Slow Divergence Integrals in Generalized Liénard Equations near Centers

Using techniques from singular perturbations we show that for any n ≥ 6 and m ≥ 2 there are Liénard equations { ˙ x = y − F(x), ˙ y = G(x)}, with F a polynomial of degree n and G a polynomial of degree m, having at least 2[ n−2 2 ] + [ m 2 ] hyperbolic limit cycles, where [·] denotes " the greatest integer equal or below " .


Introduction
The paper deals with a popular model of generalized Liénard equations ẍ + f (x) ẋ + g(x) = 0, with f and g polynomials of respective degree n − 1 and m.A representation in the phase plane of this scalar second order differential equation is given by ẋ = y ẏ = − f (x)y − g(x).
If we write G(x) = −g(x) and introduce the new variable ȳ = y + F(x), where F(x) = x 0 f (s) ds, then the above planar vector field changes into a representation of the scalar second order Liénard differential equation in the so-called Liénard plane: where we denote ȳ by y.F and G are polynomials in x of respective degree n and m.When m = 1, equation (1.1) is called a classical Liénard equation (of degree n).When m > 1, we call (1.1) a generalized Liénard equation (of type (n, m)).
Theorem 1.1.Let n ≥ 6 and m ≥ 2. Then there exist a polynomial F(x) of degree n and a polynomial G(x) of degree m so that the system of differential equations has at least 2[ n− 2  2 ] + [ m 2 ] hyperbolic limit cycles.
In [27], it has been shown that there exist generalized Liénard equations (1.1) of type (n, m), n ≥ 2 and m ≥ 2, having at least [ n+m− 2   2   ] limit cycles.Clearly, the result in Theorem 1.1 improves this lower estimate 2 ] for all n ≥ 6 and m ≥ 2 .In [20], it has been proved further that for all n ≥ 2 and m ≥ 2. On one hand, it is not hard to show that l n,m ≥ [ n+m−2

2
], with strict inequality for infinitely many pairs (n, m) (see [20]).Thus, [20] is a recent improvement of [27].On the other hand, comparing the coefficients in front of n in the expression l n,m with the coefficient in front of n in 2 , it is clear that for each fixed m ≥ 2 there exists + 1 for all n ≥ 6. Recall that it has been proved in [18] that [ 2n− 1  3 ] ≤ L(n, 2) for all n ≥ 2. To our knowledge, there are no other results on lower bounds for generalized Liénard equations beside [20] and [27] for arbitrary n and m.
In [19], new lower bounds of L(n, m) are found for many integers n and m giving the m ln m asymptotic growth of L(n, m) with some conditions on n.For small m, Theorem 1.1 improves the lower bounds of L(n, m) given in [19].For example, it has been shown in [19] for m ∈ {3, 4} and n ≥ 6.In Section 2, using well known singular perturbation techniques for planar slow-fast systems we reduce the proof of Theorem 1.1 to the computation of simple integrals which appear in an expression for slow divergence integral.In Section 3, we use mathematical induction on degree m to finish the proof of Theorem 1.1.

Singular perturbations
Theorem 1.1 will be shown using techniques from singular perturbations (see [7,8,14]).Singular perturbations arise when the coefficients of F are very large, so that after applying a rescaling, a small parameter appears in front the ẏ equation (see also [3,9,32]): In this paper, we will also use this setting, together with the assumption that Limit cycles of (2.1) are generally members of -families of limit cycles that tend to certain limit periodic sets for = 0.The limit periodic sets are called slow-fast cycles, and are of the form The second component is a heteroclinic (fast) connection for = 0, connecting two singularities on the curve of singular points y = F(x), whereas the first component is the part of the parabolic curve beneath the fast orbit (see Figure 2.1).In this paper, we will parameterize the slow-fast cycles with its rightmost x-coordinate: In order to state the principal tool that we will use in the proof, we define the fast relation, which relates an x > 0 to an L(x) < 0 so that F(x) = F(L(x)).In other words, (L(x), Y) and (x, Y) are two end points of the same fast orbit at height Y = F(x).
We then have (using [10]) the following theorem.
Theorem 2.1.Let the function x → L(x) be described as above, and consider system (2.1) with the condition (2.2) and with the extra condition Define the so-called slow divergence integral associated to Γ x : Suppose that I(x) has exactly k simple zeros, then there exists a smooth function λ = λ( ) with λ(0) = 0, so that the perturbed system has exactly k + 1 periodic orbits (provided > 0 is small enough), all of them are isolated and hyperbolic.
Choose and fix x k+1 > x k arbitrary but so that x k+1 < M and L(x k+1 ) > −M.Since the origin is a slow-fast Hopf point, the parameter λ can be used as a breaking parameter.Hence there exists a λ = λ( ) with λ(0) = 0 so that (2.4) has a limit cycle Hausdorff close to Γ x k+1 .We can refer to [10], but even early results on canards like in [1] can be used to see this statement.The cycle Γ x k+1 is considered a long canard, and when a long canard is present, smaller canard cycles are located at zeros of the above integral.In other words, there are k additional canard cycles, Hausdorff close to Γ x i , for i = 1, . . ., k.For details we refer to [10].
We note that the same conclusions can be drawn using the entry-exit relation introduced in [1] (along the long canard we have so-called "tunnel" behaviour).Here we just present a heuristic argument.When orbits are integrated inside the big canard cycle, they will either spiral inwards or spiral outwards after one iteration around the Hopf point.During one iteration, the orbits travel a distance along the critical curve.Near this curve, the orbit experiences exponential attraction towards the long canard, and it will steer away from this canard after it has experienced equally strong long repulsion (after passing the Hopf point).Orbits at the interior of the long canard cycle will be attracted to an O( )-neighbourhood of the long canard at a point (x entry , F(x entry )) and will exit this O( )-neighbourhood at a point (x exit , F(x exit )).
Before the entry point and after the exit point, the orbit more or less follows a horizontal path (fast dynamics).It is clear that the orbit is spiraling inwards when F(x exit ) < F(x entry ) and outwards when F(x exit ) > F(x entry ).From the entry-exit relation deduced as early as in [1], we know that As a consequence, at zeros of I(x), orbits go from spiraling inwards to spiraling outwards or vice-versa and therefore at each zero of I(x) there should be an additional canard cycle.
Using a perturbative approach we compute the slow divergence integral I(x) in generalized Liénard equations near centers.For a suitable choice of polynomials F and G, we show that dominant part of the slow divergence integral is an integral of a polynomial function.We will assume that where F e is even, F o is odd, F e (0) = F o (0) = g(0) = g (0) = 0 and where δ is a small perturbation parameter.Centers are obtained when δ = 0.
Proposition 2.2.The slow divergence integral (2.3) of a cycle Γ x is given under these conditions by where f e (x) := F e (x)/x.Simple zeros of I 1 (x) will persist as simple zeros of I(x), for nonzero but small δ.
Proof.We first asymptotically determine the fast relation function L(x), from its defining property . By plugging this form into the defining property we obtain so using the symmetry properties of F e and F o we find L 1 (x) = − 2F o (x) F e (x) .Next we consider G(s) ds.We obtain If we write f e (x) := F e (x) x , then In one half of the first integral appearing in I 1 we apply partial integration to obtain the result.

R. Huzak and P. De Maesschalck
Proposition 2.2 and Theorem 2.1 allow to prove the main theorem (Theorem 1.1), provided we find convenient functions f e , F o and g that satisfy the conditions and that produce an integral function I 1 with a sufficient amount of simple zeros.In the classical case (g = 0), the following result has been proven in [8].
Proposition 2.3.Let k ≥ 3.There exist an even polynomial α k of degree 2k − 2, α k (s) > 0 for all s ∈ R, and an odd polynomial β k of degree 2k − 1 and of order 3 such that the function x 0 s f e (s) ds, satisfies the conditions of Proposition 2.2 and Theorem 2.1, giving an example of classical Liénard equation of even degree n = 2k, k ≥ 3, with n − 2 hyperbolic limit cycles.
Remark 2.4.Since α k (s) > 0 for all s ∈ R, the highest order coefficient is strictly positive.Using simple rescalings we can put the highest order coefficients of α k and β k to 1.
Remark 2.5.In the next section, the general case deg G = m ≥ 2 will be treated.We will use Proposition 2.3 in the proof of Theorem 1.1 as the basis step of mathematical induction on m.
The following proposition shows that the method used in this paper cannot give more limit cycles than stated in Theorem 1.1.Proof.It is clear that deg ] zeros counting multiplicity.Since I 1 is odd and F o (0) = F o (0) = g(0) = 0, we see that I 1 has at least a triple zero at the origin.Given furthermore the symmetry, it follows that there are at most 3 Proof of Theorem 1.1 The perturbative approach presented in the previous section will be used to treat the case of even degree n (Section 3.1); the case of odd degree n (Section 3.2) will be easy to study due to hyperbolicity of limit cycles obtained in Section 3.1.

Generalized Liénard equations with n even
In this section, we prove the following statement: Generalized Liénard equations near centers 7 For each k ≥ 3 and l ≥ 1 there exists an even polynomial g k,l of degree m = 2l, with g k,l (0) = 0, such that the function If we now take and F e (x) = x 0 s f e (s) ds, and G(x) = −x + δg(x), with g(x) = g k,l (x), then the above result implies that the expression for I 1 in Proposition 2.2 has 2k − 3 + l simple zeros on {x > 0}, leading to generalized Liénard equations of type (n, m) = (2k, 2l) with 2k − 2 + l hyperbolic limit cycles (see Theorem 2.1).Noting that the expression for I 1 remains unchanged if we use g(x) = g k,l (x) + ρx 2l+1 , ρ = 0, instead of g(x) = g k,l (x), we have, again by Theorem 2.1, existence of generalized Liénard equations of type (n, m) = (2k, 2l + 1) with 2k − 2 + l hyperbolic limit cycles.
For each k ≥ 3, we use induction on l to prove the above statement.Let us assume we have an example corresponding to l, for l ≥ 0, with an even polynomial g k,l of degree 2l and g k,l (0) = 0, and with α k and β k of respective degrees 2k − 2 and 2k − 1, given in Proposition 2.3, such that H k,l has 2k − 3 + l simple zeros on {x > 0}.As a direct consequence of Proposition 2.3, this can be performed for l = 0 (g k,0 ≡ 0).For l ≥ 1, we can write g k,l = • • • + γ 0 x 2l , with γ 0 = 0.
We now state g k,l+1 (x) := g k,l (x) + γ 1 µ 2 x 2l+2 , where γ 1 = −1 for l = 0 and γ 1 = − sgn(γ 0 ) for l ≥ 1.Here sgn(x) denotes the sign function.Such a choice of g k,l+1 leads to a vector field with (n, m) = (2k, 2l + 2), i.e. 2 degrees higher in G than for µ = 0.It is clear that for small values of µ the 2k − 3 + l simple zeros of H k,l+1 that appear for µ = 0 will persist.Besides that, we show that one additional positive simple zero appears in the O(1/µ) range.It can be easily seen that Lemma 3.1.

Figure 2 . 1 :
Figure 2.1: The dynamics of (2.1) for = 0.The blue closed curve is a slow-fast cycle.
Let n = 2k + 1, k ≥ 3, and m ≥ 2. Based on Theorem 2.1 and Section 3.1, we can choose a polynomial F of degree 2k and of the form (2.5), and a polynomial G of degree m and of the R. Huzak and P. De Maesschalck form (2.6) such that the systemẋ = y − F(x) ẏ = 0 [λ 0 + G(x)] (3.1) has 2k − 2 + [ m 2 ]hyperbolic limit cycles positioned near the long canard and the simple zeros of the corresponding slow divergence integral, for some 0 and λ 0 .If we change F(x) in (3.1) by F(x) + ρx 2k+1 , for ρ sufficiently small, then the 2k − 2 + [ m 2 ] hyperbolic limit cycles persist.It follows that for degree n = 2k + 1, there are at least n −3 + [ m 2 ] = 2[ n−2 2 ] + [ m2] isolated and hyperbolic periodic orbits.Hence, we have finished the proof of Theorem 1.1 for odd degrees n ≥ 7.