ON THE ANALYTIC INTEGRABILITY OF THE LI´ENARD ANALYTIC DIFFERENTIAL SYSTEMS

. We consider the Li´enard analytic diﬀerential systems ˙ x = y , ˙ y = − g ( x ) − f ( x ) y , where f,g : R → R are analytic functions and the origin is an isolated singular point. Then for such systems we characterize the existence of local analytic ﬁrst integrals in a neighborhood of the origin and the existence of global analytic ﬁrst integrals

1. Introduction and statement of the main results.One of the more classical problems in the qualitative theory of planar analytic differential systems in R 2 is to characterize the existence of analytic first integrals in a neighborhood of an isolated singular point, and in particular the existence of a global analytic first integral when the differential system is defined in the whole R 2 .
One of the best and oldest results in this direction is the analytic nondegerate center theorem.In order to be more precise we recall some definitions.A singular point is a nondegenerate center if it is a center with eigenvalues purely imaginary.If a real planar analytic system has a nondegenerate center at the origin, then after a linear change of variables and a rescaling of the time variable, it can be written in the form: ẋ = y + X(x, y), where X(x, y) and Y (x, y) are real analytic functions without constant and linear terms defined in a neighborhood of the origin.Let U be an open subset of R 2 , H : U → R be a nonconstant analytic function and X be an analytic vector field defined on U .Then H is an analytic first integral of X if H is constant on the solutions of X ; i.e. if X H = 0.
Theorem 1 (Analytic nondegerate center theorem).The analytic differential system (1) has a nondegerate center at the origin if and only if there exists an analytic first integral defined in a neighborhood of the origin.
One of the more studied differential systems are the so-called generalized Liénard equation ẍ + f (x) ẋ + g(x) = 0, (2) which were studied by many researchers.Such dynamical systems appear very often in several branches of the sciences, such as biology, chemistry, mechanics, electronics, etc.The differential equation (2) of second order can be written as the equivalent 2-dimensional Liénard differential system of first order ẋ = y, ẏ = −g(x) − f (x)y. (3) When g(x) = x the Liénard differential systems (3) are called the classical Liénard systems.The main objective of this paper is to study the analytical integrability of the Liénard systems (3) depending on the analytic functions f and g.
In order that the origin of system (3) be a singular point we need that g(0) = 0, and since it must be isolated we need that g(x) ≡ 0. Therefore, since we want to study the local analytic integrability at the isolated singular point located at the origin in the rest of the paper we always assume that g(0) = 0, g(x) ≡ 0 and f (x) ≡ 0. ( Of course if g(x) ≡ 0 or f (x) ≡ 0 then the Liénard differential system becomes a differential equation with separable variables.The functions are useful in the study of the Liénard system (3).Through the paper Z + will denote the set of non-negative integers, Z − will denote the set of negative integers, Q + will denote the set of non-negative rational numbers and Q − will denote the set of negative rational numbers.
In the spacial case in which g(x) = − pq (p − q) 2 f (x)F (x) with p, q ∈ Z + \ {0} and p = q we have that is a global analytic first integral of system (3).
Due to technicalities we will also assume in the paper that (H0) if f (0) = 0 then g (0) = − pq (p−q) 2 f (0) 2 for some p, q ∈ Z + \ {0} and p = q.Note that when f (0 3) is integrable with first integral given in (5) but we are not considering this case in our paper.
When f (0) = g(0) = g (0) = 0 then using the Taylor expansion for the analytic functions g(x) and f (x) we can write: where 0 ≥ 2 and 1 ≥ 1.Now depending on 0 and 1 we will consider other assumptions on our systems.We will assume that: (H1) either 0 = 1 + 1; The main result of this paper is the following.
Theorem 2. System (3) under assumptions (4) and satisfying hypothesis H0 has a local analytic first integral in a neighborhood of the origin if and only if one of the following conditions hold.(a.1) Note that when 0 = 1 then we can have an analytic first integral as the following example shows.
System ( 6) satisfies 0 = 1 and it has a first integral of the form that is globally analytic when a 0 ∈ Z (taking As a Corollary of Theorem 2 we obtain the following characterization of the analytic first integrals of the so-called classical Liénard equations.In the classical Liénard system, i.e., when g(x) = x and f (x) ≡ 0, we have that (4) and hypotheses H0-H2 automatically hold.Therefore we are on the assumptions of Theorem 2. Furthermore, g (0) = 1 > 0 and then only statement (a.1) can hold.In this case, since G(x) = x 2 /2 we have that system (3) has a local analytic first integral in a neighborhood of the origin if and only if F (x) = Φ(x 2 /2), for some analytic function Φ with Φ(0) = 0.This clearly implies Theorem 4.
The paper has been divided as follows.In Section 2 we introduce some preliminary results and as a corollary as those results we prove Theorem 2(a.1).In Section 3 we prove Theorem 2 when f (0) = 0 and hypothesis H0 holds.In Section 4 we prove Theorem 2 when f (0) = g(0) = 0 and g (0) < 0. In Sections 5 we prove Theorem 2 when hypothesis H1 holds, and in finally in Section 6 we prove Theorem 2 when hypothesis H2 holds.
2. Preliminary results.In this section we shall introduce two results that will be used through the paper.Let h = h(x, y) be the vector field associated to our system (3).
The following result is due Poincaré and its proof can be found in [11], see also [4].
Theorem 5. Assume that the eigenvalues λ 1 = 0 and λ 2 = 0 at some singular point of h do not satisfy any resonance condition of the form Then system (3) has no local analytic first integrals.
The following result extends the previous one to the case that one eigenvalues is zero, see Li, Llibre and Zhang [5].Theorem 6. Assume that the eigenvalues λ 1 and λ 2 at some singular point p of h satisfy that λ 1 = 0 and λ 2 = 0. Then system (3) has no local analytic first integrals in a neighborhood of the singular p if it is isolated.
We must mention that the singular points appearing in the statements of Theorems 5 and 6 can be real or complex, but our system (3) is always real.
We have the following nice characterization of the centers at the origin for the Liénard systems (3) due to Christopher [1].
Theorem 7 (Center Theorem for analytic Liénard systems).Let f (x) and g(x) be analytic functions defined in a neighborhood of zero such that f (0) = g(0) = 0 and g (0) > 0. Then the Liénard differential system (3) has a nondegenerate center at the origin if and only if F (x) = Φ(G(x)) for some analytic function Φ(x), with Φ(0) = 0.
An immediate consequence of Theorems 1 and 7 is: Corollary 8. Let f (x) and g(x) be analytic functions defined in a neighborhood of zero such that f (0) = g(0) = 0 and g (0) > 0. Then there exists an analytic first integral defined in a neighborhood of the origin if and only if F (x) = Φ(G(x)) for some analytic function Φ(x), with Φ(0) = 0.
Note that Corollary 8 is exactly Theorem 2(a.1).If a real analytic system has a center at the origin and after a linear change of varibles and a rescaling of the time variable, it can be written in the form ẋ = y + X(x, y), ẏ = Y (x, y), where X(x, y) and Y (x, y) are real analytic functions without constant and linear terms defined in a neighborhood of the origin, then it is called a nilpotent center.

3.
Case f (0) = 0.In this section we will prove the following result.Proposition 9. Let f (x) and g(x) be analytic functions defined in a neighborhood of the origin satisfying (4) such that f (0) = 0 and hypothesis H0 holds.Then system (3) has no local analytic first integrals in a neighborhood of zero.
We separate the proof of Proposition 9 into different lemmas.
Lemma 10.Let f (x) and g(x) be analytic functions defined in a neighborhood of the origin satisfying (4) and such that f (0) = 0 and g (0) = 0. Then system (3) has no local analytic first integrals in a neighborhood of zero.
Proof.We note that system (3) has the singular point (0, 0).If X = (y, −g(x) − f (x)y), then the eigenvalues of DX(0, 0) are So one eigenvalue is zero and the other is nonzero.Since g(x) ≡ 0 and g(x) is analytic, we have that (0, 0) is an isolated singular point of system (3) and by Theorem 6, system (3) has no local analytic first integrals in a neighborhood of zero.Now we can assume that g (0) = 0. Lemma 11.Let f (x) and g(x) be analytic functions defined in a neighborhood of the origin satisfying (4) and such that f (0) = 0, g (0) = 0 and f (0) 2 /g (0) ∈ Q − .Then system (3) has no local analytic first integrals in a neighborhood of zero.
Proof.We note that system (3) has the singular point (0, 0).If X = (y, −g(x) + f (x)y) then the eigenvalues of DX(0, 0) are Clearly Suppose that there exist positive integers k 1 and k 2 such that k 1 λ 1 +k 2 λ 2 = 0. Note that by Theorem 5 if such integers do not exist the proposition is proved.Then λ 1 = −αλ 2 with α a positive rational.The two equalities of (7) become and consequently the proposition is proved.
Proof.We write g (0) = −αf (0) 2 with α ∈ Q + \{0}.Doing the rescaling (X, Y, T ) = (f (0)x, y, f (0)t) system (3) becomes of the form where we have written again (x, y, t) instead of (X, Y, T ) and where h.o.t.means terms of higher order.We assume that H = H(x, y) is a local analytic first integral in a neighborhood of the origin.We write it as H = k≥0 H k (x, y) where H k are homogeneous polynomials of degree k.We will show by induction that Then clearly from (8) we will obtain that system (3) has no local analytic first integrals, and the proof of the proposition will be done.Since H is a first integral it must satisfy Now we will do the induction.The terms of degree one in (9) must satisfy Suppose that H 1 = ax+by.Then the previous equation becomes (αx−y)b+ay = 0.So αb = 0, since α = 0 we get that ∂H 1 /∂y = b = 0. Then we also obtain ∂H 1 /∂x = 0 and thus H 1 = 0 which proves (8) for k = 1.Now we assume that ( 8) is true for k = 1, . . ., j − 1 with j ≥ 2 and we will prove it for k = j.By the induction hypothesis, the terms of order j in (9) must satisfy Therefore, either H j = 0 or H j is a first integral of the linear system ẋ = y, ẏ = αx − y.
Computing a first integral of this system, we obtain that it must be a function of .
The unique possibility for a power or function of G to be a polynomial is that Then we have that Hence, a contradiction.Since all first integrals are functions of G it is clear that H j cannot be a homogeneous polynomial of degree j.Hence H j = 0 and the induction process has ended.
From Lemmas 10, 11 and 12 it follows the proposition.
Proposition 13.Let f (x) and g(x) be analytic functions defined in a neighborhood of the origin satisfying (4) such that f (0) = 0. Then there exists an analytic first integral defined in a neighborhood of the origin if and only if F (x) = Φ(2|G(x)|) for some analytic function Φ(x).
Proof.Making the transformation (X, Y ) = (x, y + F (x), system (3) becomes where we have written again (x, y) instead of (X, Y ).Let u be the negative root of 2G.From the hypothesis on g (that is, g(0) = 0 and g (0) < 0) it is clear that it is well defined and analytic in a neighborhood of x = 0. Thus defines an invertible analytic transformation in a neighborhood of x = 0. Let x(u) denote its inverse.This transformation takes system (10) to the system Since g(x(u))/u = (−g (0)) 1/2 + O(u) is analytic in a neighborhood of the origin.We can rescale (12) by multiplying the right-hand side by u/g(x(u)) which gives We write Then system (13) becomes We introduce the change of variables with this change of variables system (14) becomes where It is clear that system ( 13) is locally integrable around the origin if and only if system ( 16) is locally integrable around the origin.Moreover, system ( 16) is locally integrable if and only if Fj = 0 for j odd, see for more details the Appendix.More precisely, the coefficients Fj for j odd are the Poincaré-Liapunov constants of the Lienard analytic differential system (16), and in the papers [2,15] it is proved that the Lienard differential systems have an analytic first integral in the neighborhood of the origin if and only if all the Poincaré-Liapunov constants are zero.In short, system (13) has a local analytic first integral in a neighborhood of the origin if and only if F is an even function.Now we express this condition in a more geometrical setting.The argument above shows that there is a local analytic first integral if and only if F (u) = F (x(u)) = φ(u 2 ) for some analytic function φ.However, u 2 = 2|G(x)| and this concludes the proof of the proposition.
Proof.System (3) becomes where h.o.t.denote the higher order terms.We claim that (17) has not global analytic first integrals.We note that the proof of the proposition will follow then from the claim.Now we shall prove the claim.
Let H = H(x, y) = k≥1 H k (x, y) be a first integral of (17), where H k is a homogeneous polynomial of degree k.Then H satisfies Note that the right-hand side of equation ( 18) has degree at least 0 .Now we will show by induction that for k ≥ 1, We have taken the criterium that for j ≥ 0, H −j = 0.If (19) holds, then since all H k−1 = 0 for k ≥ 1, and H = k≥1 H k it follows that H = 0, a contradiction with the fact that H is a global first integral of system (17).Hence the claim will be proved if we prove the induction hypothesis.
From this equation since g 0 = 0 we get that a 1 = 0 and ∂H 0 /∂x = 0, so H 1 = 0 and from equation (20) with l = 1 we get The induction hypothesis is proved for k = 2. Now we assume l ≥ 2. First we will prove by induction that for m ≥ 1, where C l,m , K l,m are positive constants depending on l and m and O(y l−2m+2 ) denote the terms of order greater or equal l − 2m + 2 in y.Furthermore, for j = 1, . . ., 0 − 2, Note that from ( 22) and ( 23) we have that Equations ( 22) and (23) will allow us to complete the proof of the induction hypothesis (19).
We first assume that l is odd.Then by ( 22) with m = (l − 1)/2 we obtain that Then computing in (18) the term of degree l + (l + 1)( 0 − 1)/2 and using (24) we get Now setting y = 0 in (27) and since g 0 = 0 we get that a l = 0. Finally, if l is even, then ( 22) with m = l/2 yields Then computing in (18) the term of degree l + (l + 2)( 0 − 1)/2 and using (24) we get Now setting y = 0 in (28) and since g 0 f 0−1 = 0 we get that a l = 0.This concludes the proof of the proposition.
6. Case H2.The main result in this section is the following.Proposition 15.System (3) with 0 ≥ 2 1 + 2 has no analytic first integrals.
Proof.System (3) becomes We claim that (29) has not global analytic first integrals.We note that the proof of the proposition will follow then from the claim.Now we shall prove the claim.Let H = H(x, y) = k≥1 H k (x, y) be a first integral of (29) where H k is a homogeneous polynomial of degree k.Then H satisfies Note that the term in (30) before ∂H/∂y has degree at least 0 .We will show by induction that for k ≥ 1, We have taken the criterium that for j ≥ 0, H −j = 0. We note that (31) clearly implies that H = 0, a contradiction with the fact that H is a global first integral of system (29).Hence the claim will be proved if we prove the induction hypothesis.
Lemma 16.For n = 1, . . ., l − 2, and for j = 1, . . ., 1 − 1, where K l,n is a positive constant depending on l, n; O(y l−1−n ) denote the terms of order greater than or equal to l − 1 − n in y and O(y l−2−n ) denote the terms of order greater than or equal to l − 2 − n in y.
Computing the degree l + 0 − 1 + ñ 1 in (30) and the induction hypothesis, we have Note that since i ≥ 1, by ( 37) and (38) with n ≤ ñ − 1 we have Furthermore we can write l + ñ 1 − i in ∂H l+ñ 1−i /∂y of the form Since 0 ≥ 2 1 + 2 and i ≥ 0 we have that and by (37) and (38) with n ≤ ñ − 3 we have that Therefore, using also the induction hypothesis, we have that , where .
Appendix.Poincaré in [10] defined the notion of a center for a real polynomial differential system in R 2 .He showed that a necessary and sufficient condition in order that a real polynomial differential system has a center at a singular point with purely imaginary eigenvalues, is that it has a local analytic first integral defined in its neighborhood.Then doing a linear change of variables and a scaling of the independent variable, we can write any polynomial differential system with a focus at the origin with purely imaginary eigenvalues (i.e.having either weak focus or a center) into the form ẋ = p(x, y) = y + p 2 (x, y) + . . .+ p m (x, y), ẏ = q(x, y) = −x + q 2 (x, y) + . . .+ q m (x, y), where p and q are complex polynomials without constant and linear terms.With the change of variables x = a, y = ib, system (44) becomes the complex differential system ȧ = ib + P (a, ib), ḃ = −ia + Q(a, ib), where P and Q are complex polynomials.Then the focus quantities V j of system (45) coincide with the saddle quantities L j of system (44).From this duality between focus quantities and saddle quantities it follows that an integrable saddle has an analytic first integral defined in a neighborhood of it.This is the reason why we call such a saddle an integrable saddle.The complex change (45) is done for showing the duality of weak focus and weak saddles.
If we apply a change of the form (45), precisely the change (15), to system (14) we get the differential system (16).Exactly system (16), which is a particular case of system (45), has been studied in [2,15] showing that it has a first integral of the form given in Lemma 17 if and only if Fj = 0 for j odd.

Theorem 4 .
System (3) with g(x) = x and f (x) ≡ 0 has a local analytic first integral in a neighborhood of the origin if and only if F (x) = F (−x).