Resonance simulation of the coupled nonlinear Mathieu's equation

Abstract

Numerous theoretical physics and chemistry problems can be modeled using Mathieu's equations (MEs). They are crucial to the theory of potential energy in quantum systems, which is equivalent to the Schrödinger equation. According to the mentioned applications, thus, the current study investigates the stability behavior of the nonlinear-coupled MEs. The analysis of the coupled harmonic resonance cases imposes two coupled solvability conditions, which leads to coupled parametric nonlinear Landau equations. In addition, a super-harmonic nonlinear resonance combination is presented. Solutions and stability criteria are discussed for each case. It's shown that resonance produces an unstable system. The transition curves are derived. Numerical calculations show the excitation of the frequency on the periodic solutions.

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AIP Advances ARTICLE pubs.aip.org/aip/adv
Resonance simulation of the coupled nonlinear
Mathieu’s equation
Cite as: AIP Advances 13, 085032 (2023); doi: 10.1063/5.0166730
Submitted: 7 July 2023 •Accepted: 9 August 2023 •
Published Online: 25 August 2023
Yusry O. El-Dib,1,a) Albandari W. Alrowaily,2, b) C. G. L. Tiofack,3,c) and S. A. El-Tantawy4,5, d)
AFFILIATIONS
1Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt
2Department of Physics, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428,
Riyadh 11671, Saudi Arabia
3Faculty of Sciences, University of Maroua, P.O. Box 814, Maroua, Cameroon
4Department of Physics, Faculty of Science, Port Said University, Port Said 42521, Egypt
5Research Center for Physics (RCP), Department of Physics, Faculty of Science and Arts, Al-Mikhwah, Al-Baha University,
Al-Baha 1988, Saudi Arabia
a)yusryeldib52@hotmail.com
b)Awalrowaily@pnu.edu.sa
c)Author to whom correspondence should be addressed: gaston.tiofack@univ-maroua.cm
d)tantawy@sci.psu.edu.eg
ABSTRACT
Numerous theoretical physics and chemistry problems can be modeled using Mathieu’s equations (MEs). They are crucial to the theory
of potential energy in quantum systems, which is equivalent to the Schrödinger equation. According to the mentioned applications, thus,
the current study investigates the stability behavior of the nonlinear-coupled MEs. The analysis of the coupled harmonic resonance cases
imposes two coupled solvability conditions, which leads to coupled parametric nonlinear Landau equations. In addition, a super-harmonic
nonlinear resonance combination is presented. Solutions and stability criteria are discussed for each case. It is shown that resonance pro-
duces an unstable system. The transition curves are derived. Numerical calculations show the excitation of the frequency on the periodic
solutions.
©2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0166730
I. INTRODUCTION
Different types of differential equations have succeeded in
modeling many real problems related to plasma physics, fluids,
optics, quantum mechanics, cosmology, biology, and chemistry, in
addition to many engineering problems and many other fields.1–10
The Mathieu equation (ME) is one of the most significant differential
equations that many researchers have dealt with due to its numerous
applications in numerous domains. This equation can be obtained
from the Schrödinger equation that has arisen in a quantum system
with potential energy. The wave functions are then connected to the
periodic eigenfunctions of the ME. In quantum mechanics, ME can
be used in any scenario where a quantum item is exposed to cosinu-
soidal potential energy.11 A general analytic solution of the nonlinear
coupled ME has not yet been found, despite the fact that it is simple
to solve numerically and that exact solutions are possible in some
circumstances.
Parametric resonance produces a motivating mathematical
challenge and plays a wide role in many applications. The nonlinear
dynamics of such problems are much created and largely undis-
covered. The simplest example of constant excitation is swinging
equipment. Generally, the equation of motion that characterized this
model is performed in the shape of the familiar ME. The coupled ME
realizes widespread use in many engineering branches, particularly
in civil, offshore, nuclear, mechanical, marine, and part engineer-
ing. This is often principally thanks to the necessary role completed
this method. Mahmoud and additional scholars conducted numer-
ical and analytical studies on the periodic resolution of bound
straightforward systems with multi-coupled nonlinear Hill’s equa-
tions.12 The well-known coupled MEs are created by formulating the
AIP Advances 13, 085032 (2023); doi: 10.1063/5.0166730 13, 085032-1
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AIP Advances ARTICLE pubs.aip.org/aip/adv
linear stability problems of surface waves (SWs) subjected to verti-
cal or horizontal oscillations. A framework of the linear analysis of
SWs was provided in Ref. 13. They obtained linear coupled MEs.
El-Dib14–17 formalizes the discussion of the complexes of those cou-
pled MEs by creating the use of size parameters. In the framework
of a Hamiltonian system, the stability of coupled and damped MEs
is examined.18–20 Bernstein et al.21 studied the dynamics of one-way
coupling in a system of nonlinear MEs.
Systems that are parametrically excited are extensively devel-
oped in the majority of engineering and physical fields. Conse-
quently, it is necessary to study these systems’ dynamic behavior.
Such systems vibrate due to a variety of mechanics, including time-
varying and explicit periodic ones. These vibrations seem in columns
manufactured from nonlinear elastic material,22 beams with a har-
monically variable length,23 beams with periodic movements of
their support,24 and floating offshore structures.25 Constant quantity
excitations occur in electrostatically driven micro-electromechanical
oscillators.26 Insensible engineering things, oscillations with con-
stant quantities have several applications, such as within the radio,
the PC, and optical maser engineering, in vivacious machines with
special style,27 and many other applications.28,29 Many fields of
physics and engineering, such as ship stability, surface wave pat-
terns on the water, and the forced motion of a swing, have a strong
foundation in constant amount resonance. The critical mass detec-
tor is elaborate as an in-plane parametrically resonant generator.30
The only mathematical model of the system with a constant quan-
tity periodic load is sometimes a linear ME. It is known that all
natural phenomena are not subjected to linear systems, but they fol-
low nonlinear behavior. Thus, for describing these phenomena, we
need to construct some nonlinear equations for this purpose, such
as the Mathieu–Duffing equation.31–33 As a result, these nonlinear
equations must be resolved in different ways because it is evident
that there is not an exact analytical solution for such nonlinear
equations.
The nonlinear form of the ME could also be thought about as a
coupling between the Duffing and linear MEs. Norris34 investigated
the bifurcation that happens to the strong nonlinearity ME. The
generalized averaging technique was employed for deriving an ana-
lytical expression for the nonlinear ME with periodic Duffing term.35
El-Dib36 thought about associating extension up to the cubic order
with every term containing sub-harmonics within the periodic term.
Ng and Rand37 used the tactic of averaging regarding the 2:1 reso-
nance within the limit of little forcing and little Duffing nonlinearity
term.Zhou et al.38 investigate a nonlinear resonance effect in the Paul
trap of a composed hexapole field, which is supposed as a perturba-
tion of the quadrupole field. An improved homotopy perturbation
method (HPM) was introduced by El-Dib39 for checking the sta-
bility of the resonance case to a damped nonlinear ME. The HPM
is ready to mix with alternative techniques like the multiple scales
method (MSM). Additionally, El-Dib40 suggested a new version to
the HPM by combining the MSM. The latter technique is effective
only for a weak nonlinear oscillators. However, the mixed method
yields the associated degree surprising result for all powerfully non-
linear oscillators. Additionally, the author applied this modification
for studying the harmonic Duffing oscillator and debates the sta-
bility criteria at the harmonic, subharmonic, and superharmonic
resonance cases.
The nonlinear coupled MEs inclusion parametric forcing of
Duffing terms were established by Napoli et al.41 because of their
experimental of a trial of electrostatically and automatically coupled
micro-cantilevers
d2
dt2xk(t)+(a2k−1+4p2k−1cos2Ωt)xk(t)
+(a2k+4p2kcos2Ωt)x3−k(t)
=(b2k−1+4q2k−1cos2Ωt)x3
k(t);k=1,2, (1)
where Ωis the external excitation frequency and tindicates the
independent variable. This governing equation could be a nonlin-
ear coupled that is well known as MEs with no restrictions on the
nonlinearity terms or the periodic forcing.
In the present analysis, an approximate analytical approach is
developed to find solutions and examine the stability behavior of
the harmonic resonance cases for the system (1). The usage of the
proposed modified multiple-scale process is defined in Ref. 41. The
essential purpose of this article is to perform the periodic solutions
and analyze the stability of a parametric forced cubic nonlinearity of
coupled Mathieu–Duffing equations.
II. MATRIX FORM OF THE COUPLED NONLINEAR
MATHIEU EQUATIONS (MEs)
For mathematical simplicity, the system (1) are introduced
within the type of ME in matrix notation as
d2
dt2Y(t)+(A+4Pcos2Ωt)Y(t)=(B+4Qcos2Ωt)Y3(t), (2)
where Aand Pare the matrix of order 2 ×2Band Qare diago-
nal matrices as defined in Ref. 42. The vector Y(t)=(x1(t)x2(t))T
could be a variable vector having two dependent variables on t. Also,
the upper T represents the vector transpose. The Duffing nonlinear
vector Y3(t)is outlined as
Y3(t)=x3
1(t)x3
2(t)T. (3)
The linear style of the above nonlinear system has been addressed in
Refs. 13–17 for the matter of SWs in fluid mechanics. The usage of
the proposed He’s multiple-scale process is defined in Ref. 43. The
homotopy system can be established as
H(t,ρ)=d2
dt2I+AY(t,ρ)+ρ4PY(t,ρ)cos2Ωt
−B+4Qcos2ΩtY3(t,ρ)=0 ; ρ∈[0,1]. (4)
It ought to be noted that the homotopy operates (4) is identical (2)
as ρ→1, apart from operating Y(t,ρ), that contains embedded the
homotopy parameter ρaccordingly, HPM, one might assume the
solution of the system (5) in an exceedingly series of ρas44–47
Y(t,ρ)=∞
∑
i=0ρiYi(t), (5)
AIP Advances 13, 085032 (2023); doi: 10.1063/5.0166730 13, 085032-2
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AIP Advances ARTICLE pubs.aip.org/aip/adv
where the details of the primary solution Y0(t)are given in Ref. 42
and the best approximation for the solution is given in Refs. 27–32.
The final solution reads
Y(t)=lim
ρ→1Y(t,ρ)=∞
∑
i=0Yi(t). (6)
The stability criteria could also be done by employing a modulat-
ing technique. El-Dib39,40 has been established a mix of the multiple
scales technique and HPM. This mixture is used to present help-
ful results to solve and discuss the stability of the excited nonlinear
equations.
III. THE HE’S MULTIPLE-SCALES TREATMENT
FOR THE NONLINEAR SYSTEM
By applying He’s multiple-scales methodology,48 we tend to
might use the two scales such Tn=ρnt,n=0,1, 2, ... The scale
T0represents a variable that is acceptable for the quick variable and
T1is for the slow one. The differential operator reads
d2
dt2(●)≡D2
0(●)+2ρD0D1+ρ2(D2
1+2D0D2)(●)+, (7)
where Dn(●)≡∂
∂Tn(●).
In addition, the vector variable Y(t)can be expanded as
Y(t,ρ)=∞
∑
i=0ρiYi(T0,T1), (8)
where the vector Yn(T0,T1)is an unknown one, which repre-
sents the correction for the main solution Y0(T0,T1). In the same
manner, the nonlinear vector Y3(t,ρ)is expanded as
Y3(t,ρ)=∞
∑
i=0ρiY3
i(T0,T1). (9)
By substituting expansions (8) and (9), into system (4), and then
collecting the coefficient of same powers of ρto zero, we get
(D2
0I+A)Y0=0, (10)
(D2
0I+A)Y1=−2D0D1+4Pcos2ΩT0Y0
+B+4Qcos2ΩT0Y3
0. (11)
As mentioned above, the solution of system (10) occupies the
following form:
Y0(T0,T1)=Rjπj(T1)eiωjT0+πj(T1)e−iωjT0, (12)
where πj(T1)and its complex conjugate πj(T1)are unknown func-
tions of integration and Rjis a constant vector, which is given
by
Rj=a2ω2
j−a1T. (13)
Now, by inserting solution (12) into the first level of perturbation
(11), we get
(D2
0I+A)Y1=−2iωjD1Rj+2P R jπj+3B+2QR3
jπ2
jπj
×eiωjT0−P Rjei(ωj+2Ω)T0+ei(ωj−2Ω)T0πj
+B+2QR3
jπ3
je3iωjT0
+3QR3
jeiωj+2ΩT0+eiωj−2ΩT0π2
jπj
+QR3
jei3ωj+2ΩT0+ei3ωj−2ΩT0π3
j+c.c. (14)
At this level of the analysis, two categories are exist: the primary class
focuses on the non-resonance case, whereas the other class addresses
resonant cases.
A. The non-resonant case
In this case, the excitation frequency Ωis supposed to be off
from ωj, 2ωj, and all sub- or super-combinations of ω1and ω2.
The elimination of the secular terms, within the non-resonance
case, of Eq. (14) ends up in
iωjRjD1πj+PRjπj−3
2B+2QR3
jπ2
jπj=0. (15)
Multiply each side of Eq. (15), from the left-side, by the transposed
vector RT
jand use the below normalized condition
RT
jRj
RT
jRj=1, (16)
we get
iD1πj+2ˆ
Pjπj−3ˆ
Bj+2ˆ
Qjπ2
jπj=0, (17)
where the following entries are used:
ˆ
Pj=SjPRj,ˆ
Bj=SjBR3
j,
ˆ
Qj=SjQR3
j, and Sj=RT
j
2ωjRT
jRj. (18)
The first-order nonlinear equation (17) has real coefficients within
the variety of Landau’s equation kind. The solution of such an equa-
tion is used to debate the stability of the matter. This equation has
the subsequent well-known precise solution49
πj(T1)=π0eiϖjT1+iϕ0, (19)
where the phase ϕ0and the amplitude π0are constants while the
characteristic exponential ϖjreads
ϖj=2ˆ
Pj−3ˆ
Bj+2ˆ
Qjπ2
0. (20)
Because the exponent ϖjrepresents a real quantity, the stability
reveals within the non-resonance case. The solution of Eq. (14) reads
AIP Advances 13, 085032 (2023); doi: 10.1063/5.0166730 13, 085032-3
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Y1(T0,T1)=−V+
jei(ωj+2Ω)T0+V−
jei(ωj−2Ω)T0πj
+3U+
jeiωj+2ΩT0+U−
jeiωj−2ΩT0π2
jπj
+Kjπ3
je3iωjT0+W+
jei3ωj+2ΩT0
+W−
jei3ωj−2ΩT0π3
j+c.c, (21)
where the vectors Kj,V±
j,U±
j, and W±
jare listed as

























Kj=(A−9ω2
jI)−1B+2QR3
j,
V±
j=A−(ωj±2Ω)2I−1P R j,
U±
j=A−(ωj±2Ω)2I−1QR3
j,
W±
j=A−(3ωj±2Ω)2I−1QR3
j.
(22)
At ρ=1, the first approximation, at the non-resonance case, can be
constructed by substituting systems (13),(21), and (19) into Eq. (9),
and for ρ=1, one gets
Y(t)=2π0Rjcos[(ωj+ϖj)t+ϕ0]+2π03π2
0U+
j−V+
j
×cos[(ωj+2Ω+ϖj)t+ϕ0]+2π03π2
0U−
j−V−
j
×cos[(ωj−2Ω+ϖj)t+ϕ0]+2π3
0Kj
×cos[3(ωj+ϖj)t+3ϕ0]+2π3
0W+
j
×cos[(3ωj+2Ω+3ϖj)t+3ϕ0]+2π3
0W−
j
×cos[(3ωj−2Ω+3ϖj)t+3ϕ0]. (23)
The uniform periodic solution (23) consists of strictly cosine
functions. Diagrammatically, the vector Y(t)is illustrated vs
the time tas shown in Fig. 1 for a system parameter having
the particulars values: a1=0.1,a2=0.2, a3=0.3,a4=0.5, p1=3.5,
p2=4.2,p3=5, p4=7, q1=15, q3=3.5, b1=10, b2=−12, whereas
the other constants have the values ϕ0=0.5, π0=0.2, and Ω=0.5.
This graph composes of the two curves x1(t)and x2(t)for the vector
Y(t). The x2(t)-curve is that the upper curve relative to the x1(t)-
curve. The two curves have common points. They are intersected at
the t-axis. The common points lie at t=0.084076 2, t=0.330 746,
t=0.577413, t=0.824 076, and t=1.07073,.... The numerical cal-
culations are drawn to the case of j=1. The solution behavior for a
FIG. 1. The perversion of the vector Y(t)vs the time t.
FIG. 2. The same system as Fig. 1, but for a long time.
FIG. 3. The x1(t)−curve of the perversion of the frequency Ω.
large time is amplified in Fig. 2.InFig. 3, the curve x1(t)is plotted
for four different values of the frequency Ω(Ω=0.1, 0.2,0.3, 0.4).
It is shown that the amplitude of the curves decreases with increasing
Ω. This indicates that the increase of Ωfor the non-resonance case
has a stabilizing influence. It seems that the variation Ωdoes not
imply the common points. This shows the helpful role in increasing
the frequency Ωat the non-resonance case.
IV. CASES OF THE HARMONIC RESONANCE
Return to the first-order level of perturbation, the review of the
right-hand facet of system (14) reveals terms, that are proportional
to exp(±iωjT0). It follows that the secular terms are made from that
is proportional to exp[±i(ω−2Ω)T0]or to exp [±i(3ωj−2Ω)T0]
According to the presence of these secular terms, there are two
second-harmonic resonance.
A. The harmonic resonant case of Ωnear ωj
To express the nearness of the frequency Ωto the natural fre-
quency ωj, we introduce the detuning parameter σin system (14) to
convert the small divisor term into a secular term as
Ω=ωj+ρσ. (24)
Hence, we write
−i(ωj−2Ω)T0=iωjT0+2iσT1,
i(3ωj−2Ω)T0=iωjT0−2iσT1.(25)
AIP Advances 13, 085032 (2023); doi: 10.1063/5.0166730 13, 085032-4
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By using system (25), the small-divisor term arising from
exp[±i(ω−2Ω)T0]in system (14) can be transformed into a sec-
ular term. Therefore, the source of secular terms is removed. At this
stage, the solvability condition given by Eq. (17) will be modified to
become
iD1πj+ˆ
Pj2πj+πje2iσT1−3ˆ
Bjπ2
jπj
−ˆ
Qjπ3
je−2iσT1+6π2
jπj+3π2
jπje2iσT1=0. (26)
The above cubic nonlinear first-order equation contains parametric
terms associated with the complex conjugate variable. The stationary
description can be sought as
πj(t)=(u0+iv0)eiσT1, (27)
where the real constant u0and v0are given by







8ˆ
Q2
ju2
0=−3ˆ
Bjˆ
Pj−2σˆ
Qj,
8ˆ
Q2
jv2
0=3ˆ
Bjˆ
Pj+4ˆ
Pjˆ
Q+2σˆ
Qj.(28)
To study the stability picture, for the present case, we perturb the
solution of Eq. (26) around the stationary solution (27) so that
πj(t)=[(u0+u1(T1))+i(v0+v1(T1))]eiσT1, (29)
where [u1(T1)+iv1(T1)]eiσtwith real functions u1(T1)and v1(T1)
refers to the deviation solution from the stationary solution
(u0+iv0)eiσtwith real constants u0and v0. Substituting Eq. (29) into
Eq. (26) and linearizing in u1(T1) and v1(T1)and by separating both
real and imaginary parts, we get
D1v1(T1)+σ−3ˆ
Pj+9ˆ
Bj+30 ˆ
Qju2
0+3ˆ
Bj+6ˆ
Qjv2
0u1
+6u0v0ˆ
Bj+2ˆ
Qjv1=0, (30)
D1u1(T1)−σ−ˆ
Pj+3ˆ
Bj+6ˆ
Qju2
0+9ˆ
Bj+6ˆ
Qjv2
0v1
−6u0v0ˆ
Bj+2ˆ
Qju1=0. (31)
For a nontrivial solution, the above-coupled equations in u1(t)and
v1(t)maybe support solutions, in the following form:







u1(T1)=αjsinΘT1,
v1(T1)=ΘcosΘT1−βjsin ΘT1,(32)
where the constants αi,βjand Θare given by
αj=σ−ˆ
Pj+3ˆ
Bj+2ˆ
Qju2
0+33ˆ
Bj+2ˆ
Qjv2
0
and
βj=6u0v0ˆ
Bj+2ˆ
Qj, (33)
Θ2=σ2+−4ˆ
Pj+12ˆ
Bj+3ˆ
Qju2
0+12ˆ
Bj+ˆ
Qjv2
0σ
+18u2
0v2
03ˆ
B2
j+4ˆ
Q2
j+12 ˆ
Bjˆ
Qj−6ˆ
Pj3ˆ
Bj+8ˆ
Qju2
0
+9ˆ
Bj+2ˆ
Qj3ˆ
Bj+10 ˆ
Qju4
0+3ˆ
Pj−ˆ
Bj−2ˆ
Qjv2
0
×ˆ
Pj−33ˆ
Bj+2ˆ
Qjv2
0. (34)
In the present case, the uniformly valid solution of Eq. (14) has the
alternative form, which may be expressed by
Y1(T0,T1)=−V+
jei(ωj+2Ω)T0πj+3U+
jeiωj+2ΩT0π2
jπj
+Kje3iωjT0+W+
jei3ωj+2ΩT0π3
j+c.c. (35)
The uniform approximate solution, in the harmonic resonance case,
is formulated for ρ→1, as
Y(t)=2Rju0+αjsinΘtcos Ωt−v0−βjsin Θt+ΘcosΘtsin Ωt
+6U+
ju0+αjsinΘt2+v0−βjsin Θt+Θcos Θt2
×u0+αjsinΘtcos 3Ωt−v0−βjsin Θt+Θcos Θtsin 3Ωt
+2u0+αjsinΘtKjcos 3Ωt+W+
jcos 5Ωt
×u0+αjsinΘt2−3v0−βjsin Θt+Θcos Θt2
−2v0−βjsinΘt+Θcos ΘtKjsin 3Ωt+W+
jsin 5Ωt
×3u0+αjsinΘt2−v0−βjsin Θt+Θcos Θt2. (36)
FIG. 4. The perversion of the vector Y(t)vs the time tthrough the harmonic
resonance case.
FIG. 5. The perversion of the vector Y(t)for a long time for the same system as
in Fig. 4.
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The periodic solution (36) consists of a mix of cosine’s and sine’s
functions. This solution is illustrated numerically within the fol-
lowing graphs, which we consider only the case j=i. The cal-
culations are computed for a system possess a1=3,a2=2, a3=1,
a4=5,p1=3.5, p2=4.2, p3=5, p4=7, q1=15, q3=3.5, b1=2, and
b2=−7, and the frequency has the value Ω=4.5. It is noted that
the frequency Ωhas a giant influence on the periodic vector Y(t)
as indicated in Fig. 4. The most-upper curve is the x2(t)-curve and
the most-lower curve is the x1(t)-curve. The disturbance due to the
impact of the frequency Ωincreases in amplitude and is repeated
for a long time as indicated in Fig. 5. The calculations are com-
puted for four different values for the frequency Ωas shown in Fig. 6
for the component x1(t), as an example, to check the impact of
increasing the frequency Ωof the harmonic resonance. It seems that
the increase of Ωleads to grow the amplitude of the disturbance.
This indicates that Ωhas a destabilizing role. This role is contrary
to that played in the case of the non-resonance case as observed
in Fig. 3.
B. Stability configuration and numerical illustration
of the harmonic resonance case
In accordance with the bounded solutions of system (32), the
right-hand side of Eq. (34) must be positive value. The simplification
of this requirement yields the following stability condition:
Qj(27Bj+2Qj)σ2+Pj81B2
j+114BjQj+28Q2
jσ
+3
4ˆ
Qj(3Bj+4Qj)27B2
j+38BjQj+Q2
j<0. (37)
Because of the relation (24), with a setting, the transition curves
describing the resonance case is wanted in terms of the excitation
frequency as
aΩ2+bΩ+c<0, (38)
where the constants (a,b,c)and read















a=ˆ
Qj27 ˆ
Bj+2ˆ
Qj,
b=ˆ
Pj81 ˆ
B2
j+114 ˆ
Bjˆ
Qj+28 ˆ
Q2
j−2ωjˆ
Qj27 ˆ
Bj+2ˆ
Qj,
c=ω2
jˆ
Qj27 ˆ
Bj+2ˆ
Qj−ωjˆ
Pj81 ˆ
B2
j+114 ˆ
Bjˆ
Qj+28 ˆ
Q2
j+3
4ˆ
Qj3ˆ
Bj+4ˆ
Qj27 ˆ
B2
j+38 ˆ
Bjˆ
Qj+ˆ
Q2
j.
(39)
The transition curves that separate between both the stable and
unstable zones are corresponding to
Ω1=−b
2a+1
2ab2−4ac, and Ω2=−b
2a−1
2ab2−4ac.
(40)
The stability condition (38) and also the transition curves as
conferred in Eq. (40) are illustrated numerically and planned in
Figs. 7–9. The region “S” refers to the stable region; meanwhile, the
region “U” denotes the unstable one (resonance region). Some para-
meters such k,ε,μ, and γare introduced to live the perversion within
FIG. 6. The disturbance of the x1(t)for a perversion of the same system as
in Fig. 4.
the matrices A,Pand also the nonlinear coefficients B and Q, respec-
tively. Such that A=kA∗,P=εP∗,Q=λQ∗, and B=γB∗. So, the
Mathieu equation (2) can be written as
d2
dt2Y(t)+(kA∗+4εP∗cos2Ωt)Y(t)=γB∗+4λQ∗cos2ΩtY3(t).
(41)
Figure 7 demonstrates the transition curves (40). Now, the goal is to
examine the influence of the perversion of the matrix Aon the stabil-
ity diagram vs the variation of the matrix P. Therefore, the frequency
FIG. 7. The plane [Ω(k)−ε]for the transition curves (40).
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FIG. 8. Stability diagram for the same system of Fig (7) with fixed the parameters
ε=λ=γ=1.
Ω(k) as a function of the parameter kis plotted contra the parameter
εfor various values of the parameter k. The numerical computa-
tions are implemented for a1=0.1k,a2=0.2k,a3=0.3k,a4=0.5k,
p1=3.5ε,p2=4.2ε,p3=5ε,p4=7ε,q1=3λ,q3=−3.5λ,b1=3γ, and
b2=5γ, where the parameters λ=γ=1. This calculation examines
the influence of the coefficient of the periodic linear term into sys-
tem (2). The graph demonstrates that the increase in the parameter
εplays a destabilizing role. The stability boundaries and the cor-
responding periodic solutions near some values of the parameter
k=0.1,k=1.0, k=1.5, k=2.0 are identical with the results in the
monograph of the single Mathieu equation.49
Figure 8 illustrates the influence of the perversion of
the matrix Avs the parameter kwith fixed parameters
ε=λ=γ=1.The numerical computations are considered for
a1=0.1k,a2=0.2k,a3=0.3k,a4=0.5k,p1=3.5ε,p2=4.2ε,p3=5ε,
p4=7ε,q1=3λ,q3=−3.5λ,b1=3γ, and b2=5γ. It is shown that the
unstable area has been embedded in the stable regions. It appears
that the increase in the parameter kleads to the enhancement of the
unstable region. This means that the increase in kplays a stabilizing
influence. In other words, the perversion of the matrix Ahas a
stabilizing influence.
In Fig. 9, the examination is made for the impress of the matrix
B(the constant part of the Duffing term) on the stability picture.
For this purpose, the frequency Ωis plotted contra the parameter
γthat measures the impact of the cubic nonlinear coefficient. The
same system is given in Fig. 7, except k=1. The graph shows that
the instability reveals for a short amount of the parameter γ. This
instability is decreased as γincreased. Large values of γhas a stabi-
lizing role. The application of the frequency Ωhas suppressed the
unstable role of small values of the parameter γ.
In Fig. 10, the transition curves (40) are plotted vs the para-
meter λthat measures the impact of the nonlinear periodic coef-
ficients. In this graph, the impact of the periodic coefficient Pof
the linear part is included. It is shown that the unstable region has
been sandwiched between stable regions. It is found that the sys-
tem becomes more stabilizing for very small λ, while increasing λ
with very small Ωleads to increase the unstable region. Addition-
ally, increasing the parameter εleads to increase the unstable region.
This means that the coefficients of the periodicity of linear as well as
the nonlinear parts play a destabilizing role.
C. The sub-harmonic resonant case of Ω, near 2ωj
This resonance case occurs due to nonlinearity. For this pur-
pose, we introduce a detuning parameter σSin system (14) to convert
the small divisor term into secular term as
Ω=2ωj+2ρσS, so wewrite −i(3ωj−2Ω)T0=iωjT0+4iσST1.
(42)
In light of Eq. (42), the cancellation of secular term results in the
following solvability condition:
iD1πj+2ˆ
Pjπj−3ˆ
Bj+2ˆ
Qjπ2
jπj−ˆ
Qjπ3
je4iσST1=0. (43)
The solution of Eq. (43) is formulated as explained in Sec. IV B. The
function πj(t)is calculated by the linear perturbation to be
πj(t)=a0+ΘcosΘt−2i2ˆ
Pj−σSsinΘteiσST1, (44)
where the constants a0and Θare given by
a2
0=2ˆ
Pj−σS
3ˆ
Bj+7ˆ
Qj, and Θ2+8ˆ
Qj2ˆ
Pj−σS2
3ˆ
Bj+7ˆ
Qj=0. (45)
The stability requires the following condition:
ˆ
Qj3ˆ
Bj+7ˆ
Qj<0. (46)
The approximate solution at the sub-harmonic resonance case reads
Y(t)=2(a0+ΘcosΘt)Rjcos 1
2Ωt−V+
jcos 5
2Ωt−V−
jcos 3
2Ωt
+42ˆ
Pj+ωj−1
2ΩRjsin 1
2Ωt−V+
jcos 5
2Ωt+V−
jcos 3
2ΩtsinΘt
+6(a0+ΘcosΘt)2+42ˆ
Pj+ωj−1
2Ω2sin2Θt








(a0+ΘcosΘt)U+
jcos 5
2Ωt+U−
jcos 3
2Ωt
+22ˆ
Pj+ωj−1
2ΩU+
jsin 5
2Ωt−U−
jsin 3
2ΩtsinΘt









+2(a0+ΘcosΘt)Kjcos 1
2Ωt+W+
jcos 5
2Ωt(a0+ΘcosΘt)2+122ˆ
Pj+ωj−1
2Ω2sin2Θt
+42ˆ
Pj+ωj−1
2ΩKjsin 1
2Ωt+W+
jsin 5
2ΩtsinΘt3(a0+Θcos Θt)2−42ˆ
Pj+ωj−1
2Ω2sin2Θt. (47)
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FIG. 9. The (Ω−γ)-plane for the same system given in Fig (7), except that k=1.
FIG. 10. The (Ω−λ)-plane for the same system given in Fig. 7, except that
k=1 and γ=1.
D. The combination of super- and
sub–super-harmonic nonlinear resonant case
System (14) is still produced in secular terms such that
exp[i(3ωj−2Ω)T0]. In the case of (3ω1−2Ω), the resonance
occurs at Ωapproaches to 1
2(3ω1+ω2). Consequently, additional
secular term corresponds to exp (iω2T0), can be obtained. On the
other hand, in the case of (3ω2−2Ω), the resonance arises whence Ω
becomes near 1
2(ω1+3ω2); accordingly, the additional secular term
can be obtained corresponds to exp (iω1T0).
Here, we express the nearness of Ωto 1
2(3ω1+ω2)by
introducing a detuning parameter δsuch that
Ω=1
2(3ω1+ω2)+3
2ρδ. Consequently,we have
−i(3ω1−2Ω)T0=iω2T0+3iδT1. (48)
In this combination case, there are no additional secular terms.
At this stage, the elimination of secular terms from system (14)
corresponds to exp(iω1T0). This still yields the same solvability
condition as obtained before in the non-resonance case. In the final
form where ρ→1, we have
id
dtπ1+2ˆ
P1π1−3ˆ
B1+2ˆ
Q1π2
1π1=0. (49)
On the other hand, and in light of Eq. (48), the following solvability
condition is imposed:
id
dπ2+2ˆ
P2π2−3ˆ
B2+2ˆ
Q2π2
2π2=ˆ
Q21π3
1e3iδt, (50)
where the notation ˆ
Qj3−j=SjQR3
3−jis used.
The first-order solution Y1without secular terms has the form
Y1=−V+
jei(ωj+2Ω)t+V−
jei(ωj−2Ω)tπj
+3U+
jeiωj+2Ωt+U−
jeiωj−2Ωtπ2
jπj
+Kjπ3
je3iωjt+W+
1ei(3ω1+2Ω)tπ3
1
+W+
2ei(3ω2+2Ω)t+W−
2ei(3ω2−2Ω)tπ3
2+c.c. (51)
The full picture of the function Y1, in the present case, can be
found by solving Eqs. (49) and (50). Equation (49) is a nonlinear
Landau equation with real coefficients, while Eq. (50) is a nonlin-
ear Landau equation in π2(t). It contains a complex conjugate cubic
nonlinear term in π1(t). The function π1(t)still yields complex con-
jugate of Eq. (49). To relax the parametric term in Eq. (50), let us
consider the transformation
π1(t)=ˆ
π1(t)eiδt. (52)
Accordingly, systems (49) and (50) reduce to
id
dt ˆ
π1+2ˆ
P1−δˆ
π1−3ˆ
B1+2ˆ
Q1ˆ
π2
1ˆ
π1=0, (53)
id
dtπ2+2ˆ
P2π2−3ˆ
B2+2ˆ
Q2π2
2π2=ˆ
Q21 ˆ
π3
1. (54)
The exact solution of Eq. (53) is given in Sec. IV C, in the
non-resonance case, and can be performed as
ˆ
π1(t)=λ0ei(ϖ1−δ)t+iψ0, (55)
where the amplitude λ0and the phase ψ0are constants. Accordingly,
Eq. (54) becomes
id
dtπ2+2ˆ
P2π2−3(ˆ
B2+2ˆ
Q2)π2
2π2=ˆ
Q21λ3
0e−3i(ϖ1−δ)t−3iψ0. (56)
Thus, the stationary state solution reads
π2(t)=re−3i(ϖ1−δ)t−3iψ0, (57)
where the amplitude r is a real constant and given by
3ˆ
B2+2ˆ
Q2r3−2ˆ
P2+3ϖ1−3δr+ˆ
Q21λ3
0=0. (58)
To discuss the stability, we may modulate the stationary solution
(57) as
π2(t)=(r+ξ(t)+iζ(t))e−3i(ϖ1−δ)t−3iψ0, (59)
where the functions ξ(t)and ζ(t)represent an infinitesimal small
deviation from the stationary-state response. The substitution of
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Eq. (59) into Eq. (56) and considering linearization of ξ(t)and ζ(t),
and then, separating the real and imaginary, we get













d
dtζ−2ˆ
P2+3ϖ1−3δ−9r2ˆ
B2+2ˆ
Q2ξ=0,
d
dtξ+2ˆ
P2+3ϖ1−3δ−3r2ˆ
B2+2ˆ
Q2ζ=0.
(60)
The above equations represent a system of linear first-order
differential equations. For the non-trivial solution, we may have
ξ(t)=ΘcosΘt,
and
ζ(t)=2ˆ
P2+3ϖ1−3δ−9r2ˆ
B2+2ˆ
Q2sin Θt, (61)
where the argument Θis given by
Θ2=2ˆ
P2+3ϖ1−3δ−3r2ˆ
B2+2ˆ
Q2
×2ˆ
P2+3ϖ1−3δ−9r2ˆ
B2+2ˆ
Q2. (62)
The stability criteria require that the right-hand side of Eq. (62) may
have a positive nature, which can be satisfied as
2ˆ
P2+3ϖ1−3δ>9r2ˆ
B2+2ˆ
Q2
and
2ˆ
P2+3ϖ1−3δ<3r2ˆ
B2+2ˆ
Q2. (63)
Because of the definition (48), the above stability conditions can be
sought in terms of the frequency Ω,
Ω>1
2(3ω1+ω2)+3ˆ
P1−ˆ
P2+9
2r2ˆ
B2+2ˆ
Q2−9
2ˆ
B1+2ˆ
Q1λ2
0
and
Ω<1
2(3ω1+ω2)+3ˆ
P1−ˆ
P2+3
2r2ˆ
B2+2ˆ
Q2−9
2ˆ
B1+2ˆ
Q1λ2
0.
(64)
The approximate solution in this resonance case reads
Y(t)=2λ0R1cos(ϖ1t+ψ0)−V+
1−3λ2
0U+
1cos[(ω1+2Ω+ϖ1)t+ψ0]−V−
1−3λ2
0U−
1cos[(ω1−2Ω+ϖ1)t+ψ0]
+2λ3
0K1cos[(3ω1+3ϖ1)t+3ψ0]+W+
1cos[(3ω1+3ϖ1+2Ω)t+3ψ0].
+2R2(r+ΘcosΘt)cos [(3ϖ1−2Ω+3ω1+2ω2)t+3ψ0]+2R22ˆ
P2−3ˆ
ϖ1−9r2ˆ
B2+2ˆ
Q2
×sinΘtsin [(3ϖ1−2Ω+3ω1+2ω2)t+3ψ0]−2(r+Θcos Θt)cos [(3ϖ1+3ω1−4Ω)t+3ψ0]
×V+
2−3U+
2(r+ΘcosΘt)2+9U+
22ˆ
P2−3ˆ
ϖ1−9r2ˆ
B2+2ˆ
Q22sin2Θt
−2 sin[(3ϖ1+3ω1−4Ω)t+3ψ0]2ˆ
P2−3ˆ
ϖ1−9r2ˆ
B2+2ˆ
Q2sin Θt
×V+
2−3U+
22ˆ
P2−3ˆ
ϖ1−9r2ˆ
B2+2ˆ
Q22sin2Θt−3U+
2(r+ΘcosΘt)2−2(r+Θcos Θt)
×cos[(3ϖ1+3ω1)t+3ψ0]V−
2−3U−
2(r+ΘcosΘt)2+9U−
22ˆ
P2−3ˆ
ϖ1−9r2ˆ
B2+2ˆ
Q22sin2Θt
−2 sin[(3ϖ1+3ω1)t+3ψ0]2ˆ
P2−3ˆ
ϖ1−9r2ˆ
B2+2ˆ
Q2sin Θt
×V−
2−3U−
22ˆ
P2−3ˆ
ϖ1−9r2ˆ
B2+2ˆ
Q22sin2Θt−3U−
2(r+ΘcosΘt)2
+2(r+ΘcosΘt)(r+Θcos Θt)2−32ˆ
P2−3ˆ
ϖ1−9r2ˆ
B2+2ˆ
Q22sin2Θt
×K2cos[(9ϖ1−6Ω+9ω1)+9ψ0]+W+
2cos[(9ϖ1−8Ω+9ω1)+9ψ0]+W−
2cos[(9ϖ1−4Ω+9ω1)+9ψ0]
+22ˆ
P2−3ˆ
ϖ1−9r2ˆ
B2+2ˆ
Q2sin Θt(r+Θcos Θt)2−2ˆ
P2−3ˆ
ϖ1−9r2ˆ
B2+2ˆ
Q22sin2Θt
×K2sin[(9ϖ1−6Ω+9ω1)+9ψ0]+W+
2sin[(9ϖ1−8Ω+9ω1)+9ψ0]+W−
2sin[(9ϖ1−8Ω+9ω1)+9ψ0]. (65)
Similar results can be obtained in the case of Ωnear to 1
2(ω1+
3ω2), by changing the role of ω1and ω2.
E. The super-harmonic coupled resonant case
In case of the nearness of Ωto the combination of super-
harmonic 1
2(ω1±ω2), the term exp[−i(ω3−j−2Ω)T0]will be
transformed to exp (iωjT0)and additional secular terms in Eq. (14)
arise. To study these contributions, we express the nearness of Ωto
1
2(ω1+ω2)by introducing a detuning parameter σcsuch as
Ω=1
2(ω1+ω2)+ρσc, thus, −i(ω3−j−2Ω)T0=iωjT0+2iσcT1.
(66)
Here, we will focus on the positive ω2sign, while the negative one
can also be used to change the sign in the results. At this end, the
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secular terms that appear in system (14) can be rearranged to introduce the below coupled nonlinear amplitude equations, in the final form
as ρ→1,
id
dtπj+2ˆ
Pjπj+ˆ
Pj3−jπ3−je2iσct
=3ˆ
Bj+2ˆ
Qjπ2
jπj+3ˆ
Qj3−jπ2
3−jπ3−je2iσct, (67)
where the notation ˆ
Pj3−j=SjPR3−jis used.
Without secular terms the first-order solution, in this case, is presented in the form
Y1(T1)=−V+
jei(ωj+2Ω)T0πj+e−i(ωj+2Ω)T0πj−V−
3−jei(ω3−j−2Ω)T0π3−j+e−i(ω3−j−2Ω)T0π3−j
+3U+
jeiωj+2ΩT0π2
jπj+e−iωj+2ΩT0π2
jπj+3U−
3−jeiω3−j−2ΩT0π2
3−jπ3−j+e−iω3−j−2ΩT0π2
3−jπ3−j
+Kjπ3
je3iωjT0+π3
je−3iωjT0+W+
jei3ωj+2ΩT0+W−
jei3ωj−2ΩT0π3
j
+W+
je−i3ωj+2ΩT0+W−
je−i3ωj−2ΩT0π3
j. (68)
The solution for the first-order function Y1can be obtained
when the coupled nonlinear amplitude Eq. (67) is solved. To solve
these nonlinear amplitude Eq. (67), we proceed as in Sec. IV D, in
which the function πj(t)is replaced by
πj(t)=(γj+Γj(t))eiσct, (69)
where γjis a real amplitude referring to the steady-state and Γj(t)
represents a small deviation from the steady-state response γj, which
is described as
2ˆ
Pj−σc−3ˆ
Bj+2ˆ
Qjγ2
jγj
+ˆ
Pj3−j−3ˆ
Qj3−jγ2
3−jγ3−j=0. (70)
For the non-trivial solution in γ1and γ2, the determination of the
above system must be zero. This imposes the following characteristic
equation:
2ˆ
P1−σc−3ˆ
B1+2ˆ
Q1γ2
12ˆ
P2−σc−3ˆ
B2+2ˆ
Q2γ2
2
−ˆ
P12 −3ˆ
Q12γ2
2ˆ
P21 −3ˆ
Q21γ2
1=0. (71)
This relation can be satisfied by the following values for γ2
1and γ2
2:

















γ2
1=2ˆ
P1−σc+ˆ
P21
3ˆ
Q21 +3ˆ
B1+2ˆ
Q1,
γ2
2=2ˆ
P2−σc+ˆ
P12
3ˆ
Q12 +3ˆ
B2+2ˆ
Q2.
(72)
The linearization in Γj(t)is governed by
id
dtΓj+2ˆ
Pj−σc−6ˆ
Bj+2ˆ
Qjγ2
jΓj
+ˆ
Pj3−j−6ˆ
Qj3−jγ2
3−jΓ3−j
=3ˆ
Bj+2ˆ
Qjγ2
jΓj+3ˆ
Qj3−jγ2
3−jΓ3−j. (73)
To express the solution of the linearized system (73), we suppose that
Γj(t)=fj(t)+igj(t), (74)
where the functions fj(t)and gj(t)are real. Equation (74) will be
separated into the following real and imaginary parts:













d
dtgj−H−
j−σcfj−H−
j3−jf3−j=0,
d
dt fj+H+
j−σcgj−H+
j3−jg3−j=0,
(75)
where the values H±
jand H±
j3−jare







H±
j=2ˆ
Pj−6ˆ
Bj+2ˆ
Qjγ2
j±3ˆ
Bj+2ˆ
Qjγ2
j,
H±
j3−j=ˆ
Pj3−j−6ˆ
Qj3−jγ2
2±3ˆ
Qj3−jγ2
2.(76)
Four homogeneous first-order differential equations in the four
unknown functions, namely, f1(t),f2(t),g1(t), and g2(t), make up
the aforementioned system.
There is at least one repeated equation for a non-trivial solu-
tion. As a result, there are at least two dependent functions. Suppose
g1(t)and g2(t), say, are the two dependent functions
g1(t)=λΘcos Θt,
g2(t)=Θcos Θt,(77)
where λand Θare constants to be determined. The substitution of
Eq. (77) into Eq. (75), on using the process of the anti-derivatives,
one gets
f1(t)=H+
12 −H+
1−σcλsinΘt
and
f2(t)=λH+
21 −H+
2−σcsin Θt(78)
AIP Advances 13, 085032 (2023); doi: 10.1063/5.0166730 13, 085032-10
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AIP Advances ARTICLE pubs.aip.org/aip/adv
where the constants λand Θcan be evaluated by substituting
Eqs. (77) and (78) into Eq. (75), the following relations between λ
and Θare imposed:
λΘ2−(H−
1−σc)(H+
1−σc)+H−
12H+
21+H+
12H−
1
−H−
12H+
2+σc(H−
12 −H+
12)=0, (79)
Θ2+λH+
21(H−
2−σc)−H−
21(H+
1−σc)
+H−
21H+
12 −(H−
2−σc)(H+
2−σc)=0. (80)
Eliminate Θ2from Eqs. (79) and (80) implies
λ2H+
21H−
2−H−
21H+
1+σc(H−
21 −H+
21)
+λH−
21H+
12 −H−
12H+
21 +(H−
1−σc)(H+
1−σc)−(H−
2−σc)
×(H+
2−σc)+H−
12H+
2−H+
12H−
1−σc(H−
12 −H+
12)=0. (81)
Eliminate λfrom Eqs. (79) and (80) yields
Θ4−Θ2(H−
1−σc)(H+
1−σc)+(H−
2−σc)(H+
2−σc)
−(H−
12H+
21 +H−
21H+
12)−H+
12H+
21 −(H+
1−σc)(H+
2−σc)
×H−
1H−
2−H−
12H−
21 −(H−
1+H−
2)σc+σ2
c=0. (82)
Note that Θ2must be positive and real in order to meet the sta-
bility criterion. This can be satisfied whence the determinant of
(82) is positive, besides the sum and the product of the two roots
of the characteristic Eq. (82) having positive values. These restric-
tions for stability are described in terms of the detuning parameter,
respectively, in the following conditions:
l0σ2
c−2l1σc+l2>0,σ2
c−1
2l3σc+l4>0,
and
σ4
c−l3σ3
c+l5σ2
c+l6σc+l7>0, (83)
where the constant coefficients l’sare listed below:
l0=H+
1+H−
1+H−
2+H+
22−4H−
12H+
21 +H−
21H+
12+4H−
21H−
12 +H+
12H+
21−4H+
1+H−
1H−
2+H+
2,
l1=2H−
12H−
21H+
1+H+
2+H+
1+H−
1H−
1H+
1−H−
2H+
2−H−
12H+
21 −H−
21H+
12
+2H+
12H+
21H−
1+H−
2+H−
2+H+
2−H−
1H+
1+H−
2H+
2−H−
12H+
21 −H−
21H+
12,
l2=4H+
12H+
21H−
1H−
2+4H−
21H−
12H+
1H+
2−4H+
12H−
12H+
21H−
21 −4H−
2H+
2H−
1H+
1+H−
1H+
1+H−
2H+
2−H−
12H+
21 −H−
21H+
122,
l3=H+
1+H−
1+H−
2+H+
2,
l4=1
2H−
1H+
1+H−
2H+
2−H−
12H+
21 −H−
21H+
12,
l5=H−
1+H+
1H−
2+H+
2+H+
1+H+
2H−
2−H−
21H−
12 +H+
12H+
21,
l6=H−
12H−
21H+
1+H+
2+H+
12H+
21H−
1+H−
2−H+
1H+
2H−
1+H−
2−H−
1H−
2H+
1+H+
2,
l7=H+
1H+
2−H+
12H+
21+H−
1H−
2−H−
12H−
21.
The first-order approximation is obtained by substituting the zero-order solution Y0(t)and the first-order solution Y1(t)into the expansion
(8) after setting ρ→1, using Eq. (69) with (74) and the solutions (78), we finally obtain
Y(t)=2(γ1+f1(t))R1cos(Ω1+)t−2λg2R1sin (Ω1+)t+2(γ2+f2(t))R2cos (Ω1−)t
−2g2R2sin(Ω1−)t−2(γ1+f1(t))V+
1cos3Ω+1
2(ω1−ω2)t+V−
1cos(Ω1−)t
−2λg2V+
1sin3Ω+1
2(ω1−ω2)t+V−
1sin(Ω1−)t+2g2V+
2sin3Ω−1
2(ω1−ω2)t−V−
2sin(Ω1+)t
−2(γ2+f2(t))V+
2cos3Ω−1
2(ω1−ω2)t+V−
2cos(Ω1+)t
+6(γ1+f1(t))(γ1+f1(t))2+λ2g2
2U+
1cos(Ω1+)t+U−
1cos(Ω1−)t
−6g1(γ1+f1(t))2+λ2g2
2U+
1sin(Ω1+)t−U−
1sin(Ω1−)t
+6(γ2+f2(t))(γ2+f2(t))2+g2
2U+
2cos(Ω1−)t+U−
2cos(Ω1+)t
−6g2(γ2+f2(t))2+g2
2U+
2sin(Ω1−)t−U−
2sin(Ω1+)t
+2(γ1+f1(t))(γ1+f1(t))2−3λ2g2
2K1cos(3Ω1+)t+W+
1cos5Ω+3
2(ω1−ω2)t+W−
1cosΩ+3
2(ω1−ω2)t
−2λg23(γ1+f1(t))2−λ2g2
2K1sin(3Ω1+)t+W+
1sin5Ω+3
2(ω1−ω2)t+W−
1sinΩ+3
2(ω1−ω2)t
AIP Advances 13, 085032 (2023); doi: 10.1063/5.0166730 13, 085032-11
© Author(s) 2023

AIP Advances ARTICLE pubs.aip.org/aip/adv
+2(γ2+f2(t))(γ2+f2(t))2−3g2
2K2cos(3Ω1−)t+W+
2cos5Ω−3
2(ω1−ω2)t+W−
2cosΩ−3
2(ω1−ω2)t
−2g23(γ2+f2(t))2−g2
2K2sin(3Ω1−)t+W+
2sin5Ω−3
2(ω1−ω2)t+W−
2sinΩ−3
2(ω1−ω2)t,
with
Ω1+=Ω+1
2(ω1−ω2),Ω1−=Ω−1
2(ω1−ω2). (84)
V. CONCLUSION
This work is focused on the investigation of the stability behav-
ior of the cubic nonlinear coupled Mathieu equations having Duffing
terms. The analysis was carried out with the usage of the matrix
form. Contrary to the conventional Mathieu equation, the matrix
form of the problem reveals that there are only two families of solu-
tions. An approximate solution at each possible resonance case is
obtained, and a stability criterion is discussed in each case. The
analysis depends on a perturbation technique. We use coupling
between the well-known multiple-scale method and the homotopy
perturbation method.30,31 The accomplishment of the second-order
disturbance that is gained from the multiple scales method has been
modulated because of the first-order perturbation of the homotopy-
multiple-scales perturbation method. In the first-order perturbation,
we discussed the non-resonance case, as well as the harmonic res-
onance case. In this study, nonlinear parametric Landau equations
have been imposed. Solutions of these coupled nonlinear equations
lead to catching the stability conditions. Graphs are depicted to
clarify the periodic solution in the non-resonance state and in the
harmonic resonance. The numerical computations showed that the
amplitude of the periodic terms, in the linear coefficients as well
the nonlinear coefficients, plays a destabilizing influence at the har-
monic resonance case. The numerical clarification of the periodic
solution showed that the frequency Ωplays a stabilizing influence in
the non-resonance case and a destabilizing effect on the harmonic
resonance case.
ACKNOWLEDGMENTS
The authors express their gratitude to Princess Nourah
bint Abdulrahman University Researchers Supporting Project No.
(PNURSP2023R378), Princess Nourah bint Abdulrahman Univer-
sity, Riyadh, Saudi Arabia.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
All authors contributed equally to this paper.
Yusry O. El-Dib: Investigation (equal); Methodology (equal);
Resources (equal); Software (equal); Supervision (equal); Valida-
tion (equal); Visualization (equal); Writing – original draft (equal).
Albandari W. Alrowaily: Data curation (equal); Formal analysis
(equal); Investigation (equal); Writing – review & editing (equal).
C. G. L. Tiofack: Conceptualization (equal); Data curation (equal);
Formal analysis (equal). S. A. El-Tantawy: Formal analysis (equal);
Investigation (equal); Supervision (equal); Writing – review &
editing (equal).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were
created or analyzed in this study.
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