LIMIT CYCLES OF PERTURBED GLOBAL ISOCHRO-
NOUS CENTER
Zouhair Diab
Department of Mathematics and Computer Science, Larbi Tebessi University, 12002 Tebessa, Algeria.
E-mail:zouhair.diab@univ-tebessa.dz
ORCID:
Maria Teresa de Bustos
Department of Applied Mathematics. University of Salamanca, Casas del Parque, 2, 37008-Salamanca, Spain.
E-mail:tbustos@usal.es
ORCID:
Miguel Ángel López
SIDIS Research Group, Department of Mathematics, Institute of Applied Mathematics in Science and Engineering
(IMACI), Polytechnic School of Cuenca, University of Castilla-La Mancha, 16071 Cuenca, Spain.
E-mail:mangel.lopez@uclm.es
ORCID:
Raquel Martínez
SIDIS Research Group, Department of Mathematics, Institute of Applied Mathematics in Science and Engineering
(IMACI), Polytechnic School of Cuenca, University of Castilla-La Mancha, 16071 Cuenca, Spain.
E-mail:Raquel.Martinez@uclm.es
ORCID:
Reception: 29/08/2022 Acceptance: 13/09/2022 Publication: 29/12/2022
Suggested citation:
Zouhair D., Maria Teresa B. , Miguel Ángel L. and Raquel M.(2022). Limit cycles of perturbed global isochronous center. 3C
Tecnología. Glosas de innovación aplicada a la pyme,11 (2), 25-36. https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed. 42 Vol. 11 N.º 2 August - December 2022
25
ABSTRACT
We apply the averaging method of first order to study the maximum number of limit cycles of the
ordinary differential systems of the form
¨x+x=ε(f1(x, y)y+f2(x, y)) ,
¨y+y=ε(g1(x, y)x+g2(x, y)) ,
where
f1
(
x, y
)and
g1
(
x, y
)are real cubic polynomials;
f2
(
x, y
)and
g2
(
x, y
)are real quadratic polynomials.
Furthermore εis a small parameter.
KEYWORDS
Limit Cycles, Averaging Method, Ordinary Differential Systems
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
1 INTRODUCTION AND STATEMENT OF THE MAIN RESULT
At the Paris International Congress of Mathematics in 1900, Hilbert presented twenty-three problems
in mathematics. Some problems are still unsolved so far, they were a challenge for all mathematicians
of that era. The second part of the well-known Hilbert’s 16th problem is to find the maximum number
of limit cycles and their position for an ordinary differential planar system of degree nof the form
˙x=ψ(x, y),
˙y=η(x, y),(1)
where
n
is a positive integer, the dot above the variables represents the first derivative with respect to
the variable
t
,
ψ
(
x, y
)and
η
(
x, y
)are real polynomials, see for instance [13,14,17]. This problem has so
far remained unresolved, for
n≥
2. Let us denote by
H
(
n
)the maximum number of limit cycles of
differential system
(1)
which is usually called a Hilbert number. For example, Chen and Wang in [3],
Shi in [22] gave the best result up to now about the lower bounds of
H
(2), which is
H
(2)
≥
4. Li, Liu,
and Yang in [15] proved
H
(3)
≥
13, Li and Li in [16] proved
H
(3)
≥
11, also Han, Wu and Bi in [10]
and Han, Zhang and Zang in [12] proved
H
(3)
≥
11. For more results about the Hilbert number, see,
for example, the paper [9] and the references therein.
There are many papers that studied number of limit cycles using several methods, including the
Poincaré–Melnikov integrals, see for instance [11]; the Poincaré return map, see for example [1]; the
Abelian integrals, see for example [4]; the averaging method, see [5, 6]; the inverse integrating factor,
see for instance [8].
In [19], Llibre and Teixeira used the averaging method of first-order for study the existence of limit
cycles of the system of second-order differential equations
¨x+x=εf(x, y),
¨y+y=εg(x, y),
where f(x, y)and g(x, y)are real cubic polynomials and εis a small parameter.
In this paper, we apply the averaging method of first-order for study the existence of limit cycles of
system of second-order differential equations
¨x+x=ε(f1(x, y)y+f2(x, y)) ,
¨y+y=ε(g1(x, y)x+g2(x, y)) ,(2)
where
f1
(
x, y
)and
g1
(
x, y
)are real cubic polynomials,
f2
(
x, y
)and
g2
(
x, y
)are real quadratic polynomials
such that
fi
(0
,
0) =
gi
(0
,
0)=0, for
i
=1
,
2, and
ε
is a small parameter. These polynomials are
expressions of the form
f1(x, y)=a1x+a2y+a3x2+a4xy +a5y2+a6x3+a7x2y+a8xy2+a9y3,
f2(x, y)=A1x+A2y+A3x2+A4xy +A5y2,
g1(x, y)=b1x+b2y+b3x2+b4xy +b5y2+b6x3+b7x2y+b8xy2+b9y3,
g2(x, y)=B1x+B2y+B3x2+B4xy +B5y2.
The system of second order differential equations
(2)
can be expressed as the following system of
first-order differential equations in the usual way:
˙x=u,
˙u=−x+ε(f1(x, y)y+f2(x, y)) ,
˙y=v,
˙v=−y+ε(g1(x, y)x+g2(x, y)) .
(3)
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed. 42 Vol. 11 N.º 2 August - December 2022
26
ABSTRACT
We apply the averaging method of first order to study the maximum number of limit cycles of the
ordinary differential systems of the form
¨x+x=ε(f1(x, y)y+f2(x, y)) ,
¨y+y=ε(g1(x, y)x+g2(x, y)) ,
where
f1
(
x, y
)and
g1
(
x, y
)are real cubic polynomials;
f2
(
x, y
)and
g2
(
x, y
)are real quadratic polynomials.
Furthermore εis a small parameter.
KEYWORDS
Limit Cycles, Averaging Method, Ordinary Differential Systems
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
1 INTRODUCTION AND STATEMENT OF THE MAIN RESULT
At the Paris International Congress of Mathematics in 1900, Hilbert presented twenty-three problems
in mathematics. Some problems are still unsolved so far, they were a challenge for all mathematicians
of that era. The second part of the well-known Hilbert’s 16th problem is to find the maximum number
of limit cycles and their position for an ordinary differential planar system of degree nof the form
˙x=ψ(x, y),
˙y=η(x, y),(1)
where
n
is a positive integer, the dot above the variables represents the first derivative with respect to
the variable
t
,
ψ
(
x, y
)and
η
(
x, y
)are real polynomials, see for instance [13,14,17]. This problem has so
far remained unresolved, for
n≥
2. Let us denote by
H
(
n
)the maximum number of limit cycles of
differential system
(1)
which is usually called a Hilbert number. For example, Chen and Wang in [3],
Shi in [22] gave the best result up to now about the lower bounds of
H
(2), which is
H
(2)
≥
4. Li, Liu,
and Yang in [15] proved
H
(3)
≥
13, Li and Li in [16] proved
H
(3)
≥
11, also Han, Wu and Bi in [10]
and Han, Zhang and Zang in [12] proved
H
(3)
≥
11. For more results about the Hilbert number, see,
for example, the paper [9] and the references therein.
There are many papers that studied number of limit cycles using several methods, including the
Poincaré–Melnikov integrals, see for instance [11]; the Poincaré return map, see for example [1]; the
Abelian integrals, see for example [4]; the averaging method, see [5, 6]; the inverse integrating factor,
see for instance [8].
In [19], Llibre and Teixeira used the averaging method of first-order for study the existence of limit
cycles of the system of second-order differential equations
¨x+x=εf(x, y),
¨y+y=εg(x, y),
where f(x, y)and g(x, y)are real cubic polynomials and εis a small parameter.
In this paper, we apply the averaging method of first-order for study the existence of limit cycles of
system of second-order differential equations
¨x+x=ε(f1(x, y)y+f2(x, y)) ,
¨y+y=ε(g1(x, y)x+g2(x, y)) ,(2)
where
f1
(
x, y
)and
g1
(
x, y
)are real cubic polynomials,
f2
(
x, y
)and
g2
(
x, y
)are real quadratic polynomials
such that
fi
(0
,
0) =
gi
(0
,
0)=0, for
i
=1
,
2, and
ε
is a small parameter. These polynomials are
expressions of the form
f1(x, y)=a1x+a2y+a3x2+a4xy +a5y2+a6x3+a7x2y+a8xy2+a9y3,
f2(x, y)=A1x+A2y+A3x2+A4xy +A5y2,
g1(x, y)=b1x+b2y+b3x2+b4xy +b5y2+b6x3+b7x2y+b8xy2+b9y3,
g2(x, y)=B1x+B2y+B3x2+B4xy +B5y2.
The system of second order differential equations
(2)
can be expressed as the following system of
first-order differential equations in the usual way:
˙x=u,
˙u=−x+ε(f1(x, y)y+f2(x, y)) ,
˙y=v,
˙v=−y+ε(g1(x, y)x+g2(x, y)) .
(3)
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed. 42 Vol. 11 N.º 2 August - December 2022
27
Note that system (3) when ε=0, is
˙x=u,
˙u=−x,
˙y=v,
˙v=−y.
(4)
We notice that the system
(4)
have a global isochronous center at the the origin, i.e. all orbits different
from the origin are 2π-periodic, for more detail see [18].
The main result of our work is the following.
Theorem 1. Using the first-order averaging method, system of second-order differential equations
(3)
, where
f1
(
x, y
)and
g1
(
x, y
)are real cubic polynomials;
f2
(
x, y
)and
g2
(
x, y
)are real quadratic
polynomials, has at most four limit cycles bifurcating from the periodic orbits of the linear center
˙x=u, ˙u=−x, ˙y=v, ˙v=−y
. Here
ε
is a small parameter. Moreover if
a4
=0and
a3
=0, the system
(3) has at most two periodic solutions.
The first-order averaging method theory, that we summarize in the sequel, can be found in a more
extended way in [2]. Similar works where the perturbations via polynomials play an important role are
for instance [7] and [20].
2
THE FIRST-ORDER AVERAGING METHOD FOR COMPUTING PERIODIC
ORBITS
Theorem 2. We consider the following two problems
˙x(t)=εF (t, x(t)) + ε2R(t, x(t),ε),(5)
and
˙y(t)=εf(y(t)),(6)
where
t∈[0,+∞)
,
x
and
y
in some open
D
of
Rn
and
ε∈(−ε0,ε
0)
is a small parameter. Moreover,
we suppose that the vector functions
F
(
t, x
)and
R
(
t, x, ε
)are
T−
periodic in the first variable and we
consider the first-order averaging function
f(y)= 1
TT
0
F(s, y)ds.
Suppose that
F
,
R
,D
xF
, and D
2
xF
are continuous and bounded by a constant
M
in
[0,∞)×D
×(−ε0,ε
0)where Mis independent of ε. Then, the statements (I)and (II)satisfied:
(I)If there exists an equilibrium point α∈Dof (6) such that
det ∂(f (y))
∂y y=p=0,
then, for
ε>
0sufficiently small, there exists an isolated
T−
periodic solution
ϕε
(
t
)of system
(5) such that ϕε(t)→0when ε→0.
(II)
If
y
=
α
( the equilibrium point ) of
(6)
is hyperbolic. Then, for
ε>
0sufficiently small, the
corresponding periodic solution of system
(5)
is unique, hyperbolic and of the same stability type
as α.
The proof of this theorem can be seen in [21, 23].
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
3 PROOF OF THEOREM 1
In this work, we consider
ρ>
0,
s>
0and writing differential system
(3)
in the new variables
(θ, ρ, s, ω)
given by
x=ρcos(θ),
u=ρsin(θ),
y=scos(θ+ω),
v=ssin(θ+ω).
we get
˙
θ=−1+εG(θ,ρ,s,ω),
˙ρ=εF1(θ,ρ,s,ω),
˙s=εF2(θ, ρ, s, ω),
˙ω=εF3(θ, ρ, s, ω),
where
G(θ,ρ,s,ω)=
1
ρcos(θ)A1ρcos(θ)+A3ρ2cos2(θ)
+scos(θ+ω)A2+(a1+A4)ρcos(θ)+a3ρ2cos2(θ)+a6ρ3cos3(θ)
+s2cos2(θ+ω)a2+A5+a4ρcos(θ)+a7ρ2cos2(θ)
+s3cos3(θ+ω)a5+a8ρcos(θ)
+s4cos4(θ+ω)a9
F1(θ,ρ,s,ω) = sin(θ)A1ρcos(θ)+A3ρ2cos2(θ)
+scos(θ+ω)A2+(a1+A4)ρcos(θ)+a3ρ2cos2(θ)+a6ρ3cos3(θ)
+s2cos2(θ+ω)a2+A5+a4ρcos(θ)+a7ρ2cos2(θ)
+s3cos3(θ+ω)a5+a8ρcos(θ)
+s4cos4(θ+ω)a9
F2(θ,ρ,s,ω) = sin(θ+ω)B1ρcos(θ)+(b1+B3)ρ2cos2(θ)+b3ρ3cos3(θ)+b6ρ4cos4(θ)
+scos(θ+ω)B2+(b2+B4)ρcos(θ)+b4ρ2cos2(θ)+b7ρ3cos3(θ)
+s2cos2(θ+ω)B5+b5ρcos(θ)+b8ρ2cos2(θ)
+s3cos3(θ+ω)b9ρcos(θ)
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed. 42 Vol. 11 N.º 2 August - December 2022
28
Note that system (3) when ε=0, is
˙x=u,
˙u=−x,
˙y=v,
˙v=−y.
(4)
We notice that the system
(4)
have a global isochronous center at the the origin, i.e. all orbits different
from the origin are 2π-periodic, for more detail see [18].
The main result of our work is the following.
Theorem 1. Using the first-order averaging method, system of second-order differential equations
(3)
, where
f1
(
x, y
)and
g1
(
x, y
)are real cubic polynomials;
f2
(
x, y
)and
g2
(
x, y
)are real quadratic
polynomials, has at most four limit cycles bifurcating from the periodic orbits of the linear center
˙x=u, ˙u=−x, ˙y=v, ˙v=−y
. Here
ε
is a small parameter. Moreover if
a4
=0and
a3
=0, the system
(3) has at most two periodic solutions.
The first-order averaging method theory, that we summarize in the sequel, can be found in a more
extended way in [2]. Similar works where the perturbations via polynomials play an important role are
for instance [7] and [20].
2
THE FIRST-ORDER AVERAGING METHOD FOR COMPUTING PERIODIC
ORBITS
Theorem 2. We consider the following two problems
˙x(t)=εF (t, x(t)) + ε2R(t, x(t),ε),(5)
and
˙y(t)=εf(y(t)),(6)
where
t∈[0,+∞)
,
x
and
y
in some open
D
of
Rn
and
ε∈(−ε0,ε
0)
is a small parameter. Moreover,
we suppose that the vector functions
F
(
t, x
)and
R
(
t, x, ε
)are
T−
periodic in the first variable and we
consider the first-order averaging function
f(y)= 1
TT
0
F(s, y)ds.
Suppose that
F
,
R
,D
xF
, and D
2
xF
are continuous and bounded by a constant
M
in
[0,∞)×D
×(−ε0,ε
0)where Mis independent of ε. Then, the statements (I)and (II)satisfied:
(I)If there exists an equilibrium point α∈Dof (6) such that
det ∂(f (y))
∂y y=p=0,
then, for
ε>
0sufficiently small, there exists an isolated
T−
periodic solution
ϕε
(
t
)of system
(5) such that ϕε(t)→0when ε→0.
(II)
If
y
=
α
( the equilibrium point ) of
(6)
is hyperbolic. Then, for
ε>
0sufficiently small, the
corresponding periodic solution of system
(5)
is unique, hyperbolic and of the same stability type
as α.
The proof of this theorem can be seen in [21, 23].
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
3 PROOF OF THEOREM 1
In this work, we consider
ρ>
0,
s>
0and writing differential system
(3)
in the new variables
(θ, ρ, s, ω)
given by
x=ρcos(θ),
u=ρsin(θ),
y=scos(θ+ω),
v=ssin(θ+ω).
we get
˙
θ=−1+εG(θ,ρ,s,ω),
˙ρ=εF1(θ,ρ,s,ω),
˙s=εF2(θ, ρ, s, ω),
˙ω=εF3(θ, ρ, s, ω),
where
G(θ,ρ,s,ω)=
1
ρcos(θ)A1ρcos(θ)+A3ρ2cos2(θ)
+scos(θ+ω)A2+(a1+A4)ρcos(θ)+a3ρ2cos2(θ)+a6ρ3cos3(θ)
+s2cos2(θ+ω)a2+A5+a4ρcos(θ)+a7ρ2cos2(θ)
+s3cos3(θ+ω)a5+a8ρcos(θ)
+s4cos4(θ+ω)a9
F1(θ,ρ,s,ω) = sin(θ)A1ρcos(θ)+A3ρ2cos2(θ)
+scos(θ+ω)A2+(a1+A4)ρcos(θ)+a3ρ2cos2(θ)+a6ρ3cos3(θ)
+s2cos2(θ+ω)a2+A5+a4ρcos(θ)+a7ρ2cos2(θ)
+s3cos3(θ+ω)a5+a8ρcos(θ)
+s4cos4(θ+ω)a9
F2(θ,ρ,s,ω) = sin(θ+ω)B1ρcos(θ)+(b1+B3)ρ2cos2(θ)+b3ρ3cos3(θ)+b6ρ4cos4(θ)
+scos(θ+ω)B2+(b2+B4)ρcos(θ)+b4ρ2cos2(θ)+b7ρ3cos3(θ)
+s2cos2(θ+ω)B5+b5ρcos(θ)+b8ρ2cos2(θ)
+s3cos3(θ+ω)b9ρcos(θ)
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed. 42 Vol. 11 N.º 2 August - December 2022
29
F3(θ,ρ,s,ω)=−cos(θ)
ρA1ρcos(θ)+A3ρ2cos2(θ)
+scos(θ+ω)A2+(a1+A4)ρcos(θ)+a3ρ2cos2(θ)+a6ρ3cos3(θ)
+s2cos2(θ+ω)a2+A5+a4ρcos(θ)+a7ρ2cos2(θ)
+s3cos3(θ+ω)a5+a8ρcos(θ)
+s4cos4(θ+ω)a9
+cos(θ+ω)
sB1ρcos(θ)+(b1+B3)ρ2cos2(θ)+b3ρ3cos3(θ)+b6ρ4cos4(θ)
+scos(θ+ω)B2+(b2+B4)ρcos(θ)
+b4ρ2cos2(θ)+b7ρ3cos3(θ)
+s2cos2(θ+ω)B5+b5ρcos(θ)+b8ρ2cos2(θ)
+s3cos3(θ+ω)b9ρcos(θ)
The previous differential system in the new independent variable θbecomes as follows
dρ
dθ =−εF1(θ, ρ, s, ω)+ε2R1(θ,ρ,s,ω,ε),
ds
dθ =−εF2(θ, ρ, s, ω)+ε2R2(θ,ρ,s,ω,ε),
dω
dθ =−εF3(θ,ρ,s,ω)+ε2R3(θ,ρ,s,ω,ε),
(7)
We observe that this system is into the normal form of the averaging method
(5)
, with
t
=
θ
and
x
=
(ρ, s, ω)
that the all assumptions of the Theorem 2 are satisfied for the system
(7)
. We compute
the average functions of the first-order associated with the system (7)
fi(ρ, s, ω)= 1
2π2π
0
Fi(θ,ρ,s,ω)dθ,
for i=1,2,3, we obtain
f1(ρ, s, ω)=ssin (ω)
83s2a5+4A2+2sa4ρcos (ω)+a3ρ2,
f2(ρ, s, ω)=−ρsin (ω)
82ρb
4scos (ω)+4B1+3ρ2b3+b5s2,
f3(ρ, s, ω)=−1
8ρs−2s3a4ρcos2(ω)+ρ3b4s−3s4a5cos(ω)
+4ρ2B1cos(ω)+2ρ3b4scos2(ω)−4s2A2cos(ω)
+3ρ2b5s2cos(ω)−3s2a3ρ2cos(ω)+3ρ4b3cos(ω)
+4ρB
2s−4sA1ρ−s3a4ρ.
If (ρ0,s
0,ω
0)is a zero of the system
fi(ρ, s, ω)=0,for i=1,2,3,(8)
such that
det ∂(f1,f2,f3)
∂(ρ, s, ω)(ρ0,s0,ω0)=0,(9)
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
then Theorem 2 assures that the system
(3)
has a periodic solutions. So, in particular a zero of
(8)
must be isolated in the set of all zeros of
(8)
. We note that the zeros of
(8)
having
sin
(
ω
)=0are
non-isolated, so we cannot apply to them the averaging theory for obtaining limit cycles. Moreover,
since the differential system
(7)
is only well defined when
s>
0and
ρ>
0
,
in the rest of this section we
will assume that
ρ>
0
,s>
0and
sin
(
ω
)
=0, and consequently we can restrict to look for the zeros of
ξ1(ρ, s, ω)=0,
ξ2(ρ, s, ω)=0,
ξ3(ρ, s, ω)=0,
(10)
satisfying (9), where
ξ1(ρ, s, ω)= 8f1
ssin(ω),
ξ2(ρ, s, ω)= −8f2
ρsin(ω),
ξ3(ρ, s, ω)=−8ρsf3.
The rest of the proof of Theorem 1 is divided into the following cases and subcases.
Case 1 If a4=0. Then, by solving the first equation ξ1=0with respect to cos (ω)we get
cos (ω)=−a3ρ2+4A2+3s2a5
2sa4ρ.
Substituting the expression of cos (ω)in the second equation, ξ2=0, we obtain
−b4a3ρ2−4b4A2−3b4s2a5+4B1a4+3b3ρ2a4+b5s2a4=0.(11)
Subcase 1.1 If b4a3−3b3a4=0. Then, from (11) we get
ρ=4B1a4−4b4A2−3b4s2a5+b5s2a4
b4a3−3b3a4
.
Substituting the expressions of
ρ
and
cos
(
ω
)in the third equation,
ξ3
=0, we obtain an equation
of the form s(A+Bs2+Cs4)
(b4a3−3b3a4)2�(b5a4−3b4a5)s2−4b4A2+4B1a4
=0,
where A
,
B
,
Care constants. From now onwards, we are going to denote by A
,
B
,
Cthis kind of
generic constants. As A+B
s2
+C
s4
is a quadratic polynomial in
s2
, which can have at most two
positive solutions for
s
, in this subcase we get two values for
ρ
,
s
and
cos
(
ω
). Observe that each
value of (
ρ, s
)provides at most two solutions for
ω
. Hence, assuming that in these four solutions
the determinant
(9)
is not zero, by Theorem 2, it follows that in this subcase we have at most
four periodic solutions of system (3).
Example 1. Let us consider the system of differential equations:
˙x=u,
˙u=−x+ε5
2x+x2+1
2xy −1
2y2+5x3−3
2x2y+y3y
+x+x2+1
2y2,
˙y=v,
˙v=−y+ε3
4y+x2−1
2xy −3
2y2+x3−xy2−y3x
+2y+x2−xy +y2.
(12)
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30
F3(θ,ρ,s,ω)=−cos(θ)
ρA1ρcos(θ)+A3ρ2cos2(θ)
+scos(θ+ω)A2+(a1+A4)ρcos(θ)+a3ρ2cos2(θ)+a6ρ3cos3(θ)
+s2cos2(θ+ω)a2+A5+a4ρcos(θ)+a7ρ2cos2(θ)
+s3cos3(θ+ω)a5+a8ρcos(θ)
+s4cos4(θ+ω)a9
+cos(θ+ω)
sB1ρcos(θ)+(b1+B3)ρ2cos2(θ)+b3ρ3cos3(θ)+b6ρ4cos4(θ)
+scos(θ+ω)B2+(b2+B4)ρcos(θ)
+b4ρ2cos2(θ)+b7ρ3cos3(θ)
+s2cos2(θ+ω)B5+b5ρcos(θ)+b8ρ2cos2(θ)
+s3cos3(θ+ω)b9ρcos(θ)
The previous differential system in the new independent variable θbecomes as follows
dρ
dθ =−εF1(θ, ρ, s, ω)+ε2R1(θ,ρ,s,ω,ε),
ds
dθ =−εF2(θ, ρ, s, ω)+ε2R2(θ,ρ,s,ω,ε),
dω
dθ =−εF3(θ,ρ,s,ω)+ε2R3(θ,ρ,s,ω,ε),
(7)
We observe that this system is into the normal form of the averaging method
(5)
, with
t
=
θ
and
x
=
(ρ, s, ω)
that the all assumptions of the Theorem 2 are satisfied for the system
(7)
. We compute
the average functions of the first-order associated with the system (7)
fi(ρ, s, ω)= 1
2π2π
0
Fi(θ,ρ,s,ω)dθ,
for i=1,2,3, we obtain
f1(ρ, s, ω)=ssin (ω)
83s2a5+4A2+2sa4ρcos (ω)+a3ρ2,
f2(ρ, s, ω)=−ρsin (ω)
82ρb
4scos (ω)+4B1+3ρ2b3+b5s2,
f3(ρ, s, ω)=−1
8ρs−2s3a4ρcos2(ω)+ρ3b4s−3s4a5cos(ω)
+4ρ2B1cos(ω)+2ρ3b4scos2(ω)−4s2A2cos(ω)
+3ρ2b5s2cos(ω)−3s2a3ρ2cos(ω)+3ρ4b3cos(ω)
+4ρB
2s−4sA1ρ−s3a4ρ.
If (ρ0,s
0,ω
0)is a zero of the system
fi(ρ, s, ω)=0,for i=1,2,3,(8)
such that
det ∂(f1,f2,f3)
∂(ρ, s, ω)(ρ0,s0,ω0)=0,(9)
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
then Theorem 2 assures that the system
(3)
has a periodic solutions. So, in particular a zero of
(8)
must be isolated in the set of all zeros of
(8)
. We note that the zeros of
(8)
having
sin
(
ω
)=0are
non-isolated, so we cannot apply to them the averaging theory for obtaining limit cycles. Moreover,
since the differential system
(7)
is only well defined when
s>
0and
ρ>
0
,
in the rest of this section we
will assume that
ρ>
0
,s>
0and
sin
(
ω
)
=0, and consequently we can restrict to look for the zeros of
ξ1(ρ, s, ω)=0,
ξ2(ρ, s, ω)=0,
ξ3(ρ, s, ω)=0,
(10)
satisfying (9), where
ξ1(ρ, s, ω)= 8f1
ssin(ω),
ξ2(ρ, s, ω)= −8f2
ρsin(ω),
ξ3(ρ, s, ω)=−8ρsf3.
The rest of the proof of Theorem 1 is divided into the following cases and subcases.
Case 1 If a4=0. Then, by solving the first equation ξ1=0with respect to cos (ω)we get
cos (ω)=−a3ρ2+4A2+3s2a5
2sa4ρ.
Substituting the expression of cos (ω)in the second equation, ξ2=0, we obtain
−b4a3ρ2−4b4A2−3b4s2a5+4B1a4+3b3ρ2a4+b5s2a4=0.(11)
Subcase 1.1 If b4a3−3b3a4=0. Then, from (11) we get
ρ=4B1a4−4b4A2−3b4s2a5+b5s2a4
b4a3−3b3a4
.
Substituting the expressions of
ρ
and
cos
(
ω
)in the third equation,
ξ3
=0, we obtain an equation
of the form s(A+Bs2+Cs4)
(b4a3−3b3a4)2�(b5a4−3b4a5)s2−4b4A2+4B1a4
=0,
where A
,
B
,
Care constants. From now onwards, we are going to denote by A
,
B
,
Cthis kind of
generic constants. As A+B
s2
+C
s4
is a quadratic polynomial in
s2
, which can have at most two
positive solutions for
s
, in this subcase we get two values for
ρ
,
s
and
cos
(
ω
). Observe that each
value of (
ρ, s
)provides at most two solutions for
ω
. Hence, assuming that in these four solutions
the determinant
(9)
is not zero, by Theorem 2, it follows that in this subcase we have at most
four periodic solutions of system (3).
Example 1. Let us consider the system of differential equations:
˙x=u,
˙u=−x+ε5
2x+x2+1
2xy −1
2y2+5x3−3
2x2y+y3y
+x+x2+1
2y2,
˙y=v,
˙v=−y+ε3
4y+x2−1
2xy −3
2y2+x3−xy2−y3x
+2y+x2−xy +y2.
(12)
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed. 42 Vol. 11 N.º 2 August - December 2022
31
Then, it can be checked that
(ρ, s, ω)=26
37,42
37,π
6
(ρ, s, ω)=26
37,42
37,11π
6
are zeros of system
(12)
with determinant
(9)
equals to
±6√3
1369
, respectively. So, this system has
two periodic solutions coming from periodic orbits of the center (4) .
Subcase 1.2
If
b4a3−
3
b3a4
=0, then
b3
=
a3b4/
(3
a4
), and we need to consider the following subcases.
Subcase 1.2.1 If b5a4−3b4a5=0, then, from (11) we obtain that
s=2
−B1a4+b4A2
b5a4−3b4a5
.
We must consider that
−B1a4
+
b4A2
=0, otherwise
s
=0and we cannot get periodic solutions.
If we substitute the expressions of
cos
(
ω
)and
s
in
ξ3
=0, we get an equation of the form
ρ
(A+B
ρ2
+C
ρ4
)=0. Since
ρ
must be positive, again in this subcase we get two values for
ρ
,
s
and cos(ω); and consequently at most four periodic solutions of system (3).
Subcase 1.2.2 b5a4−
3
b4a5
=0. Therefore, from
(11)
we must have that
−B1a4
+
b4A2
=0
,
otherwise
we do not have solutions. That is,
b5
=3
b4a5/a4
. Substituting now
cos
(
ω
)in
ξ3
=0,wegeta
continum of solutions for ρand s. So, in this case we cannot apply Theorem 2.
Case 2 a4=0. Again we need to consider the following subcases.
Subcase 2.1 a3=0. Therefore, from the first equation, ξ1=0, we get
ρ=−a3(3 s2a5+4A2)
a3
.
Of course we suppose
−a33s2a5+4A2
=0, otherwise
ρ
=0. Now, we substitute the expression
of ρin the second equation ξ2=0.
Subcase 2.1.1 b4=0. Therefore, from the second equation, ξ2=0, we get that
cos(ω)=−(b5a3−9b3a5)s2−12 b3A2+4B1a3
2b4s−a3(3 s2a5+4A2).
Substituting the expressions of ρand cos(ω)in ξ3=0, we get an equation of the form
s
a3b4−a3(3 s2a5+4A2)(A+Bs2+Cs4)=0.
Since the first factor cannot be zero, as in the previous subcases, we can get at most four periodic
solutions of system (3).
Subcase 2.1.2 b4=0.
Subcase 2.1.2.1 If b5a3−9a5b3=0, then, from the second equation, ξ2=0, we obtain
s=23b3A2−B1a3
b5a3−9a5b3
,
substituting the expressions of
ρ
and
s
in
ξ3
=0, we arrive to an equation of the form A+
B
cos
(
ω
)+C
cos2
(
ω
)=0. So, once again, we can obtain at most four solutions for
ρ
,
s
and
ω
,
and, consequently, we obtains at most four periodic solutions for system (3).
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
Subcase 2.1.2.2
If
b5a3−
9
a5b3
=0, then,
b5
=
9a5b3
a3
. Now, from
ξ2
=0, it follows that
B1a3−
3
b3A2
=0, otherwise we have no solutions. Therefore,
B1
=
3b3A2
a3
. Substituting the expression
of
ρ
in
ξ3
=0, we get a continuum of solutions. So, again, we are not in the assumptions of
Theorem 2.
Subcase 2.2 a3
=0. Looking at equation
ξ1
=0, we see that
a5
cannot be zero, otherwise
ξ1
=0
reduces to A2=0, and either we do not have solutions or we have a continuum of solutions.
Then, from ξ1=0we get
s=2
−A2
3a5
.
Substituting the expression of sin the second equation ξ2=0, we must consider the subcases:
Subcase 2.2.1 If b4=0, then, by solving the equation ξ2=0with respect to cos (ω)we get
cos(ω)=−12 B1a5+9ρ2b3a5−4b5A2
4ρb
4√−3a5A2
.
Substituting the expressions of
cos
(
ω
)and
s
in
ξ3
=0, we obtain an equation of the form
ρ(A+Bρ2)=0. Hence, as in previous subcases, system (3) has at most two periodic solutions.
Subcase 2.2.2
Assume
b4
=0. Then, the second equation
ξ2
=0is of the form A+B
ρ2
=0, so,
there is at most one positive solution for
ρ
. Hence, by substituting the value of
s
and
ρ
in
ξ3
=0,
we obtain an equation of the form A+B
cos
(
ω
)=0. Therefore, we get at most one solution for
ρ
and
cos
(
ω
). In short, putting aside those subcases where we obtain a continuum of solutions,
there is at most one solution for
s
,
ρ
and
cos
(
ω
)and so, there are at most two periodic solutions
for system (3).
In the previous case, we gave a particular solution obtained from the subcase 1.1, that is the
most general one. Now we are going to see the general solution of subcase 2.2.2, characterized by
a3=a4=b4=0.
Corollary 1. If a3=a4=b4=0then
a) If b5=0the system (10) has no solution or it has a continuum of solutions.
b) If b5=0, the solutions of system (10) are given by:
ρ2=−4B1+b5s2
3b3
s2=−4A2
3a5
cos(ω)=2(A1−B2)
b5ρs
.
Demonstração. a) If b5=0, the system (10) reduces to
4A2+3a5s2=0,
4B1+3b3ρ2=0,
4ρs(B2−A1) + cos(ω)s2(−4A2−3a5s2)+ρ2(4B1+3b3ρ2)=0
that is 4A2+3a5s2=0,
4B1+3b3ρ2=0,
4ρs(B2−A1)=0.
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3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
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32
Then, it can be checked that
(ρ, s, ω)=26
37,42
37,π
6
(ρ, s, ω)=26
37,42
37,11π
6
are zeros of system
(12)
with determinant
(9)
equals to
±6√3
1369
, respectively. So, this system has
two periodic solutions coming from periodic orbits of the center (4) .
Subcase 1.2
If
b4a3−
3
b3a4
=0, then
b3
=
a3b4/
(3
a4
), and we need to consider the following subcases.
Subcase 1.2.1 If b5a4−3b4a5=0, then, from (11) we obtain that
s=2
−B1a4+b4A2
b5a4−3b4a5
.
We must consider that
−B1a4
+
b4A2
=0, otherwise
s
=0and we cannot get periodic solutions.
If we substitute the expressions of
cos
(
ω
)and
s
in
ξ3
=0, we get an equation of the form
ρ
(A+B
ρ2
+C
ρ4
)=0. Since
ρ
must be positive, again in this subcase we get two values for
ρ
,
s
and cos(ω); and consequently at most four periodic solutions of system (3).
Subcase 1.2.2 b5a4−
3
b4a5
=0. Therefore, from
(11)
we must have that
−B1a4
+
b4A2
=0
,
otherwise
we do not have solutions. That is,
b5
=3
b4a5/a4
. Substituting now
cos
(
ω
)in
ξ3
=0,wegeta
continum of solutions for ρand s. So, in this case we cannot apply Theorem 2.
Case 2 a4=0. Again we need to consider the following subcases.
Subcase 2.1 a3=0. Therefore, from the first equation, ξ1=0, we get
ρ=−a3(3 s2a5+4A2)
a3
.
Of course we suppose
−a33s2a5+4A2
=0, otherwise
ρ
=0. Now, we substitute the expression
of ρin the second equation ξ2=0.
Subcase 2.1.1 b4=0. Therefore, from the second equation, ξ2=0, we get that
cos(ω)=−(b5a3−9b3a5)s2−12 b3A2+4B1a3
2b4s−a3(3 s2a5+4A2).
Substituting the expressions of ρand cos(ω)in ξ3=0, we get an equation of the form
s
a3b4−a3(3 s2a5+4A2)(A+Bs2+Cs4)=0.
Since the first factor cannot be zero, as in the previous subcases, we can get at most four periodic
solutions of system (3).
Subcase 2.1.2 b4=0.
Subcase 2.1.2.1 If b5a3−9a5b3=0, then, from the second equation, ξ2=0, we obtain
s=23b3A2−B1a3
b5a3−9a5b3
,
substituting the expressions of
ρ
and
s
in
ξ3
=0, we arrive to an equation of the form A+
B
cos
(
ω
)+C
cos2
(
ω
)=0. So, once again, we can obtain at most four solutions for
ρ
,
s
and
ω
,
and, consequently, we obtains at most four periodic solutions for system (3).
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
Subcase 2.1.2.2
If
b5a3−
9
a5b3
=0, then,
b5
=
9a5b3
a3
. Now, from
ξ2
=0, it follows that
B1a3−
3
b3A2
=0, otherwise we have no solutions. Therefore,
B1
=
3b3A2
a3
. Substituting the expression
of
ρ
in
ξ3
=0, we get a continuum of solutions. So, again, we are not in the assumptions of
Theorem 2.
Subcase 2.2 a3
=0. Looking at equation
ξ1
=0, we see that
a5
cannot be zero, otherwise
ξ1
=0
reduces to A2=0, and either we do not have solutions or we have a continuum of solutions.
Then, from ξ1=0we get
s=2
−A2
3a5
.
Substituting the expression of sin the second equation ξ2=0, we must consider the subcases:
Subcase 2.2.1 If b4=0, then, by solving the equation ξ2=0with respect to cos (ω)we get
cos(ω)=−12 B1a5+9ρ2b3a5−4b5A2
4ρb
4√−3a5A2
.
Substituting the expressions of
cos
(
ω
)and
s
in
ξ3
=0, we obtain an equation of the form
ρ(A+Bρ2)=0. Hence, as in previous subcases, system (3) has at most two periodic solutions.
Subcase 2.2.2
Assume
b4
=0. Then, the second equation
ξ2
=0is of the form A+B
ρ2
=0, so,
there is at most one positive solution for
ρ
. Hence, by substituting the value of
s
and
ρ
in
ξ3
=0,
we obtain an equation of the form A+B
cos
(
ω
)=0. Therefore, we get at most one solution for
ρ
and
cos
(
ω
). In short, putting aside those subcases where we obtain a continuum of solutions,
there is at most one solution for
s
,
ρ
and
cos
(
ω
)and so, there are at most two periodic solutions
for system (3).
In the previous case, we gave a particular solution obtained from the subcase 1.1, that is the
most general one. Now we are going to see the general solution of subcase 2.2.2, characterized by
a3=a4=b4=0.
Corollary 1. If a3=a4=b4=0then
a) If b5=0the system (10) has no solution or it has a continuum of solutions.
b) If b5=0, the solutions of system (10) are given by:
ρ2=−4B1+b5s2
3b3
s2=−4A2
3a5
cos(ω)=2(A1−B2)
b5ρs
.
Demonstração. a) If b5=0, the system (10) reduces to
4A2+3a5s2=0,
4B1+3b3ρ2=0,
4ρs(B2−A1) + cos(ω)s2(−4A2−3a5s2)+ρ2(4B1+3b3ρ2)=0
that is 4A2+3a5s2=0,
4B1+3b3ρ2=0,
4ρs(B2−A1)=0.
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3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed. 42 Vol. 11 N.º 2 August - December 2022
33
This system of equations does not depend on
ω
, hence either has no solution or has a continuum
of solutions.
b) If b5=0, we have the system:
4A2+3a5s2=0
4B1+3b3ρ2+b5s2=0
4ρs(B2−A1) + cos(ω)s2(−4A2−3a5s2)+ρ24B1+3b3ρ2+3b5s2=0.
From the first equation,
ξ1
=0, we obtain that
s2
=
−4A2
3a5
, and, by substituting this value in the
second equation, ξ2=0, it follows that ρ2=−4B1+b5s2
3b3.
Finally, the equation ξ3=0reduces to
ρs2(B2−A1)+b5ρs cos(ω)=0,
hence,
cos(ω)=2(A1−B2)
b5ρs .
This three equalities provides all the possible solutions of this subcase.
Example 2. In the previous corollary, if we take
A1
=
B2
, we have that
ω
=
π
2
or
ω
=
3π
2
,
and if we take the values
a5
=
−
1
,b
3
=
−
1
,b
5
=1
,A
2
=9
,B
1
=1, we obtain the following two
solutions:
(ρ, s, ω)=4
√3,2√3,π
2,
(ρ, s, ω)=4
√3,2√3,3π
2.
Observe that this subcase does not depend on the constants not listed in this example, so, we can
choose any value for them.
4 CONCLUSIONS
This paper shows that the application of averaging method of first-order it is useful for study the
existence of limit cycles of perturbated system of second-order differential equations.
We have proved that, using Theorem 2, we can obtain at most four periodic solutions of system
(3)
when
f1
(
x, y
)and
g1
(
x, y
)are real cubic polynomials, and
f2
(
x, y
)and
g2
(
x, y
)are real quadratic
polynomials. Moreover, if
a4
=0and
a3
=0, the system
(3)
has at most two periodic solutions.
We have also obtained the general solution in the case a3=a4=b4=0.
FUNDING
This paper is partially supported by the FEDER OP2014-2020 and the University of Castilla-La Mancha
under Grant 2021-GRIN-31241, and by the Junta de Comunidades de Castilla-La Mancha under grant
SBPLY/21/180501/000174.
DATA STATEMENT
This paper is not related with any data.
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
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Maoan Han and Jibin Li. Lower bounds for the Hilbert number of polynomial systems. Journal of
Differential Equations, 252(4):3278–3304, 2012.
[10] Maoan Han, Yuhai Wu, and Ping Bi. A new cubic system having eleven limit cycles. Discrete &
Continuous Dynamical Systems, 12(4):675, 2005.
[11]
Maoan Han, Junmin Yang, and Pei Yu. Hopf bifurcations for near-hamiltonian systems. Internati-
onal Journal of Bifurcation and Chaos, 19(12):4117–4130, 2009.
[12]
Maoan Han, Tonghua Zhang, and Hong Zang. On the number and distribution of limit cycles in a
cubic system. International Journal of Bifurcation and Chaos, 14(12):4285–4292, 2004.
[13]
David Hilbert. Mathematische probleme, lecture, second internat. congr. math. (paris, 1900),
nachr. ges. wiss. göttingen math. phys. kl. (1900), 253-297; english transl., bull. amer. math. soc. 8
(1902), 437-479.
[14]
Yu Ilyashenko. Centennial history of Hilbert’s 16th problem. Bulletin of the American Mathematical
Society, 39(3):301–354, 2002.
[15]
Chengzhi Li, Changjian Liu, and Jiazhong Yang. A cubic system with thirteen limit cycles. Journal
of Differential Equations, 246(9):3609–3619, 2009.
[16]
JB Li and CF Li. Distribution of limit cycles for planar cubic hamiltonian systems. Acta Math
Sinica, 28:509–521, 1985.
[17]
Jibin Li. Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. International
Journal of Bifurcation and Chaos, 13(01):47–106, 2003.
[18]
Jaume Llibre. Centers: their integrability and relations with the divergence. Applied Mathematics
and Nonlinear Sciences, 1(1):79–86, 2016.
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed. 42 Vol. 11 N.º 2 August - December 2022
34
This system of equations does not depend on
ω
, hence either has no solution or has a continuum
of solutions.
b) If b5=0, we have the system:
4A2+3a5s2=0
4B1+3b3ρ2+b5s2=0
4ρs(B2−A1) + cos(ω)s2(−4A2−3a5s2)+ρ24B1+3b3ρ2+3b5s2=0.
From the first equation,
ξ1
=0, we obtain that
s2
=
−4A2
3a5
, and, by substituting this value in the
second equation, ξ2=0, it follows that ρ2=−4B1+b5s2
3b3.
Finally, the equation ξ3=0reduces to
ρs2(B2−A1)+b5ρs cos(ω)=0,
hence,
cos(ω)=2(A1−B2)
b5ρs .
This three equalities provides all the possible solutions of this subcase.
Example 2. In the previous corollary, if we take
A1
=
B2
, we have that
ω
=
π
2
or
ω
=
3π
2
,
and if we take the values
a5
=
−
1
,b
3
=
−
1
,b
5
=1
,A
2
=9
,B
1
=1, we obtain the following two
solutions:
(ρ, s, ω)=4
√3,2√3,π
2,
(ρ, s, ω)=4
√3,2√3,3π
2.
Observe that this subcase does not depend on the constants not listed in this example, so, we can
choose any value for them.
4 CONCLUSIONS
This paper shows that the application of averaging method of first-order it is useful for study the
existence of limit cycles of perturbated system of second-order differential equations.
We have proved that, using Theorem 2, we can obtain at most four periodic solutions of system
(3)
when
f1
(
x, y
)and
g1
(
x, y
)are real cubic polynomials, and
f2
(
x, y
)and
g2
(
x, y
)are real quadratic
polynomials. Moreover, if
a4
=0and
a3
=0, the system
(3)
has at most two periodic solutions.
We have also obtained the general solution in the case a3=a4=b4=0.
FUNDING
This paper is partially supported by the FEDER OP2014-2020 and the University of Castilla-La Mancha
under Grant 2021-GRIN-31241, and by the Junta de Comunidades de Castilla-La Mancha under grant
SBPLY/21/180501/000174.
DATA STATEMENT
This paper is not related with any data.
https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
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Maoan Han, Junmin Yang, and Pei Yu. Hopf bifurcations for near-hamiltonian systems. Internati-
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Maoan Han, Tonghua Zhang, and Hong Zang. On the number and distribution of limit cycles in a
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Chengzhi Li, Changjian Liu, and Jiazhong Yang. A cubic system with thirteen limit cycles. Journal
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JB Li and CF Li. Distribution of limit cycles for planar cubic hamiltonian systems. Acta Math
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Jibin Li. Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. International
Journal of Bifurcation and Chaos, 13(01):47–106, 2003.
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Jaume Llibre. Centers: their integrability and relations with the divergence. Applied Mathematics
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https://doi.org/10.17993/3ctecno.2022.v11n2e42.25-36
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed. 42 Vol. 11 N.º 2 August - December 2022
35
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3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed. 42 Vol. 11 N.º 2 August - December 2022
36