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

1UTM Centre for Industrial and Applied Mathematics &

2Department of Mathematical Sciences, Faculty of Science,

Universiti Teknologi Malaysia,

81310 UTM Johor Bahru, Johor, Malaysia, 3Department 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

3162 Hero Waisi Salih, Zainal Abdul Aziz and Faisal Salah

- ( ) - ( ) (1)

x y

y g x f x y

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

( ) (2a)

( ) ( ) (2b)

x y

y x f x y

x y

y g x f x y

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 1n then

system (1) has at most [ ]2

n limit cycles. They proved this conjecture for 1,2n .

The conjecture for 3 n 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 4n .

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. In 1928, Liénard [15] proved that if 1 m and0

( ) ( )x

F x f s ds is a

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 3163

ii. In 1973, Rychkov [23] proved that if 1 m and ( )F x is an odd

polynomial of degree five, then system (2a) has at most two limit

cycles.

iii. In 1977, Lins et al. [1] proved that (1,1) 0H and (1,2) 1H .

iv. In 1998, Coppel [7] proved that (2,1) 1H .

v. Dumortier et al. in [9, 12] proved that (3,1) 1H .

vi. In 1997 Dumortier and Li [10] proved that (2,2) 1H

vii. In 2011 Chengzi and Llibre [14] proved that (1,3) 1H .

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 ˆ ( , ).H m n Blows and Lloyd [4], Lloyd and Lynch [18] and Lynch [19] have used

inductive arguments to prove the following results.

I. If g is odd then ˆ ( , ) [ ]2

nH m n .

II. If f is even then ˆ ( , ) .H m n n

III. If f is odd then- 2ˆ ( ,2 1) [ ] 22

mH m n

IV. If ( ) ( )eg x x g x , where eg is even then ˆ (2 ,2)H m 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 ˆ ( , )H m n of different degrees m and n :

V. 2 1ˆ ( ,2) [ ].

3

mH m

VI. 2 1ˆ (2, ) [ ]

3

nH n

.

VII. 2 2ˆ ( ,3) 2[ ]

8

mH m

for all 1 50m .

VIII. 2 2ˆ (3, ) 2[ ]

8

nH n

for all1 50n

IX. ˆ ˆ(4, ) ( ,4), 6,7,8,9H k H k k and ˆ ˆ(5,6) (6,5)H H

In 1998, Gasull and Torregrosa [13] obtained the upper bounds for ˆ ˆ ˆ ˆ(7,6) , (6,7), (7,7), and (4,20).H H H H In 2006, Yu and Han in [25] showed

3164 Hero Waisi Salih, Zainal Abdul Aziz and Faisal Salah

ˆ ˆ( , ) ( , )H m n H n m , for 4, 10,11,12,13;n m 5, 6,7,8,9; n m

6, 5,6n m , 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

(2 ,2)kH m 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 1,2,3k , and

where ( , )kH 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

2

11 11 12 12

2

21 21 22 22

- ( ( ) ( ) ) - ( ( ) ( ) ),

- - ( ( ) ( ) ) - ( ( ) ( ) ), (3)

x y g x f x y g x f x y

y x g x f x y g x f x y

where 1 1 2 2, , ,i i i ig f g f have degree , , and l k m n respectively for each 1,2i 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

-1 -1 -1 -1max{ [ ], [ ],[ ] [ ],[ ] [ ] -1,[ ] [ ]

2 2 2 2 2 2 2 2

-11,[ ] [ ]}, (4)

2 2

m l n m k m n l

k l

with-1

min{[ ],[ ]}.2 2

n k

In Alavez-Ramirez et al. [2], they considered the polynomial differential

system

2

11 12

2

21 21 22 22

- ( ) - ( ),

- - ( ( ) ( ) ) - ( ( ) ( ) ), (5)

x y g x f x

y x g x f x y g x f x y

where 1 2 2, , i i ig g f have degree ,l m and n respectively for 1,2i 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 3165

1(max{ ( ), ( ) -1}-1)

2O m n E l m

where ( ) O k is the largest odd integer k , and ( )E k is the largest even

integer k .

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

2

1( ) ( , ) ( , , ) (6)x t F t x R t x

where1 : , : (- , ) n n

f fF D R R R R D R are continuous functions, T-

periodic (of time T) in the first variable, and D is an open subset ofnR . Assume

that the following conditions hold.

i. 1F and R are locally Lipschitz with respect to x . We define

10 : nF D R as

10 1

0

1( ) ( , )

T

F z F z s dsT

ii. For a D with 10 ( ) 0F a there exists a neighbourhood V of a such that

10 ( ) 0F z for all { } z V a and 10( , , ) 0Bd F V a .

Then for | |>0 sufficiently small, there exists a T-periodic solution

(., ) 0a as

Theorem and main result

Theorem 3. Assume that for 1 k the polynomials ( )k

nf x have degree 3n ,

with 1n , 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.

3166 Hero Waisi Salih, Zainal Abdul Aziz and Faisal Salah

Consider 1k and write the polynomials 3 ( )f x appearing in (3) as

3

0

( ) .i

i

i

f x a x

By means of the change of variables cos , sin x r y r , system (3) in the

region 0 r can be written as

- sin ( , ),

-1- cos ( , ) (7)

r P r

P rr

where 3

1

0

( , ) cos sin . (8)i i

i

i

P r a r

Now taking as the new independent variable, system (7) becomes

2 2

1sin ( , ) ( ) ( , ) ( ), (9)dr

P r F rd

which is the standard form for applying the averaging theory. Then by the

averaging theory it is obtained 2

10

0

1( ) sin ( , ) .

2F r P r d

In order to calculate the exact expression of 10F the following formulas are used:

2

2 1 2

0

2

2 2

2

0

cos sin 0, 0,1,...

(10)

cos sin 0, 0,1,...

k

k

k

d k

d k

hence 3

1

100

1( ) .

2

i

i ii

i even

F r b r

Then the polynomial 10 ( )F r has at most maximum positive roots, and the

coefficients ib with even can be chosen in such a way that 10 ( )F 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 k k

n mf x , g 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 1k . The polynomials ( ), ( )f x g x appearing in (1) can be written as

3 2

0 0

( ) , ( )i i

i i

i i

f x b x g x a x

By means of the change of variables cosx r , siny r , equation (1) in the

region r >0 can be written as

- sin ( , )

(11)

-1- cos ( , ),

r P r

P rr

where

1

0 0

( , ) cos cos sin . (12)m n

i i i i

i i

i i

P r a r b r

Now taking θ as the new independent variable, equation (11) becomes

2

2

1

sin ( , ) ( )

( , ) ( ), (13)

drP r j

d

F r j

which is in the standard form of applying the averaging theory. Then by averaging

theory, we acquire

2

10

0

1( ) sin ( , ) .

2F r P r d

In order to calculate the exact expression of 10F we use the following formulas:

3168 Hero Waisi Salih, Zainal Abdul Aziz and Faisal Salah

2

0

2

2 1 2

0

2

2 2

2

0

2

3

0

cos sin 0, 0,1,...

cos sin 0, 0,1,...

(14)

cos sin 0, 0,1,...

cos sin 0, 0,1,...

k

k

k

k

k

d k

d k

d k

d k

and hence 1

10

0

1( ) .

2

ni

i i

ii even

F r b r

Then the polynomial 10 ( )F r has at most maximum positive roots, and the

coefficients ib with i even can be chosen in such a way that 10 ( )F 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 ( ) 0 for 0xg x x . If

there exists 0 0 x such that 0( ) 0 (0, );F x in x

( ) 0F x and increasing in 0( , );x

( ) ( ) ;G F

then the system has exactly one limit cycle.”

This theorem is applied for d negative, and if we choose 0 1 ,x then as a result it

is found that ( )and ( ) F x G x are 0 0

( ) , ( )

x x

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

0 1 x is chosen, then as a result it is found that ( ) and ( ) F x G x are

0 0

( ) , ( )

x x

f s ds g s ds 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