A Comparative Study of Analytical Solutions to the Coupled Van-der-Pol’s Non-linear Circuits Using the He’s Method (HPEM) and (BPES)

doi:10.4236/cs.2011.23028

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Circuits and Systems, 2011, 2, 196-200

doi:10.4236/cs.2011.23028 Published Online July 2011 (http://www.SciRP.org/journal/cs)

A Comparative Study of Analytical Solutions to the

Coupled Van-der-Pol’s Non-linear Circuits Using the

He’s Method (HPEM) and (BPES)

Hüseyin Koçak1, Ahmet Yıldırım1, Dahong Zhang2, Karem Boubaker3, Syed Tauseef Mohyud-Din4

1Department of Mathematics, Ege University, Bornova–İzmir, Turkey

2Department of Physics, South China University, Guangzhou, China

3École Supérieure des Sciences et Techniques de Tunis, University of Tunis, Tunis, Tunisia

4Department of Basic Sciences, HITEC University, Taxila Cantt, Pakistan

E-mail: [email protected], [email protected],

Received October 6, 2010; revised May 13, 2011; accepted May 20, 2011

Abstract

In this paper, the He’s parameter-expanding method (HPEM) and the 4q-Boubaker Polynomials Expansion

Scheme (BPES) are used in order to obtain analytical solutions to the non-linear modified Van der Pol’s os-

cillating circuit equation. The resolution protocols are applied to the ordinary Van der Pol equation, which

annexed to conjoint delayed feedback and delay-related damping terms. The results are plotted, and com-

pared with exact solutions proposed elsewhere, in order to evaluate accuracy.

Keywords: Van-der-Pol’s Oscillating Circuit, Delayed Feedback, Damping, BPES, HPEM, Exact Solutions,

Electrical Triode-Valve Circuit

1. Introduction

Originally, the Van der Pol’s equation was associated, in

the 1920s, with an electrical triode-valve circuit (Figure

1). In the last decades’ literature, it was the subject of

several investigations due to the panoply of dynamical

oddness as relaxation oscillations, elementary bifurca-

tions, quasiperiodicity, and chaos. Its application has

already reached nerve pulse propagation and electric

potential evolution across neural membranes.

Figure 1. Van der Pol oscillator synoptic scheme.

The actual study tries to give a theoretical supply to

the recent attempts to yield analytical solutions to this

equation, like the studies of D. D. Ganji et al. [1,2] and A.

Rajabi et al. [3] in the heat transfer domain, the investi-

gations of L. Cveticanin [4] and J. H. He [5-7] on non-

linear mechanics, fluid dynamics and oscillating systems

modelling (Figure 2).

Figure 2. A prototype of Van der Pol oscillating systems

modelling (The two integrators are Trapezoidal-type









kkkk

yhuu y



).

H. KOÇAK ET AL.

197

Among the different formulations, the well-known

standard boundary value-free Van der Pol oscillator

problem (BVFP) is given by F. M. Atay [8] by the fol-

lowing system (1):

 



 



 

,, 1

xt xtfxtxtxt

fxt xt xtxtxtk xt





 

 





 





(1)

where



is a positive parameter representing the delay,



> 0 and is the feedback gain. k

A simpler formulation is that of W. Jiang et al. [9]:

 

 



xt yt

ytxtkf xtxtyt







 







 (2)

In this study, an attempt to give analytical solution to

the nonlinear second-order Van der Pol equation annexed

to conjoint delayed feedback and delay-related damping

terms as presented by A. Kimiaeifar et al. [10]:

 

 







xt yt

ytxtxtytk xt

xxH







  















 (3)

2. Analytical Solutions Derivation

2.1. The Enhanced He’s Parameter-Expanding

Method (HPEM) Solution

The resolution protocol based on the enhanced He’s pa-

rameter-expanding method (HPEM) is founded on the

infinite serial expansions:

 

xtx t



















 







 (4a)

Substituting these expansions in the main equation

Equation (3) and processing with the standard perturba-

tion method, it has been demonstrated [10] that a solu-

tion of the kind:

 

0cosxt Ht





 (4b)

where



and



are constant, gives:

111

( )sin()( )cos()

cos( )sin( )

cos() cos()

sin()sin() 0

xt HtxtHt

Htt

kH t



 





 





 

.2. The Boubaker Polynomials Expansion

nomials expansion scheme (BPES) [11-23]. The first

step of this scheme starts by applying the

(4c)

with, as a final solution (Equation (4d)):

Scheme (BPES)-Related Solution

The resolution protocol is based on the Boubaker pol

expressions:



1λ

2kk k

tBtr



 (4e)

where 4k

B are 4k-order Boubaker polynom

the normalized time (



theials, is





0,1t), k

r are minimal

po tege

r, and

sitive roots, 0

N is a prefixed in0

1..

kkN

λ

are u poing

Consequently, it comes that:

nknownnder real coefficients.

 



yt xt λr



 



 (5)

The main advantage of these formulations (Equations (4)

and (5)) is the fact of verifying the boundary conditions in

1NBtr

t stage of

fact, due to the properties of the Boubaker polynomials

[12-18], and since

Equation (3), at the earliesresolution protocol. In

1..kN

rare the roots of

r 0

41..

kkN

B,

the following conditions stand :



d10

d2 d

Nkk

xt λAx

Btr

xt r

tN t









 

















 (6)

By introducing expressions (4) and (6) in the system

(3), and by majoring and integrating along the interval





0,1 ,





t is confined, through the coefficients

1..

kkN



, to be a weak solution of the system:





 



000

111

00 0

NNN

kk kkkkkkk

kkk

rM rPQkR

Btr

QBtrt

RBtrt

Nx NH

 





































 













(7)



4224 22

26 336 3636

cos

kk kkk

xt Ht

 











 





(4d)

198

The set of solutions

H. KOÇAK ET AL.

1, ,

ˆkkN





is the one which mini

mizes, for given values of



and the Minimum

Square function

(8)

(9)

The condition expressed by Equation (9) ensures a

non-zero solution to the system (8). The convergence of

the algorithm is tested relatively to increasing values of

e correspondent solutions are represented in Figure

3 for the data gathered in Table 1,

solutions given by F. M. Atay [8] and A. Kimiaeifar et



MS k



:



000 2

ˆˆˆ

NNN

kkk

rM rPQkR



 













kk k kk k

kk 



under the intrinsic condition:









along with the exact

. [10]. It is noted that F. M. Atay [8] demonstrated that

the presence of delay can change the amplitude of limit

cycle oscillations, or suppress them altogether through

derivative-like effects, while A. Kimiaeifar et al. [10]

elded a highly accurate solution to the same classical

Van der Pol equation with delayed feedback and a modi-

fied equation where a delayed term provides the damping.

The features of the proposed solutions [8-10] (namely

behavior at starting phase, first derivatives at limit time,

etc.) are concordant with the actually proposed results.

Figure 3. Analytical solutions plots.

Table 1. Solution parameters values.

Parameter Value



0.1

k −1.0



1.0

2.75

 0.0

Figure 4. Mean absolute error versus N0.

3. Results and Discussions

The results show a good agreement between the pro-

posed analytical solutions (Figure 3) and those of the

recent studies published elsewhere. The mean absolute

error (for) was less than 3.33% (Figure 4).

The convehe BPES-related protocol has been

recorded fs of superior to 30.

4. Conclusions

In this paper, we have used the enhanced He’s parame

ter-expanding Method (HPEM) along with thBoubaker

olynomr to ob-

c periodical solutions.

acceptable agreement

into an

istic nonlinear system. This simple

duction is carried out through the

030N

rgence of t

or the value0

ials Expansion Scheme (BPES) in ordeP

tain the Van der Pol’s characteristi

The obtained solutions were in

with those obtained from values of similarly performed

methods. The typical periodical aspect of the oscillations,

already yielded [2,10,24-27] by the enhanced He’s pa-

rameter-expanding method (HPEM) could be reproduced

using a simple and convergent polynomial approxima-

tion. This method was based on an original protocol

hich reduces the stochastic nonlinear systemw

equivalent determin

nd controllable rea

verification of the initial conditions, in the solution basic

expression, prime to launching the resolution process.

The results show that the methods are very promising

ones and might find wide applications, particularly when

exact solutions expressions are difficult to establish [28-34].

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