EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 6, No. 2, 2013, 126-136
ISSN 1307-5543 – www.ejpam.com

A Complete Classification of Liénard Equation

Halim Zeghdoudi1,3,∗, Lahsen Bouchahed1, Raouf Dridi 2

1 LaPS laboratory, Badji-Mokhtar University, Box:12, Annaba 23000-Algeria
2 Department of Mathematics, University of British Columbia, Vancouver,BC V6T 1Z1,Canada
3 Department Computing Mathematics and Physics, Waterford Institute of Technology,Waterford-
Ireland

Abstract. We consider scalar Liénard equations

ẍ(t) = f (x(t)) ẋ(t) + g(x(t)), x(t) ∈R (1)

and the diffeomorphisms ϕ : R2→ R2 in the form

ϕ(x , t) = (β(x), a.t +α(x)) (2)

where the derivative of the function β is non zero and where the real number a is non zero. The aim
result of this paper is to study the symmetries in the form given by (2) for the equation (1).

2010 Mathematics Subject Classifications: 47A63, 26A51, 45A90

Key Words and Phrases: Liénard equation, Lie’s symmetries, Characteristic sets

1. Introduction

Van der pol equation is an example of the long-standing interaction between differential
equations and the physical and biological sciences. During the development of radio and vac-
uum tubes, Liénard equations were intensely studied as they can be used to model oscillating
circuits. In 1920 the Dutch physicist Balthasar van der Pol, when he was an engineer working
for Philips Company, studied the differential equation

ẍ − ε(1− x2) ẋ + x = 0

that describes the circuit of a vacuum tube and where ε is positive parameter. A few years
after, [9] modeled the electric activity of the heart rate. POLES (in CHAOS 2007), developed
their own modified van der Pol oscillator reproducing irregular heart rate, asystole, certain
kinds of heart block, and others. In the sixties, Fitzhugh [4] and Nagumo [8] extended the

∗Corresponding author.

http://www.ejpam.com 126 c© 2013 EJPAM All rights reserved.



H. Zeghdoudi, L. Bouchahed, R.Dridi / Eur. J. Pure Appl. Math, 6 (2013), 126-136 127

van der Pol equation in a planar field as a model for action potentials of neurons. Recently,
The equation has also been utilised in seismology to model the two plates in a geological fault.

The french engineer, Liénard propose the following generalization

ẍ = f (x) ẋ + g(x) (3)

where f and g are two real-valued analytic functions.
An other example of Liénard equation is given by the Duffing’s equation, The Duffing

equation, named after Georg Duffing, is a non-linear second-order differential equation used
to model certain damped and driven oscillators. The equation is given by

ẍ +δ ẋ +αx + β x3 = γ cos(ωt)

where the (unknown) function x = x(t) is the displacement at time t, ẋ is the first deriva-
tive of x with respect to time, i.e. velocity, and ẍ is the second time-derivative of x , i.e.
acceleration. The numbers δ, α, β , γ and ω are given constants.

The equation describes the motion of a damped oscillator with a more complicated poten-
tial than in simple harmonic motion (which corresponds to the case β = δ = 0); in physical
terms, it models, for example, a spring pendulum whose spring’s stiffness does not exactly
obey Hooke’s law. The Duffing equation is an example of a dynamical system that exhibits
chaotic behavior. Moreover the Duffing system presents in the frequency response the jump
resonance phenomenon that is a sort of frequency hysteresis behaviour. The forced Duffing’s
equation, which is one of the classical oscillators first published by Duffing in 1918 .It is the
simplest oscillator displaying catastrophic jumps of amplitude and phase when the frequency
of the forcing term is taken as a gradually changing parameter. The main applications have
been in electronics, mechanic, in biology. For example, the brain is full of oscillators at micro
level, and at macro level displays jumps in sensory perception, in psychological perception, in
regulation, in switches of mood, memory, and behaviour, to say nothing of falling asleep and
waking up.

In addition, the two-dimensional autonomous dynamical system is defined by two coupled
first order differential equations of the form

ẋ = P(x , y), ẏ =Q(x , y) (4)

where P and Q are two functions of the variables x and y and the overdots denote a time
derivative. Such a dynamical system appears very often within several branches of science,
such as biology, chemistry, astrophysics, mechanics, electronics, fluid mechanics. One of the
most difficult problems connected with the study of system (4) is the question of the number
of limit cycles. A limit cycle is an isolated closed trajectory. Isolated means that the neigh-
boring trajectories are not closed; they spiral either toward or away from the limit cycle. If
all neighboring trajectories approach the limit cycle, we say that the limit cycle is stable or
attracting. Otherwise the limit cycle is unstable or, in exceptional cases, half stable. Stable
limit cycles are very important in science. They model systems that exhibit self-sustained
oscillations. For more details we can see [5]



H. Zeghdoudi, L. Bouchahed, R.Dridi / Eur. J. Pure Appl. Math, 6 (2013), 126-136 128

In this paper we consider the transformations ϕ ∈ Diffloc(R2) of the form

(t, x)→ (at +α(x), β(x))

where α and β are two real-valued function such that (ηy 6= 0). We shall see that such
transformations form a Lie pseudogroup and have the important feature of preserving periodic
solutions. Strongly aided by the computer algebraic package DIFFALG, written by François
Boulier [3], we give a complete symmetry classification of the Liénard. We shall see how
ROSENFELD-GRÖBNER allows us to discuss the structure of the symmetry Lie algebra of the
Liénard equation w.r.t f and g.

The paper is organized around three sections. The second section gives a brief description
of concept of symmetry. The aim of the third section is to present the symmetry classication
of Léinard equation and somes examples.

2. Concept of Symmetry

To define the notion of symmetry in any general information we give a group Φ operative
on the set E Now, in this context the definition of a symmetry is

Definition 1. A symmetry of the Pfaffian system E f = (M ,∆ f ) is a local diffeormorphism
ϕ ∈ Di f f loc(M) which preserves the contact structure of E f i.e.

ϕ∗(∆ f ) = ∆ f .

Symmetries in this definition are internal [1]. The set of all symmetries of a given Pfaffian
system E f is a Lie pseudogroup denoted by Aut loc(E f )⊂ Di f f loc M . Since the distribution ∆ f

is involutive, Aut loc(E f ) is the symmetry pseudogroup of a foliation. Such a pseudogroup is
infinite dimensional. And this why in practice (in order to classify), we restrict ourselves to
symmetries belonging to a certain Lie pseudogroup Φ ⊂ Di f f loc M of local diffeomorphisms
of interest. Let S f = Aut loc(E f )∩Φ denotes the Lie pseudogroup of such symmetries. Its
defining equations are given by the non linear PDE’s system

ϕ∗(∆ f ) = ∆ f et ϕ ∈ Φ (5)

where the second constraint means that ϕ fulfills the Lie defining equations of the Lie pseu-
dogroup Φ.

The non linear PDE system (5) simplifies to a linear system if we switch to the calculation
of infinitesimal generators of the Lie pseudogroup S f . Now we present briefly this technique
due to S. Lie. A good reference is the book [7] but also [5] and [2].

Let G be one-dimensional Lie group (in practice G is the additive group (R,+)). Recall that
a one-parameter transformations group on manifold M is a a map (ε, p) ∈ G×M → ϕε(p) ∈ M
satisfying ϕε+τ(p) = ϕε ◦ φτ(p) and if e is the identity element of G, ϕe is the identity
transformation. Each one-parameter transformations group ϕε induces a vector field X in the
following manner. For each p ∈ M , Xp is the tangent vector of the curve γ(ε) = ϕε(p) at



H. Zeghdoudi, L. Bouchahed, R.Dridi / Eur. J. Pure Appl. Math, 6 (2013), 126-136 129

the point p = ϕ0(p) i.e. dϕε(p)
dε
|ε=0 = Xp. The vector field X is called infinitesimal generator

associated to the one-parameter group ϕε. Conversely, to each vector field X we can associate
a “local” one-parameter transformations group. The diffeomorphism M 3 p → ϕε(p) ∈ M is
called the flow or the dynamic generated by X . [If we can take ε =∞, for each p , X is said
to be complete. If M is compact, every X is complete]. The operator

LX : Γ
�

⊗r T M ⊗s T ∗M
�

→ Γ
�

⊗r T M ⊗s T ∗M
�

defined by

LX = lim
ε→0

ϕ∗ε − Id

ε

is called the Lie derivative in the direction X and we have

ϕ∗ε = Id+ εLX +O(ε2). (6)

In particular, if Y is a vector field (Y ∈ Γ(T M)) then we have LX (Y ) = [X , Y ].
Let us go back to our symmetries: now we are looking for local one-parameter symmetry

groups. We know, such symmetries are of the form ϕε(p) = p+ εX (p) +O(ε2) for all p ∈ M
and for a certain X ∈ Γ(T M) [Of course one needs to combine this with the fact that they
also of the form (t, x)→ (at +α(x),β(x)) which is explained in Section 3]. Applying (6) to
ϕ∗ε(∆) = ∆, shows that a transformation ϕε is a symmetry of the Pfaffian system E f = (M ,∆ f )
if and only if

LX∆= 0 mod ∆. (7)

The components of the vector field X (called the infinitesimals) are now solutions of a linear
PDE’s system. The fluxes (the ϕε) are recovered by solving the system of ordinary differential
equations dϕε(p)

dε
|ε=0 = Xp with the initial condition p = ϕ0(p).

Example 1. Let be M variety of local co-ordinates (x , y1, . . . , yn). Any multi-sheet on M, of
codimension n, is localment redressable in (Ci being arbitrary constants)

y1 = C1, . . . , yn = Cn

fiber_preserving transformation ϕ ∈ Di f f loc M

(x , y1, . . . , yn)→ (ϕ0(x , y1, . . . , yn),ϕ1(x , y1, . . . , yn), . . . ,ϕn(x , y1, . . . , yn)),

where ϕiare arbitrary functions.

Remark 1 (Lie’s Classical Method). A symmetry of a differential equation is a transformation
mapping an arbitrary solution to another solution of the differential equation. The classical Lie
groups of point invariance transformations depend on continuous parameters and act on the
system’s graph space that is co-ordinatised by the independent and dependent variables. As these
symmetries can be determined by an explicit computational algorithm If a partial differential
equation (PDE) is invariant under a point symmetry, one can often find similarity solutions or
invariant solutions which are invariant under some subgroup of the full group admitted by the
PDE. These solutions result from solving a reduced equation in fewer variables.



H. Zeghdoudi, L. Bouchahed, R.Dridi / Eur. J. Pure Appl. Math, 6 (2013), 126-136 130

3. Symmetry Classification of the Liénard Equation

As announced, we consider the pseudogroup of transformations ϕ ∈ Di f f loc(R2) of the
form

(

t̄ = at +α(x), a 6= 0

x̄ = β(x), βx 6= 0
. (8)

where a ∈ R is an arbitrary constant and α, β are two arbitrary functions.

Proposition 1. Any transformation of the form (8) maps a periodic function x(t) of period T to
another periodic solution of period equals to aT .

Proof. See [10].

3.1. Generation of Lie Equations

Let us first determine the infinitesimal generators X with fluxes of the form (8). Let make
the substitution

t̄ = t + εA(x , t) +O(ε2), x̄ = x + εB(x , t) +O(ε2),

in the defining equations of the Lie pseudogroup (8):

∂ 2 t̄

∂ t2 = 0,
∂ 2 t̄

∂ x∂ t
= 0,

∂ x̄

∂ t
= 0,

∂ t̄

∂ t

∂ x̄

∂ x
6= 0.

We obtain
∂ 2A(x , t)
∂ t2 = 0,

∂ 2A(x , t)
∂ x∂ t

= 0,
∂ B(x , t)
∂ t

= 0.

This allows one to deduce that the infinitesimal generators X must be of the form

X = (λt + A(x))
∂

∂ t
+ B(x)

∂

∂ x
.

Now Lie equations are obtained by writing that the Lie derivative [X , Dt] is zero modulo
Dt =

∂
∂ t
+ p ∂

∂ x
+ ( f (x)p+ g(x)) ∂

∂ p
where p = ẋ . We obtain the ODE system























Bgx + Bx g − 2λg = 0,

Bx ,x − 2Ax f = 0,

Ax ,x = 0,

B fx + 3Ax g +λ f = 0,

λx = 0.

(9)

The system (9) depends on two arbitrary functions f and g and linear in the differential
unknowns A, B and λ.

Theorem 1. The classification below is complete.

Proof. See [10].

In the following paragraphs, we give the characteristic representations of the ideals
p

Ii .



H. Zeghdoudi, L. Bouchahed, R.Dridi / Eur. J. Pure Appl. Math, 6 (2013), 126-136 131

The generic Case

The first characteristic set is
Ax = 0, B = 0,λ= 0.

This is the generic case, there is no constraint on the functions f and g. The dimension
of the corresponding Lie algebra is equals to the number of points under the three stairs
associated to the unknowns A, B and λ. Hence equals to one. The integration shows that the
infinitesimal generator is X1 =

∂
∂ t

and the corresponding fluxes form the one-parameter Lie
group of temporal translations. Van der Pol equation ẍ − ε(1− x2) ẋ + x = 0, belongs to this
class.

Case II.

It has four subcases where the number of points under the stairs corresponding to the un-
knowns A, B and λ (i.e. the dimension of the symmetry Lie algebra) is equal to two.
If X2 = (λt + A(x)) ∂

∂ t
+ B(x) ∂

∂ x
is another vector field (different from X1 =

∂
∂ t

) then
[X2, X1] = λX1. The symmetry Lie algebra is consequently the affine algebra a(1,R) if λ 6= 0
or the abelian algebra otherwise. In both situations, it solvable and Liénard equation can be
reduced to a quadrature [6, 2].

Subcase II-1. 3gx x + 2 f fx 6= 0 and g 6= 0 The first of the four characteristic sets is


































λx = 0,

Ax =−
λ( f gx x−2 fx gx)
g(3gx x+2 f fx)

,

B =−2
λ( f 2+3gx)
3gx x+2 f fx

,

gx x x =
5gx x g f fx+6g gx x

2−2g fx
2 gx−3gx x gx

2−2 f fx gx
2

g( f 2+3gx)
fx x =

9gx x g fx−3gx x gx f+6 f fx
2 g−2 f 2 fx gx

2g( f 2+3gx)

The two last equations (in addition to the inequalities) constrain the function f and g. They
characterize the differential ideal

p

I2 ∩ { f , g}. The other equations give the functions A, B
and λ. In particular, one sees that λ furnishes a non zero structure constant. This proves:

Proposition 2. The symmetry Lie algebra in this case is isomorphic to a(1,R).

Example 2. We can consider the equation ẍ = x ẋ+x3. The infinitesimal generators are X1 =
∂
∂ t

,

X2 = t ∂
∂ t
−x ∂

∂ x
and the fluxes generated by X1 and X2 form the two-dimensional Lie group if spe-

cial affine transformations (t, x)→ (λt +µ, x
λ
). The Jacobian

�

λ 0
0 1

λ

�

of such transformation

has determinant equals to 1.



H. Zeghdoudi, L. Bouchahed, R.Dridi / Eur. J. Pure Appl. Math, 6 (2013), 126-136 132

Subcase II.2. 3gx x =−2 f fx , fx x = 0 and g 6= 0 We have the characteristic set






















λ= 0,

Ax =−
B fx

3g
,

Bx =
Bgx

g
,

fx x = 0,

gx x =−
2
3

f fx .

The first equation immediately proves

Proposition 3. The symmetry Lie algebra is the two-dimensional abelian algebra.

The integration of the two last equations, which characterize
p

I3 ∩ { f , g}, yields (the ai
are arbitrary constants)

(

f (x) = a1 x + a2,

g(x) =−1
9
a2

1 x3− 1
3
a1a2 x2+ a3 x + a4.

And this in turn yields






X1 =
∂
∂ t

,

X2 =−x a1

3a4

∂
∂ t
+
�

1+ x a3

a4
− x2 a2a1

3a4
− x3 a1

2

9a4

�

∂
∂ x

.

Subcase II.3. 3gx x − 2g2
x 6= 0, fx x 6= 0 and g 6= 0 Here the function A, B and λ satisfy











λx = 0,

Ax =
λ(−3 g f fx ,x+9 fx

2 g+ f 3 fx)
9g2 fx ,x

,

B =−λ(9 g fx+ f 3)
3g fx ,x

.

The first equation proves

Proposition 4. The symmetry Lie algebra is isomorphic to a(1,R).

The functions f and g satisfy the ODE system (
p

I4 ∩ { f , g})






f 2 =−3gx ,

gx x x x =−
1

2gx
2 g(3g gx x−2gx

2)(2g2 gx x
4+ gx gx x

2 g2 gx x x − 10gx
2 g2 gx x x

2

−9gx
2 gx x

3 g + 18gx
3 gx x g gx x x + 4gx

4 gx x
2− 8gx x x gx

5).

Example 3. In this class, we can take ẍ = x2 ẋ − 1
15

x5 for which we have X1 =
∂
∂ t

and

X2 = t ∂
∂ t
− x

2
∂
∂ x

. The corresponding fluxes form the two-parameter group of affine transforma-
tions (t, x)→ (λt +µ, xp

λ
).



H. Zeghdoudi, L. Bouchahed, R.Dridi / Eur. J. Pure Appl. Math, 6 (2013), 126-136 133

Subcase II.4. g = 0, f fx 6= 0 The last of the four subcases is










































λx = 0,

Ax =
λ
�

fx
2 fx x + f fx fx x x − 2 f fx x

2
�

2 fx
3 f

,

B =−
λ f

fx
,

fx x x x =
− f fx

3 fx x x − 6 fx x
3 f 2+ 6 f 2 fx fx x fx x x + fx x

2 f fx
2+ fx

4 fx x

f 2 fx
2

g = 0.

We deduce that the symmetry Lie algebra can not be abelian and hence:

Proposition 5. The symmetry Lie algebra is isomorphic to a(1,R).

Remark 2. Liénard equation such that g = 0 has the first integral t −
∫ x 1

∫

f (s)+C
. Hence, such

equations can not have limit cycles.

Example 4. In this case, we can take ẍ = x ẋ for which we have X1 =
∂
∂ t

and X2 = t ∂
∂ t
− x ∂

∂ x
.

The corresponding fluxes form the two-parameter group of affine transformations
(t, x)→ (λt +µ, x

λ
).

3.1.1. Third Case g 6= 0

The characteristic presentation is














































λx = 0,

Ax =−
f
�

9λg − B f 2
�

27g2 ,

Bx =
6λg − B f 2

3g
,

fx =−
f 3

9g
,

gx =
f 2

3

The number of points under the stairs of λ, A and B shows that the Lie algebra is three-
dimensional.

Proposition 6. In this case the Lie algebra is 3-dimensional and generated by










X1 =
∂
∂ t

,

X2 = (x
f (0)

3g(0) + t) ∂
∂ t
+
�

2 x − x2 f (0)2

3g(0) +
1
81

x3 f (0)4

g(0)2

�

∂
∂ x

X3 = x f (0)3

27g(0)2
∂
∂ t
+
�

1− x f (0)2

3g(0) + x2 f (0)4

27g(0)2
− x3 1

729
f (0)6

g(0)3

�

∂
∂ x

,

where f (0), g(0) denote the values of f et g at x = 0.



H. Zeghdoudi, L. Bouchahed, R.Dridi / Eur. J. Pure Appl. Math, 6 (2013), 126-136 134

Proof. See [10].

3.1.2. Fourth Case

This case corresponds to the characteristic set






















λ= 0,

Ax ,x = 0,

Bx ,x = 2 Ax f ,

fx = 0,

g = 0.

We deduce:

Proposition 7. The symmetry algebra is four-dimensional. Moreover, Liénard equation is neces-
sarily of the form

ẍ = aẋ ,

where a ∈ R. The infinitesimal generators are

X1 =
∂

∂ t
, X2 =

∂

∂ x
, X3 = x

∂

∂ x
, X4 = x

∂

∂ t
+ x2 ∂

∂ x
,

generating gl(2,R). The corresponding fluxes are

(t, x)→ (t − ln(1− εx) +µ, σ
x

1− εx
+ ν).

where ε, µ, σ and ν are the group parameters

3.1.3. Fifth Case

This case completes the classification. We have the characteristic set






















λx = 0,

Ax x = 0,

Bx x = 0,

f = 0,

g = 0.

The last two equations implies that the last differential ideal
p

I8 ∩Q{ f , g} is generated
by { f , g}. We have immediately



REFERENCES 135

Proposition 8. Liénard equation is reduced to

ẍ = 0.

The symmetry Lie algebra is generated by the vector fields

X1 =
∂

∂ t
, X2 =

∂

∂ x
, X3 = x

∂

∂ x
, X4 = x

∂

∂ t
, X5 = t

∂

∂ t
.

The corresponding fluxes form a five-parameter transformations group

(t, x)→ (λt +µ+ εx ,ρx +σ).

(λ,µ,ε,ρ,σ) are the group parameters.

ACKNOWLEDGEMENTS This work was been given European financing within the frame-
work of the program Averroés (Erasmus Mundus).

References

[1] I.M. Anderson, N. Kamran, and P.J. Olver. Internal, external, and generalized symme-
tries. Adv. Math. 100 (1), 53-100 (1993).

[2] G.W. Bluman and S. Kumei. Symmetries and differential equations. Vol. 81 of Applied
Mathematical Sciences Springer-Verlag, New York (1989).

[3] F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Representation for the radical of a finitely
generated differential ideal. In: proceedings of ISSACâĂŹ95. Montréal, Canada, pp.
158-166 (1995).

[4] R. Fitzhugh. Impulses and physiological in theoretical models of nerve membranes. Bio-
physics Journal 1, 445-466 (1961).

[5] H. Giacomini and S. Neukirch. Number of limit cycles of the Lieénard equation. Physical
Review E, 56(4), (1997).

[6] G. Joanna. Lie Symmetry Methods in Finance - An Example of the Bond Pricing Equa-
tion. Proceedings of the World Congress on Engineering Vol II, London, U.K (2008).

[7] E.R. Kolchin. Differential algebra and algebraic groups. Academic Press, New York, Pure
and Applied Mathematics, 54 (1973).

[8] J. Nagumo, S. Arimoto, and S. Yoshizawa. An active pulse transmission line simulating
nerve axon. In: Proceedings of the Institute of Radio Engineers. Vol. 50. pp. 2061-2070
(1962).



REFERENCES 136

[9] B. van der Pol and J. van der Mark. The heartbeat considered as a relaxation oscillation,
and an electrical model of the heart. The London, Edinburgh, and Dublin Philosophical
Magazine and Journal of Science, Series 7 6, 763-775 (1928).

[10] H. Zeghdoudi, R. Dridi, R.M. Remita, and L. Bouchahed. Around Complete Classifica-
tion of Liénard Equation and Application. International Journal of Pure and Applied
Mathematics, 82(3), 441-454 (2013).