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Proc. R. Soc. A (2012) 468, 2347–2360
doi:10.1098/rspa.2011.0741
Published online 11 April 2012
On the number of limit cycles of a class of
polynomial differential systems
BY JAUME LLIBRE1, *
AND
CLÀUDIA VALLS2
1 Departament
de Matemàtiques, Universitat Autònoma de Barcelona,
08193 Bellaterra, Barcelona, Catalonia, Spain
2 Departamento de Matemática, Instituto Superior Técnico, 1049–001
Lisboa, Portugal
We study the number of limit cycles of polynomial differential systems of the form
ẋ = y − g1 (x) − f1 (x)y
and
ẏ = −x − g2 (x) − f2 (x)y,
where g1 , f1 , g2 and f2 are polynomials of a given degree. Note that when g1 (x) = f1 (x) = 0,
we obtain the generalized polynomial Liénard differential systems. We provide an
accurate upper bound of the maximum number of limit cycles that the above system
can have bifurcating from the periodic orbits of the linear centre ẋ = y, ẏ = −x using the
averaging theory of first and second order.
Keywords: limit cycles; polynomial differential systems; Liénard differential systems
1. Introduction
The second part of the 16th Hilbert problem wants to find an upper bound on the
maximum number of limit cycles that a polynomial vector field of a fixed degree
can have. In this paper, we will try to give a partial answer to this problem for
the class of polynomial differential systems given by
ẋ = y − g1 (x) − f1 (x)y
and ẏ = −x − g2 (x) − f2 (x)y.
(1.1)
Note that when g1 (x) = f1 (x) = 0 coincides with generalized polynomial Liénard
differential systems. The classical polynomial Liénard differential systems are
ẋ = y
and ẏ = −x − f (x)y,
(1.2)
where f (x) is a polynomial in the variable x of degree n. For these systems, Lins
et al. (1977) stated the conjecture that if f (x) has degree n ≥ 1, then system (1.2)
has at most [n/2] limit cycles. They proved this conjecture for n = 1, 2. Recently,
the conjecture has also been proved for n = 3, see Li & Llibre (2012). For n ≥ 5,
Dumortier et al. (2007) and De Maesschalck & Dumortier (2011) showed that the
conjecture is not true for n ≥ 5. In short, at this moment, the conjecture is only
open for n = 4.
*Author for correspondence ([email protected]).
Received 22 December 2011
Accepted 12 March 2012
2347
This journal is © 2012 The Royal Society
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2348
J. Llibre and C. Valls
Many of the results on the limit cycles of polynomial differential systems
have been obtained by considering limit cycles that bifurcate from a single
degenerate singular point (i.e. from a Hopf bifurcation), which are called small
amplitude limit cycles, see Lloyd (1988). There are partial results concerning
the maximum number of small-amplitude limit cycles for Liénard polynomial
differential systems. The number of small-amplitude limit cycles gives a lower
bound for the maximum number of limit cycles that a polynomial differential
system can have.
There are many results concerning the existence of small-amplitude limit cycles
for the following generalization of the classical Liénard polynomial differential
system (1.2):
ẋ = y and ẏ = −g(x) − f (x)y,
(1.3)
where f (x) and g(x) are polynomials in the variable x of degrees n and m,
respectively. We denote by H (m, n) the maximum number of limit cycles that
system (1.3) can have. This number is usually called the Hilbert number for
system (1.3).
x
(i) In 1928, Liénard (1928) proved that if m = 1 and F (x) = 0 f (s) ds is a
continuous odd function, which has a unique root at x = a and is monotone
increasing for x ≥ a, then equations (1.3) have a unique limit cycle.
(ii) In 1973, Rychkov (1975) proved that if m = 1 and F (x) is an odd
polynomial of degree five, then equations (1.3) have at most two limit
cycles.
(iii) In 1977, Lins et al. (1977) proved that H (1, 1) = 0 and H (1, 2) = 1.
(iv) In 1998, Coppel (1998) proved that H (2, 1) = 1.
(v) Dumortier & Rousseau (1990) and Dumortier & Li (1997) proved that
H (3, 1) = 1.
(vi) In 1997, Dumortier (1997) proved that H (2, 2) = 1.
(vii) In 2012, Li & Llibre (2012) proved that H (1, 3) = 1.
Up to now and as far as we know, only for these five cases (iii)–(vii) has the
Hilbert number for system (1.3) been determined.
The maximum number of small-amplitude limit cycles for system (1.3) is
denoted by Ĥ (m, n). Blows & Lloyd (1984), Lloyd & Lynch (1988) and Lynch
(1995) have used inductive arguments in order to prove the following results:
—
—
—
—
if
if
if
if
g is odd, then Ĥ (m, n) = [n/2];
f is even, then Ĥ (m, n) = n, whatever g is;
f is odd, then Ĥ (m, 2n + 1) = [(m − 2)/2] + n; and
g(x) = x + ge (x), where ge is even, then Ĥ (2m, 2) = m.
Christopher & Lynch (1999), Lynch (1998, 1999) and Lynch & Christopher
(1999) have developed a new algebraic method for determining the Liapunov
quantities of system (1.3) and proved some other bounds for Ĥ (m, n) for different
m and n:
— Ĥ (m, 2) = [(2m + 1)/3];
— Ĥ (2, n) = [(2n + 1)/3];
Proc. R. Soc. A (2012)
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Limit cycles of differential systems
2349
— Ĥ (m, 3) = 2[(3m + 2)/8], for all 1 < m ≤ 50;
— Ĥ (3, n) = 2[(3n + 2)/8], for all 1 < m ≤ 50; and
— Ĥ (4, k) = Ĥ (k, 4), k = 6, 7, 8, 9 and Ĥ (5, 6) = Ĥ (6, 5).
In 1998, Gasull & Torregrosa (1998) obtained upper bounds for Ĥ (7, 6),
Ĥ (6, 7), Ĥ (7, 7) and Ĥ (4, 20).
In 2006, Yu & Han (2006) gave some accurate values of Ĥ (m, n) = Ĥ (n, m),
for n = 4, m = 10, 11, 12, 13; n = 5, m = 6, 7, 8, 9; n = 6, m = 5, 6; see also Llibre
et al. (2010) for a table with all the specific values.
In 2010, Llibre et al. (2010) computed the maximum number of limit cycles
H̃ k (m, n) of system (1.3) that bifurcate from the periodic orbits of the linear
centre ẋ = y, ẏ = −x, using the averaging theory of order k, for k = 1, 2, 3.
In this paper, first we consider the system
ẋ = y − 3(g11 (x) + f11 (x)y)
(1.4)
and
ẏ = −x − 3(g21 (x) + f21 (x)y),
where g11 , f11 , g21 and f21 have degree k, l, m and n, respectively, and 3 is a small
parameter.
Theorem 1.1. For |3| sufficiently small, the maximum number of limit cycles of
the generalized Liénard polynomial differential system (1.4) bifurcating from the
periodic orbits of the linear centre ẋ = y, ẏ = −x using the averaging theory of first
order is
n
(k − 1)
l1 = max
,
.
(1.5)
2
2
The proof of theorem 1.1 is given in §3.
Now we consider the system
ẋ = y − 3(g11 (x) + f11 (x)y) − 32 (g12 (x) + f12 (x)y)
(1.6)
and
ẏ = −x − 3(g21 (x) + f21 (x)y) − 32 (g22 (x) + f22 (x)y),
where g11 and g12 have degree k; f11 and f12 have degree l; g21 and g22 have degree
m; and f21 , f22 have degree n. Furthermore, 3 is a small parameter.
Theorem 1.2. For |3| sufficiently small, the maximum number of limit cycles
of the generalized Liénard polynomial differential systems (1.6) bifurcating from
the periodic orbits of the linear centre ẋ = y, ẏ = −x using the averaging theory of
second order is l3 = max{l1 , l2 }, where
(m − 1)
l
(n − 1)
m
k
m
,m +
,
+
,
+
− 1,
l2 = max m +
2
2
2
2
2
2
(n − 1)
(l − 1)
k
(l − 1)
+
+ 1,
+
,
(1.7)
2
2
2
2
with m = min{[n/2], [(k − 1)/2]}.
The proof of theorem 1.2 is given in §4.
The results that we shall use from the averaging theory of first and second
order for computing limit cycles are presented in §2.
Proc. R. Soc. A (2012)
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2350
J. Llibre and C. Valls
2. The averaging theory of first and second order
The averaging theory for studying specifically limit cycles up to first order in 3 was
developed many years ago, and can be found in Verhulst (1991), Guckenheimer &
Holmes (1990) and Buică & Llibre (2004). The averaging theory for computing
limit cycles up to second order in 3 was developed in Llibre (2002–2003) and
Buică & Llibre (2004). It is summarized as follows.
Consider the differential system
ẋ = 3F1 (t, x) + 32 F2 (t, x) + 33 R(t, x, 3),
(2.1)
where F1 , F2 : R × D → R, R : R × D × (−3f , 3f ) → R are continuous functions,
T -periodic in the first variable, and D is an open subset of Rn . Assume that the
following conditions hold.
(i) F1 (t, ·) ⊂ C 2 (D), F2 (t, ·) ⊂ C 1 (D), for all t ∈ R, F1 , F2 , R, are locally
Lipschitz with respect to x, and R is twice differentiable with respect to 3.
We define Fk0 : D → R for k = 1, 2 as
1 T
F10 (z) =
F1 (s, z) ds
T 0
1 T
and
F20 (z) =
[Dz F1 (s, z) · y1 (s, z) + F2 (s, z)] ds,
T 0
s
where
y1 (s, z) =
F1 (t, z) dt.
0
(ii) For V ⊂ D, an open and bounded set and for each 3 ∈ (−3f , 3f ) \ {0}, there
exists a3 ∈ V such that F10 (a3 ) + 3F20 (a3 ) = 0 and dB (F10 + 3F20 , V , a3 ) = 0.
Then, for |3| > 0 sufficiently small, there exists a T -periodic solution f(·, 3) of the
system such that f(0, a3 ) → a3 when 3 → 0.
The expression dB (F10 + 3F20 , V , a3 ) = 0 means that the Brouwer degree of
the function F10 + 3F20 : V → Rn at the fixed point a3 is not zero. A sufficient
condition in order that this inequality holds is that the Jacobian of the function
F10 + 3F20 at a3 is not zero.
If F10 is not identically zero, then the zeros of F10 + 3F20 are mainly the zeros of
F10 for 3 sufficiently small. In this case, the previous result provides the averaging
theory of first order.
If F10 is identically zero and F20 is not identically zero, then the zeros of F10 +
3F20 are mainly the zeros of F20 for 3 sufficiently small. In this case, the previous
result provides the averaging theory of second order.
3. Proof of theorem 1.1
We shall need the first-order averaging theory to prove theorem 1.1. We write
system (1.4) in polar coordinates (r, q) where
x = r cos q
Proc. R. Soc. A (2012)
and y = r sin q,
r > 0.
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2351
Limit cycles of differential systems
In this way, system (1.4) will become written in the standard form for applying
the averaging theory. If we write
f11 (x) =
l
f21 (x) =
ai,1 x i ,
n
ai,2 x i ,
g11 (x) =
i=0
i=0
k
m
bi,1 x i and g21 (x) =
bi,2 x i ,
i=0
i=0
(3.1)
then, system (1.4) becomes
n
m
bi,2 r i cosi q sin q
ai,2 r i+1 cosi q sin2 q +
ṙ = −3
i=0
i=0
+
l
ai,1 r
i+1
i+1
cos
q sin q +
i=0
and
k
i
i+1
bi,1 r cos
q
i=0
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
n
m
⎪
⎪
3 ⎪
ai,2 r i+1 cosi+1 q sin q +
bi,2 r i cosi+1 q ⎪
q̇ = −1 −
⎪
⎪
⎪
r i=0
⎪
i=0
⎪
⎪
⎪
⎪
l
k
⎪
⎪
⎪
2
i+1
i
i
i
⎪
⎪
ai,1 r cos q sin q −
bi,1 r cos q sin q .
−
⎭
i=0
(3.2)
i=0
Now taking q as the new independent variable, system (3.2) becomes
n
m
dr
2
i+1
i
ai,2 r cos q sin q +
bi,2 r i cosi q sin q
=3
dq
i=0
i=0
l
k
ai,1 r i+1 cosi+1 q sin q +
bi,1 r i cosi+1 q + O(32 )
+
i=0
and
1
F10 (r) =
2p
+
i=0
2p n
0
l
ai,2 r i+1 cosi q sin2 q +
i=0
ai,1 r
i+1
m
bi,2 r i cosi q sin q
i=0
cos
i+1
q sin q +
i=0
k
i
i+1
bi,1 r cos
q
dq.
i=0
Now using the expressions for the integrals in appendix A (note that ak+1 =
(2k + 1)ak ), we get
2p
[n/2]
1 2i+1
a2i,2 r
cos2i q sin2 q dq
F10 (r) =
2p i=0
0
+
Proc. R. Soc. A (2012)
1
2p
2p
[(k−1)/2]
i=0
b2i+1,1 r 2i+1
cos2i+2 q dq
0
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2352
J. Llibre and C. Valls
[n/2]
=r
i=0
[n/2]
=r
i=0
a2i,2 ai
r 2i + r
2i+1 (i + 1)!
a2i,2 ai
r 2i + r
i+1
2 (i + 1)!
[(k−1)/2]
i=0
[(k−1)/2]
i=0
b2i+1,1 ai+1 2i
r
2i+1 (i + 1)!
b2i+1,1 (2i + 1)ai 2i
r .
2i+1 (i + 1)!
(3.3)
Then, the polynomial F10 (r) has at most l1 (see (1.5)) positive roots, and we can
choose the coefficients a2i,2 and b2i+1,1 in such a way that F10 (r) has exactly l1
simple positive roots. Hence, theorem 1.1 is proved.
4. Proof of theorem 1.2
We write f11 , f21 , g11 and g21 as in (3.1), and
f12 (x) =
l
i
ci,1 x ,
f22 (x) =
i=0
n
i
ci,2 x ,
g12 (x) =
i=0
k
di,1 x
i
and g22 (x) =
i=0
m
di,2 x i .
i=0
Then, system (1.4) in polar coordinates (r, q) with r > 0 becomes
n
m
ai,2 r i+1 cosi q sin2 q +
bi,2 r i cosi q sin q
ṙ = −3
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
i=0
i=0
⎪
⎪
⎪
⎪
⎪
l
k
⎪
⎪
i+1
i+1
i
i+1
⎪
⎪
ai,1 r cos
q sin q +
bi,1 r cos
q
+
⎪
⎪
⎪
⎪
i=0
i=0
⎪
⎪
n
⎪
⎪
m
⎪
⎪
2
2
i+1
i
i
i
⎪
ci,2 r cos q sin q +
di,2 r cos q sin q ⎪
−3
⎪
⎪
⎪
⎪
i=0
i=0
⎪
⎪
⎪
⎪
l
k
⎪
⎪
⎪
i+1
i+1
i
i+1
⎪
ci,1 r cos q sin q +
di,1 r cos q
+
⎪
⎪
⎬
i=0
and
i=0
n
m
⎪
⎪
3 ⎪
ai,2 r i+1 cosi+1 q sin q +
bi,2 r i cosi+1 q ⎪
q̇ = −1 −
⎪
⎪
⎪
r i=0
⎪
i=0
⎪
⎪
⎪
⎪
l
k
⎪
⎪
⎪
2
i+1
i
i
i
⎪
⎪
ai,1 r cos q sin q −
bi,1 r cos q sin q
−
⎪
⎪
⎪
⎪
i=0
i=0
⎪
⎪
n
⎪
m
⎪
2 ⎪
e
⎪
i+1
i+1
i
i+1
⎪
ci,2 r cos q sin q +
di,2 r cos q ⎪
−
⎪
⎪
r i=0
⎪
⎪
i=0
⎪
⎪
⎪
⎪
l
k
⎪
⎪
⎪
2
i+1
i
i
i
⎪
ci,1 r cos q sin q −
di,1 r cos q sin q .
−
⎪
⎭
i=0
Proc. R. Soc. A (2012)
i=0
(4.1)
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2353
Limit cycles of differential systems
Taking q as the new independent variable, system (4.1) becomes
dr
= 3F1 (q, r) + 32 F2 (q, r) + O(33 ),
dq
where
⎫
⎪
⎪
bi,2 r cos q sin q ⎪
ai,2 r cos q sin q +
F1 (q, r) =
⎪
⎪
⎪
⎪
i=0
i=0
⎪
⎬
l
k
⎪
+
ai,1 r i+1 cosi+1 q sin q +
bi,1 r i cosi+1 q⎪
⎪
⎪
⎪
⎪
i=0
i=0
⎪
⎪
⎭
F2 (q, r) = I (r, q) + rII (r, q),
n
and
m
2
i
i+1
i
i
(4.2)
where
I (r, q) =
n
ci,2 r
i+1
m
cos q sin q +
2
i
i=0
+
di,2 r i cosi q sin q
i=0
l
ci,1 r
i+1
cos
i+1
q sin q +
k
i=0
and
II (r, q) = −
n
i=0
ai,2 r cos q sin q +
i
i
2
i=0
+
l
×
m
ai,1 r i cosi+1 q sin q +
k
i
ai,2 r cos
i+1
i=0
bi,1 r i−1 cosi+1 q
q sin q +
m
i=0
−
i=0
n
l
bi,2 r i−1 cosi q sin q
i=0
i=0
di,1 r i cosi+1 q
ai,1 r i cosi q sin2 q −
bi,2 r i−1 cosi+1 q
i=0
k
bi,1 r i−1 cosi q sin q .
i=0
In order to compute F20 (r), we need that F10 be identically zero. Then from (3.3),
⎫
a2i,2
b2i+1,1 = −
, i = 0, 1, . . . , m, ⎪
⎬
2i + 1
(4.3)
⎪
⎭
and
b2i+1,1 = a2i,2 = 0, i = m + 1, . . . , l1 .
First we compute
d
F1 (q, r) =
(i + 1)ai,2 r i cosi q sin2 q +
ibi,2 r i−1 cosi q sin q
dr
i=0
i=0
n
m
l
k
i
i+1
(i + 1)ai,1 r cos q sin q +
ibi,1 r i−1 cosi+1 q
+
i=0
Proc. R. Soc. A (2012)
i=0
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J. Llibre and C. Valls
=
m
a2i,2 r 2i cos2i q((2i + 1) sin2 q − cos2 q)
i=0
[(n−1)/2]
+
(2i + 2)a2i+1,2 r 2i+1 cos2i+1 q sin2 q
i=0
m
+
ibi,2 r i−1 cosi q sin q +
i=0
l
(i + 1)ai,1 r i cosi+1 q sin q
i=0
[k/2]
+
2ib2i,1 r 2i−1 cos2i+1 q
i=0
and y1 (q, r) =
q
0
F1 (j, r) dj. To do it, we rewrite
[(n−1)/2]
F1 (q, r) =
[(n−1)/2]
a2i+1,2 r
2i+2
2i+1
cos
q−
i=0
i=0
[n/2]
+
[n/2]
a2i,2 r 2i+1 cos2i q −
i=0
+
m
a2i,2 r 2i+1 cos2i+2 q
i=0
bi,2 r i cosi q sin q +
i=0
l
ai,1 r i+1 cosi+1 q sin q
i=0
[(k−1)/2]
+
[k/2]
b2i+1,1 r
2i+1
2i+2
cos
q+
i=0
a2i+1,2 r 2i+2 cos2i+3 q
2i + 2
m
a2i,2 r 2i+1 cos2i q −
i=0
m
i=0
m
+
b2i,1 r 2i cos2i+1 q
[(n−1)/2]
a2i+1,2 r 2i+2 cos2i+1 q −
i=0
+
i=0
[(n−1)/2]
=
a2i+1,2 r 2i+2 cos2i+3 q
i=0
bi,2 r i cosi q sin q +
l
2i + 1
a2i,2 r 2i+1 cos2i+2 q
ai,1 r i+1 cosi+1 q sin q
i=0
i=0
[k/2]
+
b2i,1 r 2i cos2i+1 q.
i=0
Then, taking into account that
1 2i
2i + 2 1
2i + 2
q
−
q = 0,
22i i
2i + 1 22i+2 i + 1
Proc. R. Soc. A (2012)
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2355
Limit cycles of differential systems
and using the integrals of appendix A, we obtain
[(n−1)/2]
y1 (q, r) =
a2i+1,2 r
i+1
2i+2
i=0
+
m
a2i,2 r
2i+1
i+1
i=0
+
b̃i,l sin(2lq)
l=1
m
l
bi,2 i
ai,1 i+1
r (1 − cosi+1 q) +
r (1 − cosi+2 q)
i
+
1
i
+
2
i=0
i=0
[k/2]
+
g̃i,l sin((2l + 1)q)
l=0
b2i,1 r 2i
i=0
i
gi,l sin((2l + 1)q),
l=0
where
⎧
bi,l − 2(i + 1)bi+1,l
⎪
⎪
, 0 ≤ l ≤ i,
⎨
(2i + 1)
b̃i,l =
⎪
−2(i + 1)bi+1,i+1
⎪
⎩
,
l = i + 1.
(2i + 1)
g̃i,l =
gi,l − gi+1,l , 0 ≤ l ≤ i,
l = i + 1,
−gi+1,i+1 ,
Again, using the integrals of appendix A, we conclude that
III =
2p
0
d
F1 (q, r)y1 (q, r) dq = rP1 (r 2 ),
dr
where P1 (r 2 ) is equal to
m [(m−1)/2]
i=0
Ai,j (r 2 )i+j +
j=0
m [l/2]
[(n−1)/2] [m/2]
Bi,j (r 2 )i+j +
i=0 j=0
[(n−1)/2] [(l−1)/2]
+
i=0
j=0
i=0
j=0
j=0
i=0
j=0
Ci,j (r 2 )i+j
[k/2] [(l−1)/2]
Ei,j (r 2 )i+j−1 +
i=0 j=0
Gi,j (r 2 )i+j +
Proc. R. Soc. A (2012)
i=0
i=0
j=0
2 i+j+1
Ji,j (r )
+
[l/2] m
i=0 j=0
i=0
[(m−1)/2] m
[(l−1)/2] [(n−1)/2]
+
[k/2] [m/2]
Di,j (r 2 )i+j+1 +
[m/2] [(n−1)/2]
+
Fi,j (r 2 )i+j
j=0
[m/2] [k/2]
Hi,j (r 2 )i+j +
Ii,j (r 2 )i+j−1
i=0 j=0
[(l−1)/2] [k/2]
2 i+j
Ki,j (r )
+
i=0
j=0
Li,j (r 2 )i+j ,
Downloaded from http://rspa.royalsocietypublishing.org/ on July 12, 2017
2356
J. Llibre and C. Valls
with
Ai,j =
pa2i,2 b2j+1,2 ai+j+1
,
2i+j+1 (i + j + 2)!
Ci,j = −
Ei,j =
p(2i + 2)a2i+1,2 b2j,2 ai+j+1
,
(2j + 1)2i+j+1 (i + j + 2)!
j+1
p(2i + 2)a2i+1,2 a2j+1,1 ai+j+2
,
(2j + 3)2i+j+2 (i + j + 3)!
Di,j = −
p2ib2i,1 b2j,2 ai+j+1
,
(2j + 1)2i+j (i + j + 1)!
Gi,j = p
pa2i,2 a2j,1 ai+j+1
,
2i+j+1 (i + j + 2)!
Bi,j =
p2ib2i,1 a2j+1,1 ai+j+2
,
(2j + 3)2i+j+1 (i + j + 2)!
Fi,j =
j+1
(2i + 1)b2i+1,2 a2j,2 b̃j,s Ki,s ,
Hi,j = p
2ib2i,2 a2j+1,2 g̃j,s Ci,s ,
s=0
Ii,j = p
j
s=1
j+1
(2i + 2)a2i+1,1 a2j+1,2 g̃j,s Ci,s ,
Ji,j = p
2ib2i,2 b2j,1 gj,s Ci,s ,
s=0
s=0
j
Li,j = p
(2i + 2)a2i+1,1 b2j,1 gj,s Ci,s .
j+1
Ki,j = p
(2i + 1)a2i,1 a2j,2 b̃j,s Ki,s
and
s=1
s=0
Then, P1 (r 2 ) is a polynomial in the variable r 2 of degree l2 , see (1.7). In the
notation of (4.2), we have that
2p
2p
[n/2]
I (r, q) dq =
0
c2i,2 r
cos2i q sin2 q dq
2i+1
0
i=0
2p
[(k−1)/2]
+
d2i+1,1 r
2i+1
i=0
[n/2]
= pr
i=0
cos2i+2 q dq
0
c2i,2 ai 2i
r + pr
i
2 (i + 1)!
[(k−1)/2]
d2i+1,1 ai (2i + 1) 2i
r = rP2 (r 2 ),
2i (i + 1)!
i=0
where P2 is a polynomial in the variable r 2 of degree l1 .
Furthermore,
2p
II (r, q) dq = −2
0
m
n 2p
ai,2 bj,2 r
0
i=0 j=0
+
l
n 2p
+
k
m i=0 j=0
cosi+j q sin4 q dq
ai,2 aj,1 r i+j
0
i=0 j=0
Proc. R. Soc. A (2012)
cosi+j+1 q sin2 q dq
i+j−1
2p
bi,2 bj,1 r
cosi+j q sin2 q dq
i+j−2
0
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Limit cycles of differential systems
−
n
l 2p
ai,1 aj,2 r
cosi+j+2 q sin2 q dq
i+j
0
i=0 j=0
+2
k
l 2p
ai,1 bj,1 r
m
k cosi+j+1 q sin2 q dq
i+j−1
0
i=0 j=0
−
2357
2p
bi,1 bj,2 r
i+j−2
cosi+j+2 q dq.
(4.4)
0
i=0 j=0
Using relation (4.3), we get
2p
II (r, q) dq
0
= −2
m [(m−1)/2]
i=0
2p
a2i,2 b2j+1,2 r
cos2i+2j+2 q sin2 q dq
2i+2j
0
j=0
2p
[(n−1)/2] [m/2]
−2
i=0
−
m [l/2]
a2i+1,2 b2j,2 r
cos2i+2j+2 q sin2 q dq
2i+2j
0
j=0
2p
a2i,2 a2j,1 r
cos2i+2j q sin2 q cos(2q) dq
2i+2j
0
i=0 j=0
2p
[(n−1)/2] [(l−1)/2]
−
i=0
j=0
0
2p
[m/2] [k/2]
−
b2i,2 b2j,1 r
2i+2j−2
cos2i+2j q cos(2q) dq
0
i=0 j=0
2p
[(m−1)/2] [(k−1)/2]
−
i=0
−2
b2i+1,2 b2j+1,1 r
a2j,2 2i+2j
r
a2i,1
2j + 1
j=0
i=0
j=0
cos2i+2j+2 q cos(2q) dq
0
2p
cos2i+2j+2 q sin2 q dq
0
2p
[(l−1)/2] [k/2]
+2
2i+2j
j=0
[l/2] m
i=0
cos2i+2j+2 q sin2 q cos(2q) dq
a2i+1,2 a2j+1,1 r 2i+2j+2
a2i+1,1 b2j,1 r
cos2i+2j+2 q sin2 q dq,
2i+2j
0
where the third and fourth (respectively, fifth and sixth) lines come from taking
together the second and fourth (respectively, third and sixth) lines of (4.4). We
Proc. R. Soc. A (2012)
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2358
write
J. Llibre and C. Valls
2p
0
II (r, q) dq = P3 (r 2 ), where P3 is equal to
P3 (r ) =
2
m [(m−1)/2]
i=0
+
[(n−1)/2] [m/2]
2 i+j
Āi,j (r )
+
j=0
m [l/2]
C̄ i,j (r )
+
j=0
B̄ i,j (r 2 )i+j
i=0
j=0
D̄ i,j (r 2 )i+j+1
[(l−1)/2] [k/2]
2 i+j−1
Ē i,j (r )
+
i=0
i=0 j=0
F̄ i,j (r 2 )i+j ,
j=0
pa2i,2 b2j+1,2 ai+j+1
pa2i+1,2 b2j,2 ai+j+1
i+j +1
−1 +
, B̄ i,j = − i+j
,
Āi,j = i+j
2i + 1
2 (i + j + 2)!
2 (i + j + 2)!
pa2i,2 a2j,1 ai+j
2(i + j) + 1
C̄ i,j = − i+j
,
i+j −1+
2 (i + j + 2)!
2i + 1
D̄ i,j = −
and
i=0
[m/2] [k/2]
where
[(n−1)/2] [(l−1)/2]
2 i+j
i=0 j=0
+
F̄ i,j =
p(i + j)a2i+1,2 a2j+1,1 ai+j+1
,
2i+j+1 (i + j + 3)!
p(i + j)b2i,2 b2j,1 ai+j
2i+j−1 (i + j + 1)!
Ē i,j = −
pa2i+1,1 b2j,1 ai+j+1
.
2i+j (i + j + 2)!
Then, P3 (r 2 ) is a polynomial in the variable r 2 of degree l2 , see (1.7). Then,
2pF20 = r(P1 (r 2 ) + P2 (r 2 ) + P3 (r 2 )).
Then, to find the real positive roots of F20 , we must find the zeros of a polynomial
in r 2 of degree l3 . This yields that F20 has at most l3 real positive roots. Moreover,
we can choose the coefficients ai,1 , ai,2 , bi,1 , bi,2 , ci,1 , ci,2 , di,1 , di,2 in such a way that
F20 has exactly l3 real positive roots. Hence, the theorem is proved.
J.L. has been supported by the grants MICINN/FEDER MTM 2009-03437, CIRIT 2009SGR 410
and ICREA Academia. C.V. is supported by the grant AGAUR PIV-DGR-210, by FCT through
PTDC/MAT/117106/210 and by CAMGDS, Lisbon.
Appendix A. Formulae
In this appendix, we recall some formulae that will be used during the paper, for
more details, see Abramowitz & Stegun (1964). For i ≥ 0, we have
2p
2p
2p
2
2i+1
i
cos
q sin q dq = 0,
cos q sin q dq = 0,
cos2i+1 q dq = 0,
0
2p
cos2i q dq =
0
pai
,
2i−1 i!
ai = 1 · 3 · 5 · · · (2k − 1),
Proc. R. Soc. A (2012)
2p
0
0
cos2i q sin2 q dq =
0
pai
,
+ 1)!
2i (i
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2359
Limit cycles of differential systems
2p
cos2i q cos(2q) dq =
0
2p
piai
,
+ 1)!
2i−1 (i
p(i − 1)ai
,
2i (i + 2)!
0
q
i 1 2i
1 2i
1
2i
sin(2lq)
cos f df = 2i
q + 2i
i
+
l
2
2
l
i
0
cos2i q sin2 q cos(2q) dq =
l=1
i
1 2i
bi,l sin(2lq),
= 2i
q+
i
2
q
2i+1
cos
0
1
f df = 2i
2
=
i
i l=0
l=1
1
2i + 1
sin((2l + 1)q)
i − l 2l + 1
gi,l sin((2l + 1)q),
l=0
q
1
(1 − cosi+1 q),
i
+
1
0
q
q
2i+1
cos
q sin q sin((2l + 1)q) dq = cos2i q sin q sin(2lq) dq = 0,
cosi f sin f df =
0
2p
cos q sin((2l + 1)q) dq =
2p
i
0
2p
0
2p
0
2p
and
l ≥ 0,
0
cosi q sin(2lq) dq = 0,
0
cosi q sin2 q sin((2l + 1)q) dq =
2p
l ≥ 0,
cosi q sin2 q sin(2lq) dq = 0,
l ≥ 0,
0
cos2i q sin q sin((2l + 1)q) dq = pCi,l ,
cos2i+1 q sin q sin(2lq) dq = pKi,l ,
l ≥ 0,
l ≥ 1,
0
where Ci,j and Ki,l are non-zero constants.
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