Circuits and Systems, 2011, 2, 196-200
doi:10.4236/cs.2011.23028 Published Online July 2011 (http://www.SciRP.org/journal/cs)
Copyright © 2011 SciRes. 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: hkocak.ege@gmail.com, ahmet.yildirim@ege.edu.tr,
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
11
2
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):
 

 



 
2
,,
,, 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]:
 
 




21
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]:
 
 




2
0
0
1
00
00
xt yt
ytxtxtytk xt
xxH
xx

  



(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:
 
 
0
0
n
n
n
n
n
n
xtx t
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:
2
111
32
( )sin()( )cos()
cos( )sin( )
cos() cos()
sin()sin() 0
xt HtxtHt
Htt
kH t
kH t
 



 




.2. The Boubaker Polynomials Expansion
y-
nomials expansion scheme (BPES) [11-23]. The first
step of this scheme starts by applying the
(4c)
with, as a final solution (Equation (4d)):
2
Scheme (BPES)-Related Solution
The resolution protocol is based on the Boubaker pol
expressions:


4
0
1λ
2kk k
k
0
1
N
x
tBtr
N

(4e)
where 4k
B are 4k-order Boubaker polynom
the normalized time (
theials, is
0,1t), k
r are minimal
po tege
4k
B
r, and
sitive roots, 0
N is a prefixed in0
1..
kkN
λ
are u poing
Consequently, it comes that:
nknownnder real coefficients.
 

04
1
0
2d
kk
kk
k
yt xt λr
Nt

(5)
The main advantage of these formulations (Equations (4)
and (5)) is the fact of verifying the boundary conditions in
d
1NBtr
t stage of
fact, due to the properties of the Boubaker polynomials
[12-18], and since
Equation (3), at the earliesresolution protocol. In
0
1..kN
k
rare the roots of
k
r 0
41..
kkN
B,
the following conditions stand :



0
0
0
01
0
4
1
0
00
10
d
d10
d2 d
N
k
tk
Nkk
kk
k
tt
xt λAx
N
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 ,
x
t is confined, through the coefficients
0
1..
kkN
, to be a weak solution of the system:





 

000
0
2
111
2
14
2
0
14
0
1
4
0
1
4
0
00 0
1
0
dd
d
dd
d
d
d
NNN
kk kkkkkkk
kkk
kk
k
kk
k
kkk
kkk
N
k
k
rM rPQkR
Btr
Mt
t
Btr
Pt
t
QBtrt
RBtrt
Nx NH
 





 

(7)

4224 22
2
26 336 3636
cos
kk kkk
xt Ht
k
 




 




(4d)
Copyright © 2011 SciRes. CS
198
The set of solutions
H. KOÇAK ET AL.
0
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
al
yi
k

,
MS k
:



000 2
2
1
,
ˆˆˆ
MS
NNN
kkk
k
k
rM rPQkR
 



11
kk k kk k
kk

under the intrinsic condition:
0
0
1
ˆ
N
k
k
NH

0
N.
Th
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
0
x
2.75
0
x
0.0
0
N
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
N
-
e
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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