Applied Mathematical Sciences, Vol. 8, 2014, no. 64, 3161 - 3171
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.43147
Maximum Number of Limit Cycles of Cubic
Liénard Differential System
Hero Waisi Salih1,2, Zainal Abdul Aziz1,2 and Faisal Salah1,3
1
2
UTM Centre for Industrial and Applied Mathematics &
Department of Mathematical Sciences, Faculty of Science,
Universiti Teknologi Malaysia,
81310 UTM Johor Bahru, Johor, Malaysia,
3
Department of Mathematics, Faculty of Science,
University of Kordofan, Elobid, Sudan
Copyright © 2014 Hero Waisi Salih, Zainal Abdul Aziz and Faisal Salah. This is an open access
article distributed under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The number of limit cycles of the cubic Liénard polynomial differential system of
the form x  y, y   g ( x)  f ( x) y is examined, where f ( x) is a polynomial of
degree three and g ( x), a polynomial of degree one and two. The accurate upper
bound of the maximum number of limit cycles of this Liénard differential system
is obtained. By using the first order averaging theory, this system is shown to
bifurcate from the periodic orbits of the linear center x  y, y  -x . The maximum
number of limit cycles of the differential system is found to be unique.
Mathematics Subject Classification: 34C05, 34C07, 37G15
Keywords: Limit cycles, Liénard differential system, Averaging theory,
Uniqueness theorem.
Introduction
The second part of the 16th Hilbert’s problem aims to find an upper bound on the
maximum number of limit cycles of the class of all polynomial vector fields with
a fixed degree. This work attempts to give a partial answer to this problem for the
class of Liénard polynomial differential system
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Hero Waisi Salih, Zainal Abdul Aziz and Faisal Salah
x y
y  - g ( x) - f ( x) y
(1)
where f ( x) and g ( x) , are polynomials of degree n, m respectively, and by
applying one of the five theorems of uniqueness from [24] (Theorem 2 of Sabatini
and Villari [24]).
The classical Liénard polynomial differential system is given by
x y
y   x  f ( x) y
x y
y   g ( x)  f ( x) y
(2a)
(2b)
where f ( x) is a polynomial of degree n .
In 1977 for the system (1), Lins et al. [1] stated that, if f ( x) has degree n  1 then
n
system (1) has at most [ ] limit cycles. They proved this conjecture for n  1, 2 .
2
The conjecture for n  3 has been proved recently by Chengzi and Llibre in
[14]. For n  5 the conjecture is shown to be invalid, see De Maesschalck and
Dumortier [8] and Dumortier et al. [11]. Thus it remains to be realized whether
the conjecture is true or not for n  4 .
Several conclusions on the limit cycles of polynomial differential systems
have been obtained by considering a Hopf bifurcation, which are known as small
amplitude limit cycles, see for instance [3]. There are partial results concerning
the maximum number of small amplitude limit cycles for Liénard polynomial
differential systems. Of course, the number of small amplitude limit cycles gives a
lower bound for the maximum number of limit cycles of a polynomial differential
system.
There are various results concerning the existence of small amplitude limit
cycles for the generalized Liénard polynomial differential system (1). H (m, n)
denotes the number of large amplitude limit cycles that system (1) can have. This
number is usually called the Hilbert number for system (1). The following is a list
of previous research outputs related to H (m, n).
i.
x
In 1928, Liénard [15] proved that if m  1 and F ( x)   f ( s)ds is a
0
continuous odd function, which has a unique root at x  a and is
monotone increasing for x  a , and then system (2a) has a unique limit
cycle.
Cubic Liénard differential system
ii.
iii.
iv.
v.
vi.
vii.
3163
In 1973, Rychkov [23] proved that if m  1 and F ( x) is an odd
polynomial of degree five, then system (2a) has at most two limit
cycles.
In 1977, Lins et al. [1] proved that H (1,1)  0 and H (1, 2)  1 .
In 1998, Coppel [7] proved that H (2,1)  1 .
Dumortier et al. in [9, 12] proved that H (3,1)  1 .
In 1997 Dumortier and Li [10] proved that H (2, 2)  1
In 2011 Chengzi and Llibre [14] proved that H (1,3)  1 .
To the best of our knowledge, the determination of the number of limit cycles are
obtained only for the five cases ((iii)–(vii)) of the Hilbert number for system (1).
The maximum number of small amplitude limit cycles for system (1) is denoted
by Hˆ (m, n). Blows and Lloyd [4], Lloyd and Lynch [18] and Lynch [19] have used
inductive arguments to prove the following results.
II.
n
If g is odd then Hˆ (m, n)  [ ] .
2
If f is even then Hˆ (m, n)  n.
III.
If f is odd then Hˆ (m, 2n  1)  [
I.
IV.
m-2
] 2
2
If g ( x)  x  g e ( x) , where g e is even then Hˆ (2m, 2)  m
Christopher and Lynch [6, 20, 21, and 22] have formulated a new method for
finding the Liapunov quantities of system (1) and proved some other bounds for
Hˆ (m, n) of different degrees m and n :
V.
VI.
VII.
VIII.
IX.
2m  1
Hˆ (m, 2)  [
].
3
2n  1
Hˆ (2, n)  [
] .
3
2m  2
Hˆ (m,3)  2[
] for all 1  m  50 .
8
2n  2
Hˆ (3, n)  2[
] for all 1  n  50
8
Hˆ (4, k )  Hˆ (k , 4), k  6,7,8,9 and Hˆ (5, 6)  Hˆ (6,5)
In 1998, Gasull and Torregrosa [13] obtained the upper bounds for
Hˆ (7,6) , Hˆ (6,7), Hˆ (7,7), and Hˆ (4, 20). In 2006, Yu and Han in [25] showed
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Hero Waisi Salih, Zainal Abdul Aziz and Faisal Salah
for n  4, m  10,11,12,13;
n  5, m  6,7,8,9;
Hˆ (m, n)  Hˆ (n, m) ,
n  6, m  5,6 , and refer also [22] for a table with all the specific values.
In 2010 Llibre et al. [16] calculated the maximum number of limit cycles
H k (2m, 2) of system (1) which bifurcates from the periodic orbits of the linear
center x  y, y  -x , via the averaging theory of order k , for k  1, 2,3 , and
where H k (m, n) is the lower number of limit cycles which can bifurcate from the
periodic orbit of a linear center.
In [17] the authors studied using the averaging theory of first and second
order of the more general system
x  y -  ( g11 ( x)  f11 ( x) y) -  2 ( g12 ( x)  f12 ( x) y),
y  - x - ( g 21 ( x)  f 21 ( x) y) -  2 ( g 22 ( x)  f 22 ( x) y),
(3)
where g1i , f1i , g 2i , f 2i have degree l , k , m and n respectively for each i  1, 2 and
 is a small parameter. Using the averaging method of first and second order, they
proved the following result.
Theorem 1. (See [17]) For | | sufficiently small, the maximum number of limit
cycles of the generalized Liénard polynomial differential system (3) bifurcates
from the periodic orbits of the linear center x  y, y  - x, using the averaging
theory of second order, then
m -1
l n -1
m k
m
n -1 l -1
  max{  [
],   [ ],[
]  [ ],[ ]  [ ] -1,[
] [ ]
2
2
2
2 2
2
2
2
k
l -1
 1,[ ]  [ ]},
(4)
2
2
n k -1
]}.
with   min{[ ],[
2
2
In Alavez-Ramirez et al. [2], they considered the polynomial differential
system
x  y -  g11 ( x) -  2 f12 ( x),
y  - x -  ( g 21 ( x)  f 21 ( x) y) -  2 ( g 22 ( x)  f 22 ( x) y),
(5)
where g1i , g 2i , f 2i have degree l , m and n respectively for i  1, 2 and  is a small
parameter. They proved the following result.
Theorem 2. (See [2]), For | | sufficiently small the maximum number of limit
cycles of the generalized Liénard polynomial differential system (5) bifurcates
from the periodic orbits of the linear center x  y, y  - x, using the averaging
theory of third order, then
Cubic Liénard differential system
where O(k )
integer  k .
3165
1
(max{O(m  n), E (l  m) -1}-1)
2
is the largest odd integer  k , and E(k ) is the largest even
The present article investigates system (1), where the maximum number of
limit cycles is obtained by using the averaging theory.
The first order averaging theory
The averaging theory for studying precisely the limit cycles in  was developed
in [5]. It is summarized as follows. Consider
x(t )   F1 (t , x)   2 R(t , x,  )
(6)
where F1 : D  R  R n , R : R  D  (- f ,  f )  R n are continuous functions, Tperiodic (of time T) in the first variable, and D is an open subset of R n . Assume
that the following conditions hold.
i.
F1 and R are locally Lipschitz with respect to x . We define F10 : D  R n as
T
1
F10 ( z )   F1 ( z, s)ds
T 0
ii. For a  D with F10 ( a )  0 there exists a neighbourhood V of a such that
F10 ( z )  0 for all z V {a} and d B ( F10 , V , a )  0 .
Then for | |> 0 sufficiently small, there exists a T-periodic solution
 (.,  )  a as   0
Theorem and main result
Theorem 3. Assume that for k  1 the polynomials f nk ( x) have degree n  3 ,
with n  1, then for a sufficiently small parameter | | and by using the averaging
theory of first order, the maximum number of limit cycles of the Liénard system
(3) is one. The limit cycle bifurcates from the periodic orbits of the linear
center x  y, y  -x .
Proof
It is based on the first-order averaging theory, which is presented in the previous
section.
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Hero Waisi Salih, Zainal Abdul Aziz and Faisal Salah
Consider k  1
f 3 ( x ) appearing in (3) as
and write the polynomials
3
f ( x)   ai x i .
i 0
By means of the change of variables x  r cos , y  r sin  , system (3) in the
region r  0 can be written as
r  - sin  P(r ,  ),

  -1- cos  P(r , )
(7)
r
where
3
P(r , )   ai r i 1 cosi  sin  .
(8)
i 0
Now taking  as the new independent variable, system (7) becomes
dr
  sin  P(r , )   ( 2 )  F1 (r , )   ( 2 ),
d
(9)
which is the standard form for applying the averaging theory. Then by the
averaging theory it is obtained
2
1
F10 (r ) 
sin  P(r ,  )d .
2 0
In order to calculate the exact expression of F10 the following formulas are used:
2
 cos
2 k 1
0
2
 cos
0
2k




 0, k  0,1,...


 sin 2  d  0, k  0,1,...
 sin  d   2 k
2
(10)
hence
F10 (r ) 
1
2
3
 b r
i 0
i even
i
i
i 1
.
Then the polynomial F10 (r ) has at most maximum positive roots, and the
coefficients bi with even can be chosen in such a way that F10 (r ) has exactly
simple positive roots or one simple positive root. Hence, by the averaging theory,
theorem 3 is shown.
Theorem 4: Assume that for k = 1 the polynomials f nk  x  , g mk  x  are of degrees
3, 2 respectively. By using the first order averaging theory with unique limit cycle
Cubic Liénard differential system
3167
then for  sufficiently small, the maximum number of limit cycles of the Lienard
system (1) bifurcates from the periodic orbits of the linear center x = y, y = -x .
Proof
This is based on the first-order averaging theory presented in the previous section.
Consider k  1 . The polynomials f ( x), g ( x) appearing in (1) can be written as
3
2
i 0
i 0
f ( x)   bi xi , g ( x)   ai x i
By means of the change of variables x  r cos  , y  r sin  , equation (1) in the
region r >0 can be written as
r  - sin  P(r ,  )





  -1- cos  P(r , ), 

r

(11)
where
m
n
i 0
i 0
P(r , )   ai r i cosi   bi r i 1 cos  i sin .
(12)
Now taking θ as the new independent variable, equation (11) becomes
dr
  sin  P(r ,  )  j ( 2 )
d
  F1 (r ,  )  j ( 2 ),
(13)
which is in the standard form of applying the averaging theory. Then by averaging
theory, we acquire
1
F10 (r ) 
2
2
 sin  P(r, )d .
0
In order to calculate the exact expression of F10 we use the following formulas:
3168
Hero Waisi Salih, Zainal Abdul Aziz and Faisal Salah
2


0

2

2 k 1
2

cos

sin

d


0,
k

0,1,...
0


2

2k
2
0 cos  sin  d   2k  0, k  0,1,...

2

k
3
0 cos  sin  d  0, k  0,1,... 
1 n
and hence F10 (r ) 
bi i r i 1.

2 i 0
k
 cos  sin  d  0,
k  0,1,...
(14)
i even
Then the polynomial F10 (r ) has at most maximum positive roots, and the
coefficients bi with i even can be chosen in such a way that F10 (r ) has exactly
simple positive roots or one simple positive root. Hence, by the averaging theory,
the theorem is proven.
Use of the uniqueness theorem
In order to verify the uniqueness of the maximum number of limit cycles
attained in the abovementioned theorems 3 and 4, one of the five theorems in [24]
is used.
In this work the system (2a) applies theorem 2 of the five theorems in [24],
where the statement of the theorem 2 in [24] is given as follows.
“Assume f ( x) even, g ( x) odd, and xg ( x)  0 for x  0 . If
 there exists x0  0 such that F ( x)  0 in (0, x0 );
 F ( x)  0 and increasing in ( x0 , );
 G()  F ()  ;
then the system has exactly one limit cycle.”
This theorem is applied for d negative, and if we choose x0  1 , then as a result it
is found that F ( x)and G( x) are
x
x
0
0
 f (s)ds,  g (s)ds
respectively. Thus the system
(2a) has a unique maximum number of limit cycles.
Similarly for system (2b), the abovementioned theorem 2 in [24] is applied
accordingly.
Cubic Liénard differential system
3169
This theorem is applied for d negative, with factors of g ( x) are positive and if
x0  1 is chosen, then as a result it is found that F ( x) and G( x) are
x
x

f ( s)ds,  g ( s)ds
0
0
respectively. Thus the system (2b) has a unique maximum
number of limit cycles.
Conclusion
The number of limit cycles of the cubic Liénard polynomial differential system of
the form x  y, y   g ( x)  f ( x) y is computed where f ( x) , is a polynomial of
degree three and g ( x) is polynomial of degree one and two. In particular, two
main theorems (theorem (3) and (4)) are proved to accomplish this objective.
Thus an accurate upper bound of the maximum number of limit cycles of this
differential system is obtained. By using the first order averaging theory, the
system is shown to bifurcate from the periodic orbits of the linear
center x  y, y  -x . The maximum number of limit cycles of the cubic Liénard
polynomial differential system is found to be unique via the G. Sansone’s
uniqueness theorem [24].
Acknowledgement
This research is partially funded by MOE FRGS Vot No. R.J130000.7809.4F354
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Received: March 3, 2014