Int. J. Anal. Appl. (2022), 20:73

Unusual Nonpolynomial Van der Pol Oscillator Equations With Exact Harmonic and

Isochronous Solutions

Kolawolé Kêgnidé Damien Adjaï, Marcellin Nonti, Jean Akande, Marc Delphin Monsia∗

Department of Physics, University of Abomey-Calavi, Abomey-Calavi, 01.BP.526, Cotonou, Benin

∗Corresponding author: monsiadelphin@yahoo.fr

Abstract. We do not know Van der Pol-type equations with nonlinear restoring force having explicitly an

exact periodic solution. We present, for the first time, nonpolynomial Van der Pol oscillator equations

that do not satisfy the classical existence theorems. We exhibit their exact harmonic and isochronous

solutions and prove the existence of limit cycles by using averaging theory. We also present first

integrals and exact solutions of polynomial Van der Pol-Duffing equations to show that they do not

have any limit cycle. Additionally, we prove that the damped Duffing-type equations are equivalent to

the conservative Duffing equations exhibiting nonoscillatory solutions.

1. Introduction

The Lienard equation:

ẍ + f (x)ẋ +g(x)=0, (1.1)

where overdot is the derivative with respect to time and f (x) and g(x) are functions of x, is one of the

most important autonomous second-order differential equations. This importance results in the fact

that it often occurs in mathematical modelling of physical and engineering systems. Consequently,

Equation (1.1) has been widely investigated in the literature. A celebrated equation of this type is the

Van der Pol oscillator [1–3]:

ẍ +β(x2 −1)ẋ +x =0, (1.2)

where β � 0, mentioned having a unique limit cycle only in the light of qualitative theory of differential
equations and existence theorems [1–3] since it has no known exact and general solutions. Thus,

Received: Nov. 12, 2022.

2010 Mathematics Subject Classification. 34A05, 34C07, 34C25, 34C29, 65L06.
Key words and phrases. Van der Pol-Duffing equation, nonpolynomial Van der Pol-type oscillator, damped Duffing

equation, first integrals, exact harmonic and limit cycle solutions, existence theorem.

https://doi.org/10.28924/2291-8639-20-2022-73
ISSN: 2291-8639

© 2022 the author(s).

https://doi.org/10.28924/2291-8639-20-2022-73


2 Int. J. Anal. Appl. (2022), 20:73

the result known currently under the name of the Lienard-Levinson-Smith theorem [1–3] has been

intensively applied to investigate the existence of limit cycles for the equations of type (1.1). In this

way the Van der Pol-Duffing equation [4] is:

ẍ +β(x2 −1)ẋ +γx +αx3 =0, (1.3)

where α, β and γ are constants and has been considered for a long time as a self-excited oscillator.

However, similar to the Van der Pol equation, Equation (1.3) is not integrable in general. Recently,

Udwadia and Cho [5] succeeded in showing that equation (1.3) in the form:

ẍ +β(x2 −1)ẋ −3α
(
1+
3α

β3

)
x +αx3 =0, (1.4)

does not have any limit cycle by explicitly calculating its general solution. In [6], the author established

a first integral of equations of type (1.3) but in terms of special functions, namely, in terms of hyper-

geometric functions. Monsia et al. [4], for the first time, successfully derived the time-independent

first integral of a more general form of equation (1.4) in terms of elementary functions such that it

became possible to obtain the Lagrangian of equations of type (1.3). The above explicitly shows that

the Van der Pol equation with polynomial nonlinear restoring force can have no limit cycle. Moreover,

it is worth mentioning that the exact and general solution depicted by Udwadia and Cho [5] is not

periodic. Recently, Akande et al. [7] successfully showed that the Lienard equation:

ẍ −
q1x√
q22 −x2

ẋ =0, (1.5)

admits exact harmonic and isochronous periodic solutions, while this equation does not satisfy the

classical existence theorems. This fact is also observed for the Lienard equations:

ẍ ±q3
√
q4q5 −q6q7x2ẋ +q4q7x +q4q7x2 −q3q4q5 =0, (1.6)

studied by Akplogan et al. [8], where the parameters qi, i = 1, ....,7 are arbitrary constants. In this

situation, let us consider the following equation with Van der Pol damping called Van der Pol-type

equation:

ẍ +β(x2 −a)ẋ +γx +λ(c1 −c2x2)p =0, (1.7)

where β � 0, γ � 0, a � 0, λ, c1, c2, and p are constants. When λ = 0, Equation (1.7) reduces
to the Van der Pol equation (1.2) for γ = 1. Equation (1.7) generalizes the equation given in [ [1]

p.146] as an exercise for c1 = 0, and p = 1. If c1 = 0 and p =
3
2
, Equation (1.7) becomes the Van

der Pol-Duffing equation (1.3) mentioned above. For c2 = 0, Equation (1.7) takes the form of the

biased Van der Pol equation, which appears in [ [9] p.287]. To the best of our knowledge, such a

Van der Pol equation with nonpolynomial restoring force has not been previously investigated in the

literature. From this perspective, the question is to ask if Equation (1.7) can exhibit harmonic and

limit cycle oscillations. Thus, the objective is to show the nonexistence of limit cycles for Van der Pol

equations with polynomial restoring force and to prove the existence of the harmonic and limit cycle



Int. J. Anal. Appl. (2022), 20:73 3

solutions of Equation (1.7). To that end, we first present some results for the Van der Pol equations

with polynomial restoring force (Section 2) and second, explicitly prove the existence of the exact

harmonic and limit cycle solutions of Equation (1.7) (Section 3). Finally, the results are compared to

numerical solutions using the fourth-order Runge-Kutta algorithm, and we give a conclusion for the

work.

2. Van der Pol equations with polynomial restoring force

In this part, we present time-independent first integrals allowing the determination of Lagrangian

and exact solutions of Van der Pol-type equations with polynomial restoring forces.

2.1. Van der Pol-Duffing equations. Consider the equation:

ẍ − (k1x2 +3k2)ẋ +2k22x −2k
2
1x
5 =0, (2.1)

where k1 and k2 are arbitrary parameters. A first integral of this equation can be written as follows:

ẋ = k2x +k1x
3. (2.2)

Putting k1 =−µ and 3k2 = k, one can, from Equation (2.1), arrive at:

ẍ +(µx2 −k)ẋ +
2k2

9
x −2µ2x5 =0, (2.3)

which is a quintic Duffing-Van der Pol equation.

Consider the equation:

ẍ +(3k1x
2 −2k2)ẋ +k22x −k1k2x

3 =0, (2.4)

which denotes the cubic Duffing-Van der Pol equation. The first integral of Equation (2.4) has the

form:

ẋ = k2x −k1x3. (2.5)

The cubic-quintic Duffing-Van der Pol equation:

ẍ +k1(x
2 −1−2k1)ẋ +2k31x +2k

2
1(1−k1)x

3 −2k21x
5 =0, (2.6)

has the time-independent first integral:

k1x
5 = k1x

3 −x2ẋ. (2.7)

The first integral of the equation:

ẍ −k2(1−x2)ẋ −k1(k1 +k2)x +
k1k2
3
x3 −k1k3 =0, (2.8)

can be written as follows:

ẋ =(k1 +k2)x −
k2
3
x3 +k3, (2.9)

where k3 is an arbitrary parameter.



4 Int. J. Anal. Appl. (2022), 20:73

Consider the equation:

ẍ −k1(1−x2)ẋ +k2(k1 −k2)x −
k1k2
3
x3 =0. (2.10)

Its first integral can be read:

ẋ =(k1 −k2)x −
k1
3
x3. (2.11)

2.2. Generalized Van der Pol-type equation. Now, consider the more general Van der Pol-type

equation with polynomial restoring force:

ẍ +

(
k2
k3
`+

k1
k3
x1−`

)
ẋ +
(`−1)k21

k23
x3−2` +

2(`−1)k1k2
k23

x2−` +
(`−1)k22

k23
x =0. (2.12)

One can verify that Equation (2.12) has the time-independent first integral:

ẋ =−
1

k3

(
k1x

2−` +k2x

)
, (2.13)

where ` is an arbitrary parameter. It suffices to put ` = (2−n) into Equation (2.12) to recover the
general form considered in [6]. The author derived the first integral of this type of Equation (2.12)

in terms of hypergeometric functions. Equation (2.12) with ` = (2− n) has also been investigated
by Chandrasekar et al. [10]. The first integrals derived by these authors [10] using the so-called

generalized extended Prelle-Singer method are functions of time. The general solution of Equation

(2.12), as can be verified, using the first integral (2.13), becomes:

x(t)=

[
1

k2

(
−k1 +e

−k2
k3
(`−1)(t+K)

)] 1
`−1
, (2.14)

where ` 6= 1, and K is an integration constant. An interesting case of Equation (2.12) consists of
putting ` =−1 to obtain:

ẍ +

(
k1
k3
x2 −

k2
k3

)
ẋ −
2k21
k23
x5 −

4k1k2

k23
x3 −

2k22
k23
x =0. (2.15)

The Van der Pol-Duffing equation (2.15) is equivalent to the conservative cubic-quintic Duffing

equation:

ẍ −
3k21
k23
x5 −

4k1k2

k23
x3 −

k22
k23
x =0, (2.16)

obtained by using the first integral (2.13), where ` = −1. In this regard, the general solution of
Equation (2.15) or (2.16), taking into consideration Equation (2.14), can read:

x(t)=
(k2)

1
2[

e
2k2
k3
(t+K) −k1

]1
2

. (2.17)



Int. J. Anal. Appl. (2022), 20:73 5

This result shows that the widely studied cubic-quintic Duffing equation is (2.12) in fact a pseudo-

oscillator. From Equation (2.13), the more general Van der Pol-type equation with polynomial non-

linear restoring force becomes equivalent to the generalized Duffing equation:

ẍ −
k1k2

k23
(3− `)x2−` −

k21
k23
(2− `)x3−2` −

k22
k23
x =0. (2.18)

Conservative equation (2.18) also has the general solution (2.14). It is worth noting that the

conservative cubic-quintic Duffing equation (2.16) is equivalent by using the first integral (2.13) to

the dissipative Lienard equation:

ẍ +
4k2
k3
ẋ −
3k21
k23
x5 +

3k22
k23
x =0, (2.19)

with the general solution (2.17). Equation (2.19) is also known as the damped-quintic Duffing equa-

tion.The general solution (2.17) of Equation (2.19) contradicts the results of existence of the bounded

periodic solutions exhibited in [11]. The above shows that the Van der Pol-type equations with poly-

nomial restoring force do not have any limit cycle. That being so, we can establish the exact general

harmonic solution and limit cycle of Equation (1.7).

3. Exact harmonic and limit cycle solutions

3.1. Exact harmonic and isochronous solutions. In this part, we look for Equation (1.7), an exact

harmonic solution:

x(t)= Acos(wt +ϕ), (3.1)

where A, w and ϕ are arbitrary parameters. Thus, substituting Equation (3.1) into Equation (1.7)

yields, after a few algebraic calculations, the results are c1 = a = A2, γ = w2, c2 =1, λ =±βw, and
p = 3

2
. The arbitrary constant ϕ can be determined by using initial conditions. From these integrability

conditions, the desired Van der Pol-type equation (1.7) with nonlinear restoring force takes the form:

ẍ +β(x2 −A2)ẋ +w2x ±βw(A2 −x2)
3
2 =0. (3.2)

Comparing Equation (3.2) with Lienard equation (1.1) yields f (x) = β(x2 − A2) and g(x) =
w2x ±βw(A2 −x2)

3
2 . Hence, for g(x)= w2x −βw(A2 −x2)

3
2 , Equation (3.2) is written:

ẍ +β(x2 −A2)ẋ +w2x −βw(A2 −x2)
3
2 =0, (3.3)

which admits the exact and general solution (3.1). For g(x)= w2x +βw(A2−x2)
3
2 , Equation (3.2)

becomes:

ẍ +β(x2 −A2)ẋ +w2x +βw(A2 −x2)
3
2 =0, (3.4)

and has the exact and general solution:

x(t)= Asin(wt +ϕ1), (3.5)



6 Int. J. Anal. Appl. (2022), 20:73

where ϕ1 is an arbitrary constant that can be calculated by the application of initial conditions.

3.2. Existence theorem analysis. As seen, the functions f (x) and g(x) do not satisfy the conditions

required by usual theorems for the existence of a centre at the origin [1–3,12]. For example, according

to Theorem 11.3 of [ [1] p.390], the origin is a centre for the Lienard equation (1.1) when f (x) and

g(x) are odd, and g(0) = 0. However, the previous expression of f (x) is not odd but rather even.

The previous formulas of g(x) are not odd, and g(0) 6=0 while the general solutions (3.1) and (3.5)
show that the origin is an isochronous centre for Equations (3.3) and (3.4).

3.3. Phase plane analysis. Consider Equation (3.3). Then, the equivalent dynamical system can be

written as follows: 

ẋ = y

ẏ =−β(x2 −A2)y −w2x +βw(A2 −x2)
3
2 .

(3.6)

The equilibrium points are given by y =0, and:

βw(A2 −x2)
3
2 −w2x =0. (3.7)

One can easily observe that x = 0 does not satisfy Equation (3.7). Therefore, for the qualitative

theory of differential equations, Equation (3.3) cannot have the origin as an equilibrium point, which

contradicts the exact and general harmonic solutions (3.1). According to Equation (3.7), x = 0,

when:

βwA3 =0. (3.8)

Equation (3.8) holds only, as w 6=0, and A 6=0, when β =0. In this case, Equation (3.3) reduces
to the linear harmonic oscillator equation:

ẍ +w2x =0. (3.9)

The previous analysis also holds for Equation (3.4). The above shows that the classical theorems

for the existence of isochronous centres clearly exclude a number of Lienard equations, as seen in

several previous papers [7,8].

3.4. Application of averaging method for equation (3.3). In this part, we investigate the existence

of a limit cycles for Equation (3.3) using the averaging method. Equation (3.3) can be written as

follows:

ẍ +w2x +β

[
(x2 −A2)ẋ −w(A2 −x2)

3
2

]
=0. (3.10)

The equation (3.10) has the form:

ẍ +w2x = εF(ẋ,x), (3.11)



Int. J. Anal. Appl. (2022), 20:73 7

where F(ẋ,x) = −(x2 − A2)ẋ + w(A2 − x2)
3
2 and β = ε, such that ε is small parameter, that is,

0 < ε � 1, as required in the application of the averaging method. Now, we seek for Equation (3.11)
the solution of the form:

x(t)= r(t)cos

(
wt +φ(t)

)
, (3.12)

under the initial conditions:

x(0)= A, ẋ(0)=0, (3.13)

such that:

ẋ(t)=−wr(t)sin
(
wt +φ(t)

)
, (3.14)

and

ṙ(t)cos

(
wt +φ(t)

)
− r(t)φ̇(t)sin

(
wt +φ(t)

)
=0. (3.15)

Hence, knowing that:

(A2 −x2)
3
2 =(A2 −

r2(t)

2
)

[
1−

3r2(t)

4(A2 − r
2(t)
2
)
cos(2wt +2φ(t))

]
, (3.16)

one can, after a little algebraic manipulation, obtain:

F(ẋ,x)= w

[
A2 −

r2(t)

2

]3
2

−
3

4
w

[
A2 −

r2(t)

2

]1
2

r2(t)cos(2wt +2φ(t))+

1

4
wr3(t)sin(3wt +3φ(t))+wr(t)

[
r2(t)

4
−A2

]
sin(wt +φ(t)).

(3.17)

From:

ṙ(t)=−
ε

2πw

∫ 2π
0

F

[
r(t)cos(wt +φ(t)),−wr(t)sin(wt +φ(t))

]
sin(wt +φ(t))d(wt +φ(t)),

(3.18)

and

φ̇(t)=−
ε

2πwr(t)

∫ 2π
0

F

[
r(t)cos(wt+φ(t)),−wr(t)sin(wt+φ(t))

]
cos(wt+φ(t))d(wt+φ(t)).

(3.19)

After a few algebra, it results in:

ṙ(t)=
εr(t)

8

[
4A2 − r2(t)

]
(3.20)

Integrating, after separation of variables, yields:

r(t)=
2A[

1+e−εA
2(t+K)

]1
2

, (3.21)



8 Int. J. Anal. Appl. (2022), 20:73

where K is a constant of integration. Using the initial conditions (3.13), the following is obtained:

r(t)=
2A[

1+3e−εA
2t

]1
2

, (3.22)

where e−εA
2K =3. Now, from Equation (3.19):

φ̇(t)=0,

that is:

φ(t)= ϕ0, (3.23)

where ϕ0 is a constant. In this context, the desired solution (3.12) takes the form:

x(t)=
2A[

1+3e−εA
2t

]1
2

cos(wt +ϕ0). (3.24)

When A = w =1, and ϕ0 =0, the solution (3.23) becomes:

x(t)=
2

√
1+3e−εt

cos(t). (3.25)

It is worth mentioning that the formula (3.25) is the solution obtained for the Van der Pol oscillator

equation:

ẍ +ε(x2 −1)ẋ +x =0, (3.26)

by Strogatz [ [9], p.225] using the averaging method.

4. Numerical applications

To do so, it is first necessary to determine the constants ϕ and ϕ1 from the initial conditions.

4.1. Solution (3.1) in terms of x0 and v0. From the general initial conditions x(0)= x0 and ẋ(0)=

v0, one can obtain, using the general solution (3.1), the system of algebraic equations:

x0 = Acosϕ, v0 =−wAsinϕ, (4.1)

which yields the constant:

ϕ = arctan

(
−

v0
wx0

)
. (4.2)

Using this expression, the general solution (3.1) takes the form:

x(t)= Acos

[
wt −arctan

(
v0
wx0

)]
. (4.3)

Figure 1 shows the graphical comparison of the result (4.3) in the circles line with the solution

obtained by numerical integration of Equation (3.3) in solid line where A = 1, w = 1, β = 0.01,

x0 =1, and v0 =0.001.



Int. J. Anal. Appl. (2022), 20:73 9

Figure 1. Comparison of solution (4.3) to the numerical solution of Equation

(3.3).Typical values are A =1, w =1, β =0.01, x0 =1, and ϑ =0.001.

4.2. Solution (3.12) in terms of x0 and v0. Under the general initial conditions that x(0)= x0 and

ẋ(0)= v0, solution (3.5) leads to the system of algebraic equations:

x0 = Asinϕ1, v0 = wAcosϕ1, (4.4)

such that the constant ϕ1 is defined as:

ϕ1 = arccotan

(
v0
wx0

)
. (4.5)

From this, the general solution (3.5) can be rewritten in the form:

x(t)= Asin

[
wt +arccot

(
v0
wx0

)]
. (4.6)

The graphical comparison of this solution (4.6) in the circles line with the result obtained by

numerical integration of Equation (3.4) is depicted in Figure 2, where A = 1, w = 1, β = 0.01,

x0 =1, and v0 =0.001.

Figure 2. Comparison of solution (4.6) to the numerical solution of Equation

(3.4).Typical values are A =1, w =1, β =0.01, x0 =1, and ϑ =0.001.



10 Int. J. Anal. Appl. (2022), 20:73

5. Conclusion

We have presented exceptional nonpolynomial Van der Pol oscillator equations in this paper. We

have exhibited their exact harmonic and limit cycle solutions while they do not satisfy classical theorems

for the existence of at least one periodic solution. We have proven that the Van der Pol-Duffing-type

equations are equivalent to the conservative Duffing equations so that they could not admit limit cycle

oscillations. Additionally, we have shown that the damped quintic Duffing equations are equivalent

to the conservative cubic-quintic Duffing equations. It was for the first time such results have been

obtained in the literature.

Authors’ Contributions: All authors have equal contributions in this paper.

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

References

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[2] R.E. Mickens, Oscillations in Planar Dynamic Systems, World Scientific, Singapore, 1996. https://doi.org/10.

1142/2778.

[3] M. Cioni, G. Villari, An Extension of Dragilev?s Theorem for the Existence of Periodic Solutions of the Liénard

Equation, Nonlinear Anal.: Theory Meth. Appl. 127 (2015), 55-70. https://doi.org/10.1016/j.na.2015.06.026.

[4] M. D. Monsia, J. Akande, K.K.D. Adjaï, On the Non-Existence of Limit Cycles of the Duffing-Van Der Pol Type

Equations, (2021). https://doi.org/10.6084/m9.figshare.14547501.v1.

[5] F.E. Udwadia, H. Cho, First Integrals and Solutions of Duffing-van Der Pol Type Equations, J. Appl. Mech. 81

(2013), 034501. https://doi.org/10.1115/1.4024673.

[6] T. Stachowiak, Hypergeometric First Integrals of the Duffing and Van Der Pol Oscillators, J. Differ. Equ. 266 (2019),

5895-5911. https://doi.org/10.1016/j.jde.2018.10.049.

[7] J. Akande, K.K.D. Adjaï, A.B. Yessoufou, et al. Exact and Sinusoidal Periodic Solutions of Lienard Equation Without

Restoring Force, (2021). https://doi.org/10.6084/M9.FIGSHARE.14546019.

[8] A.R.O. Akplogan, K.K.D. Adjaï, J. Akande, et al. Modified Van Der Pol-Helmohltz Oscillator Equation With Exact

Harmonic Solutions, (2021). https://doi.org/10.21203/rs.3.rs-730159/v1.

[9] S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering,

Westview Press, Cambridge, 2001.

[10] V.K. Chandrasekar, S.N. Pandey, M. Senthilvelan, et al. A Simple and Unified Approach to Identify Integrable

Nonlinear Oscillators and Systems, J. Math. Phys. 47 (2006), 023508. https://doi.org/10.1063/1.2171520.

[11] E.E. Obinwanne, A.R. Okeke, On Application of Lyapunov and Yoshizawa’s Theorems on Stability, Asymptotic

Stability, Boundaries and Periodicity of Solution of Duffing’s Equation, Asian J. Appl. Sci. 2 (2014), 970-975.

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10.1006/jdeq.1998.3520.

https://doi.org/10.1142/2778
https://doi.org/10.1142/2778
https://doi.org/10.1016/j.na.2015.06.026
https://doi.org/10.6084/m9.figshare.14547501.v1
https://doi.org/10.1115/1.4024673
https://doi.org/10.1016/j.jde.2018.10.049
https://doi.org/10.6084/M9.FIGSHARE.14546019
https://doi.org/10.21203/rs.3.rs-730159/v1
https://doi.org/10.1063/1.2171520
https://doi.org/10.1006/jdeq.1998.3520
https://doi.org/10.1006/jdeq.1998.3520

	1. Introduction
	2. Van der Pol equations with polynomial restoring force
	2.1. Van der Pol-Duffing equations
	2.2. Generalized Van der Pol-type equation

	3. Exact harmonic and limit cycle solutions
	3.1. Exact harmonic and isochronous solutions
	3.2. Existence theorem analysis
	3.3. Phase plane analysis
	3.4. Application of averaging method for equation (3.3)

	4. Numerical applications
	4.1. Solution (3.1) in terms of x0 and v0.
	4.2. Solution (3.12) in terms of x0 and v0.

	5. Conclusion
	References