Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 271, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
LIMIT CYCLES BIFURCATING FROM THE PERIODIC
ANNULUS OF CUBIC HOMOGENEOUS POLYNOMIAL
CENTERS
JAUME LLIBRE, BRUNO D. LOPES, JAIME R. DE MORAES
Abstract. We obtain an explicit polynomial whose simple positive real roots
provide the limit cycles which bifurcate from the periodic orbits of any cubic
homogeneous polynomial center when it is perturbed inside the class of all
polynomial differential systems of degree n.
1. Introduction and statement of main results
One of the main goals in the qualitative theory of real planar differential systems
is the determination of their limit cycles. It is well known that perturbing the
periodic orbits of a center often produces limit cycles, see for instance [1, 2, 12].
One of the first in studying these perturbations was Pontrjagin [10]. These last
years this problem has been studied by many authors see the second part of the
book [5] and the hundreds of references quoted there.
Hilbert in 1900 was interested in the maximum number of the limit cycles that
a polynomial differential system of a given degree can have. This problem is the
well-known 16-th Hilbert problem, which together with the Riemann conjecture are
the two problems of the famous list of 23 problems of Hilbert which remain open.
See for more details [7, 13].
There exist several methods to study the number of limit cycles that bifurcate from the periodic annulus of a center, such as the Poincaré return map, the
Poincaré-Melnikov integrals, the Abelian integrals, the inverse integrating factor,
and the averaging theory. In the plane all of them are essentially equivalent.
There are few works trying to study this problem for homogeneous cubic polynomial differential systems. Our main objetive will be to solve this problem for the
cubic homogeneous polynomial differential systems.
In [6] the authors classified all the cubic homogeneous polynomial differential systems. In [8] the authors proved that any real planar cubic homogeneous polynomial
differential system having a center can be written as
ẋ = ax3 + (b − 3αµ)x2 y − axy 2 − αy 3 = P (x, y),
ẏ = αx3 + ax2 y + (b + 3αµ)xy 2 − ay 3 = Q(x, y),
(1.1)
2010 Mathematics Subject Classification. 37C10, 34H99.
Key words and phrases. Polinomial vector field; limit cycle; averaging method; periodic orbit;
isochronous center; homogeneous cubic centers.
c
2015
Texas State University.
Submitted October 8, 2014. Published October 21, 2015.
1
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J. LLIBRE, B.D. LOPES, J.R. DE MORAES
EJDE-2015/271
with α ∈ {−1, 1}, a, b, µ ∈ R and µ > −1/3, after doing an affine change of variables
and a rescaling of the time.
It is known that the maximum number of limit cycles which bifurcate from the
periodic orbits of a cubic homogeneous center (1.1) using perturbations of first order
inside the class of all polynomial differential systems of degree n is [(n − 1)/2], see
for details statement (c) of Theorem A of [9]. Here [x] denotes the integer part
function of x.
The objetive of this work is to provide an explicit polynomial whose real positive
simple zeros gives the exact number of limit cycles which bifurcate, at first order
in the perturbation parameter, from the periodic orbits of any cubic homogeneous
center (1.1). More precisely consider the system.
ẋ = ax3 + (b − 3αµ)x2 y − axy 2 − αy 3 + εp(x, y),
ẏ = αx3 + ax2 y + (b + 3αµ)xy 2 − ay 3 + εq(x, y),
(1.2)
where
p(x, y) =
n
X
pi (x, y),
q(x, y) =
i=0
n
X
qi (x, y),
(1.3)
i=0
pi , qi are homogeneous polynomials of degree i, and ε is a small parameter.
Define the following functions
f1 (θ) = −a sin4 θ + a cos4 θ + (b − 3αµ + α) sin θ cos3 θ
+ (b + 3αµ − α) sin3 θ cos θ,
g1 (θ) = α 6µ sin2 θ cos2 θ + sin4 θ + cos4 θ ,
Z θ f (s) 1
ds ,
k(θ) = exp
g
(s)
1
0
Bi (θ) = Q(cos θ, sin θ)pi (cos θ, sin θ) − P (cos θ, sin θ)qi (cos θ, sin θ).
In sequel we state our main result where the function M (θ) is defined in (3.2),
we do not provide it here due to its length.
Theorem 1.1. For |ε| > 0 sufficiently small and for every positive simple zero r0∗
of the polynomial
n−1
[ 2 ]
Z 2π
1 X
2k
r0 F(r0 ) =
r0
A2k+1 (θ)dθ,
2π
0
k=0
where
Ai (θ) =
Bi (θ)k(θ)i−2
,
g1 (θ)2 M (θ)
for i = 1, 2, . . . , [ n−1
2 ], the perturbed systems (1.2) has a limit cycle bifurcating from
the periodic orbit r(θ, r0∗ ) = k(θ)r0∗ of the period annulus of the center (1.1) using
the averaging theory of first order. In particular the perturbed systems (1.2) has at
most [ n−1
2 ] limit cycles.
Theorem 1.1 is proved in Section 3. In Section 4 we provide an example that
illustrates Theorem 1.1 with n = 5. We obtain two limit cycles.
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2. Preliminaries
In this section we give some known results that we shall need for proving Theorem
1.1. Consider a system in the form
ẋ = F0 (t, x) + εF1 (t, x) + ε2 F2 (t, x, ε),
(2.1)
n
where ε 6= 0 is sufficiently small and the functions F0 , F1 : R × Ω → R and
F2 : R × Ω × (−ε0 , ε0 ) → Rn are C 2 functions, T −periodic in the first variable and
Ω is an open subset of Rn . We assume that the unperturbed system
ẋ = F0 (t, x)
(2.2)
has a submanifold of periodic solutions of dimension n.
Consider x(t, z, ε) the solution of system (2.2) such that x(0, z, ε) = z. The
linearization of the unperturbed system along a periodic solution x(t, z, 0) is given
by
ẏ = Dx F0 (t, x(t, z, 0))y.
(2.3)
In sequel we denote by Mz (t) the fundamental matrix of the linearized system
(2.3) such that Mz (0) is the identity.
We suppose that there exists an open set V with Cl(V ) ⊂ Ω such that for
each z ∈ Cl(V ), x(t, z, 0) is T −periodic, where x(t, z, 0) denotes the solution of the
unperturbed system (2.2). Here Cl(V ) denotes the closure of V . We have that the
set Cl(V ) is isochronous for system (2.2), i.e. it is formed only by periodic orbits
with period T .
The next result is the averaging theorem for studying the bifurcation of T periodic solutions of system (2.1) from the periodic solutions x(t, z, 0) contained in
Cl(V ) of system (2.2) when |ε| > 0 is sufficiently small. See [3] for a proof. For
more details on the averaging theory see [4] and the book [11].
Theorem 2.1 (Perturbations of an isochronous set). We assume that there exists
an open and bounded set V with Cl(V ) ⊂ Ω such that for each z ∈ Cl(V ), the
solution x(r, z, 0) is T −periodic. Consider the function F : Cl(V ) → Rn
Z
1 T −1
Mz (t)F1 (t, x(t, z, 0))dt.
(2.4)
F(z) =
T 0
Then the following statements hold.
(i) If there exists a ∈ V with F(a) = 0 and det((∂F/∂z)(a)) 6= 0 then there
exists a T −periodic solution x(t, ε) of system (2.1) such that x(0, ε) → a
when ε → 0.
(ii) The kind of the stability of the periodic solution x(t, ε) is given by the eigenvalues of the Jacobian matrix ((∂F/∂z)(a)).
3. Proof of Theorem 1.1
The next result follows easily.
Lemma 3.1. Let Pk (x, y) and P3 (x, y) be homogeneous polynomials of degree k
and 3 respectively, where (x, y) ∈ R2 . Thus in polar coordinates x = r cos θ and
y = r sin θ we have
Pk (r cos θ, r sin θ)P3 (r cos θ, r sin θ)
= (−1)k+1 Pk (r cos(θ + π), r sin(θ + π)P3 (r cos(θ + π), r sin(θ + π)).
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J. LLIBRE, B.D. LOPES, J.R. DE MORAES
EJDE-2015/271
Now we pass system (1.2) to polar coordinates taking x = r cos θ, y = r sin θ and
we obtain
ṙ = r3 f1 (θ) + ε(cos θ p(r cos θ, r sin θ) + sin θ q(r cos θ, r sin θ)),
1
θ̇ = r2 g1 (θ) + ε (cos θ q(r cos θ, r sin θ) − sin θ p(r cos θ, r sin θ)).
r
Note that g1 (θ) 6= 0 for all θ ∈ [0, 2π]. Thus we take the quotient ṙ/θ̇ and we get
the differential equation
dr
= F0 (r, θ) + εF1 (r, θ) + O(ε2 ),
dθ
(3.1)
in the standard form for applying the averaging theory of first order, where
F0 (r, θ) =
F1 (r, θ) =
f1 (θ)
r,
g1 (θ)
1
(Q(r cos θ, r sin θ)p(r cos θ, r sin θ)
r5 g1 (θ)2
− P (r cos θ, r sin θ)q(r cos θ, r sin θ)).
Note that the differential equation (3.1) satisfies the assumptions of Theorem 2.1.
Consider r(θ, r0 ) the periodic solution of the differential equation ṙ = rf1 (θ)/g1 (θ)
such that r(0, r0 ) = r0 . Solving this differential equation we obtain
r(θ, r0 ) = k(θ)r0 = r0 ek1 (θ) k2 (θ),
where
θ
−1 √tan θ
(3µ + R + 1) tan−1 √tan
aα (−3µ + R − 1) tan
3µ−R
3µ+R
√
√
+
,
k1 (θ) = −
2R
3µ − R
3µ + R
√
1 ( b −1)
R−αb
αb+R
k2 (θ) = sec2 θ(3µ − R) 4R (3µ + R) 4R tan2 θ + 3µ − αR 4 R
− b+R
× tan2 θ + 3µ + αR 4R ,
p
R = 9µ2 − 1.
Solving the variational equation (2.3) for our differential equation (3.1) we get
that the fundamental matrix of (2.3) is
tan θ p
M (θ) = − 2aα(−3µ + R − 1) 3µ + R tan−1 √
3µ − R
tan θ p
p
− 2aα 3µ − R(3µ + R + 1) tan−1 √
+ α 3µ − R
3µ + R
p
p
2
× 3µ + R(b − αR) log tan θ + 3µ − R − α 3µ − R
p
p
p
(3.2)
× 3µ + R(b + αR) log tan2 θ + 3µ + R + 3µ − R 3µ + R
p
p
× (R − αb) log(3µ − R) + 3µ − R 3µ + R(αb + R) log(3µ + R)
p
p
+ 2R 3µ − R 3µ + R log sec2 θ
p
p
p
p
+ 4R 3µ − R 3µ + R
4R 3µ − R 3µ + R .
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CUBIC HOMOGENEOUS POLYNOMIAL CENTERS
5
Note that M (θ) does not depend on r0 . Using the polynomials p and q given in (1.3)
and system (1.1) we have that the integrant of the integral (2.4) for our differential
equation is
M −1 (θ)F1 (θ, r(θ, r0 )) =
F1 (θ, r(θ, r0 ))
h(r, θ)
= 5
,
M (θ)
r g1 (θ)2 M (θ)
(3.3)
where
h(r, θ) = Q(r cos θ, r sin θ)p(r cos θ, r sin θ) − P (r cos θ, r sin θ)q(r cos θ, r sin θ).
Since p and q are sum of homogeneous polynomials (see eq. (1.3)), we can rewrite
the equality (3.3) as follows
M −1 (θ)F1 (θ, r(θ, r0 )) =
n
X
i=0
=
n
X
Bi (θ)
r(θ, r0 )i−2
g1 (θ)2 M (θ)
r0i−2
i=0
=
n
X
Bi (θ)(ek1 (θ) k2 (θ))i−2
g1 (θ)2 M (θ)
r0i−2 Ai (θ).
i=0
Computing the integral (2.4) we have
Z 2π
Z
n
1
1 X i−2 2π
F(r0 ) =
r0
M −1 (θ)F1 (θ, r(θ, r0 )dθ =
Ai (θ)dθ.
2π 0
2π i=0
0
If i is even then Lemma 3.1 implies Bi (θ) = −Bi (θ + π). Thus since k1 (θ) =
k1 (θ + π), k2 (θ) = k2 (θ + π) and M (θ) = M (θ + π) we easily obtain
Z 3π/2
Z 3π/2
Bi (θ)(ek1 (θ) k2 (θ))i−2
dθ
Ai (θ)dθ =
g1 (θ)2 M (θ)
π
π
Z π/2
Bi (θ + π)(ek1 (θ+π) k2 (θ + π))i−2
dθ
=
g1 (θ + π))2 M (θ + π))
0
Z π/2
Bi (θ)(ek1 (θ) k2 (θ))i−2
=
−
dθ
g1 (θ)2 M (θ)
0
Z π/2
=−
Ai (θ)dθ,
0
Z
2π
Z
2π
Ai (θ)dθ =
3π/2
3π/2
π
Z
=
π/2
π
Bi (θ)(ek1 (θ) k2 (θ))i−2
dθ
g1 (θ)2 M (θ)
Bi (θ + π)(ek1 (θ+π) k2 (θ + π))i−2
dθ
g1 (θ + π))2 M (θ + π))
Z
−
=
π/2
Z
Bi (θ)(ek1 (θ) k2 (θ))i−2
dθ
g1 (θ)2 M (θ)
π
=−
Ai (θ)dθ.
π/2
6
J. LLIBRE, B.D. LOPES, J.R. DE MORAES
EJDE-2015/271
Therefore for i even we have
2π
Z
Ai (θ)dθ = 0.
0
Analogously if i is odd then Lemma 3.1 implies Bi (θ) = Bi (θ + π). Thus we easily
can check that
Z π/2
Z 3π/2
Z π
Z 2π
Ai (θ)dθ =
Ai (θ)dθ and
Ai (θ)dθ =
Ai (θ)dθ.
0
π
π/2
So we have
Z
2π
π
Z
Ai (θ)dθ 6= 0,
Ai (θ)dθ = 2
0
3π/2
0
because Ai is a π−periodic even function. So the function F can be written in the
following way
[ n−1
Z 2π
2 ]
1 X
F(r0 ) =
r02k−1
A2k+1 (θ)dθ.
(3.4)
2π
0
k=0
Note that the coefficients A2k+1 (θ) in (3.4) are linearly independent because the
polynomials pi and qi are linearly independents. Thus the averaged function F has
at most [(n − 1)/2] simple zeros which correspond to the limit cycles of system (1.2)
and Theorem 1.1 is proved.
4. Example
In this section we present an example that illustrates Theorem 1.1. Consider the
cubic polynomial homogeneous center
ẋ = −y 3 ,
ẏ = x3 ,
and its perturbation
ẋ = −y 3 + ε(a1 x + a2 x3 + a3 x5 ),
ẏ = x3 .
(4.1)
Passing system (4.1) to the polar coordinates we get
ṙ = r3 (sin θ cos3 θ − sin3 θ cos θ) + εr cos2 θ(a1 + a2 r2 cos2 θ + a3 r4 cos4 θ),
θ̇ = r2 (sin4 θ + cos4 θ) − ε sin θ cos θ(a1 + a2 r2 cos2 θ + a3 r4 cos4 θ).
Taking the quotient ṙ/θ̇ we obtain the following system in the standard form of
Theorem 2.1 for applying the averaging theory
dr
= F0 (r, θ) + εF1 (r, θ) + O(ε2 ),
dθ
where
r sin θ cos3 θ − r sin3 θ cos θ
,
sin4 θ + cos4 θ
cos4 θ(a1 + a2 r2 cos2 θ + a3 r4 cos4 θ)
F1 (r, θ) =
.
r(sin4 θ + cos4 θ)2
F0 (r, θ) =
Thus for system (4.2) we have
√
k(θ) = p
4
2
cos(4θ) + 3
,
(4.2)
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CUBIC HOMOGENEOUS POLYNOMIAL CENTERS
7
1
(− log(cos(4θ) + 3) + 4 + log 4),
4
and the integrant of the integral (2.4) of system (4.2) is
M (θ) =
A(θ) + B(θ)r02 + C(θ)r04
,
r0
where
√
32 2 cos4 θ
,
(cos(4θ) + 3)7/4 (− log(cos(4θ) + 3) + 4 + log 4)
√
64 2 cos6 θ
B(θ) = a2
,
(cos(4θ) + 3)9/4 (− log(cos(4θ) + 3) + 4 + log 4)
√
128 2 cos8 θ
.
C(θ) = a3
11/4
(− log(cos(4θ) + 3) + 4 + log 4)
(cos(4θ) + 3)
A(θ) = a1
Computing numerically the integral (2.4) for system (4.2) we obtain
F(r0 ) =
3.72731 . . . a1 + 3.34745 . . . a2 r02 + 3.10284 . . . a3 r04
.
r0
Taking
−1
2
−0.1
, a2 =
anda3 =
,
3.72731 . . .
3.34745 . . .
3.10284 . . .
it is easy to check that the function F has two positive simple zeros given by
a1 =
r0∗ = 0.716357 . . .
and r0∗∗ = 4.41439 . . .
which correspond to two limit cycles of the perturbed system (4.1) with ε 6= 0
sufficiently small.
Acknowledgements. The first author is partially supported by a MINECO/FEDER
grant number MTM2009-03437, by an AGAUR grant number 2014SGR-568, by an
ICREA Academia, two FP7+PEOPLE+2012+IRSES numbers 316338 and 318999,
and FEDER-UNAB10-4E-378. The second author is supported by CAPES/GDU
- 7500/13-0.
References
[1] A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maı̆er; Theory of bifurcations of
dynamic systems on a plane, Halsted Press [A division of John Wiley & Sons], New YorkToronto, Ont., 1973, Translated from the Russian.
[2] T. R. Blows, L. M. Perko; Bifurcation of limit cycles from centers and separatrix cycles of
planar analytic systems, SIAM Rev., 36 (1994), 341–376.
[3] A. Buică, J. P. Françoise, J. Llibre; Periodic solutions of nonlinear periodic differential
systems with a small parameter, Commun. Pure Appl. Anal., 6 (2007), 103–111.
[4] A. Buică, J. Llibre; Averaging methods for finding periodic orbits via Brouwer degree, Bull.
Sci. Math., 128 (2004), 7–22.
[5] C. Christopher, C. Li; Limit cycles in differential equations, Birkhauser, Boston, 1999.
[6] A. Cima, J. Llibre; Algebraic and topological classification of the homogeneous cubic vector
fields in the plane, J. Math. Anal. Appl., 147 (1990), 420–448.
[7] C. Hilbert; Mathematische probleme, Lecture, Second Internat. Congr. Math. (Paris, 1900),
Nachr. Ges. Wiss. G”ttingen Math. Phys. KL. (1900), 253–297; English transl., Bull. Amer.
Math. Soc. 8 (1902), 437–479; Bull. (New Series) Amer. Math. Soc. 37 (2000), 407–436.
[8] C. Li, J. Llibre; Cubic homogeneous polynomial centers, Publ. Mat., 58 (2014), 297–308.
[9] W. Li, J. Llibre, J. Yang, Z. Zhang; Limit cycles bifurcating from the period annulus of
quasi–homogeneous centers, J. Dynam. Differential Equations, 21 (2009), 133–152.
8
J. LLIBRE, B.D. LOPES, J.R. DE MORAES
EJDE-2015/271
[10] L. S. Pontrjagin; Über autoschwingungssysteme, die den hamiltonshen nahe liegen,
Physikalische Zeitschrift der Sowjetunion, 6 (1934), 25–28.
[11] J. A. Sanders, F. Verhulst; Averaging methods in nonlinear dynamical systems, Applied
Mathematical Sciences, vol. 59, Springer, New York, 2007.
[12] Y. Shao, K. Wu; Bifurcation of limit cycles for cubic reversible systems, Electron. J. Differential Equations, 2014 (2014), No. 96, 10 pp.
[13] S. Smale; Mathematical problems for next century, Math. Intelligencer, 20 (1998), 7–15.
Jaume Llibre
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra,
Barcelona, Catalonia, Spain
E-mail address: jllibre@mat.uab.cat
Bruno D. Lopes
Departamento de Matemática - IBILCE-UNESP, Rua C. Colombo, 2265, CEP 15054-000
S. J. Rio Preto, São Paulo, Brazil
E-mail address: brunodomicianolopes@gmail.com
Jaime R. de Moraes
Curso de Matemática - UEMS, Rodovia Dourados-Itaum Km 12, CEP 79804-970 Dourados, Mato Grosso do Sul, Brazil
E-mail address: jaime@uems.br