Original Article
Nonlinear oscillators dynamics using
optimal and modified homotopy
perturbation method
Journal of Low Frequency Noise,
Vibration and Active Control
2024, Vol. 43(4) 1469–1480
© The Author(s) 2024
DOI: 10.1177/14613484241272253
journals.sagepub.com/home/lfn
Tapas Roy1, Biswanath Rath2, Jihad Asad3, Dilip K Maiti1, Pravanjan Mallick2 and
Rabab Jarrar3
Abstract
This study investigates the several strongly nonlinear oscillators (Van der Pol, Duffing and Rayleigh) for which we have
applied the most powerful and advanced optimal semi-analytical technique: optimal and modified homotopy perturbation
method (OM-HPM) for the convergent semi-analytical series solution. The numerical simulation demonstrates the high
accuracy of the OM-HPM, which is straightforward, does not require any domain decomposition, special transformation,
or pade approximations to get the convergent series solution. The key features for the high accuracy of the OM-HPM lies
on the best optimal auxiliary linear operator. Therefore, OM-HPM offering a valuable tool for engineers and researcher to
analyze the complex nonlinear oscillator.
Keywords
Optimal homotopy perturbation method, series solution, analytical solution, oscillators
Introduction
Many scientific fields, including physics, engineering, and structures including thin-walled structures1 and active structure2
use nonlinear oscillators. Many nonlinear differential equations appear in these fields and studying these equations is a
major issue. Lots of efforts have been made to explore them in literature, and numerous analytical and numerical techniques
have been proposed and employed to solve such problems.3–7 In literature, several analytical methods have been utilized
since an exact solution could be too complex to be applied in real-world situations. For instance, the Hamiltonian
approach,8,9 the variational iteration method,10–12 the homotopy perturbation method,13–17 the variational approach,18–21 in
addition to many other methods.22–25
Nonlinear oscillators come in two varieties: weakly and extremely nonlinear. Weak nonlinearity is indicated by a small
coefficient of the nonlinear term; however, some authors have claimed that this is not always enough and that the amplitude
and starting conditions must also be taken into consideration. Weak nonlinearity can be quantified by treating it as a
disturbance of the corresponding linear oscillator. There are several perturbation techniques available, but not all of them
yield consistently reliable results. Modifications have been developed to alleviate these shortcomings.26,27
Numerous researchers have shown interest in a number of notable oscillators that have important applications in physics
and engineering.28–32 For example, the excited spring pendulum has been studied in detail utilizing the so-called multi-scale
approach, which enables the authors to estimate the equations of motion for the system.29 Additionally, the Van der Poland
Duffing oscillators have been considered in a number of papers.29–31 It is clear from these references that several
1
Department of Applied Mathematics, Vidyasagar University, WB, India
Department of Physics, Maharaja Sriram Chandra Bhanja Deo University, Baripada, Odisha, India
3
Department of Physics, Faculty of Applied Sciences, Palestine Technical University- Kadoorie, Palestine
2
Corresponding author:
Jihad Asad, Department of Physics, Faculty of Applied Sciences, Palestine Technical University- Kadoorie, Java Stree, Tulkarm P 305, Palestine.
Email: j.asad@ptuk.edu.ps
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License
(https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/
en-us/nam/open-access-at-sage).
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Journal of Low Frequency Noise, Vibration and Active Control 43(4)
perturbation approaches were investigated in order to approximate the solution to the dynamic equations of the Van der Pol
oscillator.29,30 On the other hand, the Duffing oscillator was also studied in References 29–32, where the authors used the
effective analytical approach known as the fourth-order Runge–Kutta method to determine the solution and compare it with
a numerical solution.
Semi-analytical approaches, which combine analytical and numerical techniques, are becoming more and more common
for approximating and solving nonlinear problems. They are useful for comprehending and handling complicated nonlinear
occurrences in a variety of domains because they provide a balance between accuracy and computing efficiency. Among
such semi-analytical approaches, homotopy perturbation method (HPM)13,14,32 is one of the most powerful for solving the
highly nonlinear oscillators and is widely use to solve such problems. However, the accuracy and convergence of the HPM
series solution are slower for few problems.33,34 Even sometimes HPM33,34 fails to provide us the convergent series solution
for few highly nonlinear problems. Moreover, others semi-analytical methods including variational iteration method
(VIM)35 and differential transformation method (DTM)34,35 are also fails to provide convergent series solutions. In recent
literature, there is most powerful and advance technique: an optimal and modified homotopy perturbation method (OMHPM).36 Which is nothing but the optimal form of the classical HPM13,14,32 and it is well established for highly nonlinear
differential equations to get the convergence series solution. However, it is seen that the accuracy and convergence of the
HPM series solution are slower for few problems.33,34 Even sometimes HPM33,34 fails to provide us the convergent series
solution for some of the highly nonlinear problems, particularly, for the van der pol oscillator. Several researchers11,37–41 put
effort to solve such van der pol oscillator using other contemporary semi-analytical methods including variational iteration
method (VIM),35 optimal VIM,42 differential transformation method (DTM).37,38 Unfortunately, they failed to provide
convergent series solutions. In the literature, we have added an advanced semi-analytical method, namely, Optimal and
Modified Homotopy Perturbation Method (OM-HPM).36 The traditional Homotopy Perturbation Method
(HPM)3,13–16,19,24–29 is updated by defining the linear operator as an auxiliary linear operator, and then optimized this
operator by minimizing the residual error. Unlike traditional methods, OM-HPM36 does not require any auxiliary parameters or functions. In addition, it can be directly applied to singular or non-singular highly nonlinear ordinary differential
equations without decomposition, special transformations, or pade approximation. Therefore, in this study, our aim is to
analyze the dynamics of some of the strong nonlinear oscillators, namely, duffing oscillator (DO), Rayleigh oscillator (RO),
and van der Pol oscillator (vdPO), using this advance optimal analytical technique.
The rest of this paper is organized as follows: In section 2, we present the methodology of the method used. In section 3,
analytical and numerical methods have been applied to solve three oscillators (Duffing, Van der Pol and Rayleigh). Finally,
we close our paper by a conclusion.
Methodology
In order to solve the considered nonlinear problems, we adopted the recently proposed optimal and modified homotopy
perturbation method (OM-HPM).36 The method is follows:
For the nonlinear differential equation of the form
N ½x ¼ A½x þ gðtÞ ¼ 0, t 2 V
(1)
∂x
B x,
¼ 0, t 2 Γ
∂t
(2)
with the boundary condition
where N , nonlinear operator; A, general differential operator; B, boundary operator; Γ, boundary of the domain V; and g
and x are the known analytic and unknown functions, respectively. The zero-th order homotopy equation as
ð1 pÞ L½Fðt, pÞ x0 ðtÞ þ p N ½Fðt, pÞ ¼ 0
(3)
where p 2 ½0, 1, embedding parameter; Fðt, pÞ, unknown; x0 ðtÞ, initial approximation; and L, auxiliary linear operator of
the form
Roy et al.
1471
L½xðtÞ ¼
dnx
d n1 x
d n2 x
dx
n
þ ð1Þ a0 xðtÞ
n þ an1 n1 þ an2 n2 þ :::::: þ a1
dt
dt
dt
dt
(4)
with
ð1Þ k1
Bk s1 , 1!s2 , 2!s3 , :::::::, ð1Þ ðk 1Þ!sk ,
k!
P
where Bk is the Bell’s polynomial,8 sk ¼ ni¼1 λi k and a0 ¼ ∏ni¼1 λi , λi ’s are the roots of the auxiliary equation of
L½xðtÞ ¼ 0. The auxiliary linear operator L satisfied the property L½xðtÞ ¼ 0 for xðtÞ ¼ 0. For p ¼ 0, we get from equation
(3):
k
ank ¼
Fðt, 0Þ ¼ x0 ðtÞ:
Also when p ¼ 1 from equation (3) we have
Fðt, 1Þ ¼ xðtÞ,
this represents the solution of the given equation (1). As p increase from 0 to 1, the initial solution x0 ðtÞ continuously
deforms to the final solution xðtÞ of the problem. That is, the Taylor series
Xþ∞
Fðt, pÞ ¼ x0 ðtÞ þ k¼1 xk ðtÞpk
(5)
converges when p ¼ 1. Therefore, the series solution of the form
Xþ∞
xðtÞ ¼ x0 ðtÞ þ
xn ðtÞ,
n¼1
(6)
which must satisfy the equation N ½xðtÞ ¼ 0.
If we differentiate the zero-th order homotopy equation (3) by m-times with respect to p and dividing it by m! finally if we
set p ¼ 0, we have the so-called mth order deformation equation as
L½xm ðtÞ χ m xm1 ðtÞ ¼ Rm1 ð!
x m1 ðtÞÞ
(7)
i n x
ðtÞp
m
m¼1
(8)
where
1 ∂
Rk ð!
x ðtÞÞ ¼
N
k! ∂pk
k
hXþ∞
p¼0
and
χk ¼
0,
1,
k ≤ 1;
k > 1:
(9)
Therefore, from (7) we have particular solution as
x*m ðtÞ ¼ χ m xm1 ðtÞ L1 Rm1 ð!
x m1 ðtÞÞ
(10)
Hence, the general solution becomes
xm ðtÞ ¼ xm* ðtÞ þ
Xn
k¼1
ck fk ðtÞ
(11)
where, n is the order of given equation (1), ck are constant coefficients, and fk ðtÞ are non-zero solution of L½xðtÞ ¼ 0. As
noted in form of the auxiliary linear operator equation (4), the linear operator L½xðtÞ contains the auxiliary roots λi of
L½xðtÞ ¼ 0, i ¼ 1, 2, ::, n. Now, if all are not simultaneously equal to zero and known. Then, the initial approximation as
well as finial series solution contains those unknowns λi . To compute these unknown roots λi , i ¼ 1, 2, ::, n we minimize the
square residual error 4ðλi Þ define as
Z
h Xm
i2
4ðλi Þ ¼ N
x
dx
(12)
k ðtÞ
k¼0
V
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Journal of Low Frequency Noise, Vibration and Active Control 43(4)
where xk ðtÞ is the kth order homotopy approximation.
In order to reduce the computational time, we utilize the discretised form of the square residual which is based on newton
quotes quadrature rule (simpson’s 13 composite rule) as
4ðλi Þ ≈
h Xm hXm i2 o
i 2
i2
h Xk2 n hXm N
x
þ
4N
x
þ
N
xi t2j
i t2j2
i t2j1
j¼1
i¼0
i¼0
i¼0
3
(13)
Where fk ðηÞ is the homotopy approximation of the kth order, h ¼ ba
k , k ¼ even number. Therefore, minimizeing the square
residual 4ðλi Þ, defined in equation (13), we compute unknowns λi . Thereby we ensure the convergence of our mth order
series solution xm ðtÞ (equation (11)). Readers are referred to our previous study33 for more detailed discussion on the
necessary and sufficient condition for the convergence of the computed series solution.
Analytical and numerical illustration
Duffing oscillator
00
x þ x þ x3 ¼ 0
xð0Þ ¼ 1, x0 ð0Þ ¼ 0
(14)
As equation (14) is of 2nd order differential equation, so from equation (4) we have the auxiliary linear operator of the
form as
L½xðtÞ ¼ x00 ðλ1 þ λ2 Þx0 þ λ1 λ2 x
(15)
For optimal homotopy perturbation based analytical solution at least one root λ1 or λ2 is to be remain unknown, which
will be obtained optimally using residual minimization. Let λ1 ≠ 0 be unknown and λ2 ¼ 0 : There are other numerious
possibilities for the selection of auxiliary linear operator. One may keep both λ1 and λ2 as unknown. In that case
computational time will be more.31,32 Therefore, equation (15) becomes
L½xðtÞ ¼ x00 λ1 x0
(16)
Now, for the periodic solution we consider the initial approximation as x0 ðtÞ ¼ A cosðωtÞ Using the mth order deformation
equation defined in equation (7) with
Xn
Xm1
Rm1 ð!
x m1 ðtÞÞ ¼ x00m1 þ xm1 þ
xmn1
xk xnk
(17)
n
k¼0
and with the help of symbolic software we compute successive homotopy perturbation approximations. The first order
approximation is
x1 ðtÞ ¼ A 3A2 ω cos½3tω ω2 þ λ21 þ A2 sin½3tωλ1 ω2 þ λ21 þ 3ω 4 þ 3A2 4ω2 cos½tω 9ω2 þ λ21 þ
12ω ω2 þ λ21 9ω2 þ λ21
3 4 þ 3A2 4ω2 sin½tωλ1 9ω2 þ λ21 12etλ1 ω ω2 9 þ 7A2 9ω2 þ 1 þ A2 ω2 λ21
(18)
Now, the condition for which it becomes periodic is that the coefficient of cos[ωt should be zero. Therefore,
8 A 4 þ 3A2 4ω2
>
>
¼0
>
<
4ðω2 þ λ21 Þ
>
>
A 4 þ 3A2 4ω2 λ1
>
:
¼0
4ωðω2 þ λ21 Þ
(19)
Roy et al.
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Solving equation (19) we have amplitude A ¼ 1 and frequency ω ¼ 12 4 þ 3A2. Substituiting the value of amplitude A
we have ω ¼ 1:322875.
Now substituting the value of amplitude A and frequency ω in the equation (20) we have the first order OM-HPM series
solution as
xðtÞ ¼ x0 ðtÞ þ x1 ðtÞ
(20)
pffiffi
pffiffi
pffiffi
pffiffi
pffiffiffi
pffiffiffi
3 7 etλ1 þ 63 cos 27t þ Cos 3 27t þ 2 sin 3 27t λ1 þ 12 7cos 27t λ21
pffiffiffi
xðtÞ ¼
3 7ð63 þ 4λ21 Þ
(21)
Therefore, using the minimization technique on 4ðλi Þ in (13), we have the minimum square residual as 3:79662 × 104 at
λ1 ¼ 0:3078049. Hence the final solution becomes
pffiffiffi
pffiffiffi
pffiffiffi
7t
3 7t
3 7t
0:3078049t
þ 0:9999 cos
þ 0:0157781 cos
xðtÞ ¼ 0:015778e
0:0012237 sin
(22)
2
2
2
As the first order solution give us a sufficiently enough small residual error, the higher order may be avoided here. It is
important to emphasized that all the analyses given in six have been considered as dimensionless. To confirm the solution
accuracy we present Fig-1 where we compare the computed OM-HPM series based analytical solution with those of
numerical solution obtained by 4th order Rungee-Kutta method.
From Figure 1 it is clear that first order OM-HPM 36 analytical series solution is almost matching with the numerical
solution obtained by RK4 method. Therefore, for this problem first order OM-HPM approximation is sufficient. Then we
continue upto 5th order OM-HPM approximations. Table 1 represnts the different order of approximations, optimal value
of λ1 and the minimum square residual. From Table 1 one can easily see that the error diminishes with the order of
0
approximations. Figure 2 is presented to show the phase portrait for velocity x ðtÞ versus the solution x(t) at different
order of OM-HPM approximations. This problem was solved by He34 in 1998 to certify his proposed HPM and He
obtained the relative percentage error upto 5.4%. It may be noted that there is no guarantee of convergence of the solution
Figure 1. Comparison of the computed 1st order OM-HPM 32 analytical solution with the numerical solution by RK4. (a). Time series
0
solution of x(t) versus t. (b). Time series of velocity x ðtÞ versus t.
Table 1. Optimal table for duffing oscillator (14).
Order m
Optimal λ1
Min error
1
3
5
0:3078049
0:56026707
1:478902
3:79662 × 104
8:54681 × 106
1:23794 × 107
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Journal of Low Frequency Noise, Vibration and Active Control 43(4)
0
Figure 2. Phase portrait for velocity x ðtÞ versus the solution x(t) for the equation (14)) at different order of approximations.
by HPM .3,8,13–20,27,29,32 Then Tao et al.9 applied VIM with matrix Lagrange’s multiplier to solve this duffing oscillator
and obtained a series based semi-analytical solution but there was no improvement on the solution. While by our recently
proposed OM-HPM36 one can get accurate and convergent solution, and this efficiency of OM-HPM and inefficiency of
HPM/VIM lies on the choice of optimal linear operator. We found the maximum relative percentage error as 1.303%. We
also presented the square residual in Table 1 to clarify the high efficiency of our advanced technique OM-HPM36 as order
increases.
From Figure 2 it is clear that both the lower (m = 1) and higher (m = 5) order OM-HPM approximation are close to each
other and providing limit cycle, that is, stable periodic motion over time, forming a closed trajectory in the phase space.
Van der Pol oscillator
00
x ðtÞ þ xðtÞ x0 ðtÞ 1 bx2 ðtÞ ¼ 0
xð0Þ ¼ 1, x0 ð0Þ ¼ 0
(23)
Several researchers attempted to solve this van der pol oscillator analytically using the HPM,13,34,35 HAM,40,41 and
VIM.11,35 Ozis et al.34 and others 13,35 solved using the classical HPM and obtained series solution with relative percentage
error upto 37.5%. Here, we are interested to apply our recently proposed advanced technique OM-HPM.34 Similarly, using
the linear operator (17), the initial approximation x0 ðtÞ ¼ A cosðωtÞ, and the mth order deformation equation as
Xn
Xm1
Rm1 ð!
x m1 ðtÞÞ ¼ x00m1 þ xm1 x0m1 þ b n x0mn1 k¼0 xk xnk
(24)
we compute successive homotopy perturbation approximations. The first order approximation is of the form
tλ
1
2
tλ1
5
3
b
ð
1
þ
e
Þω
3ω
A
36
3
þ
A
36e 1 36etλ1 ω2
2
2
12ωλ1 ðω2 þ λ1 Þð9ω2 þ λ1 Þ
þ36 1 þ ω2 cos½tω þ9ð4 þ A2 b ω sin½tω þ A2 bω sin½3tω λ1 ω2 ð 27ð 4þA2 b ω cos½tω A2 bω cos½3tω
þ4 30 3etλ1 þ A2 bð 10 þ 3etλ1 Þ ω þ27 1 þ ω2 sin½tω Þλ21 3ω 4etλ1 4etλ1 ω2
þ4 1 þ ω2 cos½tω þð4 þ A2 b ω sin½tω þ A2 bω sin½3tωÞλ31 þ 3 4 þ A2 b ω cos½tω þA2 bω cos½3tω
(25)
4ðð3 þ A2 b ω þ 3 1 þ ω2 Þsin½tωÞÞλ41
x1 ðtÞ ¼
Therefore, for the periodic solution we have
Roy et al.
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8 A 4 4ω2 þ 4 þ A2 b λ1
>
>
¼0
>
<
4ðω2 þ λ21 Þ
>
>
A 4 þ A2 b ω2 þ4 1 þ ω2 λ1
>
:
¼0
4ωðω2 þ λ21 Þ
(26)
Solving we get amplitude A ¼ p2ffiffib and frequency ω ¼ 1. For b = 1 the limiting amplitude is A = 2 and frequency ω as
1. Substituiting the value of computed amplitude A and frequency ω, we have the first order OM-HPM series
solution as
2 9ð 1 þ etλ1 Þ 3ð 9 cos½t þ sin½3tÞλ1 þð 1 þ cos½3tÞλ21 þ3 cos½tλ31
xðtÞ ¼
(27)
3λ1 ð9 þ λ21 Þ
Therefore, we have the minimum square residual for b = 1 as 4:066553 × 102 at the λ1 ¼ 2:4889329: Again substituting
the optimal value of the auxiliary root λ1 into the equation (27) we have the final solution as
xðtÞ ¼ 0:009989 9 1 þ e3:3262438t 110:4036Cos½t þ 11:063897ð 1 þ cos½3tÞ þ9:97873ð9cos½t þ sin½3t
(28)
However, here the residual error at 1st order is not satisfactory. Here, we continue upto 5th order approximations and
prepare the optimal table as
Therefore, from Table 2 we have the best optimal auxiliary linear operator as.
L½xðtÞ ¼ x00 þ 1:2530439x0
(29)
Now, substituting the optimal value of the auxiliary roots λ1 we have the 5th order approximation solution of the form
xðtÞ ¼ x0 ðtÞ þ x1 ðtÞ þ x2 ðtÞ þ x3 ðtÞ þ x4 ðtÞ þ x5 ðtÞ
(30)
The solutions of the equation (23) are shown in the Figure 3. In the Figure 3 we compare our computed OM-HPM
solution with the numerical solution by RK4. Figure 3(a) presents the solution x(t) while Figure 3(b) represents the
0
velocity x ðtÞ. From Figure 3(a) to (b) it is seen that our 1 st order approximation solution is not much well but our 3 rd
th
and 5 order approximations are very close to the numerical solution and we have the relative error as 3.04%, 1.93%
and 0.84% for 1st , 3 rd and 5th approximations, respectively. Eigoli et al 41 solved this oscillator using HAM and found
that to achieve good accuracy one need 25 th order approximations. Moreover, when Huang 35 applied VIM for this
oscillator, he found that VIM gives divergent solution. Therefore, Huang 35 improved the scheme for the VIM by
adding HPM 3,13,14 and Laplace transformation and shown that 6 th order solution by improved VIM is agree with
numerical solution. On the other hand, from Table 2 and Figure 3(a) to (b) it is noteworthy that our OM-HPM 34 gives
better solution at only 1 st order approximation. Therefore, in comparison to the classical HPM, HAM, 41 VIM,35 and
improved VIM,35 our OM-HPM 36 is much more efficient to handle such strongly nonlinear oscillators.
The phase portrait at the different order (m = 1 and 5) of OM-HPM approximations series solutions for the oscillator (23)
is shown in the Figure 4. We have seen that there is a closed trajectory surrounding the origin. Hence, the system exhibits
limit cycle behavior. The system does not settle down to equilibrium but rather to a periodic solution.
Table 2. Optimal table for Van der Pol oscillator (23).
Order m
Optimal λ1
Min error
1
3
5
2:4889329
1:8649311
1:2530439
4:066553 × 102
7:37648 × 104
2:06427 × 105
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Journal of Low Frequency Noise, Vibration and Active Control 43(4)
Figure 3. Comparison of the computed analytical solution with the numerical solution by RK4 at different order of approximations (m =
0
1,3 and 5) for the limiting case b = 1. (a). Time series solution of x(t) (b). Time series of velocity x ðtÞ.
0
Figure 4. Phase portrait for velocity x ðtÞ versus the solution x(t) for the equation (23) at different order of approximations.
Rayleigh oscillator
2
x00 þ x x0 1x0 ¼ 0
xð0Þ ¼ 1, x0 ð0Þ ¼ 0
(31)
Following the similar process with the linear operator (17), the initial approximation x0 ðtÞ ¼ A cosðωtÞ and the mth order
deformation equation
Xn
Xm1
Rm1 ð!
x m1 ðtÞÞ ¼ x00m1 þ xm1 x0m1 þ
x0mn1
x0 x0
(32)
n
k¼0 k nk
we compute successive homotopy perturbation approximations. The first order approximation is
Roy et al.
1477
1
tλ1
5
2 2
x1 ðtÞ ¼
þ3ω3 36etλ1 þ 36etλ1 ω2
2
2 Að36ð 1 þ e Þω 3 þ 2A ω
2
2
12ωλ1 ðω þ λ1 Þð9ω þ λ1 Þ
36 1 þ ω2 cos½tω 9ωð4 þ 3A2 ω2 sin½tω þA2 ω3 sin½3tωÞλ1 ω2 ð27ω 4 þ 3A2 ω2 cos½tω
þA2 ω3 cos½3tω þ 4 ω 30 þ 3etλ1 þ 20A2 ω2 þ27 1 þ ω2 sin½tω Þλ21 þ3ω 4etλ1 þ 4etλ1 ω2
4 1 þ ω2 cos½tω þð4ω 3A2 ω3 sin½tω þ A2 ω3 sin½3tωÞλ31 þ 3ω 4 þ 3A2 ω2 cos½tω A2 ω3 cos½3tω
þ4ð3ω 2A2 ω3 3 1 þ ω2 sin½tω Þλ41
Similarly for the periodic solution we have
8 A 4 4ω2 þ 4 þ 3A2 ω2 λ1
>
>
¼0
>
<
4ðω2 þ λ21 Þ
>
>
A 4ω2 3A2 ω4 4 1 þ ω2 λ1
>
:
¼0
4ωðω2 þ λ21 Þ
(33)
(34)
Solving we have
2
A ¼ pffiffiffi and ω ¼ 1
3
Substituting the value of computed amplitude A and frequency ω we have the first order OM-HPM series solution as
xðtÞ ¼ x0 ðtÞ þ x1 ðtÞ
2
3
10
4
4
4
tλ1
tλ1
12ð 1 þ e Þ þ4sin½3tλ1 4 3 þ 3e
þ 3 cos½3t λ1 þ 4sin½3tλ1 þ 3 3 cos½3t λ41
2cos½t
pffiffiffi
xðtÞ ¼ pffiffiffi þ
3
6 3λ1 ð1 þ λ21 Þð9 þ λ21 Þ
(35)
(36)
The minimum square residual is computed as 1:1618723 × 103 at the λ1 ¼ 1:77395597: Again substituiting the
optimal value of the auxiliary root λ1 into the equation (36) we have final solution as
xðtÞ 0:07232426 þ 0:053587e1:77395597t þ 1:1547005 cos½t þ 0:0187371499 cos½3t þ 0:03168706 sin½3t
(37)
Figure 5. Comparison of the computed analytical solution of equation (33) with the numerical solution by RK4 at different order of
0
approximations (a) time series solution of x(t) (b) time series of velocity x ðtÞ.
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Journal of Low Frequency Noise, Vibration and Active Control 43(4)
0
Figure 6. Phase portrait for velocity x ðtÞ versus the solution x(t) for the equation (24) at different order of approximations.
In similar process we successively compute up to 5th order OM-HPM series solutions and the solutions are shown in Figures
5 and 6. In Figure 5(a) we compare our computed OM-HPM position solution with numerical RK4 solution at different
order of approximations while in Figure 5(b) we compare velocity. The phase portrait of the oscillator (31) is shown in
Figure 6 at the different order of approximation solutions.
Similar to the Van der Pol oscillator, here also arise a closed trajectory surrounding the origin. Hence, the system exhibits
limit cycle, that is, stable periodic motion over time.
Conclusions
In this paper, we applied the optimal and modified homotopy perturbation method (OM-HPM) technique to provide
convergent semi-analytical solutions for three different models: duffing oscillator (DO), Rayleigh oscillator (RO)
and van der Pol oscillator (vdPO). The high accuracy of this technique is attributed to the selection of optimal
auxiliary linear operator, which makes it a practical tool for engineers and researchers. It has been shown that out of
the three models discussed, only the DO is conservative in nature and other two are non-conservative in nature.
Surprisingly all the three oscillators are stable oscillators. Our method is suitable as long as the phases remains
closed in nature, thereby justifying the stability. Present method will not give encouraging results on unstable cases
reflecting open nature of phase portrait. It may be recommended that this advanced approach OM-HPM can be
applied to other strongly nonlinear oscillators to explore the complex dynamics. However, OM-HPM need more
CPU time than traditional HPM.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.
References
1. Rama G, Marinkovic D, and Zehn M. High performance 3-node shell element for linear and geometrically nonlinear analysis of
composite laminates. Compos B Eng 2018; 151: 118–126.
2. Milić P, Marinković D, Klinge S, et al. Reissner-Mindlin based isogeometric finite element formulation for piezoelectric active
laminated shells. Tech Gaz 2023; 30(2): 416–425. DOI: 10.17559/TV-20230128000280.
3. Ismail GM, Abul-Ez M, Zayed M, et al. Analytical accurate solutions of nonlinear oscillator systems via coupled homotopy
variational approach. Alex Eng J 2022; 61: 5051–5058.
4. Chen G. Applications of a generalized Galerkin’s method to non-linear oscillations of two-degree-of-freedom systems. J Sound Vib
1987; 119(2): 225–242.
Roy et al.
1479
5. Ladygina YV. AI. Manevich free oscillations of anon-linear cubic system with two-degrees-of-freedom and close natural frequencies. J Appl Math Mech 1993; 57(2): 257–266.
6. Cveticanin L. Vibrations of a coupled two degree-of-freedom system. J Sound Vib 2001; 247(2): 279–292.
7. Cveticanin L. The motion of a two-mass system with non-linear connection. J Sound Vib 2002; 252(2): 361–369.
8. He JH. Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 2006; 20: 1141–1199.
9. He JH. Variational iteration method–a kind of non-linear analytical technique: some examples. Int J Nonlinear Mech 1999; 34:
699–708.
10. He JH. Variational iteration method–some recent results and new interpretations. J Comput Appl Math 2007; 207: 3–17.
11. Tao ZL, Chen GH, and Chen YH. Variational iteration method with matrix Lagrange multiplier for nonlinear oscillators 2019. DOI:
10.1177/1461348418817868.
12. Anjum N and He JH. Laplace transform: making the variational iteration method easier. Appl Math Lett: 134–138.
13. He JH. Homotopy perturbation method with an auxiliary term. Abstr Appl Anal 2012; 38(3-4): 857612, ArticleNumber.
14. He JH. Homotopy perturbation method with two expanding parameters. Indian J Phys 2014; 88: 193–196.
15. Adamu MY and Ogenyi P. Parameterized homotopy perturbation method. Nonlinear Sci Lett A 2017; 8: 240–243.
16. El-Dib YO. Multiple scales homotopy perturbation method for nonlinear oscillators. Nonlinear Sci Lett A 2017; 8: 352–364.
17. He J, He C, and Alsolami A. A good initial guess for approximating nonlinear oscillatorsby the homotopy perturbation method.
Facta Univ – Ser Mech Eng 2023; 21(1): 021–029. DOI: 10.22190/FUME230108006H.
18. Song HY. A modification of homotopy perturbation method for a hyperbolic tangent oscillator arising in nonlinear packaging
system. DOI: 10.1177/1461348418822135.
19. Pasha SA, Nawaz Y, and Arif MS. The modified homotopy perturbation method with an auxiliary term for the nonlinear oscillator
with discontinuity. 2019. DOI: 10.1177/0962144X18820454.
20. Yu DN, He JH, and Garcıa AG. Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators. J low Freq
Noise Vibr Act Control, 2019. DOI: 10.1177/1461348418811028.
21. LiX X and He CH. Homotopy perturbation method coupled with the enhanced perturbation method. J Low Freq Noise Vib Act
Control 2019. DOI: 10.1177/1461348418800554.
22. Nadeem M and Li F. He–Laplace method for nonlinear vibration systems and nonlinear wave equations. J Low Freq Noise Vib Act
Control 2019. DOI: 10.1177/1461348418818973.
23. Li F and Nadeem M. He–Laplace method for nonlinear vibration in shallow water waves. J Low Freq Noise Vib Act Control 2019.
DOI: 10.1177/1461348418817869.
24. Suleman M, Lu D, Yue C, et al. He–Laplace method for general nonlinear periodic solitary solution of vibration equations. J Low
Freq Noise Vib Act Control 2019. DOI: 10.1177/1461348418816266.
25. He JH. Variational approach for nonlinear oscillators. Chaos, Solit Fractals 2007; 34: 1430–1439.
26. Nayhef A and Mook D. New York: John Wiley; 1979. 2. Burton TD. A perturbation method for certain non-linear oscillators. Int J
Non Lin Mech 1984; 19: 397–407.
27. Burton TD. A perturbation method for certain non-linear oscillators. Int J Non Lin Mech 1984; 19: 397–407.
28. Markov N, Dmitriev V, Maltseva S, et al. Application of the nonlinear oscillations theory to the study of non-equilibrium financial
market. Financial Assetsand Investing 2016; 7: 5–19.
29. Alhejaili Salas WA and Hand Tag-Eldin Tantawy ESA. On perturbative methods for analyzing third-order forced Van-der Pol
oscillators. Symmetry 2023; 15: 89.
30. Alhejaili Salas WAH and El-Tantawy SA. Analytical and numerical study on forced and damped complex duffing oscillators.
Mathematics 2022; 10: 4475.
31. Salas AH, Abu Hammad M, Alotaibi BM, et al. Analytical and numerical approximations to some coupled forced damped duffing
oscillators. Symmetry 2022; 14: 2286.
32. He JH. The simpler, the better: analytical methods for nonlinear oscillators and fractional oscillators. J Low Freq Noise Vib Act
Control 2019; 38: 1252–1260.
33. Roy T and Maiti DK. General approach on the best fitted linear operator and basis function for homotopy methods and application to
strongly nonlinear oscillators. Math Comput Simulat 2024; 219: 100–121. DOI: 10.1016/j.matcom.2024.01.004.
34. Ozis T and Yıldırım A. A note on He’s homotopy perturbation method for van der Pol oscillator with very strong nonlinearity. Chaos
Solit. Fractals 2006; 34: 989–991. DOI: 10.1016/j.chaos.2006.04.013.
35. Huang Y and Liu HK. A new modification of the variational iteration method for van der pol equations. Appl Math Model 2013; 37:
8118–8130. DOI: 10.1016/j.apm.2013.03.033.
36. Roy T and Maiti DK. An optimal and modified homotopy perturbation method for strongly nonlinear differential equations.
Nonlinear Dynam 2023; 111: 15215–15231. DOI: 10.1007/s11071-023-08662-w.
1480
Journal of Low Frequency Noise, Vibration and Active Control 43(4)
37. Rath B, Erturk VS, Asad J, et al. An asymmetric model two-dimensional oscillator . J. Low Freq. Noise Vib. Act. Control 2024; 0:
1–11.
38. Kavyanpoor M and Shokrollahi S. Challenge on solutions of fractional Van Der Pol oscillator by using the differential transform
method. Chaos Solit. Fractals 2017; 98: 44–45.
39. He J-H. Homotopy perturbation technique. Comput Methods Appl Mech Eng 1999; 178: 257–262.
40. Li Y, Nohara BT, and Liao S. Series solutions of coupled Van der Pol equation by means of homotopy analysis method. J Math Phys
2010; 51: 063517.
41. Eigoli AK and Khodabakhsh M. A homotopy analysis method for limit cycle of the van der Pol oscillator with delayed amplitude
limiting. Appl Math Comput 2011; 217: 9404–9411. DOI: 10.1016/j.amc.2011.04.029.
42. Ain QT, Nadeem M, Karim S, et al. Optimal variational iteration method for parametric boundary value problem. AIMS Mathematics
2022; 7: 16649–16656. DOI: 10.3934/math.2022912.