CUBO A Mathematical Journal

Vol.10, N
o
¯ 03, (115–132). October 2008

Limit Cycles of Liénard-Type Dynamical Systems

Valery A. Gaiko∗

Department of Mathematics,

Belarusian State University of Informatics and Radioelectronics,

L. Beda Str. 6-4, Minsk 220040 – Belarus

email: valery.gaiko@yahoo.com

ABSTRACT

In this paper, using geometric properties of the field rotation parameters, we present

a solution of Smale’s Thirteenth Problem on the maximum number of limit cycles

for Liénard’s polynomial system, generalize the obtained results for special classes of

polynomial systems, and complete the global qualitative analysis of a piecewise linear

dynamical system approximating a Liénard-type polynomial system with an arbitrary

number of finite singularities.

RESUMEN

En este art́ıculo, usando propriedades geometricas del campo de rotación de parametros,

nosotros presentamos una solución del problema trece de Smale sobre el número máximo

de ciclos ĺımite para el sistema polinomial de Liénard, generalizamos los resultados

obtenidos para clases especiales de sistemas polinomiales, y completamos el analisis

cualitativo global de un sistema dinamico linear por pedazos aproximando un sistema

polinomial de tipo Liérnard con un número arbitrario finito de singularidades.

∗This work was supported by the Netherlands Organization for Scientific Research (NWO).



116 Valery A. Gaiko CUBO
10, 3 (2008)

Key words and phrases: Planar polynomial dynamical system, Liénard’s polynomial system,

generalized Liénard’s cubic system, piecewise linear Liénard-type dynamical system, Hilbert’s six-

teenth problem, Smale’s thirteenth problem, field rotation parameter, bifurcation, limit cycle.

Math. Subj. Class.: 34C05, 34C07, 34C23, 37G05, 37G10, 37G15.

1 Introduction

We consider planar dynamical systems

.
x = Pn(x,y),

.
y = Qn(x,y), (1.1)

where Pn(x,y) and Qn(x,y) are polynomials with real coefficients in the real variables x, y and

not greater than n degree. First of all, we consider a special case of (1.1): a classical Liénard’s

polynomial system of the form

.
x = y,

.
y = −x + µ1 y + µ2 y

2 + µ3 y
3 + . . . + µ2k y

2k + µ2k+1 y
2k+1. (1.2)

The main problem of qualitative theory of such systems is Hilbert’s Sixteenth Problem on the

maximum number and relative position of their limit cycles, i. e., closed isolated trajectories of

(1.1). This problem was formulated as one of the fundamental problems for mathematicians of

the XX century, however it has not been solved even in the simplest (quadratic, cubic, etc.) cases

of the polynomial systems. In this paper, we suggest a new geometric approach to solving the

problem in the case of Liénard’s system (1.2). In this special case, it is considered as Smale’s

Thirteenth Problem becoming one of the main problems for mathematicians of the XXI century

[16], [20].

In Section 2 of this paper, applying a canonical system with field rotation parameters and using

geometric properties of the spirals filling the interior and exterior domains of limit cycles, we present

a solution of Smale’s Thirteenth Problem for Liénard’s polynomial system (1.2). In Section 3, by

means of the same geometric approach, we generalize the obtained result and present a solution

of Hilbert’s sixteenth problem on the maximum number of limit cycles surrounding a singular

point for an arbitrary polynomial system. In Section 4, we consider generalized Liénard’s cubic

system with three finite singularities, for which the developed geometric approach can complete

its global qualitative analysis: in particular, it easily solves the problem on the maximum number

of limit cycles in their different distribution. In this section, we give also an alternative proof of

the main theorem for the generalized Liénard’s system applying the Wintner–Perko termination

principle for multiple limit cycles. In Section 5, by means of the same principle, we complete the

global qualitative analysis of a piecewise linear dynamical system approximating a Liénard-type

polynomial system with an arbitrary number of finite singularities.



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Limit Cycles of Liénard-Type Dynamical Systems 117

2 Liénard’s polynomial system

System (1.2) and more general Liénard’s systems have been studied in numerous works (see, for

example, [2], [16], [17], [19], [20]). It is easy to see that (1.2) has the only finite singularity: an

anti-saddle at the origin. At infinity, system (1.2) for k > 1 has two singular points: a node at the

“ends” of the y-axis and a saddle at the “ends” of the x-axis. For studying the infinite singularities,

the methods applied in [2] for Rayleigh’s and van der Pol’s equations and also Erugin’s two-isocline

method developed in [10] can be used. Following [10], we will study limit cycle bifurcations of (1.2)

by means of a canonical system containing only the field rotation parameters of (1.2). It is valid

the following theorem.

Theorem 2.1. Liénard’s polynomial system (1.2) with limit cycles can be reduced to the canonical

form
.
x = y ≡ P,

.
y = −x + µ1 y + y

2 + µ3 y
3 + . . . + y2k + µ2k+1 y

2k+1 ≡ Q, (2.1)

where µ1, µ3, . . . , µ2k+1 are field rotation parameters of (2.1).

Proof. Vanish all odd parameters of (1.2),

.
x = y,

.
y = −x + µ2 y

2 + µ4 y
4 + . . . + µ2k y

2k, (2.2)

and consider the corresponding equation

dy

dx
=

−x + µ2 y
2 + µ4 y

4 + . . . + µ2k y
2k

y
≡ F(x,y). (2.3)

Since F(x,−y) = −F(x,y), the direction field of (2.3) (and the vector field of (2.2) as well) is

symmetric with respect to the x-axis. It follows that for arbitrary values of the parameters µ2,

µ4, . . . , µ2k system (2.2) has a center at the origin and cannot have a limit cycle surrounding this

point. Therefore, without loss of generality, all even parameters of system (1.2) can be supposed

to be equal, for example, to one: µ2 = µ4 = . . . = µ2k = 1 (they could be also supposed to be

equal to zero).

To prove that the rest (odd) parameters rotate the vector field of (2.1), let us calculate the

following determinants:

∆µ1 = PQ
′

µ1
− QP ′µ1 = y

2 ≥ 0,

∆µ3 = PQ
′

µ3
− QP ′µ3 = y

2 ≥ 0,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∆µ2k+1 = PQ
′

µ2k+1
− QP ′µ2k+1 = y

2 ≥ 0.

By definition of a field rotation parameter [4], for increasing each of the parameters µ1, µ3, . . . ,

µ2k+1, under the fixed others, the vector field of system (2.1) is rotated in positive direction



118 Valery A. Gaiko CUBO
10, 3 (2008)

(counterclockwise) in the whole phase plane; and, conversely, for decreasing each of these param-

eters, the vector field of (2.1) is rotated in negative direction (clockwise).

Thus, for studying limit cycle bifurcations of (1.2), it is sufficient to consider canonical system

(2.1) containing only its odd parameters, µ1, µ3, . . . , µ2k+1, which rotate the vector field of (2.1).

The theorem is proved. 2

By means of canonical system (2.1), let us study global limit cycle bifurcations of (1.2) and

prove the following theorem.

Theorem 2.2. Liénard’s polynomial system (1.2) has at most k limit cycles.

Proof. According to Theorem 2.1, for the study of limit cycle bifurcations of system (1.2), it is

sufficient to consider canonical system (2.1) containing only the field rotation parameters of (1.2):

µ1, µ3, . . . , µ2k+1.

Vanish all these parameters:

.
x = y,

.
y = −x + y2 + y4 + . . . + y2k. (2.4)

System (2.4) is symmetric with respect to the x-axis and has a center at the origin. Let us input

successively the field rotation parameters into this system beginning with the parameters at the

highest degrees of y and alternating with their signs. So, begin with the parameter µ2k+1 and let,

for definiteness, µ2k+1 > 0 :

.
x = y,

.
y = −x + y2 + y4 + . . . + y2k + µ2k+1 y

2k+1. (2.5)

In this case, the vector field of (2.5) is rotated in positive direction (counterclockwise) turning the

origin into a nonrough unstable focus.

Fix µ2k+1 and input the parameter µ2k−1 < 0 into (2.5):

.
x = y,

.
y = −x + y2 + y4 + . . . + µ2k−1 y

2k−1 + y2k + µ2k+1 y
2k+1. (2.6)

Then the vector field of (2.6) is rotated in opposite direction (clockwise) and the focus immediately

changes the character of its stability (since its degree of nonroughness decreases and the sign of

the field rotation parameter at the lower degree of y changes) generating a stable limit cycle.

Under further decreasing µ2k−1, this limit cycle will expand infinitely, not disappearing at infinity

(because of the parameter µ2k+1 at the higher degree of y).

Denote the limit cycle by Γ1, the domain outside the cycle by D1, the domain inside the cycle

by D2 and consider logical possibilities of the appearance of other (semi-stable) limit cycles from a

“trajectory concentration” surrounding the origin. It is clear that, under decreasing the parameter

µ2k−1, a semi-stable limit cycle cannot appear in the domain D2, since the focus spirals filling this

domain will untwist and the distance between their coils will increase because of the vector field

rotation.



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Limit Cycles of Liénard-Type Dynamical Systems 119

By contradiction, we can also prove that a semi-stable limit cycle cannot appear in the domain

D1. Suppose it appears in this domain for some values of the parameters µ
∗

2k+1
> 0 and µ∗

2k−1
< 0.

Return to initial system (2.4) and change the inputting order for the field rotation parameters.

Input first the parameter µ2k−1 < 0 :

.
x = y,

.
y = −x + y2 + y4 + . . . + µ2k−1 y

2k−1 + y2k. (2.7)

Fix it under µ2k−1 = µ
∗

2k−1
. The vector field of (2.7) is rotated clockwise and the origin turns into

a nonrough stable focus. Inputting the parameter µ2k+1 > 0 into (2.7), we get again system (2.6),

the vector field of which is rotated counterclockwise. Under this rotation, a stable limit cycle Γ1

will immediately appear from infinity, more precisely, from a separatrix cycle of the Poincaré circle

form containing infinite singularities of the saddle and node types [2]. This cycle will contract,

the outside spirals winding onto the cycle will untwist and the distance between their coils will

increase under increasing µ2k+1 to the value µ
∗

2k+1
. It follows that there are no values of µ∗

2k−1
< 0

and µ∗
2k+1

> 0, for which a semi-stable limit cycle could appear in the domain D1.

This contradiction proves the uniqueness of a limit cycle surrounding the origin in system (2.6)

for any values of the parameters µ2k−1 and µ2k+1 of different signs. Obviously, if these parameters

have the same sign, system (2.6) has no limit cycles surrounding the origin at all.

Let system (2.6) have the unique limit cycle Γ1. Fix the parameters µ2k+1 >0, µ2k−1 <0 and

input the third parameter, µ2k−3 > 0, into this system:

.
x = y,

.
y = −x + y2 + . . . + µ2k−3 y

2k−3 + y2k−2 + . . . + µ2k+1 y
2k+1. (2.8)

The vector field of (2.8) is rotated counterclockwise, the focus at the origin changes the character of

its stability and the second (unstable) limit cycle, Γ2, immediately appears from this point. Under

further increasing µ2k−3, the limit cycle Γ2 will join with Γ1 forming a semi-stable limit cycle, Γ12,

which will disappear in a “trajectory concentration” surrounding the origin. Can another semi-

stable limit cycle appear around the origin in addition to Γ12? It is clear that such a limit cycle

cannot appear either in the domain D1 bounded on the inside by the cycle Γ1 or in the domain

D3 bounded by the origin and Γ2 because of the increasing distance between the spiral coils filling

these domains under increasing the parameter µ2k−3.

To prove impossibility of the appearance of a semi-stable limit cycle in the domain D2 bounded

by the cycles Γ1 and Γ2 (before their joining), suppose the contrary, i. e., for some set of values of

the parameters, µ∗
2k+1

> 0, µ∗
2k−1

< 0, and µ∗
2k−3

> 0, such a semi-stable cycle exists. Return to

system (2.4) again and input first the parameters µ2k−3 > 0 and µ2k+1 > 0 :

.
x = y,

.
y = −x + y2 + . . . + µ2k−3 y

2k−3 + y2k−2 + y2k + µ2k+1 y
2k+1. (2.9)

Both parameters act in a similar way: they rotate the vector field of (2.9) counterclockwise turning

the origin into a nonrough unstable focus.

Fix these parameters under µ2k−3 = µ
∗

2k−3
, µ2k+1 = µ

∗

2k+1
and input the parameter µ2k−1 < 0

into (2.9) getting again system (2.8). Since, by our assumption, this system has two limit cycles



120 Valery A. Gaiko CUBO
10, 3 (2008)

for µ2k−1 > µ
∗

2k−1
, there exists some value of the parameter, µ12

2k−1
(µ∗

2k−1
< µ12

2k−1
< 0), for

which a semi-stable limit cycle, Γ12, appears in system (2.8) and then splits into a stable cycle,

Γ1, and an unstable cycle, Γ2, under further decreasing µ2k−1. The formed domain D2 bounded

by the limit cycles Γ1, Γ2 and filled by the spirals will enlarge since, on the properties of a field

rotation parameter, the interior unstable limit cycle Γ2 will contract and the exterior stable limit

cycle Γ1 will expand under decreasing µ2k−1. The distance between the spirals of the domain D2

will naturally increase, what will prevent the appearance of a semi-stable limit cycle in this domain

for µ2k−1 < µ
12

2k−1
.

Thus, there are no such values of the parameters, µ∗
2k+1

> 0, µ∗
2k−1

< 0, and µ∗
2k−3

> 0, for

which system (2.8) would have an additional semi-stable limit cycle. Obviously, there are no other

values of the parameters µ2k+1, µ2k−1, and µ2k−3 for which system (2.8) would have more than

two limit cycles surrounding the origin. Therefore, two is the maximum number of limit cycles for

system (2.8). This result agrees with [19], where it was proved for the first time that the maximum

number of limit cycles for Liénard’s system of the form

.
x = y,

.
y = −x + µ1 y + µ3 y

3 + µ5 y
5 (2.10)

was equal to two.

Suppose that system (2.8) has two limit cycles, Γ1 and Γ2 (this is always possible if µ2k+1 ≫

−µ2k−1 ≫ µ2k−3 > 0), fix the parameters µ2k+1, µ2k−1, µ2k−3 and consider a more general system

than (2.8) (and (2.10)) inputting the fourth parameter, µ2k−5 < 0, into (2.8):

.
x = y,

.
y = −x + y2 + . . . + µ2k−5 y

2k−5 + y2k−4 + . . . + µ2k+1 y
2k+1. (2.11)

Under decreasing µ2k−5, the vector field of (2.11) will be rotated clockwise and the focus at the

origin will immediately change the character of its stability generating the third (stable) limit

cycle, Γ3. Under further decreasing µ2k−5, Γ3 will join with Γ2 forming a semi-stable limit cycle,

Γ23, which will disappear in a “trajectory concentration” surrounding the origin; the cycle Γ1 will

expand infinitely tending to the Poincaré circle at infinity.

Let system (2.11) have three limit cycles: Γ1, Γ2, Γ3. Could an additional semi-stable limit

cycle appear under decreasing µ2k−5, after splitting of which system (2.11) would have five limit

cycles around the origin? It is clear that such a limit cycle cannot appear either in the domain

D2 bounded by the cycles Γ1 and Γ2 or in the domain D4 bounded by the origin and Γ3 because

of the increasing distance between the spiral coils filling these domains under decreasing µ2k−5.

Consider two other domains: D1 bounded on the inside by the cycle Γ1 and D3 bounded by the

cycles Γ2 and Γ3. As before, we will prove impossibility of the appearance of a semi-stable limit

cycle in these domains by contradiction.

Suppose that for some set of values of the parameters µ∗
2k+1

> 0, µ∗
2k−1

< 0, µ∗
2k−3

> 0, and

µ∗
2k−5

< 0, such a semi-stable cycle exists. Return to system (2.4) again, input first the parameters

µ2k−5 < 0, µ2k−1 < 0 and then the parameter µ2k+1 > 0 :

.
x = y,

.
y = −x+y2+. . .+µ2k−5y

2k−5 +. . .+µ2k−1y
2k−1 +y2k +µ2k+1y

2k+1
. (2.12)



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Limit Cycles of Liénard-Type Dynamical Systems 121

Fix the parameters µ2k−5, µ2k−1 under the values µ
∗

2k−5
, µ∗

2k−1
, respectively. Under increasing

µ2k+1, the node at infinity will change the character of its stability, the separatrix behavior of the

infinite saddle will be also changed and a stable limit cycle, Γ1, will immediately appear from the

Poincaré circle at infinity [2]. Fix µ2k+1 under the value µ
∗

2k+1
and input the parameter µ2k−3 > 0

into (2.12) getting system (2.11).

Since, by our assumption, (2.11) has three limit cycles for µ2k−3 < µ
∗

2k−3
, there exists some

value of the parameter µ23
2k−3

(0 < µ23
2k−3

< µ∗
2k−3

) for which a semi-stable limit cycle, Γ23, appears

in this system and then splits into an unstable cycle, Γ2, and a stable cycle, Γ3, under further

increasing µ2k−3. The formed domain D3 bounded by the limit cycles Γ2, Γ3 and also the domain

D1 bounded on the inside by the limit cycle Γ1 will enlarge and the spirals filling these domains

will untwist excluding a possibility of the appearance of a semi-stable limit cycle there.

All other combinations of the parameters µ2k+1, µ2k−1, µ2k−3, and µ2k−5 are considered in

a similar way. It follows that system (2.11) has at most three limit cycles. If we continue the

procedure of successive inputting the odd parameters, µ2k−7, . . . , µ3, µ1, into system (2.4), it is

possible first to obtain k limit cycles (µ2k+1 ≫ −µ2k−1 ≫ µ2k−3 ≫ −µ2k−5 ≫ µ2k−7 ≫ . . .) and

then to conclude that canonical system (2.1) (i. e., Liénard’s polynomial system (1.2) as well) has

at most k limit cycles. The theorem is proved. 2

3 An arbitrary polynomial system

Let us consider an arbitrary polynomial system

.
x = Pn(x,y,µ1, . . . ,µk),

.
y = Qn(x,y,µ1, . . . ,µk), (3.1)

where Pn and Qn are polynomials in the real variables x, y and not greater than n degree containing

k field rotation parameters, µ1, . . . ,µk, and having an anti-saddle at the origin. Generalizing the

main result of the previous section, we will prove the following theorem.

Theorem 3.1. Polynomial system (3.1) containing k field rotation parameters and having a

singular point of the center type at the origin for the zero values of these parameters can have at

most k − 1 limit cycles surrounding the origin.

Proof. Vanish all parameters of (3.1) and suppose that the obtained system

.
x = Pn(x,y, 0, . . . , 0),

.
y = Qn(x,y, 0, . . . , 0) (3.2)

has a singular point of the center type at the origin. Let us input successively the field rotation

parameters, µ1, . . . ,µk, into this system.

Suppose, for example, that µ1 > 0 and that the vector field of the system

.
x = Pn(x,y,µ1, 0, . . . , 0),

.
y = Qn(x,y,µ1, 0, . . . , 0) (3.3)



122 Valery A. Gaiko CUBO
10, 3 (2008)

is rotated counterclockwise turning the origin into a stable focus under increasing µ1.

Fix µ1 and input the parameter µ2 into (3.3) changing it so that the field of the system

.
x = Pn(x,y,µ1,µ2, 0, . . . , 0),

.
y = Qn(x,y,µ1,µ2, 0, . . . , 0) (3.4)

would be rotated in opposite direction (clockwise). Let be so for µ2 < 0. Then, for some value of

this parameter, a limit cycle will appear in system (3.4). There are three logical possibilities for

such a bifurcation: 1) the limit cycle appears from the focus at the origin; 2) it can also appear from

some separatrix cycle surrounding the origin; 3) the limit cycle appears from a so-called “trajectory

concentration”. In the last case, the limit cycle is semi-stable and, under further decreasing µ2,

it splits into two limit cycles (stable and unstable), one of which then disappears at (or tends

to) the origin and the other disappears on (or tends to) some separatrix cycle surrounding this

point. But since the stability character of both a singular point and a separatrix cycle is quite

easily controlled [10], this logical possibility can be excluded. Let us choose one of the two other

possibilities: for example, the first one, the so-called Andronov–Hopf bifurcation. Suppose that, for

some value of µ2, the focus at the origin becomes non-rough, changes the character of its stability

and generates a stable limit cycle, Γ1.

Under further decreasing µ2, three new logical possibilities can arise: 1) the limit cycle Γ1

disappears on some separatrix cycle surrounding the origin; 2) a separatrix cycle can be formed

earlier than Γ1 disappears on it, then it generates one more (unstable) limit cycle, Γ2, which joins

with Γ1 forming a semi-stable limit cycle, Γ12, disappearing in a “trajectory concentration” under

further decreasing µ2; 3) in the domain D1 outside the cycle Γ1 or in the domain D2 inside Γ1,

a semi-stable limit cycle appears from a “trajectory concentration” and then splits into two limit

cycles (logically, the appearance of such semi-stable limit cycles can be repeated).

Let us consider the third case. It is clear that, under decreasing µ2, a semi-stable limit cycle

cannot appear in the domain D2, since the focus spirals filling this domain will untwist and the

distance between their coils will increase because of the vector field rotation. By contradiction, we

can prove that a semi-stable limit cycle cannot appear in the domain D1. Suppose it appears in

this domain for some values of the parameters µ∗
1
> 0 and µ∗

2
< 0. Return to initial system (3.2)

and change the inputting order for the field rotation parameters. Input first the parameter µ2 < 0 :

.
x = Pn(x,y,µ2, 0, . . . , 0),

.
y = Qn(x,y,µ2, 0, . . . , 0). (3.5)

Fix it under µ2 = µ
∗

2
. The vector field of (3.5) is rotated clockwise and the origin turns into a

unstable focus. Inputting the parameter µ1 > 0 into (3.5), we get again system (3.4), the vector

field of which is rotated counterclockwise. Under this rotation, a stable limit cycle, Γ1, will appear

from some separatrix cycle. The limit cycle Γ1 will contract, the outside spirals winding onto this

cycle will untwist and the distance between their coils will increase under increasing µ1 to the value

µ∗
1
. It follows that there are no values of µ∗

2
< 0 and µ∗

1
> 0, for which a semi-stable limit cycle

could appear in the domain D1.

The second logical possibility can be excluded by controlling the stability character of the



CUBO
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Limit Cycles of Liénard-Type Dynamical Systems 123

separatrix cycle [10]. Thus, only the first possibility is valid, i. e., system (3.4) has at most one

limit cycle.

Let system (3.4) have the unique limit cycle Γ1. Fix the parameters µ1 > 0, µ2 < 0 and

input the third parameter, µ3 > 0, into this system supposing that µ3 rotates its vector field

counterclockwise:

.
x = Pn(x,y,µ1,µ2,µ3, 0, . . . , 0),

.
y = Qn(x,y,µ1,µ2,µ3, 0, . . . , 0). (3.6)

Here we can have two basic possibilities: 1) the limit cycle Γ1 disappears at the origin; 2) the second

(unstable) limit cycle, Γ2, appears from the origin and, under further increasing the parameter µ3,

the cycle Γ2 joins with Γ1 forming a semi-stable limit cycle, Γ12, which disappears in a “trajectory

concentration” surrounding the origin. Besides, we can also suggest that: 3) in the domain D2

bounded by the origin and Γ1, a semi-stable limit cycle, Γ23, appears from a “trajectory concentra-

tion”, splits into an unstable cycle, Γ2, and a stable cycle, Γ3, and then the cycles Γ1, Γ2 disappear

through a semi-stable limit cycle, Γ12, and the cycle Γ3 disappears through the Andronov–Hopf

bifurcation; 4) a semi-stable limit cycle, Γ34, appears in the domain D2 bounded by the cycles Γ1,

Γ2 and, for some set of values of the parameters, µ
∗

1
, µ∗

2
, µ∗

3
, system (3.6) has at least four limit

cycles.

Let us consider the last, fourth, case. It is clear that a semi-stable limit cycle cannot appear

either in the domain D1 bounded on the inside by the cycle Γ1 or in the domain D3 bounded by

the origin and Γ2 because of the increasing distance between the spiral coils filling these domains

under increasing the parameter µ3. To prove impossibility of the appearance of a semi-stable limit

cycle in the domain D2, suppose the contrary, i. e., for some set of values of the parameters, µ
∗

1
> 0,

µ∗
2
< 0, and µ∗

3
> 0, such a semi-stable cycle exists. Return to system (3.2) again and input first

the parameters µ3 > 0, µ1 > 0 :

.
x = Pn(x,y,µ1,µ3, 0, . . . , 0),

.
y = Qn(x,y,µ1,µ3, 0, . . . , 0). (3.7)

Fix these parameters under µ3 = µ
∗

3
, µ1 = µ

∗

1
and input the parameter µ2 < 0 into (3.7) getting

again system (3.6). Since, by our assumption, this system has two limit cycles for µ2 > µ
∗

2
, there

exists some value of the parameter, µ12
2

(µ∗
2
< µ12

2
< 0), for which a semi-stable limit cycle, Γ12,

appears in system (3.6) and then splits into a stable cycle, Γ1, and an unstable cycle, Γ2, under

further decreasing µ2. The formed domain D2 bounded by the limit cycles Γ1, Γ2 and filled by

the spirals will enlarge, since, on the properties of a field rotation parameter, the interior unstable

limit cycle Γ2 will contract and the exterior stable limit cycle Γ1 will expand under decreasing µ2.

The distance between the spirals of the domain D2 will naturally increase, what will prevent the

appearance of a semi-stable limit cycle in this domain for µ2 < µ
12

2
.

Thus, there are no such values of the parameters, µ∗
1
> 0, µ∗

2
< 0, µ∗

3
> 0, for which system

(3.6) would have an additional semi-stable limit cycle. Therefore, the fourth case cannot be realized.

The third case is considered absolutely similarly. It follows from the first two cases that system

(3.6) can have at most two limit cycles.



124 Valery A. Gaiko CUBO
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Suppose that system (3.6) has two limit cycles, Γ1 and Γ2, fix the parameters µ1 > 0, µ2 < 0,

µ3 > 0 and input the fourth parameter, µ4 < 0, into this system supposing that µ4 rotates its

vector field clockwise:

.
x = Pn(x,y,µ1, . . . ,µ4, 0, . . . , 0),

.
y = Qn(x,y,µ1, . . . ,µ4, 0, . . . , 0). (3.8)

The most interesting logical possibility here is that when the third (stable) limit cycle, Γ3, appears

from the origin and then, under preservation of the cycles Γ1 and Γ2, in the domain D3 bounded

on the inside by the cycle Γ3 and on the outside by the cycle Γ2, a semi-stable limit cycle, Γ45,

appears and then splits into a stable cycle, Γ4, and an unstable cycle, Γ5, i. e., when system (3.8)

for some set of values of the parameters, µ∗
1
, µ∗

2
, µ∗

3
, µ∗

4
, has at least five limit cycles. Logically, such

a semi-stable limit cycle could also appear in the domain D1 bounded on the inside by the cycle

Γ1, since, under decreasing µ4, the spirals of the trajectories of (3.8) will twist and the distance

between their coils will decrease. On the other hand, in the domain D2 bounded on the inside by

the cycle Γ2 and on the outside by the cycle Γ1 and also in the domain D4 bounded by the origin

and Γ3, a semi-stable limit cycle cannot appear, since, under decreasing µ4, the spirals will untwist

and the distance between their coils will increase. To prove impossibility of the appearance of a

semi-stable limit cycle in the domains D3 and D1, suppose the contrary, i. e., for some set of values

of the parameters, µ∗
1
> 0, µ∗

2
< 0, µ∗

3
> 0, and µ∗

4
< 0, such a semi-stable cycle exists. Return to

system (3.2) again, input first the parameters µ4 < 0, µ2 < 0 and then the parameter µ1 > 0 :

.
x = Pn(x,y,µ1,µ2,µ4, 0, . . . , 0),

.
y = Qn(x,y,µ1,µ2,µ4, 0, . . . , 0). (3.9)

Fix the parameters µ4, µ2 under the values µ
∗

4
, µ∗

2
, respectively. Under increasing µ1, a separatrix

cycle is formed around the origin generating a stable limit cycle, Γ1. Fix µ1 under the value µ
∗

1
and

input the parameter µ3 > 0 into (3.9) getting system (3.8).

Since, by our assumption, system (3.8) has three limit cycles for µ3 < µ
∗

3
, there exists some

value of the parameter µ23
3

(0 < µ23
3
< µ∗

3
) for which a semi-stable limit cycle, Γ23, appears in this

system and then splits into an unstable cycle, Γ2, and a stable cycle, Γ3, under further increasing

µ3. The formed domain D3 bounded by the limit cycles Γ2, Γ3 and also the domain D1 bounded

on the inside by the limit cycle Γ1 will enlarge and the spirals filling these domains will untwist

excluding a possibility of the appearance of a semi-stable limit cycle there.

All other combinations of the parameters µ1, µ2, µ3, and µ4 are considered in a similar way. It

follows that system (3.8) has at most three limit cycles. If we continue the procedure of successive

inputting the field rotation parameters, µ5, µ6, . . . , µk, into system (3.2), it is possible to conclude

that system (3.1) can have at most k − 1 limit cycles surrounding the origin. The theorem is

proved. 2



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Limit Cycles of Liénard-Type Dynamical Systems 125

4 Generalized liénard’s cubic system

In [13], we considered generalized Liénard’s cubic system of the form:

.
x = y,

.
y = −x + (λ − µ) y + (3/2) x2 + µxy − (1/2) x3 + αx2y . (4.1)

This system has three finite singularities: a saddle (1, 0) and two antisaddles — (0, 0) and (2, 0). At

infinity system (4.1) can have either the only nilpotent singular point of fourth order with two closed

elliptic and four hyperbolic domains or two singular points: one of them is a hyperbolic saddle and

the other is a triple nilpotent singular point with two elliptic and two hyperbolic domains. We

studied global bifurcations of limit and separatrix cycles of (4.1), found possible distributions of its

limit cycles and carried out a classification of its separatrix cycles. We proved also the following

theorems.

Theorem 4.1. The foci of system (4.1) can be at most of second order.

Theorem 4.2. System (4.1) has at least three limit cycles.

Using the results obtained in [13] and applying the approach developed in this paper, we can

easily prove a much stronger theorem.

Theorem 4.3. System (4.1) has at most three limit cycles with the following their distributions :

((1, 1), 1), ((1, 2), 0), ((2, 1), 0), ((1, 0), 2), ((0, 1), 2), where the first two numbers denote the numbers

of limit cycles surrounding each of two anti-saddles and the third one denotes the number of limit

cycles surrounding simultaneously all three finite singularities.

Theorem 4.3 agrees, for example, with the earlier results by Iliev and Perko [15], but it does

not agree with a quite recent result by Dumortier and Li [5] published in the same journal. The

authors of both papers use very similar methods: small perturbations of a Hamiltonian system.

In [15], the zeros of the Melnikov functions are studied and, in particular, it is proved that at

most two limit cycles can bifurcate from either the interior or exterior period annulus of the

Hamiltonian under small parameter perturbations giving a generalized Liénard system. In [5],

zeros of the Abelian integrals are studied and it is “proved” that at most four limit cycles can

bifurcate from the exterior period annulus. Thus, Dumortier and Li “obtain” a configuration of

four big limit cycles surrounding three finite singularities together with the fifth small limit cycle

which surrounds one of the anti-saddles.

The result by Dumortier and Li [5] also does not agree with the Wintner–Perko termination

principle for multiple limit cycles [10], [18]. Applying the method as developed in [3], [7]–[13],

we can show that system (4.1) cannot have either a multiplicity-three limit cycle or more than

three limit cycles in any configuration. That will be another proof of Theorem 4.3 (the same

approach can be applied to proving Theorems 2.2 and 3.1 as well). But first let us formulate the

Wintner–Perko termination principle [18] for the polynomial system

.

x = f (x, µ), (4.2µ)



126 Valery A. Gaiko CUBO
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where x ∈ R2; µ ∈ Rn; f ∈ R2 (f is a polynomial vector function).

Theorem 4.4 (Wintner–Perko termination principle). Any one-parameter family of multi-

plicity-m limit cycles of relatively prime polynomial system (4.2µ) can be extended in a unique way

to a maximal one-parameter family of multiplicity-m limit cycles of (4.2µ) which is either open or

cyclic.

If it is open, then it terminates either as the parameter or the limit cycles become unbounded;

or, the family terminates either at a singular point of (4.2µ), which is typically a fine focus of mul-

tiplicity m, or on a (compound ) separatrix cycle of (4.2µ), which is also typically of multiplicity m.

The proof of this principle for general polynomial system (4.2µ) with a vector parameter

µ ∈ Rn parallels the proof of the planar termination principle for the system

.
x = P(x,y,λ),

.
y = Q(x,y,λ) (4.2λ)

with a single parameter λ ∈ R (see [10], [18]), since there is no loss of generality in assuming that

system (4.2µ) is parameterized by a single parameter λ; i. e., we can assume that there exists an

analytic mapping µ(λ) of R into Rn such that (4.2µ) can be written as (4.2 µ(λ)) or even (4.2λ)

and then we can repeat everything, what had been done for system (4.2λ) in [18]. In particular,

if λ is a field rotation parameter of (4.2λ), the following Perko’s theorem on monotonic families of

limit cycles is valid.

Theorem 4.5. If L0 is a nonsingular multiple limit cycle of (4.20), then L0 belongs to a one-

parameter family of limit cycles of (4.2λ); furthermore:

1) if the multiplicity of L0 is odd, then the family either expands or contracts monotonically

as λ increases through λ0;

2) if the multiplicity of L0 is even, then L0 bifurcates into a stable and an unstable limit cycle

as λ varies from λ0 in one sense and L0 disappears as λ varies from λ0 in the opposite sense; i. e.,

there is a fold bifurcation at λ0.

Proof of Theorem 4.3. The proof is carried out by contradiction. Suppose that system (4.1)

with three field rotation parameters, λ, µ, and α, has three limit cycles around, for example, the

origin (the case when limit cycles surround another focus is considered in a similar way). Then we

get into some domain in the space of these parameters which is bounded by two fold bifurcation

surfaces forming a cusp bifurcation surface of multiplicity-three limit cycles.

The corresponding maximal one-parameter family of multiplicity-three limit cycles cannot be

cyclic, otherwise there will be at least one point corresponding to the limit cycle of multiplicity

four (or even higher) in the parameter space. Extending the bifurcation curve of multiplicity-

four limit cycles through this point and parameterizing the corresponding maximal one-parameter

family of multiplicity-four limit cycles by a field rotation parameter, according to Theorem 4.5, we



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Limit Cycles of Liénard-Type Dynamical Systems 127

will obtain a monotonic curve which, by the Wintner–Perko termination principle (Theorem 4.4),

terminates either at the origin or on some separatrix cycle surrounding the origin. Since we know

absolutely precisely at least the cyclicity of the singular point (Theorem 4.1) which is equal to

two, we have got a contradiction with the termination principle stating that the multiplicity of

limit cycles cannot be higher than the multiplicity (cyclicity) of the singular point in which they

terminate.

If the maximal one-parameter family of multiplicity-three limit cycles is not cyclic, on the same

principle (Theorem 4.4), this again contradicts to Theorem 4.1 not admitting the multiplicity of

limit cycles higher than two. Moreover, it also follows from the termination principle that either

the ordinary separatrix loop or the eight-loop cannot have the multiplicity (cyclicity) higher than

two (in that way, it can be proved that the cyclicity of three other separatrix cycles [13] is at most

two). Therefore, according to the same principle, there are no more than two limit cycles in the

exterior domain surrounding all three finite singularities of (4.1). Thus, system (4.1) cannot have

either a multiplicity-three limit cycle or more than three limit cycles in any configuration. The

theorem is proved. 2

5 A piecewise linear dynamical system

Consider a Liénard-type dynamical system

.
x = y − ϕ(x),

.
y = β − αx − y, α > 0, β > 0, (5.1)

where ϕ(x) is a piecewise linear function containing k dropping sections and approximating an

arbitrary polynomial of degree 2k + 1. The line β − αx − y = 0 and the curve y = ϕ(x) can

be considered as the isoclines of zero and infinity, respectively, for the corresponding equation.

Such systems and equations may occur, for example, when tunnel diode circuits and some other

problems are studied (see [1], [2], [6], [14]).

Suppose that the ascending sections of system (5.1) have an inclination k1 > 0 and the

descending (dropping) sections have an inclination k2 < 0. Then the phase plane of (5.1) can be

divided onto 2k + 1 parts in every of which (5.1) is a linear system: the ascending sections are in

k + 1 strip regions (I,III,V, . . . , 2K + 1) and the descending sections are in other k such regions

(II,IV,V I, . . . , 2K). The parameters k1, k2, and also α can be considered as rotation parameters

for the sewed vector field of (5.1) (see [2], [10]).

System (5.1) can have an odd number of simple singular points: 1, 3, 5, . . . , 2k + 1. If (5.1) has

the only singular point, this point will be always an antisaddle (center, focus or node). A focus

(node) will be always stable in odd regions and unstable in even regions if k2 > 1. If system (5.1)

has 2k + 1 singularities, then k of them are saddles (they are in even regions) and k + 1 others

are antisaddles (foci or nodes) which are always stable (they are in odd regions). The pieces of

the straight lines β = x2i−1α + y2i−1 and β = x2iα + y2i (i = 1, 2, . . . ,k), where (x2i−1,y2i−1) and



128 Valery A. Gaiko CUBO
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(x2i,y2i) are the coordinates of the upper and lower corner points of the curve ϕ(x), respectively,

form a discriminant curve separating the domains in the plane (α,β), where α ≤ k2, with different

numbers of singular points. The points of the discriminant curve correspond to the sewed singular-

ities of saddle-focus or saddle-node type (α < k2) and its corner points correspond to the unstable

equilibrium segments (α = k2) which coincide with the dropping sections of the curve y = ϕ(x).

In the case when k2 < 1, closed trajectories cannot exist and only bifurcations of singular

points are possible in system (5.1). Therefore, we will consider further only the case when k2 > 1

and (k1 − 1)
2 < 4k2 giving various bifurcations and, first of all, the bifurcations of limit cycles.

Studying all such bifurcations (local and global), we will give a proof of the following theorem.

Theorem 5.1. System (5.1) with k dropping sections and 2k + 1 singular points can have at most

k + 2 limit cycles, k + 1 of which surround the foci one by one and the last, (k + 2)-th, limit cycle

surrounds all of the singular points of (5.1).

Proof of Theorem 5.1. To prove the theorem, we will study both local and global bifurcations

of limit cycles. The limit cycle of system (5.1) will be called small if it belongs to at most two

adjoining regions; the cycle will be called big if it belongs to at least three adjoining regions.

5.1 Local bifurcations

Following [1], we will study first stability of the singular points on the line of sewing. Suppose that

the straight line β − αx − y = 0 passes through the corner point (x1,y1) of the curve y = ϕ(x) on

the boundary of regions I, II and that α > (k2 + 1)
2/4. Then the region I (II) will be filled by

the pieces of trajectories of the stable (unstable) focus.

Introduce positive coordinates S0 (lower (x1,y1)) and S1 (upper (x1,y1)) on the line of sewing

of regions I and II; S2 (lower (x2,y2)) and S3 (upper (x2,y2)) on the line of sewing of regions

II and III, etc. The maps S0 → S1 along the trajectories of region I and S1 → S0 along the

trajectories of region II are written as follows:

S1 = S0e
πσ1/ω1, S̄0 = S1e

πσ2/ω2, (5.2)

where σi, ωi (i = 1, 2) are the real and imaginary parts of the roots of the characteristic equation

for a singular point of regions I, II, respectively.

The singular point (x1,y1) will be a sewed center (S̄0 = S0) iff σ1/ω1 + σ2/ω2 = 0, i. e., when

α = α∗ ≡ (1 −k1/k2)/(k2 −k1 + 2). The sewed focus (x1,y1) will be stable (S̄0 < S0) when α > α
∗

and unstable (S̄0 > S0) when α < α
∗.

Consider the return map S0 → S̄0 along the trajectories of regions I and II. For region I, we



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Limit Cycles of Liénard-Type Dynamical Systems 129

will have

S0 =
δ0

sin ω1τ1
(ω1 cos ω1τ1 − σ1 sin ω1τ1 − ω1e

−σ1τ1 ) ≡ δ0ζ(τ1),

S1 =
δ0

sin ω1τ1
(ω1 cos ω1τ1 + σ1 sin ω1τ1 − ω1e

σ1τ1 ) ≡ δ0χ(τ1),

(5.3)

where δ0 is the distance from the boundary of regions I, II to the singular point; ζ and χ are

monotonic functions. The return map along the trajectories of region II has a similar form.

Calculation of the first derivative for the return map gives

dS̄0
dS0

=
S0

S̄0
e2(σ1τ1+σ2τ2), (5.4)

where τi (i = 1, 2) is motion time along the trajectories of regions I, II, respectively; σi = (1+ki)/2

(i = 1, 2).

Studying the return map S0 → S̄0 by means of (5.4), we prove that at most one limit cycle

can exist in regions I and II (see also [1]). The same result can be obtained for regions III and

IV,. . . , 2K − 1 and 2K.

Consider now the map S̄0 = f(S0) sewed of two pieces: S̄0 = ξ(S0) along the trajectories in

regions I, II, . . . , 2K and S̄0 = ψ(S0) along the trajectories in all regions, I, II, . . . , 2K, 2K + 1.

The map S0 → S1 in region I is given by (5.3). The maps S1 → S3, S3 → S5, . . . , S2k−1 → S2k−2

(S2k−1 → S2k+1, S2k+1 → S2k, S2k → S2k−2), S2k−2 → S2k−4, . . . , S2 → S0 have similar forms.

The derivatives for the functions ξ(S0), ψ(S0) are given by the following expressions, respec-

tively:
dS̄0
dS0

=
S0

S̄0
e
2(σ1(τ1+τ

+

3
+τ
−

3
+...+τ2k−1)+σ2(τ

+

2
+τ
−

2
+...+τ

+

2k−2
+τ
−

2k−2
))
, (5.5)

dS̄0
dS0

=
S0

S̄0
e2(σ1(τ1+τ

+

3
+τ
−

3
+...+τ2k+1)+σ2(τ

+

2
+τ
−

2
+...+τ

+

2k
+τ
−

2k
)), (5.6)

where τ1, τ2k−1, τ2k+1 are motion times in regions I, 2K − 1, 2K + 1 and τ
+

2i
(τ−

2i
), τ+

2i+1
(τ−

2i+1
),

i = 1, 2, . . . ,k, are motion times in the upper (lower) parts of regions II,III, . . . , 2K, respectively.

Studying the return map S̄0 = f(S0) by means of (5.5) and (5.6), we prove that at most two

limit cycles can be generated by the boundary of the domain filled by closed trajectories of (5.1)

and that these two limit cycles can be only outside the boundary.

Suppose that a part of the straight line β − αx − y = 0 coincides with a dropping section of

(5.1), for example, with the first one (α = k2). The dropping section of (5.1) will be an unstable

equilibrium segment and regions I, II (because of the condition (k1 − 1)
2 < 4k2) will be filled by

trajectories of the stable foci. It is easy to obtain an explicit expression for the map of the half-line

S0 into itself:

S̄0 = S0 e
2πσ1/ω1 + δ(k2 − 1)(1 + e

πσ1/ω1 ), (5.7)

where δ is the width of regions II.



130 Valery A. Gaiko CUBO
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This map has the only stable fixed point, and we can show that two stable foci surrounded by

unstable limit cycles (one by one) are generated from the ends of the equilibrium segment under

the rotation of the line β − αx − y = 0 (see also [1]).

The simplest type of separatrix cycles of (5.1) is a so-called eight-loop formed by two ordinary

saddle loops. In the case of 2k + 1 simple singular points, a separatrix cycle can contain k + 1

saddle loops, the first and the last of which are ordinary loops with one rough saddle on each and

the k − 1 others are separatrix digons with two rough saddles on each. Such a separatrix cycle

will be called nondegenerate. In the cases when the straight line β − αx − y = 0 passes through

the corner points of the curve y = ϕ(x), we will have degenerate separatrix cycles of lips-type

containing one or two sewed saddle-nodes. It is clear that the bifurcations of separatrix cycles do

not depend on the parameter β (see [1]). The separatrix cycles can be formed or destroyed only

under a variation of the parameter α. The character of their stability will be determined by the

sign of the saddle quantities which are always positive in our case, when the saddles are inside or

on the boundary of even regions II, IV, . . . , 2K and k2 > 1 (the corresponding theorems are valid

for the piecewise linear dynamical systems as well [2]). It follows that the separatrix cycles of (5.1)

are always unstable (inside and outside) and, under a variation of α, a nondegenerate separatrix

cycle can generate at most k + 1 small unstable limit cycles inside its loops (digons) or the only

big unstable limit cycle outside it.

5.2 Global bifurcations

Now we are able to consider the global bifurcations of limit cycles. Suppose again that the zero

isocline β−αx−y = 0 passes through the corner point (x1,y1) of the infinite isocline y = ϕ(x) and

that α > α∗. In this case, the only singular point in the phase plane is a sewed stable focus and

all trajectories of (5.1) tend to it when t → +∞. For decreasing α (k2 < α < α
∗), the sewed focus

becomes unstable and a stable limit cycle is generated from the boundary curve of the domain

filled by closed trajectories (immediately after passing the value α∗ by the parameter α).

For α = k2, the first dropping section of (5.1) will coincide with a part of the straight line

β − αx − y = 0 and an unstable equilibrium segment will appear inside the stable limit cycle.

If we rotate the line β − αx − y = 0 around an interior point of the segment (changing both

of the parameters, α and β), two unstable limit cycles surrounding stable foci (one by one) will

be generated from the ends (x1,y1) and (x2,y2) of the equilibrium segment. Under the further

rotation of the line β − αx − y = 0, it will pass first through the next corner point, (x4,y4), and

then, successively, through the points (x6,y6), . . . , (x2k,y2k). Every time, the corner point becomes

a sewed saddle-node generating an unstable limit cycle surrounding a stable focus. So, we will get

a piecewise linear system with 2k + 1 singular points having at least k + 1 small unstable limit

cycles surrounding the stable foci (one by one) inside a big stable limit cycle, k + 2, surrounding

all of the singular points.

Under the further rotation of the zero isocline, all k + 1 small limit cycles simultaneously



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Limit Cycles of Liénard-Type Dynamical Systems 131

disappear in a separatrix cycle consisting of k + 1 loops (digons), this separatrix cycle generates a

big (unstable) limit cycle which combines with another big (stable) limit cycle of (5.1) forming a

semi-stable (double) limit cycle which finally disappears in a so-called trajectory condensation.

Let us prove that system (5.1) cannot have more than k + 2 limit cycles. The proof is carried

out by contradiction by means of the Wintner–Perko termination principle [2], [10], [18]. Since a

small limit cycle is always unique in the corresponding strip regions, suppose that system (5.1)

with three field rotation parameters, k1, k2, and α, has three big limit cycles. Then we get into

some domain in the space of these parameters which is bounded by two fold bifurcation surfaces

forming a cusp bifurcation surface of multiplicity-three limit cycles [10], [18].

The corresponding maximal one-parameter family of multiplicity-three limit cycles cannot be

cyclic, otherwise there will be at least one point corresponding to the limit cycle of multiplicity four

(or even higher) in the parameter space. Extending the bifurcation curve of multiplicity-four limit

cycles through this point and parameterizing the corresponding maximal one-parameter family of

multiplicity-four limit cycles by a field rotation parameter, for example, by the parameter α, we will

obtain a monotonic curve which, by the Wintner–Perko termination principle, terminates either

at the boundary curve of the domain filled by closed trajectories of (5.1) or on some degenerate

separatrix cycle of (5.1) [10], [18].

Since we know at least the cyclicity of the boundary curve which is equal to two, we have

got a contradiction with the termination principle stating that the multiplicity of limit cycles

cannot be higher than the multiplicity (cyclicity) of the end bifurcation points in which they

terminate [10], [18].

If the maximal one-parameter family of multiplicity-three limit cycles is not cyclic, using

the same principle, this again contradicts with the cyclicity result for the boundary curve not

admitting the multiplicity of limit cycles to be higher than two. Moreover, it also follows from the

termination principle that the degenerate separatrix cycles of (5.1) cannot have the multiplicity

(cyclicity) higher than two. Therefore, according to the same principle, there are no more than

two big limit cycles in the exterior domain outside the boundary curve of (5.1).

The same results can be obtained by means of the new geometric methods developed in [12].

The phase portraits and bifurcation diagrams for system (5.1) will be similar to that which were

constructed in [1], [2]. Thus, system (5.1) with 2k + 1 singular points cannot have more than

k + 2 limit cycles, i. e., k + 2 is the maximum number of limit cycles of such system and the

obtained distribution (k + 1 small limit cycles plus a big limit cycle) is the only possibility for their

distribution. The theorem is proved. 2

Received: May 2008. Revised: June 2008.



132 Valery A. Gaiko CUBO
10, 3 (2008)

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∑

2

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aix

2i+1

is equal to two, Diff. Equat., 11 (1975), 301–302.

[20] S. Smale, Mathematical problems for the next century, Math. Intellig., 20 (1998), 7–15.


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