Applied Mathematics, 2011, 2, 1221-1224
doi:10.4236/am.2011.210170 Published Online October 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
On Canard Homoclinic of a Liénard
Perturbation System
Makoto Hayashi
Department of Mathematics, College of Science and Technology,
Nihon University, Chiba, Japan
E-mail: mhayashi@penta.ge.cst.nihon-u.ac.jp
Received June 8, 2011; revised August 17, 2011; accepted August 24, 2011
Abstract
The classification on the orbits of some Liénard perturbation system with several parameters, which is rela-
tion to the example in [1] or [2], is discussed. The conditions for the parameters in order that the system has a
unique limit cycle, homoclinic orbits, canards or the unique equilibrium point is globally asymptotic stable
are given. The methods in our previous papers are used for the proofs.
Keywords: Liénard System, Canards, Limit Cycles, Homoclinic Orbits, Global Asymptotic Stability
1. Introduction
We shall consider the Liénard perturbation system

32
3
32
,
x
x
xy
ykxa


 


 
(P1)
where a,
,
and k are positive real numbers.
System (P1) has a unique equilibrium point at 33
x
a
and the uniqueness of solutions of initial value problems
for the system is guaranteed. In [1], it has been given that
the unique equilibrium point of System (P1) for the case
a = 0 and 1
 is a global attractor but unstable. In
[2], when 1
and a = 0, the result that System (P1)
has the special orbit called “a Canard Homoclinic” has
been announced by the method of non standard. Our aim
is to classify the orbits of System (P1) completely by the
values of the parameters. Thus, we improve the results of
the papers [1,2].
Our main results are the following
Theorem 1.1. System (P1) has homoclinic orbits lo-
cally if and only if (a = 0 and 8k
).
Then the system has no limit cycles.
Theorem 1.2. System (P1) has a unique limit cycle if
and only if 01a.
Specially, if
and a are sufficiently small, the
orbit is called a Canard Limit Cycle.
Theorem 1.3. The unique equilibrium point (0, 0) for
System (P1) is globally asymptotic stable if and only if
one of the following
0a
or
0a
and
8k
or 1a
is satisfied.
In Section 2, we shall see that System (P1) is trans-
formed to a usual Liénard system (see System (P3)) with
the unique equilibrium point at the origin. In Section 3,
the existence of the homoclinic orbit of the system will
be discussed by using the method in [3]. In virtue of this
result, the interesting fact that both the limit cycle and
the homoclinic orbit of the system cannot coexist is giv-
en. If a > 0, a and
are sufficiently small, it has
been well-known b y E. Benoît ([4]) that System (P3) has
the orbit called “a Canard”. The orbit changes to the ho-
moclinic orbit for the system as a = 0. So the orbit has
been called “a Canard Homoclinic” ([2]). In Section 4,
the fact that the system has at most one limit cycle will
be proved by using the method of [5]. When 01a
,
a and
are sufficiently small, the orbit‘Canard’ spi-
rals to a unique limit cycle of the system. We call the
orbit “a Canard Limit Cycle”. In Section 5, it shall be
seen from the facts of Section 3 and Section 4 that the
unique equilibrium point for the system with the pa-
rameters in Theorem 1.3 is globally asymptotic stable.
Finally, a phase portrait of System (P1) with respect to
Theorem 1.1 will be presented in Section 6.
2. Transformation to a Liénard System
By using the transformation tt
,
x
x and
M. HAYASHI
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yy for System (P1), the system is changed to the
following

32
3
32
.
x
x
xy
ykxa

 


 
(P2)
Moreover, using the transformation
x
x
 and

2
2yy
 for
satisfying the equation
30a
, System (P2) is transformed to the system
 


2
22
23216 1
6
33.
xy xxx
ykxx x


 
 
(P3)
System (P3) has a unique equilibrium poin t (0, 0) and the
uniqueness of solutions of initial value problems is also
guaranteed.
We set
 

2
()232161 ,
6
Fxx xx




22
g33xkxxx


and
 
0d.
x
Gx g

From the value of
, the graph of the characteristic
curve

y
Fx is divided into four cases;

1.. 1,ie a


10..01,ie a
 

0.. 0ie a
 and

0.. 0.ie a

We will note that the situation of the graph is deeply
concerned in the qualitativ e prop erty of the orbits.
3. Proof of Theorem 1.1
In System (P3), a trajectory is said to be a homoclinic
orbit if its
- and
- limit sets are the origin. If Sys-
tem (P3) has a homoclinic orbit,

0Fx (or
0Fx
)
in the neighborhood of the origin is necessary. Thus we
have the assumption

0.. 0iea
 in this section.
Consider a function
x
with the condition
[C1]


1,00and 0CxFxx
 
 
for 0.x
The following resul t has been given in [3] .
Lemma 3.1. System (P3) with 0
has homoclinic
orbits if and only if there exists a function
x
with
[C1] such that
[C2]
 
0Fx x

and
 
0
x
Fx xgx



for 0.x
As the supplement function in the above lemma, take
2
x
rx
with 8k
and 2
48rk
 
 .
Then we have
 

22312 0
6
Fxxx xr

and



323123 0
3
xFxx gx
xrx rrk

 



for
03122.xr

Thus, we see that the conditions [C1] and [C2] are satis-
fied. Hence System (P3) has (l ocal) homoclinic orbits.
Moreover, the following is known by the Corollary 3
in [3].
Lemma 3.2. If the conditions [C1] and [C2] hold for
1
xx
 and 10,x
then System (P3) with 0
has homoclinic orbits locally, but no limit cycles.
Taking 1
x
in the proof of Lemma 3.1, we see
that the above lemma holds. Therefore, the proof of
Theorem 1.1 is completed now.
When a > 0, a and
are sufficiently small, it has
been well-known from [4] that System (P3) has the orbit
called “a Canard”. The orbit “Canard” changes to the
mentioned homoclinic orbit above as a = 0. So the orbit
has been called “a Canard Homoclinic” by [2]. Thus, we
see that there exists a canard hom oclinic in System (P1).
4. Proof of Theorem 1.2
We shall assume the condition 10
  (i.e. 01a
).
Then we can easily check that System (P3) has at least
one limit cycle. In facts, the unique equilibrium point
(0, 0) is a unstable focus by (0) 0F and the all orbits
are uniformly ultimately bounded (for the details see [6]).
Thus, by the well-known Poincaré-Bendixson theorem,
the system has a limit cycle (for instance see [7]).
The following is a useful method ([5]) in order to
guarantee that a Liénard system has at most one limit
cycle.
Lemma 4.1. If there exists a constant 0m such
that

0FxGx mFxgx
for

0,x System
(P3) has at most one limit cycle.
We have
 
2;, ,
12
k
FxGx mFxgxxxm


where
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 





43
2
2
3
;,3436 124 1
1531 221
685343
18 121 .
x
mmx mx
mx
mx
m

 
 




 
Let 12m and 14
 in

;,xm
. Then we
have

2
2
113 3
;,0 0.
244 8
xxx x



 





Thus, we see from Lemma 4.1 that System (P3) has at
most one limit cycle. So we conclude that System (P3)
has a unique limit cycle if –1 < α < 0.
Conversely, suppose that System (P3) has a limit cycle.
Then if System (P3) doesn’t satisfy the condition –1 < α
< 0, this contradicts to the existence of the limit cycle by
Theorem 1.1 and the proof of Theorem 1.3 (see Section
5).
Therefore, the proof of Theorem 1.2 is completed now.
In virtue of E. Benoît ([4]), if a > 0, a and
are
sufficiently small, then System (P3) has the orbit called
“a Canard”. Then the orbit “Canard” spirals to a unique
limit cycle of System (P3). So we call the special limit
cycle “a Canard Limit Cycle”.
5. Proof of Theorem 1.3
In this section, we shall assume the condition 1
or
0
(i.e. 0a or 1a). The following is a powerful
method (see [8]) to prove the non-existence of non-trivial
closed orbits of a Liénard system.
Lemma 5.1. If the curve
 
,
F
xGx has no in-
tersecting points with itself, then System (P3) has no non-
trivial closed orbits.
From the situation of the graph

,
y
Fx we shall
prove the theorem by dividing into four cases;
i) 012,

ii) 12 ,
iii) 32 1,

iv) 32.

First, we shall discuss the case i). The discussion is
similar to the method shown in [9].
Let

1, 2
i
pi
be the solutions of the equation

0Fx and
 
12
0.pp

 Now we consider
the equation
 
Fx F
for
11.p
 

This equation has two roots other than .x
Let

11
uu
and

22
uu
denote these roots. Then
we have

12
1up
  and 2
012.u
 
From a property of the curve
 

,
F
xGx , we shall
show that, if
12
F
uFu for

11,p
 

then
21
0.GuGu
From
12
F
uFu we have
 
22
1122 12
2321610.uuuuuu


Thus we get
 



22
21 211212
222
112212
()
4
46.
k
GuGuuuu uuu
uuuuuu

 
 
Since 1
u and 2
u are solutions of the equation
,Fx F
we have
12
32 1
2
uu
 and
12 31.uu


Thus we get
22 2
12 9
33 .
4
uu


By substituting 12
uu
and 22
12
uu to

21
Gu Gu,
we have
 
21 21
,
4
k
GuGuuu L
 
where

32
332 94278.L
  
 
Then we have
0L
for all
and
0L
27 80
. Thus we get
0L
for 0.
From these facts and 21
0uu
, if 0
, we conclude
that
21
GuGu G
 for

11p

.
Namely, the curve
,
F
xGx has no intersecting
points with itself. This means that System (P3) with
012
has no non-trivial closed orbits.
Similarly we can check that

0L
for 12
and
0L
for 1.
Thus we have

21
0Gu Gu
for 12
or 1.
Therefore, we conclude that
System (P3) also has no non-trivial closed orbits for the
another cases ii), iii) and iv).
We say that the equilibrium point

0,0E is globally
asymptotically stable if E is stable and every orbit of
System (P3) tends to E. We will see the global asymp-
totic stability of E from the following conditions:
[i] all orbits of System (P3) are bounded in the future,
[ii] System (P3) has no non-trivial closed orbits,
[iii] System (P3) has no homoclinic orbits,
[iv] E is asymptotic stable.
The condition [i], [ii] or [iii] has been checked in Sec-
tion 3, Section 4 or the mentioned fact above. So we
shall check the condition [iv].
Suppose that E is not stable. Then, by checking the
direction of the vector
 
,yFx gx for System
(P3), we have that every positive semi-trajectories of
System (P3) starting in the neighborhood of E keep on
rotating around E and go away from E. Hence, by the
M. HAYASHI
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y
0
2
1
4
yx
32
32
x
x
y
x
Figure 1. a = 0, λ = 1 and ε = k = 1/100.
fact [i] and the Poincaré-Bendixson theorem, the system
has a closed orbit. This contradicts to the fact [ii]. It fol-
lows from the direction of the vector field that E is as-
ymptotically stable.
Conversely, suppose that E is globally asymptotic sta-
ble. Then we see from Theorem 1.1 and 1.2 that System
(P3) must satisfy the condition in Theorem 1.3.
Therefore, the proof of Theorem 1.3 is completed
now.
Remark. In the case of (0
and 80)k
 or
1
 , E is a non-hyperbolic equilibrium point. We see
from the fact
 
21
0Gu Gu
and [10] that E is a
stable focus. Thus, a unique equilibrium point of System
(P1) cannot be “Center”.
6. A Numerical Example
We shall present a phase portrait of System (P1). We
consider the example of the case 0,a 1
and
1 100.k
Then we have 8.k
Thus we shall
see that the system has a homoclinic orbit, but no limit
cycles as is shown in the Figure 1.
7. References
[1] D. Changming, “The Homoclinic Orbits in the Liénard
Plane,” Journal of Mathematical Analysis and Applica-
tions, Vol. 191, 1995, pp. 26-39.
[2] B. Rachid, “Canards Homocliniques,” the 7-th Interna-
tional Colloqium on Differential Equations in Bulgaria,
Plovdiv, 1997.
[3] M. Hayashi, “A Geometric Condition for the Existence of
the Homoclinic Orbits of Liénard Systems,” International
Journal of Pure and Applied Mathematics, Vol. 66, No.1,
2011, pp. 53-60.
[4] E. Benoît, “Sy stè me s Lents-Rapides Dans 3
et Leurs a
Canard Orbits,” Asterisque, Vol. 109-110, 1983, pp. 159-
191.
[5] Z. Zhang, et al., “Qualitative Theory of Differential Equ-
ations,” Translate Mathematical Monographs in AMS,
Vol. 102, 1992, p. 236.
[6] J. Graef, “On the Generalized Liénard Equation with
Negative Damping,” Journal of Differential Equations,
Vol. 12, 1992, pp. 34-62.
doi:10.1016/0022-0396(72)90004-6
[7] M. Hayashi, “On Uniqueness of the Closed Orbit of the
Liénard System,” Mathematica Japonicae, Vol. 46, No. 3,
1997, pp. 371-376.
[8] A. Gasull and A. Guillamon, “Non-Existence of Limit
Cycles for Some Predator-Prey Systems,” Proceedings of
Equadiff’91, World Scientific, Singapore, 1993, pp. 538-
546.
[9] M. Hayashi, “Non-Existence of Homoclinic Orbits and
Global Asymptotic Stability of FitzHugh-Nagumo Sys-
tem,” Vietnam Journal of Mathematics, Vol. 27, No. 4,
1999, pp. 335-343.
[10] J. Sugie, “Non-Existence of Periodic Solutions for the
FitzHugh Nerve System,” Quarterly Journal of Applied
Mathematics, Vol. 49, 1991, pp. 543-554.