DULAC-CHERKAS FUNCTIONS FOR GENERALIZED LIÉNARD SYSTEMS

Dulac-Cherkas functions can be used to derive an upper bound for the number of limit cycles of planar autonomous differential systems including criteria for the non-existence of limit cycles, at the same time they provide information about their stability and hyperbolicity. In this paper, we present a method to construct a special class of Dulac-Cherkas functions for generalized Lienard systems of the type dx dt = y, dy dt = ∑l j=0 hj(x)y j with l ≥ 1. In case 1 ≤ l ≤ 3, linear differential equations play a key role in this process, for l ≥ 4, we have to solve a system of linear differential and algebraic equations, where the number of equations is larger than the number of unknowns. Finally, we show that Dulac-Cherkas functions can be used to construct generalized Lienard systems with any l possessing limit cycles.


Introduction
The problem of estimating the number of limit cycles for two-dimensional systems of autonomous differential equations dx dt = P (x, y), dy dt = Q(x, y) (1.1) in some open region G ⊂ R 2 represents one of the famous problems formulated by D.Hilbert [6].This problem is still open.There are several approaches to attack this problem, including intentions to weaken it [7].One known method to estimate the number of limit cycles of (1.1) from above is the method of Dulac function [2].Here, the upper bound on the number of limit cycles also depends essentially on the connectivity of the region G. Frequently, this method is used to establish that system (1.1) has in some simply connected region no limit cycle or in a doubly connected region at most one limit cycle.
EJQTDE, 2011 No. 35 p. 1 The method of Dulac function has been generalized into different directions.One promising generalization is due to the first author who introduced in 1997 a function which we call now Dulac-Cherkas function that not only permits to get an upper bound for the number of limit cycles but also provides an information about their stability (see [1]).The problem of construction of such a function has been investigated by the first and the second author in [4] with respect to the Liénard system with g(0) = 0.In that paper, it has been shown that linear differential equations combined with the method of linear programming can be used to determine Dulac-Cherkas functions.Recently, Gasull and Giacomini used in [3] principally the same method to estimate the number of limit cycles for the Kukles system The paper is organized as follows: In section 2 we recall some definitions and known results.In section 3 we present an algorithm to construct a special class of Dulac-Cherkas for 1 ≤ l ≤ 3. We show in section 4 how this algorithm can be applied also in case l ≥ 4 in order to derive conditions on the functions h i implying that the corresponding system (1.4) has at most one limit cycle or no limit cycle.In the last section we demonstrate how a Dulac-Cherkas function can be used to construct a generalized Liénard system having a unique limit cycle surrounding the origin.EJQTDE, 2011 No. 35 p. 2

Preliminaries
First we recall the definition of a Dulac function.
Definition 2.1.Let P, Q ∈ C 1 (G, R), let X be the vector field defined by (1.1).A function ∂y does not change sign in G and vanishes only on a set N of measure zero, where no oval (closed curve homeomorphic to a circle) in N is a limit cycle.
The existence of a Dulac function implies the following estimate on the number of limit cycles of system (1.1) in G. (1.1) in G, then (1.1) has not more than p − 1 limit cycles in G.
The method of Dulac function has been generalized in different ways.One possibility is to admit that B is not necessarily C 1 at any equilibrium provided the number of equilibria is finite in G.This generalization has been proposed by the third author in 1968 (see [8]).Another generalization is due to the first author (see [1]).The corresponding generalized Dulac function, which we call Dulac-Cherkas function, is defined as follows.
Remark 2.1.Condition (2.1) can be relaxed by assuming that Φ may vanish in G on a set of measure zero, and that no oval of this set is a limit cycle of (1.1).
For the sequel we introduce the subset W of G by The following three theorems can be found in [1].Theorem 2.2 has been generalized in [5] by the second and the third authors as follows.
Theorem 2.4.Let G be a p-connected region, let Ψ be a Dulac-Cherkas function of (1.1) in G such that W has s ovals in G. Then system (1.1) has at most p − 1 + s limit cycles in G, any existing limit cycle is hyperbolic.
Remark 2.2.In [5] it has been also shown that the differentiability conditions of Ψ in Theorem 2.4 can be weakened in the same manner as in case of a Dulac function.
The problem to construct a Dulac-Cherkas function has been solved by the first author for the Liénard system (1.2).He uses as Ψ the function where α is an appropriate constant and G is defined by G(x) := x 0 g(σ)dσ.According to this choice of Ψ, the curve Ψ(x, y) = 0 has at most one oval.Moreover, we get from (2.1) and (1.2) Thus, Φ does not depend on y, and applying Theorem 2.4 we get the result: EJQTDE, 2011 No. 35 p. 4 Theorem 2.5.Suppose f, g : R → R to be continuous.Additionally, we assume that there is a constant α such that the function Φ 1 defined by does not change sign in R and vanishes only at finitely many points.Then system (1.2) has at most one limit cycle Γ, and, if Γ exists, it is hyperbolic.
In the case g(x) ≡ x, f (x) ≡ µ(x 2 − 1) system (1.2) represents the van der Pol equation, and we get that is, all conditions of Theorem 2.5 and of Theorem 2.3 are fulfilled for µ = 0, and the curve Ψ(x, y) = 0 consists in R 2 of the circle O := {(x, y) ∈ R 2 : y 2 + x 2 = 1}.Thus, we get the known result: Proposition 2.2.The van der Pol equation has for any µ = 0 at most one limit cycle Γ(µ) which is located outside the region bounded by the circle O. Γ(µ) is hyperbolic and stable (unstable) for µ > 0 (µ < 0).
We note that in case of Liénard system (1.2), to the Dulac-Cherkas function Ψ in (2.2) there belongs a function Φ defined in (2.1) that does not depend on y for a special value of k.The advantage of eliminating the variable y from the function Φ consists in the fact that the inequality (2.1) must be fulfilled only on some interval.Hence, this approach makes it easier to check the validity of inequality (2.1) or to derive conditions guaranteeing that (2.1) is satisfied.This reasoning stimulated the first and second authors to develop in [4] an algorithmic way for constructing a Dulac-Cherkas function Ψ for the Liénard system (1.2) in the form where the coefficient functions Ψ j can be determined by means of linear differential equations such that the corresponding function Φ in (2.1) does not depend on y.Additionally, the problem to derive conditions such that Φ is either positive or negative in the considered region was formulated as a problem of linear programming.
In [3] Gasull and Giacomini consider the class of planar autonomous systems (1.3), where the functions h i : R → R, 0 ≤ i ≤ 2, are continuous.This system represents a generalized Liénard system.They also look for a Dulac-Cherkas function in the form (2.4) and prove that to any given positive integer n there is a function Ψ as in (2.4) and a special value k such that the corresponding function Φ does not depend on y, and that the functions Ψ j can be determined by solving linear differential equations.They did not mention that this approach in case of the Liénard system (1.2) has been introduced by the first and second author in [4], probably, they were not aware of that paper.
In the next section we consider the generalized Liénard system (1.4) and describe an algorithm to find a function Ψ and a number k such that the corresponding function Φ in (2.1) does not depend on y.

Construction of a class of Dulac-Cherkas functions Ψ
for (1.4) in case 1 ≤ l ≤ 3 We consider the vector field X l (x, y) defined by the differential system (1.4) in some region G ⊂ R 2 .For the Dulac-Cherkas function Ψ(x, y) of (1.4) in G we make the ansatz (2.4) with n ≥ 2. In what follows we describe an algorithm to determine the functions Ψ j (x) in (2.4) and the constant k such that the corresponding function Φ(x, y) determined by does not depend on y.We show that this algorithm works generically in the considered case.If we put (2.4) into the right hand side of (3.1) and take into account EJQTDE, 2011 No. 35 p. 6 that the vector field X l is determined by (1.4) we get For the sequel we represent Φ(x, y) in the form where Φ i (x) are functions of the known coefficient functions h 0 (x), ..., h l (x), of the unknown coefficient functions Ψ 0 (x), ..., Ψ n (x), of their first derivatives Ψ ′ 0 (x), ..., Ψ ′ n (x), and of k.Concerning the highest power m of y in (3.3) we get from (3.2) Our goal is to determine the functions Ψ j (x), j = 0, ..., n, and the real number k in such a way that we have and if Φ 0 (x) vanishes only at finitely many points of x, then Ψ is a Dulac-Cherkas function of (1.4) in G.
From (3.2)-(3.4)we get that for l = 1 and l = 2 the relations (3.5) represent a system of n + 1 linear differential equations to determine the n + 1 functions Ψ j , j = 0, ..., n.In case l = 1 this system reads EJQTDE, 2011 No. 35 p. 7 It is easy to see that this system can be solved successively by simple quadratures, starting with Ψ n .The general solution depends on n + 1 integration constants and on the constant k as well as on the functions h i .An appropriate choice of these constants leads to efficient conditions on the functions h i such that Ψ is a Dulac-Cherkas function for (1.4) in G.
We look for a Dulac-Cherkas function in the form Ψ(x, y) = Ψ 0 (x) + Ψ 1 (x)y + Ψ 2 (x)y 2 (3.10) with Ψ 2 (x) ≡ 0. Putting n = 2 in (3.8) we obtain the following system of differential equations (3.11) Setting k = −2 we get from the first two equations where c 2 and c 1 are real constants.Putting c 1 = 0 we obtain from the last differential equation in (3.11) where c 0 is any real constant.Thus, we have To guarantee the validity of one of the inequalities Φ 0 (x) ≤ 0, Φ 0 (x) ≥ 0, we impose on h 0 and h 1 the following assumption.
(H). h 0 , h 1 : R → R are continuous and such that there is a constant c * 0 ensuring that the function does not change sign in R, where Φ0 (x) vanishes only in finite many points x k .
Applying Theorem 2.4 we get the result: Proposition 3.1.Suppose hypothesis (H) to be valid.Then system (3.9) has at most one limit cycle in the finite part of the phase plane.
If system (3.9) has a limit cycle, then it is hyperbolic.
We note that Proposition 3.1 coincides with Theorem 2.5.
In case l = 2 we get the system This system can also be integrated successively by solving inhomogeneous linear differential equations, starting with Ψ n .In order to derive efficient conditions guaranteeing the validity of the inequality (3.7) we EJQTDE, 2011 No. 35 p. 9 have to choose k and the integration constants appropriately.
Next we consider the case l = 3.From (3.2) and (3.3) we obtain The first equation is an algebraic equation which determines according to (1.5) and (2.5) the constant k uniquely as k = − n 3 .The remaining equations represent a system of n + 1 linear differential equations.Its general solution depends on n + 1 integration constants which can be chosen appropriately in order to derive efficient conditions on the functions h i satisfying the validity of the inequality (3.7).

Construction of Cherkas-Dulac functions in case l ≥ 4
In case l ≥ 4, system (3.5)consist of n+1 linear differential equations and l−2 algebraic equations to determine k and the functions Ψ 0 , ..., Ψ n such that Φ does not depend on y.Thus, this system has generically no solution.In what follows we show that under additional conditions on the functions h i system (3.5) has a nontrivial solution satisfying the inequality (3.7).We demonstrate this approach by considering the system dx dt = y, dy dt = h 0 (x) + h 1 (x)y + h 2 (x)y 2 + h 3 (x)y 3 + h 4 (x)y 4 .
This system of differential and algebraic equations can be solved for Ψ 0 (x), Ψ 1 (x) and Ψ 2 (x) only under additional conditions on the coefficient functions h i (x).In what follows we describe an algorithm to determine the Dulac-Cherkas function in such a way that the corresponding function Φ depends only on the variable x.We describe this EJQTDE, 2011 No. 35 p. 11 approach under the additional assumption (A 2 ).There is a real constant κ = 0 such that h 3 (x) ≡ κ h 4 (x) = 0 ∀x ∈ R. (4.9)By (4.9) and (4.3) we get from (4.4) Taking into account (4.9) and (4.10) we obtain from (4.5) A function Ψ 2 satisfying the differential equations (4.12) and has also to obey the homogeneous equation Thus, we have where c = 0 by (4.3).From (4.9), (4.3), (4.11) and (4.16) we get that the functions Ψ 1 and Ψ 2 never take the value zero.A solution of (4.14) satisfies the differential equation (4.12)only if the relation and d is any real constant.If we substitute the function Ψ 2 (x) defined in (4.16) and the function Ψ 0 (x) defined in (4.21) into (4.17)we get the relation  In order to guarantee that Ψ defined in (4.2) is a Dulac-Cherkas function we have to require that the function Φ 0 defined in (3.6) satisfies (3.7).Thus, we have the following result.Theorem 4.1.Consider system (4.1)under the assumptions (A 1 ) and (A 2 ).Additionally we suppose (A 3 ).There are constants c, d, κ and the functions h 0 , h 1 , h 2 , h 4 are such that (i). the function  4.22), respectively.Then system (4.1) has at most one limit cycle in R 2 .If system (4.1) has a limit cycle, then it is hyperbolic.By Corollary 2.1, the existence of a limit cycle under the assumptions of Theorem 4.1 requires that the set W defined by Ψ 0 (x) + Ψ 1 (x)y + Ψ 2 (x)y 2 = 0 contains an oval surrounding the origin.That means especially that the quadratic equation Ψ 0 (0) + Ψ 1 (0)y + Ψ 2 (0)y 2 = 0 must have negative and positive roots.By (4.21), (4.11), (4.16) it holds Thus, we have and the following result is valid.
We note that the assumption (4.26) implies Taking into account (4.26),(4.33)and (4.9), relation (4.23) takes the form Using this relation we obtain from (4.30) Hence, analogously to Theorem 4.1 and Theorem 4.2 we have Theorem 4.3.Consider system (4.1)under the assumptions (A 1 ) and (A 2 ).Additionally we suppose: There are real constants c, d, κ and functions h 0 , h 1 , h 2 , h 4 such that (i). the relations (4.33) and (4.34) are valid for x ∈ R. (ii). the function Φ0 defined in (4.35) is positive or negative semidefinite on R and vanishes only in finitely many points.
Then system (4.1) has at most one limit cycle in R 2 .If system (4.1) has a limit cycle, then it hyperbolic.If we additionally suppose dc > 0, then system (4.1) has no limit cycle in R 2 .
For the following we additionally assume (4.38) The initial value problem (4.38) can be used to determine h 1 (x) as a function of h 0 (x).Its explicit solution reads

Additionally we assume:
The functions h 1 , h 2 , h 4 are defined by (4.33),(4.39)(4.36).There are real constants c, d, κ, λ and the function h 0 is such that the function Φ0 defined in (4.35) is positive or negative semidefinite on R and vanishes only in finitely many points.Then system (4.1) has at most one limit cycle in R 2 .If system (4.1) has a limit cycle, then it is hyperbolic.If we additionally require dc > 0, then system (4.1) has no limit cycle in R 2 .
Taking into account assumption (A 1 ) and using (4.30) we get from (4.31) Thus, we have the following corollary.The following example shows that the interval I can coincide with the real axis.We consider system (4.1)under the assumptions (A 1 )−(A 4 ) of Theorem 4.4.As function h 0 we choose Then, the function h 1 reads Thus, we have Especially, we obtain from (4.42) and (4.43)We consider the generalized Liénard system (1.4) with l = 5 (5.1) Our aim is to determine the functions h i , 0 ≤ i ≤ 5, by means of a Dulac-Cherkas function Ψ such that (5.1) has a unique limit cycle surrounding the origin.We note that the following algorithm is not restricted to the case l = 5.
For Ψ we use the ansatz where we assume First we determine the constant k and the functions h i such that the equations (3.5) hold, that is, the function Φ defined in (3.1) has the form Φ(x, y) ≡ Φ 0 (x) = Ψ 1 (x)h 0 (x) + kΨ 0 (x)h 1 (x).In that case the corresponding oval O 1 := {(x, y) ∈ R 2 : x 2 + 0.01xy + y 2 = 1} represents a slightly perturbed unit circle.In order to ensure that Φ 0 has constant sign and that system (5.1) has a unique limit cycle we have to choose the function h 5 in an appropriate way.For the following, we set h 5 (x) ≡ 10 −3 .
(5.17) Thus, using (5.16) and (5.17), we get from (5.6)-( 5 Thus, in case that the functions h 0 -h 5 are determined by the relations (5.17 has unique real root at x = 0, where Φ ′ 0 (x) is positive for x < 0 and negative for x > 0. Taking into account Φ 0 (0) = −7/4000 we can conclude that Φ 0 (x) is negative for any x.Thus, the function Ψ(x, y) ≡ x 2 − 1 + 0.01xy + y 2 is a Dulac-Cherkas function for system (5.1) in R 2 .In what follows we construct a Bendixson annular region containing at least one limit cycle.As inner boundary we can use the oval O 1 which intersect all trajectories of (5.1) transversally by Theorem 2.1, they enter the region bounded by O 1 for increasing t.As outer boundary we can choose the circle x 2 +y 2 = 9.It can be verified that a trajectory of (5.1) which meets this circle intersects it transversally, and leaves the annulus for increasing t.Thus we have the following result Theorem 5.1.System (5.1) with h i defined by (5.17) -( 5.22) has a unique limit cycle which is hyperbolic and unstable.

Acknowledgement
The second author acknowledgements the financial support by DAAD and the hospitality of the Institute of Mathematics of Humboldt University Berlin.

2 j=0ΨRemark 4 . 1 . 5 j=0Φ
).The functions h i : R → R, 0 ≤ i ≤ 4, are continuous and such that h 4 is not identically zero.Our aim is to construct a Dulac-Cherkas function in the form Ψ(x, y) = In case that Ψ has the form (4.2), the set W defined by Ψ(x, y) = 0 contains at most one oval surrounding the origin.Taking into account Corollary 2.1, we can conclude that if Ψ has the form (4.2) and is a Dulac-Cherkas function for system (4.2) in R 2 , then system (4.1) has at most one limit cycle.To the function Ψ with the representation (4.2) there belongs by (3.1)-(3.4) the function Φ with the representation Φ(x, y) = j (x)y j .Our goal is to determine the Dulac-Cherkas function Ψ in such a way that Φ i (x) ≡ 0 for i = 1, • • • , 5. Taking into account (4.1) we get from (3.5) and (3.2) the relations 2 equation (4.18) takes the form

Corollary 4 . 1 .
Under the assumptions of Theorem 4.4 and under the additional condition interval I containing the origin such that system (4.1) has in the region I × R at most one limit cycle.EJQTDE, 2011 No. 35 p. 16 .40) Setting κ = λ = c = d = 1 ) > 0 ∀x ∈ R, (4.45) and we can conclude that h 1 (x) has a unique minimum at x = x m .From h ′ 1 (x m ) = − 389 48 e − xm 6 + 8 = 0 and from (4.42) we get