- Email: [email protected]
On the number of limit cycles of a class of polynomial systems of Liénard type
J. Math. Anal. Appl. 408 (2013) 775–780
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
On the number of limit cycles of a class of polynomial systems of Liénard type✩ Guifeng Chang a , Tonghua Zhang b , Maoan Han a,∗ a
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
b
Mathematics, H38, FEIS, Swinburne University of Technology, Hawthorn, VIC. 3122, Australia
article
abstract
info
In this paper we study a planar system of the form x˙ = pk (y), y˙ = −gm (x) − ε fn (x)y, where pk (y) is a polynomial of degree k in y, and fn (x), gm (x) are polynomials in x with degree of n and m, respectively. Let H (m, n, k) denote the maximal number of limit cycles having odd multiplicities of this system. We give some lower bounds of H (m, n, k). © 2013 Elsevier Inc. All rights reserved.
Article history: Received 7 January 2013 Available online 28 June 2013 Submitted by Junping Shi Keywords: Limit cycle Abelian integral Hamiltonian function Hilbert number
1. Introduction and main results The second part of Hilbert’s 16th problem asks for the maximal number H (m) of limit cycles and their relative positions of planar polynomial vector fields of degree m. In the past decades, there have been many studies on the problem, various kinds of particular systems have been investigated and many research articles have been published. However, up to now, this problem is so difficult and elusive that it is still open even for m = 2. If let H ∗ (m) denote the maximal number of limit cycles of the following planar near-Hamiltonian polynomial system of degree m ∗ x˙ = Hy + ε Pm (x, y),
y˙ = −Hx + ε Qm∗ (x, y),
∗ (x, y) and Qm∗ (x, y) are polynomials of degree m, H (x, y) is a polynomial of degree no more where ε is a small parameter, Pm than m + 1; then we have H (m) ≥ H ∗ (m) and also H ∗ (m) is known as the Hilbert number for the above mentioned nearHamiltonian system. Although there are only few results on the upper bounds of H (m) and H ∗ (m), many studies on their lower bounds can be found in the open literature, to name but a few in what follows. Authors of [1,9] proved that H (2) ≥ 4 independently. For cubic systems Li and Liu [6] and Li, Liu and Yang [7] obtained that H (3) ≥ H ∗ (3) ≥ 13, respectively. For general systems Christopher and Lloyd [2] proved
lim sup
m→∞
H (m)
(m + 1) ln(m + 1) 2
≥
1 2 ln 2
and Han and Li [3] proved lim inf
m→∞
H (m) 1 ≥ . (m + 1)2 ln(m + 1) 2 ln 2
✩ The project was supported by National Natural Science Foundation of China (11271261) and FP7-PEOPLE-2012-IRSES-316338.
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (M. Han).
0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.06.043
776
G. Chang et al. / J. Math. Anal. Appl. 408 (2013) 775–780
Let H1 (n, m) denote the maximal number of limit cycles with each having an odd multiplicity of the following system: x˙ = y,
y˙ = −gm (x) − ε fn (x)y,
where fn (x) and gm (x) are polynomials in x of degree n and m, respectively. Llibre, Mereu and Teixeira [8] proved H1 (n, m) ≥
n+m−1
2
,
and then Han, Tian and Yu [5] obtained an improvement of the above result H1 (n, m) ≥ max
m−2
+
3
2n + 1 3
n−2 2m + 1 , + . 3
3
Recently, the authors [4] obtained the following results: 1 (1) H1 (n, 4) ≥ H1 (n, 3) ≥ 2[ n− ] + [ n−2 1 ], n ≥ 3; 4 1 (2) H1 (n, 6) ≥ H1 (n, 5) ≥ 2[ n− ] + [ n−2 1 ], n ≥ 5; 3
(3) H1 (n, 7) ≥
− 9 for n ≥ 7. In general, for any integer m ≥ 7 there exists γm > 0 such that 1 ln(m + 2) n − γ m , n ≥ m. −
3 n 2
H1 (n, m) ≥
2 ln 2
3
In particular, lim lim inf
m→∞ n→∞
H1 (n, m) n ln(m + 2)
≥
1 2 ln 2
.
(4) For any integer r ≥ 0, lim inf
H1 (m ± r , m) m ln m
m→∞
≥
1 2 ln 2
.
(5) For m = 2p+1 − 1, p ≥ 1, we have H1 (m, m) ≥ H1 (m − 1, m) ≥
(m + 1) ln(m + 1) 2 ln 2
+ 1.
(6) For all m ≥ 3 we have H1 (m, m) ≥ H1 (m − 1, m) ≥
(m + 2) ln(m + 2) 3 ln 2
−
m+2 3
1+
ln 3
ln 2
+ 1.
(7) For any positive integers r ≥ 1 and k ≥ 2 there exist constants Bk and B¯ k satisfying lim
k→∞
Bk ln k
= lim
k→∞
B¯ k ln k
=−
1 2 ln 2
such that H1 (m − r , m) ≥
(m + 1) ln(m + 1) 2 ln 2
+ Bk (m + 1) + 1
for m = 2p k − 1, p ≥ 1, and H1 (m − r , m) ≥
(m + 2) ln(m + 2) 2 ln 2
+ B¯ k (m + 2) + 1
for m = 2p (k + 1) − 2, p ≥ 1. Motivated by the method used in [7,4], in this paper we consider the following planar system: x˙ = pk (y),
y˙ = −gm (x) − ε fn (x)y,
(1.1)
where pk (y) is a polynomial in y with degree k, fn (x) and gm (x) are polynomials in x of degree n and m, respectively and ε is a small parameter. Let H (m, n, k) denote the maximal number of limit cycles with an odd multiplicity of system (1.1) for ε sufficiently small and for all possible pk , gm and fn . Then our main result can be stated as follows. Theorem 1.1. Assume that there exist fn and gm , where fn (x) and gm (x) are polynomials of degree of n and m respectively, such that the system x˙ = y,
y˙ = −gm (x) − ε fn (x)y
has H1 (n, m) limit cycles for 0 < |ε| ≪ 1 each having an odd multiplicity. Then, (1) If k = 2l+1 − 1 ≥ 3, then H (m, n, k) ≥ k+2 1 H1 (n, m); (2) If k ≥ 2, k ̸∈ {2l+1 − 1|l = 1, 2, . . .}, then H (m, n, k) ≥ k+4 2 H1 (n, m).
G. Chang et al. / J. Math. Anal. Appl. 408 (2013) 775–780
777
By using Theorem 1.1 and [4], we can immediately make some conclusions on the lower bound of H (m, n, k). In particular, we have the following corollary. Corollary 1.1. For system (1.1) we have (i) For all m ≥ 3, H (m, m, k) ≥ H (m, m − 1, k) ≥ Nk
(m + 2) ln(m + 2) 2 ln 2
−
m+2 3
1+
ln 3 ln 2
+1 ,
where Nk =
k+1
if k = 2l+1 − 1, l ≥ 1,
2
k + 2
otherwise. 4 (ii) For polynomial system (1.1) of degree m with k = m = n + 1,
H (m) 1 ≥ , 2 (m + 1) ln(m + 1) 4 ln 2 1 H (m) ≥ lim inf m→∞ (m + 1)2 ln(m + 1) 8 ln 2 lim sup
m→∞
where H (m) = H (m − 1, m, m). The rest of the paper is organized as follows. In Section 2, we will introduce and prove some preliminary results. Then in Section 3 of the paper, we will prove our main results. 2. Preliminary results Consider a planar Hamiltonian system of the form
x˙ = p(y), y˙ = −g (x) − ε f (x)y,
(2.1)
where f (x) and g (x) are polynomials in x, p(y) is a polynomial in yandε is a small parameter. We make a hypothesis, denoted by A(k), about system (2.1). A(k): (i) It has k limit cycles with each having an odd multiplicity when 0 < ε ≪ 1, which corresponds to k zeros of the first order Melnikov function I ( h) =
f (x)ydx
Lh
with an odd multiplicity, where L(h) : H (x, y) = h, h ∈ J, is a family of closed orbits of system (2.1)|ε=0 with J an open interval. (ii) These k limit cycles are located in a compact set of R2 , which is independent of ε . Hence, there is a y∗0 < 0 such that all of these k limit cycles are between the lines y = ±y∗0 . For y0 < y∗0 , letting v = y − y0 , system (2.1) can then be rewritten as the following:
x˙ = p(v + y0 ), v˙ = −(g (x) + ε f (x)y0 ) − ε f (x)v.
(2.2)
By the assumptions we know that system (2.2) also has k limit cycles with each having an odd multiplicity when 0 < ε ≪ 1 and they are located in the region v > 0. Further letting v = w 2 in (2.2) yields
x˙ = 2w p(w 2 + y0 ), w ˙ = −(g (x) + ε f (x)y0 ) − ε f (x)w2 .
(2.3)
Then we can easily see that for ε = 0 systems (2.1) and (2.3) have the following Hamiltonian functions respectively: H (x, y) = G(x) +
y
p(t )dt , 0
and
H (x, w) = G(x) +
w
2w p(y0 + w 2 )dw
0
where G(x) =
x
g (u)du, 0
with the following relation holds H (x, w) = H (x, y) − H (0, y0 ) for y = y0 + w 2 .
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G. Chang et al. / J. Math. Anal. Appl. 408 (2013) 775–780
Next, instead of (2.3) we consider system
x˙ = 2w p(w 2 + y0 ), w ˙ = −g (x) − ε f (x)w.
(2.4)
Let
I j ( h) =
Lj ( h)
f (x)w dx
denote the Abelian integral of system (2.4), where Lj ( h) : H (x, w) = h, (−1)j w < 0, with h ∈ J = J − H (0, y0 ) = {h − H (0, y0 ) : h ∈ J } and j = 1, 2. Then we have the following lemma. Lemma 2.1. For |y0 | ≫ 1,
1 I j ( h) = (−1)j−1 √ I (h) + O 2 |y0 |
1
|y0 |
, h = h − H (0, y0 ) ∈ J, h ∈ J.
Proof. Notice that w 2 = y − y0 . Then for j = 1, |y| ≤ |y∗0 | ≪ |y0 | we have
w=
√
y − y0 = |y0 |
1 2
1+
21
y
= |y0 |
|y0 |
1 2
ai
i
y
|y0 |
i≥0
with a0 = 1,
1
a1 =
2
,
an = (−1)n−1
1 (2n − 3)!! 2n
n!
,
n ≥ 2.
Since when ε = 0, system (2.4) has the same family of closed orbits with system (2.1), the Abelian integral of system (2.4) corresponding to the family {L1 ( h)| h ∈ J } is given by I 1 ( h) =
L1 ( h)
f (x)w dx
f (x)|y0 |
=
1 2
1+
Lh
1 2
f (x) |y0 | +
= Lh
1 2
f (x)|y0 | dx +
1 y 2 |y0 |
1 2
√
y
|y0 |
+
ai
+ |y0 |
dx
1 2
|y0 |
y
y
1
φ
y
|y0 | 1 2
dx
y
dx + |y0 | f (x) √ 2 Lh |y0 | |y0 | y 1 1 I ( h) + f (x)y2 φ dx , = √ |y0 | Lh |y0 | 2 |y0 | where φ(u) = 2 i≥2 ai ui−2 = 2a2 + O(|u|), which implies that for |y0 | ≫ 1 1 1 I 1 (h) = √ I (h) + O . |y0 | 2 |y0 |
=
Lh
f (x)
1
i
|y0 | 2
i≥2 1 2
y
Lh
2
2
φ
y
|y0 |
dx
(2.5)
(2.6)
In fact, by mathematical induction, we can verify that 1 (2n − 3)!! 2n
n!
≤
1 8
,
for n ≥ 2,
(2.7)
which implies that
f (x)y φ 2
Lh
dx = |y0 | ≤ y
i−2 y i−1 1 (2i − 3)!! 2f (x)y (−1) dx i i ! 2 | y | 0 Lh i ≥2 i −2 1 (2i − 3)!! y (−1)i−1 2f (x)y2 dx i i ! 2 | y | 0 Lh i ≥2 i−2 1 y ≤ f (x)y2 dx. 4 i≥2 |y0 | Lh 2
(2.8)
G. Chang et al. / J. Math. Anal. Appl. 408 (2013) 775–780
779
Noticing that Lh is a closed curve located in a compact set, there is a positive number M independent of h such that max{|x|, |y|} ≤ M along the curve. Then from (2.8) by using the Mean Value Theorem for Definite Integrals we know that there are ξ ∈ (−M , M ) and N > 0 such that
f (x)y φ 2
Lh
i−2 y 2 f (x)y dx | y | 0 Lh i −2 M 3 |f (ξ )| M ≤ 2 |y0 | i≥2 i−2 M 3 |f (ξ )| 1 ≤
1 dx ≤ |y0 | 4 i ≥2 y
2
i≥2
2
≤N
(2.9)
when |y0 | > 2M. Eqs. (2.8) and (2.9) yield Eq. (2.6). Similarly, we can prove for j = 2
1 I ( h) + O I 2 ( h) = − √ 2 |y0 | This completes the proof.
1
|y0 |
.
From Lemma 2.1 we have immediately the following lemma. Lemma 2.2. Let (2.1) satisfy the assumption A(k). Then there exists y0 < 0 such that (2.4) satisfies A(2k). In fact, by A(k), the function I (h) has k zeros, say h1 < · · · < hk , each having an odd multiplicity. Further by Lemma 2.1, there exists y¯ 0 < y∗0 such that for y0 = y¯ 0 , each of the functions ¯I1 ( h) and ¯I2 ( h) has also k zeros, say h1j < · · · < hkj , j = 1, 2. Each of these zeros has an odd multiplicity. Therefore, we obtain 2k limit cycles of (2.4), each having an odd multiplicity. Moreover, for fixed y¯ 0 and sufficiently small ε , these 2k limit cycles are in a compact set which is independent of ε . Therefore, (2.4) satisfies assumption A(2k). 3. The proof of our main results In this section we will give the detailed proof of our main results listed in Section 1. First we consider a polynomial Liénard system of the form x˙ = y,
y˙ = −gm (x) − ε fn (x)y
(3.1)
where ε is sufficiently small, fn (x) and gm (x) are as before. Since by [4] there exist gm and fn such that system (3.1) satisfies A(H1 (m, n)), we assume that all limit cycles of system (3.1) are located between the lines y = ±y1 with −y1 ≫ 1. Then letting w12 = y − y1 gives (3.1) in the following form:
x˙ = 2w1 (w12 + y1 ), w˙1 = −g1,m (x) − εfn (x)w12
(3.2)
where g1,m (x) = gm (x) + ε fn (x)y1 . Next, instead of investigating system (3.2), we consider the following system:
x˙ = 2w1 (w12 + y1 ) ≡ p3 (w1 ), w ˙ 1 = −gm (x) − ε fn (x)w1 ,
(3.3)
which has exactly the same unperturbed system as that of (3.2). By using Lemma 2.2 we know that there exists y1 < 0 such that (3.3) satisfies A(2H1 (m, n)). Without loss of generality, we can also assume that all these limit cycles are located between lines w1 = ±w1,0 with −w1,0 ≫ 1. Then letting w22 = w1 − w10 gives (3.3) in the following form:
x˙ = 22 w2 (w22 + w10 )((w22 + w10 )2 + y1 ), w ˙ 2 = −g2,m (x) − εfn (x)w22 ,
(3.4)
where g2,m (x) = gm (x) + ε fn (x)w10 . As before, instead of investigating system (3.4), we consider
x˙ = 22 w2 (w22 + w10 )((w22 + w10 )2 + y1 ) ≡ p7 (w2 ), w ˙ 2 = −gm (x) − ε fn (x)w2 ,
(3.5)
which has exactly the same unperturbed system as that of (3.4). By using Lemma 2.2 we know that there exists w10 < 0 such that (3.5) satisfies A(22 H1 (m, n)). Again, without loss of generality, we can assume that all these limit cycles are located
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G. Chang et al. / J. Math. Anal. Appl. 408 (2013) 775–780
between lines wl = ±wl,0 with −wl,0 ≫ 1, l ≥ 3. Then we repeat the process as what we did to system (3.1). After l times, we end up with the following system:
x˙ = pk (wl ), w ˙ l = −gm (x) − ε fn (x)wk ,
which satisfies A(2l H1 (m, n)), where the degree of pk (wl ) is k = 2l+1 − 1. Note that 2l = k+2 1 . Then the first conclusion of Theorem 1.1 follows. For the second conclusion, let k ̸∈ {2l+1 − 1|l = 1, 2, . . .}. Then there exists l ≥ 0 such that 2l+1 − 1 < k ≤ 2l+2 − 2. Thus, by the first conclusion H (m, n, k) ≥ H (m, n, 2l+1 − 1) ≥ 2l H1 (n, m) ≥
k+2 4
H1 (n, m).
This completes the proof of Theorem 1.1. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L. Chen, M. Wang, The relative position and the number of limit cycles of a quadratic differential system, Acta Math. Sinica 22 (1979) 751–758. C.J. Christopher, N.G. Lloyd, Polynomial systems: a lower bound for the Hilbert numbers, Proc. R. Soc. Lond. Ser. A 450 (1995) 219–224. M. Han, J. Li, Lower bounds for the Hilbert number of polynomial systems, J. Differential Equations 252 (2012) 3278–3304. M. Han, V.G. Romanovski, On the number of limit cycles of polynomial Liénard systems, Nonlinear Anal. RWA (2012) http://dx.doi.org/10.1016/j. nonrwa.2012.11.002. M. Han, Y. Tian, P. Yu, Small-amplitude limit cycles of polynomial Liénard systems, Sci. China Math. (2013) 1–14. http://dx.doi.org/10.1007/s11425013-4618-9. J. Li, Y. Liu, New results on the study of Zq -equivariant planar polynomial vector fields, Qual. Theory Dyn. Syst. 9 (2010) 167–219. C. Li, C. Liu, J. Yang, A cubic system with thirteen limit cycles, J. Differential Equations 246 (2009) 3609–3619. J. Llibre, A.C. Mereu, M.A. Teixeira, Limit cycles of the generalized polynomial Liénard differential equations, Math. Proc. Cambridge Philos. Soc. 148 (2010) 363–383. S. Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sin. 23 (1980) 153–158.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
On the number of limit cycles of a class of polynomial systems of Liénard type✩ Guifeng Chang a , Tonghua Zhang b , Maoan Han a,∗ a
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
b
Mathematics, H38, FEIS, Swinburne University of Technology, Hawthorn, VIC. 3122, Australia
article
abstract
info
In this paper we study a planar system of the form x˙ = pk (y), y˙ = −gm (x) − ε fn (x)y, where pk (y) is a polynomial of degree k in y, and fn (x), gm (x) are polynomials in x with degree of n and m, respectively. Let H (m, n, k) denote the maximal number of limit cycles having odd multiplicities of this system. We give some lower bounds of H (m, n, k). © 2013 Elsevier Inc. All rights reserved.
Article history: Received 7 January 2013 Available online 28 June 2013 Submitted by Junping Shi Keywords: Limit cycle Abelian integral Hamiltonian function Hilbert number
1. Introduction and main results The second part of Hilbert’s 16th problem asks for the maximal number H (m) of limit cycles and their relative positions of planar polynomial vector fields of degree m. In the past decades, there have been many studies on the problem, various kinds of particular systems have been investigated and many research articles have been published. However, up to now, this problem is so difficult and elusive that it is still open even for m = 2. If let H ∗ (m) denote the maximal number of limit cycles of the following planar near-Hamiltonian polynomial system of degree m ∗ x˙ = Hy + ε Pm (x, y),
y˙ = −Hx + ε Qm∗ (x, y),
∗ (x, y) and Qm∗ (x, y) are polynomials of degree m, H (x, y) is a polynomial of degree no more where ε is a small parameter, Pm than m + 1; then we have H (m) ≥ H ∗ (m) and also H ∗ (m) is known as the Hilbert number for the above mentioned nearHamiltonian system. Although there are only few results on the upper bounds of H (m) and H ∗ (m), many studies on their lower bounds can be found in the open literature, to name but a few in what follows. Authors of [1,9] proved that H (2) ≥ 4 independently. For cubic systems Li and Liu [6] and Li, Liu and Yang [7] obtained that H (3) ≥ H ∗ (3) ≥ 13, respectively. For general systems Christopher and Lloyd [2] proved
lim sup
m→∞
H (m)
(m + 1) ln(m + 1) 2
≥
1 2 ln 2
and Han and Li [3] proved lim inf
m→∞
H (m) 1 ≥ . (m + 1)2 ln(m + 1) 2 ln 2
✩ The project was supported by National Natural Science Foundation of China (11271261) and FP7-PEOPLE-2012-IRSES-316338.
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (M. Han).
0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.06.043
776
G. Chang et al. / J. Math. Anal. Appl. 408 (2013) 775–780
Let H1 (n, m) denote the maximal number of limit cycles with each having an odd multiplicity of the following system: x˙ = y,
y˙ = −gm (x) − ε fn (x)y,
where fn (x) and gm (x) are polynomials in x of degree n and m, respectively. Llibre, Mereu and Teixeira [8] proved H1 (n, m) ≥
n+m−1
2
,
and then Han, Tian and Yu [5] obtained an improvement of the above result H1 (n, m) ≥ max
m−2
+
3
2n + 1 3
n−2 2m + 1 , + . 3
3
Recently, the authors [4] obtained the following results: 1 (1) H1 (n, 4) ≥ H1 (n, 3) ≥ 2[ n− ] + [ n−2 1 ], n ≥ 3; 4 1 (2) H1 (n, 6) ≥ H1 (n, 5) ≥ 2[ n− ] + [ n−2 1 ], n ≥ 5; 3
(3) H1 (n, 7) ≥
− 9 for n ≥ 7. In general, for any integer m ≥ 7 there exists γm > 0 such that 1 ln(m + 2) n − γ m , n ≥ m. −
3 n 2
H1 (n, m) ≥
2 ln 2
3
In particular, lim lim inf
m→∞ n→∞
H1 (n, m) n ln(m + 2)
≥
1 2 ln 2
.
(4) For any integer r ≥ 0, lim inf
H1 (m ± r , m) m ln m
m→∞
≥
1 2 ln 2
.
(5) For m = 2p+1 − 1, p ≥ 1, we have H1 (m, m) ≥ H1 (m − 1, m) ≥
(m + 1) ln(m + 1) 2 ln 2
+ 1.
(6) For all m ≥ 3 we have H1 (m, m) ≥ H1 (m − 1, m) ≥
(m + 2) ln(m + 2) 3 ln 2
−
m+2 3
1+
ln 3
ln 2
+ 1.
(7) For any positive integers r ≥ 1 and k ≥ 2 there exist constants Bk and B¯ k satisfying lim
k→∞
Bk ln k
= lim
k→∞
B¯ k ln k
=−
1 2 ln 2
such that H1 (m − r , m) ≥
(m + 1) ln(m + 1) 2 ln 2
+ Bk (m + 1) + 1
for m = 2p k − 1, p ≥ 1, and H1 (m − r , m) ≥
(m + 2) ln(m + 2) 2 ln 2
+ B¯ k (m + 2) + 1
for m = 2p (k + 1) − 2, p ≥ 1. Motivated by the method used in [7,4], in this paper we consider the following planar system: x˙ = pk (y),
y˙ = −gm (x) − ε fn (x)y,
(1.1)
where pk (y) is a polynomial in y with degree k, fn (x) and gm (x) are polynomials in x of degree n and m, respectively and ε is a small parameter. Let H (m, n, k) denote the maximal number of limit cycles with an odd multiplicity of system (1.1) for ε sufficiently small and for all possible pk , gm and fn . Then our main result can be stated as follows. Theorem 1.1. Assume that there exist fn and gm , where fn (x) and gm (x) are polynomials of degree of n and m respectively, such that the system x˙ = y,
y˙ = −gm (x) − ε fn (x)y
has H1 (n, m) limit cycles for 0 < |ε| ≪ 1 each having an odd multiplicity. Then, (1) If k = 2l+1 − 1 ≥ 3, then H (m, n, k) ≥ k+2 1 H1 (n, m); (2) If k ≥ 2, k ̸∈ {2l+1 − 1|l = 1, 2, . . .}, then H (m, n, k) ≥ k+4 2 H1 (n, m).
G. Chang et al. / J. Math. Anal. Appl. 408 (2013) 775–780
777
By using Theorem 1.1 and [4], we can immediately make some conclusions on the lower bound of H (m, n, k). In particular, we have the following corollary. Corollary 1.1. For system (1.1) we have (i) For all m ≥ 3, H (m, m, k) ≥ H (m, m − 1, k) ≥ Nk
(m + 2) ln(m + 2) 2 ln 2
−
m+2 3
1+
ln 3 ln 2
+1 ,
where Nk =
k+1
if k = 2l+1 − 1, l ≥ 1,
2
k + 2
otherwise. 4 (ii) For polynomial system (1.1) of degree m with k = m = n + 1,
H (m) 1 ≥ , 2 (m + 1) ln(m + 1) 4 ln 2 1 H (m) ≥ lim inf m→∞ (m + 1)2 ln(m + 1) 8 ln 2 lim sup
m→∞
where H (m) = H (m − 1, m, m). The rest of the paper is organized as follows. In Section 2, we will introduce and prove some preliminary results. Then in Section 3 of the paper, we will prove our main results. 2. Preliminary results Consider a planar Hamiltonian system of the form
x˙ = p(y), y˙ = −g (x) − ε f (x)y,
(2.1)
where f (x) and g (x) are polynomials in x, p(y) is a polynomial in yandε is a small parameter. We make a hypothesis, denoted by A(k), about system (2.1). A(k): (i) It has k limit cycles with each having an odd multiplicity when 0 < ε ≪ 1, which corresponds to k zeros of the first order Melnikov function I ( h) =
f (x)ydx
Lh
with an odd multiplicity, where L(h) : H (x, y) = h, h ∈ J, is a family of closed orbits of system (2.1)|ε=0 with J an open interval. (ii) These k limit cycles are located in a compact set of R2 , which is independent of ε . Hence, there is a y∗0 < 0 such that all of these k limit cycles are between the lines y = ±y∗0 . For y0 < y∗0 , letting v = y − y0 , system (2.1) can then be rewritten as the following:
x˙ = p(v + y0 ), v˙ = −(g (x) + ε f (x)y0 ) − ε f (x)v.
(2.2)
By the assumptions we know that system (2.2) also has k limit cycles with each having an odd multiplicity when 0 < ε ≪ 1 and they are located in the region v > 0. Further letting v = w 2 in (2.2) yields
x˙ = 2w p(w 2 + y0 ), w ˙ = −(g (x) + ε f (x)y0 ) − ε f (x)w2 .
(2.3)
Then we can easily see that for ε = 0 systems (2.1) and (2.3) have the following Hamiltonian functions respectively: H (x, y) = G(x) +
y
p(t )dt , 0
and
H (x, w) = G(x) +
w
2w p(y0 + w 2 )dw
0
where G(x) =
x
g (u)du, 0
with the following relation holds H (x, w) = H (x, y) − H (0, y0 ) for y = y0 + w 2 .
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Next, instead of (2.3) we consider system
x˙ = 2w p(w 2 + y0 ), w ˙ = −g (x) − ε f (x)w.
(2.4)
Let
I j ( h) =
Lj ( h)
f (x)w dx
denote the Abelian integral of system (2.4), where Lj ( h) : H (x, w) = h, (−1)j w < 0, with h ∈ J = J − H (0, y0 ) = {h − H (0, y0 ) : h ∈ J } and j = 1, 2. Then we have the following lemma. Lemma 2.1. For |y0 | ≫ 1,
1 I j ( h) = (−1)j−1 √ I (h) + O 2 |y0 |
1
|y0 |
, h = h − H (0, y0 ) ∈ J, h ∈ J.
Proof. Notice that w 2 = y − y0 . Then for j = 1, |y| ≤ |y∗0 | ≪ |y0 | we have
w=
√
y − y0 = |y0 |
1 2
1+
21
y
= |y0 |
|y0 |
1 2
ai
i
y
|y0 |
i≥0
with a0 = 1,
1
a1 =
2
,
an = (−1)n−1
1 (2n − 3)!! 2n
n!
,
n ≥ 2.
Since when ε = 0, system (2.4) has the same family of closed orbits with system (2.1), the Abelian integral of system (2.4) corresponding to the family {L1 ( h)| h ∈ J } is given by I 1 ( h) =
L1 ( h)
f (x)w dx
f (x)|y0 |
=
1 2
1+
Lh
1 2
f (x) |y0 | +
= Lh
1 2
f (x)|y0 | dx +
1 y 2 |y0 |
1 2
√
y
|y0 |
+
ai
+ |y0 |
dx
1 2
|y0 |
y
y
1
φ
y
|y0 | 1 2
dx
y
dx + |y0 | f (x) √ 2 Lh |y0 | |y0 | y 1 1 I ( h) + f (x)y2 φ dx , = √ |y0 | Lh |y0 | 2 |y0 | where φ(u) = 2 i≥2 ai ui−2 = 2a2 + O(|u|), which implies that for |y0 | ≫ 1 1 1 I 1 (h) = √ I (h) + O . |y0 | 2 |y0 |
=
Lh
f (x)
1
i
|y0 | 2
i≥2 1 2
y
Lh
2
2
φ
y
|y0 |
dx
(2.5)
(2.6)
In fact, by mathematical induction, we can verify that 1 (2n − 3)!! 2n
n!
≤
1 8
,
for n ≥ 2,
(2.7)
which implies that
f (x)y φ 2
Lh
dx = |y0 | ≤ y
i−2 y i−1 1 (2i − 3)!! 2f (x)y (−1) dx i i ! 2 | y | 0 Lh i ≥2 i −2 1 (2i − 3)!! y (−1)i−1 2f (x)y2 dx i i ! 2 | y | 0 Lh i ≥2 i−2 1 y ≤ f (x)y2 dx. 4 i≥2 |y0 | Lh 2
(2.8)
G. Chang et al. / J. Math. Anal. Appl. 408 (2013) 775–780
779
Noticing that Lh is a closed curve located in a compact set, there is a positive number M independent of h such that max{|x|, |y|} ≤ M along the curve. Then from (2.8) by using the Mean Value Theorem for Definite Integrals we know that there are ξ ∈ (−M , M ) and N > 0 such that
f (x)y φ 2
Lh
i−2 y 2 f (x)y dx | y | 0 Lh i −2 M 3 |f (ξ )| M ≤ 2 |y0 | i≥2 i−2 M 3 |f (ξ )| 1 ≤
1 dx ≤ |y0 | 4 i ≥2 y
2
i≥2
2
≤N
(2.9)
when |y0 | > 2M. Eqs. (2.8) and (2.9) yield Eq. (2.6). Similarly, we can prove for j = 2
1 I ( h) + O I 2 ( h) = − √ 2 |y0 | This completes the proof.
1
|y0 |
.
From Lemma 2.1 we have immediately the following lemma. Lemma 2.2. Let (2.1) satisfy the assumption A(k). Then there exists y0 < 0 such that (2.4) satisfies A(2k). In fact, by A(k), the function I (h) has k zeros, say h1 < · · · < hk , each having an odd multiplicity. Further by Lemma 2.1, there exists y¯ 0 < y∗0 such that for y0 = y¯ 0 , each of the functions ¯I1 ( h) and ¯I2 ( h) has also k zeros, say h1j < · · · < hkj , j = 1, 2. Each of these zeros has an odd multiplicity. Therefore, we obtain 2k limit cycles of (2.4), each having an odd multiplicity. Moreover, for fixed y¯ 0 and sufficiently small ε , these 2k limit cycles are in a compact set which is independent of ε . Therefore, (2.4) satisfies assumption A(2k). 3. The proof of our main results In this section we will give the detailed proof of our main results listed in Section 1. First we consider a polynomial Liénard system of the form x˙ = y,
y˙ = −gm (x) − ε fn (x)y
(3.1)
where ε is sufficiently small, fn (x) and gm (x) are as before. Since by [4] there exist gm and fn such that system (3.1) satisfies A(H1 (m, n)), we assume that all limit cycles of system (3.1) are located between the lines y = ±y1 with −y1 ≫ 1. Then letting w12 = y − y1 gives (3.1) in the following form:
x˙ = 2w1 (w12 + y1 ), w˙1 = −g1,m (x) − εfn (x)w12
(3.2)
where g1,m (x) = gm (x) + ε fn (x)y1 . Next, instead of investigating system (3.2), we consider the following system:
x˙ = 2w1 (w12 + y1 ) ≡ p3 (w1 ), w ˙ 1 = −gm (x) − ε fn (x)w1 ,
(3.3)
which has exactly the same unperturbed system as that of (3.2). By using Lemma 2.2 we know that there exists y1 < 0 such that (3.3) satisfies A(2H1 (m, n)). Without loss of generality, we can also assume that all these limit cycles are located between lines w1 = ±w1,0 with −w1,0 ≫ 1. Then letting w22 = w1 − w10 gives (3.3) in the following form:
x˙ = 22 w2 (w22 + w10 )((w22 + w10 )2 + y1 ), w ˙ 2 = −g2,m (x) − εfn (x)w22 ,
(3.4)
where g2,m (x) = gm (x) + ε fn (x)w10 . As before, instead of investigating system (3.4), we consider
x˙ = 22 w2 (w22 + w10 )((w22 + w10 )2 + y1 ) ≡ p7 (w2 ), w ˙ 2 = −gm (x) − ε fn (x)w2 ,
(3.5)
which has exactly the same unperturbed system as that of (3.4). By using Lemma 2.2 we know that there exists w10 < 0 such that (3.5) satisfies A(22 H1 (m, n)). Again, without loss of generality, we can assume that all these limit cycles are located
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G. Chang et al. / J. Math. Anal. Appl. 408 (2013) 775–780
between lines wl = ±wl,0 with −wl,0 ≫ 1, l ≥ 3. Then we repeat the process as what we did to system (3.1). After l times, we end up with the following system:
x˙ = pk (wl ), w ˙ l = −gm (x) − ε fn (x)wk ,
which satisfies A(2l H1 (m, n)), where the degree of pk (wl ) is k = 2l+1 − 1. Note that 2l = k+2 1 . Then the first conclusion of Theorem 1.1 follows. For the second conclusion, let k ̸∈ {2l+1 − 1|l = 1, 2, . . .}. Then there exists l ≥ 0 such that 2l+1 − 1 < k ≤ 2l+2 − 2. Thus, by the first conclusion H (m, n, k) ≥ H (m, n, 2l+1 − 1) ≥ 2l H1 (n, m) ≥
k+2 4
H1 (n, m).
This completes the proof of Theorem 1.1. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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