normalizing coordinates (z, w) near the semi-hyperbolic saddle points s ± , transformingL n a into the equivalent differential system For readability, in the notation of D and R i , i = 1, 2 we left out the dependence on the parameter a. Clearly is C ∞ at w = 0 and its jet of infinite order at w = 0 is given by j ∞ ( ) 0 (w) ≡ R 2 (0) , the so-called breaking coefficient. If R 2 (0) = 0, then there is no connection between s + and s − in the finite plane for the considered parameter a ∈ R n−1 , and hence there are no large amplitude limit cycles for parameter values sufficiently close to a. 5. Since there is no explicit expression known for the connection 2 in the finite plane, there is almost no chance to perform an asymptotic analysis of R 2 . Therefore, by taking the derivative and an algebraic manipulation, we replace the study of zeroes of (·, a) to zeroes of 1 (·, a) in such a way that zeroes of 1 (·, a) correspond to solutions of ∂ ∂w (·, a) = 0. By Rolle's Theorem the number of zeroes w ↓ 0 of (·, a) is bounded by 1 plus the number of zeroes w ↓ 0 of 1 (·, a) , where the zeroes are counted with multiplicity. The principal part of the so-called reduced difference map 1 can be written as: where R 1 (w, a) = wR 1 (w, a) . 6. After some calculations one obtains the asymptotic expansion forR 1 : for some C 0 > 0. Furthermore, if a 0 2l−1−2i = 0, ∀0 ≤ i ≤ k − 1, then there exists C k > 0 such that this asymptotic expansion reduces tō (a) If L n a 0 has no center at (0, 0) , then there exists 0 ≤ k ≤ l − 1 such that a 0 2l−1−2i = 0, ∀0 ≤ i ≤ k − 1 and a 0 2l−1−2k = 0. Then by continuity k+1 has no zeroes bifurcating from w = 0 for a sufficiently close to a 0 . By Rolle's Theorem it then follows that there are at most k + 1 large amplitude limit cycles for a → a 0 .
However 1 is not analytic at w = 0, and therefore this fact does not imply automatically that 1 ·, a 0 ≡ 0. However if all a 0 2l−1−2k = 0, 1 ≤ k ≤ l then L n a 0 satisfies a symmetry property implying that L n a 0 has a center at (0, 0) , and hence 1 ·, a 0 ≡ 0. Applying Taylor's Theorem 1 can locally be written as: By the following rescaling of the parameter we can reduce this degenerate case to the regular case: In terms of the rescaled parameter (b, ρ) = (b 1 , . . . , b 2l−1 , ρ) we then have For each b ∈ R n−1 fixed with l i=1 b 2 2i−1 = 1, we can apply the divisionderivation algorithm to¯ 1 and find that there are at most l +1 large amplitude limit cycles that bifurcate from for a → a 0 .

Putting a Boundary on the Space of Liénard Equations
Following the idea of Roussarie in [30] we put a boundary on the Liénard family L n a a∈R n−1 to end up with a compact family. More precisely we imbed L n a a∈R n−1 in the family S n a,ε that is defined as S n a,ε ↔ẋ = y − F n a (x) ,ẏ = −εx for ε > 0.
Clearly L n a ≡ S n a,1 . Furthermore S n a,0 is a singular family of planar vector fields, having the graph of y = F n a (x) full of non-isolated singularities; the vector fields of the family S n a,ε for small ε > 0 are called slow-fast Liénard equations. For later use let us introduce the natural projections on R n = R n−1 ×R by π 1 : R n−1 × R → R n−1 : (a, ε) → a and π 2 : The family S n a,ε a∈R n−1 ,ε>0 is quasi-homogeneous; as a consequence for all (a, ε) and (x,ȳ) fixed, there exist 1-parameter groups of diffeomorphisms T τ (x,ȳ) = τx, τ nȳ and (4.5) U τ (a, ε) = τ n−1 a 1 , τ n−2 a 2 , . . . , τ a n−1 , τ 2n−2 ε for a = (a 1 , . . . , a n−1 ) . (4.6) The phase portraits of S n U τ (b,δ) and S n b,δ thus are diffeomorphic via T τ , and so S n and S n b,δ have the same number of limit cycles in the plane. Let · be the Euclidean norm on R n−1 , and define the unit disc and its boundary respectively by ≡ a ∈ R n−1 : a ≤ 1 and ∂ ≡ a ∈ R n−1 : a = 1 .
Then for any ε > 0 the family S n a,ε a∈ is equivalent to the bounded Liénard family L n b b∈K ε via τ (ε) , where K ε ≡ π 1 U τ (ε) (a, ε) : a ≤ 1 and τ (ε) ≡ 2n+2 1/ε for ε > 0. (4.7) In particular one has π 2 U τ (ε) (a, ε) = 1 and for ε ↓ 0 the diameter of K ε tends to ∞, see Fig. 5. Indeed notice that the 1-parameter groups of smooth diffeomorphisms {U τ (a, ε) , τ > 0} for fixed values (a, ε) can be defined by the differential equations in parameter space R n . Furthermore the orbits of (4.6) are transverse to the hyperspace {ε = 1} ; therefore and since the equations for the parameter variable a ∈ R n−1 are independent from the one for ε > 0, we can study the evolution of the disc and its boundary ∂ in the projected space R n−1 by π 1 after time log τ (ε) , which is strictly increasing to +∞ when ε ↓ 0 (see Fig. 5). Here a(·, a 0 ) denotes the flow defined by (4.8) with initial condition a = a 0 ; the time log(τ (ε)) is necessary to pass from ε to 1. Since (4.8) describes a linear expansion the compact set K ε ↑ R n−1 for ε ↓ 0, see (4.7) Fig. 6 Compactification of classical Liénard equations {L n b : b ∈ R n−1 }. The constants ε i > 0, i = 0, 1 are assumed to be ordered as ε 0 < ε 1 . The bounded Liénard family {L n b : b ∈ K ε 0 } is equivalent to {S n a,ε 0 : a ∈ } and the unbounded Liénard family {L n b : b ∈ R n−1 \ K ε 0 } is equivalent to {S n a,ε : a ∈ ∂ , 0 ≤ ε < ε 0 } Therefore for any fixed ε 0 > 0 the family S n a,ε 0 a∈ is equivalent to a 'bounded Liénard family' L n b b∈K ε 0 and the family S n a,ε a∈∂ ,0≤ε<ε 0 is equivalent to its complement in the space of Liénard equations L n b b∈R n−1 , i.e. L n b b∈R n−1 \K ε 0 , which is unbounded (see Figs. 5 and 6).
It follows from the results in Sects. 4.3 and 4.4 that for any fixed ε 0 > 0 there exists an integer N (n, ε 0 ) such that the number of limit cycles of the bounded Liénard family L n b b∈K ε 0 is uniformly bounded by N (n, ε 0 ) in the plane. Therefore to complete Smale's 13th Problem it suffices to solve the finiteness problem for the compact family of polynomial vector fields S n a,ε a∈∂ ,0≤ε≤ε 0 , where ε 0 > 0 can be taken as small as needed. Notice that for ε 0 ↓ 0 limit cycles of S n a,ε shrink to the origin. As a consequence, using the compactness of ∂ and the localization method of Roussarie, Smale's 13th Problem is reduced to a singular ε-perturbation problem: the cyclicity problem of the singular point (0, 0) of the singular vector field S n a 0 ,0 inside the family (S n a 0 ,ε ) 0≤ε≤ε 0 for all a 0 ∈ ∂ , where ε 0 is sufficiently small (but independent of a 0 ).

Conclusions and Some Generalizations
In Sects. 4.1, 4.2, 4.3, 4.4 and 4.5 we have sketched the proof of Theorem 2.1, i.e., how Hilbert's 16th Problem is solved for bounded classical Liénard equations, and how Smale's 13th Problem is reduced to some cyclicity problems for slow-fast systems. Besides, this proof includes an independent proof of the Dulac Theorem for classical Liénard equations.
In [19] the Hilbert number for the bounded family of classical Liénard equations of even degree n in case the origin is a focus, say H * B L (1, n) , is estimated in terms of different parameters using the Growth-and-Zeroes Theorem provided by Ilyashenko and Yakovenko and the result on large amplitude limit cycles from [4].
For a complete study of large amplitude limit cycles for the family of generalized Liénard equations, i.e., (1.2) when g (x) = x, the characterization of an unbounded center is needed; this is detailed in [5] where complete results only are obtained for certain subfamilies of generalized Liénard equations.