Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 195, 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 FOR DISCONTINUOUS GENERALIZED LIENARD POLYNOMIAL DIFFERENTIAL EQUATIONS
JAUME LLIBRE, ANA CRISTINA MEREU
Abstract. We divideR2 into sectorsS1, . . . , Sl, withl >1 even, and define a discontinuous differential system such that in each sector, we have a smooth generalized Lienard polynomial differential equation ¨x+fi(x) ˙x+gi(x) = 0, i= 1,2 alternatively, wherefiandgiare polynomials of degreen−1 andm respectively. Then we apply the averaging theory for first-order discontinuous differential systems to show that for anynandmthere are non-smooth Lienard polynomial equations having at least max{n, m}limit cycles. Note that this number is independent of the number of sectors.
Roughly speaking this result shows that the non-smooth classical (m= 1) Lienard polynomial differential systems can have at least the double number of limit cycles than the smooth ones, and that the non-smooth generalized Lienard polynomial differential systems can have at least one more limit cycle than the smooth ones.
1. Introduction
A large number of problems from mechanics and electrical engineering, theory of automatic control, economy, impact systems among others cannot be described with smooth dynamical systems. This fact has motivated many researchers to the study of qualitative aspects of the phase space of non-smooth dynamical systems.
One of the main problems in the qualitative theory of real planar continuous and discontinuous differential systems is the determination of their limit cycles. The non-existence, existence, uniqueness and other properties of limit cycles have been studied extensively by mathematicians and physicists, and more recently also by chemists, biologists, economists, etc (see for instance the books [2, 5, 24]). This problem restricted to continuous planar polynomial differential equations is the well known 16th Hilbert’s problem [10]. Since this Hilbert’s problem turned out a strongly difficult one Smale [23] particularized it to Lienard polynomial differential equations in his list of problems for the present century.
The classical Lienard polynomial differential equations
¨
x+f(x) ˙x+g(x) = 0, (1.1)
where f(x) and g(x) =xgoes back to [11]. The dot denotes differentiation with respect to the time t. This second-order differential equation (1.1) can be written
2000Mathematics Subject Classification. 34C29, 34C25, 47H11.
Key words and phrases. Limit cycles; non-smooth Li´enard systems; averaging theory.
c
2013 Texas State University - San Marcos.
Submitted May 7, 2013. Published September 3, 2013.
1
as the following first-order differential system inR2
˙
x=y−F(x),
˙
y=−g(x), (1.2)
whereF(x) =Rx 0 f(s)ds.
Many results on the number of limit cycles has been obtained for the continuous generalized polynomial differential equations (1.1) beingf(x) andg(x) polynomials in the variable x of degrees n−1 and m respectively. The continuous classical Lienard polynomial differential equations (1.1) were studied in 1977 by Lins, de Melo and Pugh [12] who stated the conjecture:
Iff(x) has degree n−1>0 andg(x) =x, then (1.1) has at most [(n−1)/2] limit cycles.
Here [z] denotes the integer part function ofz∈R. They also proved the conjecture forn= 2,3. Forn= 4 this conjecture has been proved in 2012 (see [13]). Forn≥7 Dumortier, Panazzolo and Roussarie proved that this conjecture is not true in [7], they show that these differential equations can have [(n−1)/2] + 1 limit cycles.
Recently De Maesschalck and Dumortier proved in [22] that the classical Lienard equation of degreen≥6 can have [(n−1)/2] + 2 limit cycles. The conjecture for n= 5 is still open.
Results on the number of limit cycles for continuous generalized Lienard polyno- mial differential equations can be found in [14] where the authors show that there are differential equations (1.1) having at least [(n+m−2)/2] limit cycles. See also [1, 3, 6, 8, 18, 19, 20, 21, 25].
The objective of this work is to star the study of the number of limit cycles for a kind of discontinuous generalized Lienard polynomial differential systems.
Here we shall play with many straight lines of discontinuities through the origin of coordinates and with two different continuous generalized Lienard polynomial differential systems located alternatively in the sectors defined by these straight lines.
A similar work but with only one classical Lienard polynomial differential system and only one straight line of discontinuity was studied in [16] obtaining [n/2] limit cycles, instead of the [(n−1)/2] of the continuous classical Lienard polynomial differential equation obtained in [12].
Now we shall define the discontinuous generalized Lienard polynomial differential system that we will study. We consider the functionh:R2→Rdefined by
h(x, y) =
l 2−1
Y
k=0
y−tan α+2kπ l
x , wherel >1 even. The set
h−1(0) =∪k=02l−1{(x, y) :y = tan α+2kπ l
x},
divides R2 into l sectors, S1, S2, . . . , Sl, i.e. h−1(0) is the product ofl/2 straight lines passing through the origin of coordinates dividing the plane in sectors of angle 2π/l.
In this work we study the maximum number of limit cycles given by the averaging theory of first order, which can bifurcate from the periodic orbits of the linear
center ˙x=y, ˙y=−x, perturbed inside the following class of discontinuous Lienard polynomial differential systems
X˙ =Z(x, y) =
(Y1(x, y) ifh(x, y)>0,
Y2(x, y) ifh(x, y)<0, (1.3) where
Y1(x, y) =
y−εF1(x)
−x−εg1(x)
, Y2(x, y) =
y−εF2(x)
−x−εg2(x)
, (1.4)
where εis a small parameter, andFi(x) and gi(x), for i= 1,2 are polynomials in the variablex, and degreesnandmrespectively. System (1.3) can be written using the sign function as
X˙ =Z(x, y) =G1(x, y) + sgn(h(x, y))G2(x, y), (1.5) whereG1(x, y) =12(Y1(x, y) +Y2(x, y)) andG2(x, y) = 12(Y1(x, y)−Y2(x, y)).
Our main result reads as follows.
Theorem 1.1. Assume that for i = 1,2 the polynomials Fi(x) and gi(x) have degreen≥1andm≥1respectively, and thatl >1is even. Then for|ε|sufficiently small there are discontinuous Lienard polynomial differential systems (1.3)having at least max{n, m} limit cycles bifurcating from the periodic orbits of the linear centerx˙ =y,y˙=−x.
Taking into account Theorem 1.1 and roughly speaking, we can say that the non- smooth classical Lienard polynomial differential systems can have at least max{n,1}
limit cycles; i.e. the double number of limit cycles than the smooth ones which at least have [(n−1)/2] + 2 for n ≥ 6. Comparing the mentioned result from [14], that smooth generalized Lienard polynomial differential systems have at least [(n+m−1)/2] limit cycles with Theorem 1.1, we can say that the non-smooth generalized Lienard polynomial differential systems can have at least one more limit cycle than the smooth ones. Of course all these comparisons are done with the present known results.
2. Averaging theory for first-order discontinuous differential systems
The first-order averaging theory developed for discontinuous differential systems in [15] is presented in this section. It is summarized as follows.
Theorem 2.1. We consider the discontinuous differential system
x0(t) =εF(t, x) +ε2R(t, x, ε), (2.1) with
F(t, x) =F1(t, x) + sgn(h(t, x))F2(t, x), R(t, x, ε) =R1(t, x, ε) + sgn(h(t, x))R2(t, x, ε),
whereF1, F2:R×D→Rn,R1, R2:R×D×(−ε0, ε0)→Rn andh:R×D→R are continuous functions, T–periodic in the variablet andD is an open subset of Rn. We also suppose that his a C1 function having0 as a regular value. Denote by M=h−1(0), byΣ ={0} ×D *M, byΣ0 = Σ\M 6=∅, and its elements by z≡(0, z)∈ M./
Define the averaged functionf :D→Rn as f(x) =
Z T
0
F(t, x)dt.
We assume the following three conditions.
(i) F1, F2, R1, R2 andhare locally L-Lipschitz with respect to x;
(ii) for a ∈ Σ0 with f(a) = 0, there exist a neighborhood V of a such that f(z)6= 0for all z∈V\{a}anddB(f, V, a)6= 0, (i.e. the Brouwer degree of f atais not zero).
(iii) If ∂h/∂t(t0, z0) = 0for some(t0, z0)∈ M, then h∇xh, F1i2− h∇xh, F2i2
(t0, z0)>0.
Then, for |ε| > 0 sufficiently small, there exists a T-periodic solution x(·, ε) of system (2.1)such that x(t, ε)→aasε→0.
Remark 2.2. We note that if the functionf(z) isC1and the Jacobian of f ata is not zero, then dB(f, V, a)6= 0. For more details on the Brouwer degree see [4]
and [17].
3. Proof of Theorem 1.1
The discontinuous Lienard differential systems (1.3) in polar coordinates (r, θ) become
˙
r=−ε(cosθ Fi(rcosθ) + sinθ gi(rcosθ)), θ˙=−1 + ε
r(sinθ Fi(rcosθ)−cosθ gi(rcosθ)),
with i = 1 if sgn(h(rcosθ, rsinθ)) > 0 and i = 2 if sgn (h(rcosθ, rsinθ)) <
0. Taking the angle θ as new independent variable the discontinuous differential systems become
˙
r=ε(cosθ Fi(rcosθ) + sinθ gi(rcosθ)) +O(ε2). (3.1) This discontinuous differential system is studied under the assumptions of Theorem 2.1, taking
t=θ, T = 2π, x=r, M=h−1(0) =∪k=02l−1{(θ, r) :θ=α+2kπ
l , r >0}.
So according to Theorem 2.1 we must study the zeros of the averaged function f(r) =
l
X
k=1
hZ α+(2k−1)πl
α+2(k−1)πl
(cosθ F1(rcosθ) + sinθ g1(rcosθ))dθ
+
Z α+2kπl
α+(2k−1)πl
(cosθ F2(rcosθ) + sinθ g2(rcosθ))dθi .
(3.2)
Denoting F1(x) =
n
X
i=0
aixi, F2(x) =
n
X
i=0
bixi, g1(x) =
m
X
i=0
cixi, g2(x) =
m
X
i=0
dixi we have
f(r) =
l
X
k=1
hZ α+(2k−1)πl
α+2(k−1)πl
Xn
i=0
airicosi+1θ+
m
X
i=0
ciricosiθsinθ dθ
+
Z α+2kπl
α+(2k−1)πl
Xn
i=0
biricosi+1θ+
m
X
i=0
diricosiθsinθ dθi
.
To calculate the exact expression off(r) we use [9, formulae 2.513 3 and 2.513 4]:
Z
cos2mθ dθ= 1 22m
2m m
θ+ 1 22m−1
m−1
X
j=0
2m j
sin(2m−2j)θ 2m−2j , Z
cos2m+1θ dθ= 1 22m
m
X
j=0
2m+ 1 j
sin(2m−2j+ 1)θ 2m−2j+ 1 . Thus we have
f(r) =f1(r) +f2(r) +f3(r) +f4(r), where
f1(r) =
l
X
k=1
Z α+(2k−1)πl
α+2(k−1)πl
Xn
i=0
airicosi+1θ dθ
=
l
X
k=1
h Xn
i=1, iodd
airih 1 2i+1
i+ 1 (i+ 1)/2
π l + 1
2i
i−1 2
X
j=0
i+ 1 j
ϕi,j,k]
+
n
X
i=0, ieven
airi 2i
i 2
X
j=0
i+ 1 j
ϕi,j,k
i
=
l
X
k=1
[
n
X
i=1, iodd
airi 2i+1
i+ 1 (i+ 1)/2
π l +
n
X
i=0
airi 2i
[2i]
X
j=0
i+ 1 j
ϕi,j,k], with
ϕi,j,k= sin
(i−2j+ 1)
α+(2k−1)πl
−sin
(i−2j+ 1)
α+2(k−1)πl
i−2j+ 1 6= 0;
f2(r) =
l
X
k=1
Z α+(2k−1)πl
α+2(k−1)πl
Xm
i=0
ciricosiθsinθ dθ=−
l
X
k=1 m
X
i=0
ciri i+ 1φi,k, with
φi,k= cosi+1
α+(2k−1)π l
−cosi+1
α+2(k−1)π l
6= 0;
f3(r) =
l
X
k=1
Z α+(2k−1)πl
α+2(k−1)πl
Xn
i=0
biricosi+1θ dθ
=
l
X
k=1
h Xn
i=1, iodd
biri 2i+1
i+ 1 (i+ 1)/2
π l +
n
X
i=0
biri 2i
[2i]
X
j=0
i+ 1 j
ψi,j,k
i ,
with ψi,j,k=
sin (i−2j+ 1) α+2kπl
−sin
(i−2j+ 1) α+(2k−1)πl
i−2j+ 1 6= 0;
f4(r) =
l
X
k=1
Z α+2kπl
α+(2k−1)πl
Xm
i=0
diricosiθsinθ dθ=−
l
X
k=1 m
X
i=0
diri i+ 1ζi,k, with
ζi,k= cosi+1 α+2kπ l
−cosi+1 α+(2k−1)π l
6= 0.
Thus f(r) =
l
X
k=1
h Xn
i=1, iodd
ri 2i+1
i+ 1 (i+ 1)/2
π
l(ai+bi) +
n
X
i=0
ri 2i
[2i]
X
j=0
i+ 1 j
(aiϕi,j,k+biψi,j,k)−
m
X
i=0
ri
i+ 1(ciφi,k+diζi,k)i . The functionf(r) is a polynomial in the variablerof degree max{n, m}therefore f(r) has at most max{n, m}positive roots. Ifr∗is a simple zero off(r); i.e. f(r∗) = 0 and drdf
r=r∗6= 0, then the Brouwer degreedB(f, V, r∗)6= 0 being V a convenient open neighborhood ofr∗(see Remark 2.2). We can choose the coefficientsai,bi,ci
edi in such a way thatf(r) has exactly max{n, m} simple positive roots. Hence Theorem 1.1 is proved.
4. Examples
In this section we illustrate Theorem 1.1 by studying the existence of 2π-periodic solutions for two non-smooth Lienard polynomial differential systems.
Example 4.1. We consider l = 2 and α= 0. Thus the function h: R2 →R is defined byh(x, y) =y andh−1(0) ={(x, y)∈R2:y= 0}. System (1.3) becomes
X˙ =Z(x, y) =
(Y1(x, y) ify >0,
Y2(x, y) ify <0, (4.1) where
F1= 1 +x+x2+ 1 9π−1
x3, F2= 1 + 11 12π−1
x+x2+x3, g1= 7
8+x+5
8x2, g2= 1 +x+x2. Thus we have
Y1(x, y) =
y−ε 1 +x+x2+ 9π1 −1 x3
−x−ε 78 +x+58x2
, Y2(x, y) =
y−ε 1 + 12π11 −1
x+x2+x3
−x−ε(1 +x+x2)
. The averaging function (3.2) is
f(r) = Z π
0
(cosθ F1(rcosθ) + sinθ g1(rcosθ))dθ +
Z 2π
π
(cosθ F2(rcosθ) + sinθ g2(rcosθ))dθ
=−6 + 11r−6r2+r3.
The zeros of f(r) are r = 1, r = 2 and r = 3, and they are simple. Hence, by Theorem 1.1 it follows that forε6= 0 sufficiently small the discontinuous differential system (4.1) has three periodic solutions.
Example 4.2. We considerl = 4 andα=π/4. Thus the function h:R2→Ris defined byh(x, y) = (y−x)(y+x) andh−1(0) ={(x, y) :y=x}∪{(x, y) :y=−x}.
System (1.3) becomes
X˙ =Z(x, y) =
(Y1(x, y) if (y−x)(y+x)>0,
Y2(x, y) if (y−x)(y+x)<0, (4.2) whereF1=x2,F2= 12√
2π+72
√ 2
5 x2,g1= 1+x+x2+x3andg2=−88πx−32π3 x3. Thus we have
Y1(x, y) =
y−εx2
−x−ε(1 +x+x2+x3)
, Y2(x, y) =
y−ε 12√ 2π+72
√ 2 5 x2
−x−ε −88πx−32π3 x3
. The averaging function (3.2) is
f(r) = Z 3π/4
π/4
(cosθ F1(rcosθ) + sinθ g1(rcosθ))dθ +
Z 5π/4
3π/4
(cosθ F2(rcosθ) + sinθ g2(rcosθ))dθ +
Z 7π/4
5π/4
(cosθ F1(rcosθ) + sinθ g1(rcosθ))dθ +
Z 9π/4
7π/4
(cosθ F2(rcosθ) + sinθ g2(rcosθ))dθ
=−6 + 11r−6r2+r3.
The zeros of f(r) are r = 1, r = 2 and r = 3, and they are simple. Hence, by Theorem 1.1 it follows that forε6= 0 sufficiently small the discontinuous differential system (4.2) has three periodic solutions.
Acknowledgments. The first author is partially supported by a MICINN/FEDER grant MTM 2008–03437, an AGAUR grant number 2009SGR-0410, an ICREA Academia, and FP7-PEOPLE-2012-IRSES-316338 and 318999. The second author is partially supported by a FAPESP-BRAZIL grant 2012/20884-8. Both authors are also supported by the joint project CAPES–MECD grant PHB-2009-0025-PC.
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Jaume Llibre
Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
E-mail address:[email protected]
Ana Cristina Mereu
Department of Physics, Chemistry and Mathematics, UFSCar, 18052-780, Sorocaba, SP, Brazil
E-mail address:[email protected]
