Dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing equation

Abstract

In this paper, the computation schemes for periodic solutions of the forced fractional-order Mathieu-Duffing equation are derived based on incremental harmonic balance (IHB) method. The general forms of periodic solutions are founded by the IHB method, which could be useful to obtain the periodic solutions with higher precision. The comparisons of the approximate analytical solutions by the IHB method and numerical integration are fulfilled, and the results certify the correctness and higher precision of the solutions by the IHB method. The dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing equation is investigated by the IHB method. Then, the effects of the excitation frequency, fractional order, fractional coefficient, and nonlinear stiffness coefficient on the complex dynamical behaviors are analyzed. At last, the detailed results are summarized and the conclusions are made, which present some useful information to analyze and/or control the dynamical response of this kind of system.

Full text

Dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing equation
Shao-Fang Wen, Yong-Jun Shen, Xiao-Na Wang, Shao-Pu Yang, and Hai-Jun Xing
Citation: Chaos 26, 084309 (2016); doi: 10.1063/1.4959149
View online: http://dx.doi.org/10.1063/1.4959149
View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/26/8?ver=pdfcov
Published by the AIP Publishing
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Dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing
equation
Shao-Fang Wen,
1
Yong-Jun Shen,
2,a)
Xiao-Na Wang,
3
Shao-Pu Yang,
2
and Hai-Jun Xing
2
1
Transportation Institute, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
2
Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
3
Hebei Vocational College of Rail Transportation, Shijiazhuang 052165, China
(Received 7 March 2016; accepted 18 May 2016; published online 2 August 2016)
In this paper, the computation schemes for periodic solutions of the forced fractional-order
Mathieu-Duffing equation are derived based on incremental harmonic balance (IHB) method. The
general forms of periodic solutions are founded by the IHB method, which could be useful to obtain
the periodic solutions with higher precision. The comparisons of the approximate analytical
solutions by the IHB method and numerical integration are fulfilled, and the results certify the
correctness and higher precision of the solutions by the IHB method. The dynamical analysis of
strongly nonlinear fractional-order Mathieu-Duffing equation is investigated by the IHB method.
Then, the effects of the excitation frequency, fractional order, fractional coefficient, and nonlinear
stiffness coefficient on the complex dynamical behaviors are analyzed. At last, the detailed results
are summarized and the conclusions are made, which present some useful information to analyze
and/or control the dynamical response of this kind of system. Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4959149]
Recently, fractional calculus was paid more and more
attention from researchers in different fields and had
become an international hot topic. Comparing with the
traditional integer-order counterpart, the fractional-
order system may be much closer to the nature of real
world and has more advantages. A nonlinear fractional-
order system may exhibit considerably complex dynamic
behaviors such as switch in response stability, quasiperi-
odic and chaotic motion. Especially the strongly nonlin-
ear oscillators were so difficult to solve that many
scholars had been working on it. In this paper, a strongly
nonlinear fractional-order oscillator is studied, i.e., the
strongly nonlinear fractional-order Mathieu-Duffing
equation, and the incremental harmonic balance (IHB)
method is used to obtain the periodic solutions of the
oscillator with higher precision. The comparisons
between the numerical results and the approximate ana-
lytical solutions based on the IHB method verify the
excellent accuracy of the IHB method. The effects of the
system parameters on the amplitude-frequency curves
are presented, which will be useful to design and control
these kinds of fractional systems.
I. INTRODUCTION
Although fractional calculus had been proposed for more
than 300 years, its applications in physical and engineering
fields were just a recent research focus.
1–3
Comparing with
the traditional integer-order counterpart, the fractional-order
system may be much closer to the nature of real world
and has more advantages. In the engineering field with
fractional-order derivative, the effects of the parameters in
the fractional-order derivative on a dynamical system were
important and interesting.
4,5
A lot of researchers in some rele-
vant fields had applied the fractional-order models to solve
the complicate dynamical problems.
6–14
In the nonlinear fractional-order dynamical systems,
there may be a lot of strong nonlinearities such as large
deformation in mechanical and civil engineering or strongly
nonlinear terms in control engineering.
15
At this time, the
system cannot be regarded as a perturbation one for its
derived linear system, because the strong nonlinearities will
seriously affect the response of the periodic solution. Even in
some parameter cases, the solution may be chaotic, so that
those classical perturbation methods for weakly nonlinear
system will not work for those strongly nonlinear prob-
lems.
16–19
Some numerical integration methods, such as
Runge-Kutta method, could be used to solve the strongly
nonlinear problem, but the convergence speeds of those
numerical methods were generally slow. It is also obvious
that those numerical integration methods are not suitable to
analyze the whole dynamical behaviors if the system param-
eters are changed in wide ranges.
The incremental harmonic balance (IHB) method is a
powerful semi-analytical method with many advantages. For
example, it is capable of dealing with both strongly and
weakly nonlinear systems, and its convergence speed is very
quick. Moreover, one could obtain periodic solutions with
higher precision by easily adding the orders of periodic solu-
tions in the IHB method. That means the IHB method is very
feasible in different nonlinear systems. The Formulas about
linear and typical nonlinear terms of integer-order differen-
tial equations had been deduced in many references, Refs.
20–24, and the general forms of the periodic solutions for
those terms were founded, which were useful to obtain the
solutions with higher precision. For example, Shen et al.
23
had studied the dynamical properties of a spur gear pair with
a)
Author to whom correspondence should be addressed. Electronic mail:
shenyongjun@126.com
1054-1500/2016/26(8)/084309/8/$30.00 Published by AIP Publishing.26, 084309-1
CHAOS 26, 084309 (2016)
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2016 11:11:43

time-varying stiffness and backlash based on the IHB
method, and the formulas about parametric excitation term
were deduced. Lau and Zhang
24
investigated nonlinear
vibrations of piecewise-linear systems by the IHB method,
and an explicit formulation has been worked out. According
to the authors’ knowledge, those works on system dynamics
only respect to the traditional integer-order system, but the
iteration scheme of IHB method for fractional-order nonlin-
ear oscillator has not been studied. In the following part, the
research on this problem is carried out.
In this paper, the strongly nonlinear fractional-order
Mathieu-Duffing equation is presented in Section II. Then the
IHB computation scheme for periodic solutions of fractional-
order Mathieu-Duffing equation is derived, and the general
forms in the computation scheme for a fractional-order non-
linear oscillator are founded based on the IHB method in
Section III. Section IV presents the comparisons between the
numerical results with the approximate analytical solutions
by the IHB method, and the effects of the system parameters
on the amplitude-frequency curves are also given in this
section. At last, the detailed results are summarized and the
conclusions are made.
II. FRACTIONAL-ORDER MATHIEU-DUFFING
EQUATION
The considered fractional-order Mathieu-Duffing equa-
tion is shown as
d2xt
ðÞ
dt2þC1
dxt
ðÞ
dtþcþ2ecos xt
ðÞ
xt
ðÞ
þC2x3t
ðÞþK1Dpxt
ðÞ½
¼Fcos x1t;(1)
where c,C
1
,C
2
,F,x
1
, and 2ecos xtare the linear stiffness
coefficient, linear viscous damping coefficient, nonlinear
stiffness coefficient, excitation amplitude, excitation fre-
quency, and the periodic time-varying stiffing coefficient,
respectively. K1Dp½xðtÞ is the p-order derivative of xðtÞto t
with fractional coefficient K1(K1>0) and fractional order
is restricted as 0 <p<1. There are several definitions for
fractional-order derivative such as Gr€
unwald-Letnikov,
Riemann-Liouville, and Caputo definitions.
1–4,25,26
Under
wide senses, they are equivalent for most mathematical func-
tions. Without generality, Caputo’s definition is adopted in
Eq. (1) with the form as
Dpxt
ðÞ½
¼1
C1p
ðÞ
ðt
0
x0u
ðÞ
tu
ðÞ
pdu;(2)
where CðyÞis the Gamma function satisfying Cðyþ1Þ
¼yCðyÞ, and the fractional order meets n1<p<nwhile
nis a natural number.
The relationship between the forced and parametric
excitation frequencies meets x1mx, where mis a natural
number. This means there may exist several different reso-
nance cases in Eq. (1). Moreover, there is no limitation on
the parameter values such as small damping coefficient,
weak nonlinearity, and excitation amplitude. Accordingly,
Eq. (1) is a strongly nonlinear fractional-order system with
large viscous damping coefficient, and the traditional
perturbation-based methods for a weakly nonlinear system
could not present satisfactory results. Eq. (1) must be solved
by those methods appropriate for the strongly nonlinear
system.
III. THE APPROXIMATE ANALYTICAL SOLUTION
BY THE IHB METHOD
Letting s¼xt, Eq. (1) becomes
x2€
xþxC1_
xþðcþ2ecos sÞxþC2x3þK1xpDp½xðtÞ
¼F1cos ms:(3)
The symbol • denotes the derivative with respect to s. The
periodic solution taking N-order harmonic terms for Eq. (3)
could be expressed as
x0¼a0þX
N
n¼1
½ancosðnsÞþbnsinðnsÞ;(4)
and accordingly
Dx¼Da0þX
N
n¼1
½DancosðnsÞþDbnsinðnsÞ:(5)
Substituting x¼x0þDxinto Eq. (3), expanding all the
terms into Taylor series, and omitting the higher-order terms
of the small increment Dx, one could get
x2D€
xþxC1D_
xþðcþ2ecos sÞDxþ3C2x2
0DxþK1xpDp
s½Dx
¼fx2€
x0þf0F1cos msþK1xpDp
s½x0g;(6)
where f0¼xC1_
x0þðcþ2ecos sÞx0þC2x3
0.
Introducing the vectors
X¼½1;cos s;cos 2s; ::: cos ns;sin s;sin 2s; ::: sin ns;(7a)
A0¼½a0;a1; :::an;b1;b2; :::bnT;(7b)
DA¼½Da0;Da1; :::Dan;Db1;Db2; :::DbnT;(7c)
one could obtain
x0¼XA0;Dx¼XDA:(7d)
Applying the Galerkin’s procedure
24
for Eq. (6), one may
get the explicit forms for the vector A0. In the integration
procedure, the fractional-order derivative is an aperiodic
function, so that one could select the time terminal Tas T
¼1 and obtain the integration results for the fractional-
order derivative. For the other periodic functions, one could
simply select the time terminal as T¼2p. Accordingly, the
following equations could be established:
1
2pð2p
0
dDx
ðÞ
x2D€
xþxC1D_
xþcþ2ecos s
ðÞ
Dx

þ3C2x2
0Dxdsþ1
TðT
0
dDx
ðÞ
K1xpDp
sDx
½

ds
¼1
2pð2p
0
dDx
ðÞ
x2€
x0þf0F1cos ms

ds
1
TðT
0
dDx
ðÞ
K1xpDp
sx0
½

ds:(8)
084309-2 Wen et al. Chaos 26, 084309 (2016)
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Rearranging Eq. (8), one could get
dDA
ðÞ
T1
2pð2p
0
xTx2€
xþxC1_
xþcþ2ecoss
ðÞ
x

(
þ3C2x2
0xds)DAþdDA
ðÞ
T1
TðT
0
xTK1xpDp
sx
½

ds
()
DA
¼dDA
ðÞ
T1
2pð2p
0
x2_
xT_
xA0f0xTþF1cosms

ds
()
dDA
ðÞ
T1
TðT
0
K1xpDp
sx0
½
xTds
()
:(9)
Furthermore, one could obtain the 2 Nþ1 linearized equa-
tions about DA
MDA¼R;(10)
where
M¼M1þMp
2;(11a)
M1¼1
2pð2p
0
xTx2€
xþxC1_
xþcþ2ecos s
ðÞ
xþ3C2x2
0x

ds;
(11b)
Mp
2¼1
TðT
0
xTK1xpDp
sx
½

ds;(11c)
R¼R1þRq
1þRp
2;(11d)
R1¼1
2pð2p
0
x2_
xT_
xA0f0xT

ds;(11e)
Rq
1¼1
2pð2p
0
F1cos msds;(11f)
Rp
2¼1
TðT
0
K1xpDp
sx0
½
xT
no
ds:(11g)
Here, M1and R1are the Jacobi matrix and corrective vector
for the integer parts in Eq. (10) based on the IHB method,
and they had been deduced in some Refs. 20–24.
Considering different m, the explicit forms of different val-
ues Rq
1are
Rq
10 ¼F1;
0;
m¼0
m6¼ 0;
(12a)
Rq
1i¼
F1
2;
0;
m¼i
m6¼ i;i¼1; :::; N:
8
<
:
(12b)
Mp
2and Rp
2are the Jacobi matrix and corrective vector for the
fractional-order derivative based on the IHB method, and the
explicit forms can be solved
Mp
2¼½M11p½M12 p
½M21p½M22 p
"#
;Rp
2¼
Rp
10
Rp
1i
Rp
2i
2
6
6
43
7
7
5
;(13)
where
M11
½
p
ij¼lim
T!1
1
TðT
0
K1xpcos isDp
scos js
½
ds;
i¼0;1; :::; N;j¼0;1; :::; N;(14a)
M12
½
p
ij¼lim
T!1
1
TðT
0
K1xpcos isDp
ssin js
½
ds;
i¼0;1; :::; N;j¼1; :::; N;(14b)
M21
½
p
ij¼lim
T!1
1
TðT
0
K1xpsin isDp
scos js
½
ds;
i¼1; :::; N;j¼0;1; :::; N;(14c)
M22
½
p
ij¼lim
T!1
1
TðT
0
K1xpsin isDp
ssin js
½
ds;
i¼1; :::; N;j¼1; :::; N;(14d)
Rp
10 ¼lim
T!1 1
TðT
0
K1xpDp
sx0
½

ds;(14e)
Rp
1i¼lim
T!1 1
TðT
0
K1xpDp
sx0
½
cos is

ds;i¼1; :::; N;
(14f)
Rp
2i¼lim
T!1 1
TðT
0
K1xpDp
sx0
½
sin is

ds;i¼1; :::; N:
(14g)
According to Caputo’s definition in Eq. (2), Eq. (14a)
becomes
M11
½
p
ij ¼lim
T!1
1
TðT
0
K1xpcos isDp
scos js
½
ds
¼lim
T!1
1
TðT
0
K1xpcos is1
C1p
ðÞ
ðs
0
jsin ju
su
ðÞ
pdu

ds:
(15)
Letting s¼suand du ¼ds, Eq. (15) becomes
M11
½
p
ij ¼jK1xp
C1p
ðÞ
lim
T!1
1
TðT
0
cos isðs
0
sin jsjs
ðÞ
spds

ds
¼jK1xp
C1p
ðÞ
lim
T!1
1
TðT
0
cos issin jsðs
0
cos js
spds

ds
jK1xp
C1p
ðÞ
lim
T!1
1
TðT
0
cos iscos jsðs
0
sin js
spds

ds:
(16)
Defining the first part as A1and the second part as A2in Eq.
(16), and integrating them by parts, respectively, one could
obtain
A1¼jK1xp
C1p
ðÞ
lim
T!1
1
TðT
0
1
2sin iþj
ðÞ
ssin ij
ðÞ
s
½

ðs
0
cosjs
spdsds
¼jK1xp
2C1p
ðÞ
lim
T!1
1
T
cos iþj
ðÞ
s
iþj
ðÞ
cos ij
ðÞ
s
ij
ðÞ

ðs
0
cosjs
spds
T
0
jK1xp
2C1p
ðÞ
lim
T!1
1
TðT
0
cos iþj
ðÞ
s
iþj
ðÞ
cos ij
ðÞ
s
ij
ðÞ

cosjs
spds;
(17)
084309-3 Wen et al. Chaos 26, 084309 (2016)
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A2¼jK1xp
C1p
ðÞ
lim
T!1
1
TðT
0
1
2cos iþj
ðÞ
sþcos ij
ðÞ
s
½
ðs
0
sin js
spds

ds
¼jK1xp
2C1p
ðÞ
lim
T!1
1
T
sin iþj
ðÞ
s
iþj
ðÞ
þsin ij
ðÞ
s
ij
ðÞ

ðs
0
sin js
spds T
0jK1xp
2C1p
ðÞ
lim
T!1
1
TðT
0
sin iþj
ðÞ
s
iþj
ðÞ
þsin ij
ðÞ
s
ij
ðÞ

sin js
spds:
(18)
In the calculating procedure for equations in this paper, the value of sin 0s
0and cos 0s
0should be 1 and 0, respectively. For
simplicity, one could define the first and second parts in Eq. (17) as BA11 and BA12 and the first and second parts in Eq. (18) as
BA21 and BA22, respectively. We introduce two important formulae, which had been deduced in Refs. 27 and 28
lim
T!1 ðT
0
sin jt
ðÞ
tpdt ¼jp1C1p
ðÞ
cos pp
2

;(19a)
lim
T!1 ðT
0
cos jt
ðÞ
tpdt ¼jp1C1p
ðÞ
sin pp
2

:(19b)
Substituting Eq. (19b) into BA11 and integrating BA12, we obtain
BA11 ¼
K1xpjpsin pp
2

2lim
T!1
1
T
cos iþj
ðÞ
T
iþj
ðÞ
cos ij
ðÞ
T
ij
ðÞ

¼0;(20a)
BA12 ¼ jK1xp
2C1p
ðÞ
lim
T!1
1
TðT
0
cos iþj
ðÞ
scos js
iþj
ðÞ
spcos ij
ðÞ
scos js
ij
ðÞ
sp

ds¼0:(20b)
Substituting Eq. (19a) into BA21, and integrating BA22 , we obtain
BA21 ¼
K1xpjpcos pp
2

2lim
T!1
1
T
sin iþj
ðÞ
T
iþj
ðÞ
þsin ij
ðÞ
T
ij
ðÞ

¼
0;i6¼ j
K1xpjpcos pp
2

2;i¼j;
8
>
>
<
>
>
:
(21a)
BA22 ¼ jK1xp
2C1p
ðÞ
lim
T!1
1
TðT
0
sin iþj
ðÞ
ssin js
iþj
ðÞ
spþsin ij
ðÞ
ssin js
ij
ðÞ
sp

ds¼0:(21b)
Combining Eqs. (16)–(18),(20), and (21), one could get
M11
½
p
ij¼
0;i6¼ j
K1xpjpcos pp
2

2;i¼j:
8
>
>
<
>
>
:
(22)
Through some similar procedures, the other terms in Eq. (14) could also be deduced. The fractional-order parts of the ele-
ments in Mp
2and Rp
2are expressed by
M11
½
p
ij ¼dijK1xpip
2cos pp
2

;i¼0;1;…;N;j¼0;1;…;N;
M12
½
p
ij ¼dijK1xpip
2sin pp
2

;i¼0;1; :::; N;j¼1; :::; N;
M21
½
p
ij ¼dij K1xpip
2sin pp
2

;i¼1; :::; N;j¼0;1; :::; N;
M22
½
p
ij ¼dijK1xpip
2cos pp
2

;i¼1; :::; N;j¼1; :::; N;
Rp
10 ¼0;
Rp
1i¼K1xpai
ip
2cos pp
2

þbi
ip
2sin pp
2

;i¼1; :::; N;
Rp
2i¼K1xpai
ip
2sin pp
2

þbi
ip
2cos pp
2

;i¼1; :::; N;(23)
where dij is Kronecker’s notation. Based on the above equations, one could easily get the periodic solutions of a strongly non-
linear fractional-order Mathiue-Duffing oscillator with higher precision. That is to say, according to an initial guess of A0the
084309-4 Wen et al. Chaos 26, 084309 (2016)
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increment DAcould be solved and then the next initial value
A0would be obtained. After some iterations, the steady-state
A0could be obtained if the increment DAis smaller than the
given error estimate.
IV. NUMERICAL SIMULATIONS
A. Comparison between the approximate analytical
solution and numerical results
In strongly nonlinear case of the original system (1),
those classical perturbation-based methods for weakly non-
linear system cannot work.
16–19
In order to verify the feasi-
bility and precision of IHB method, we present the
numerical results, and the adopted numerical scheme is
1–4
Dp½xðtlÞ  hpX
l
j¼0
Cp
jxðtljÞ;(24)
where tl¼lh is the time sample points, his the sample step,
and Cp
jis the fractional binomial coefficient with the iterative
relationship as
Cp
0¼1;Cp
j¼11þp
j

Cp
j1:(25)
Here, we select h¼0.001, and the total computation time is
300 s. After omitting the temporary response in frontal 150 s,
we take the peak value of the posterior 150 s response as the
steady-state amplitude of numerical results. The comparisons
of the solutions by the IHB method and numerical solutions
are given in Figs. 1–4. The amplitude-frequency curves by
the IHB method are denoted by the solid line, and the numer-
ical solutions are shown by the circles in the following
figures.
The five-order approximately analytical solution based
on the IHB method is presented to compare with the numeri-
cal results in Figs. 1–4. Some illustrative examples are stud-
ied herein as defined by the basic system parameters c¼2,
C1¼0:5, C2¼0.5, F¼1, K1¼0:3, p¼0:5, e¼0:5, and
x10inFig.1,x1xin Fig. 2, and x12xin Fig. 3,
respectively. In Fig. 4, the parameters C1¼0.1, C2¼2,
x1x, and the other system parameters are the same as
those in Figs. 1–3. It could be concluded that the results by
the IHB method agree very well with the numerical results
and the IHB method could present satisfactory precision,
whenever the nonlinear terms are weak or strong. One could
obtain solutions with much higher precision by easily adding
the orders in the IHB method. From the observation of Figs.
1–3, it could be found that not only the primary resonance is
reflected but also the parametric resonance is also proved to
exist in this system. From the observation of Fig. 4, it could
FIG. 1. The amplitude-frequency curves with constant excitation x
1
¼0.
FIG. 2. The amplitude-frequency curves with excitation frequency x
1
¼x.
FIG. 3. The amplitude-frequency curves with excitation frequency x
1
¼2x.
FIG. 4. The amplitude-frequency curves with smaller damping coefficient.
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be found that not only the primary and parametric resonances
are reflected but also the super-harmonic and sub-harmonic
resonances are also proved to exist in numerical results.
Obviously, the existence condition for sub-harmonic reso-
nance is related with the damping coefficient, and small
damping coefficient makes the emergence of sub-harmonic
resonance more easily. Accordingly, we could use the
approximately analytical solution to investigate the primary,
parametric, and super-harmonic resonances at the same time.
Unfortunately, the sub-harmonic resonance could not be
reflected in the results by the IHB method. This is due to the
solution forms in Eq. (4), and one could obtain the solution
for sub-harmonic resonance by other revised solution forms.
This will be our next work in the future.
B. Effects of the system parameters
When the excitation frequency x
1
mxis changed, i.e.,
mis selected as different natural numbers, the amplitude-
frequency curves are shown in Figs. 5and 6. The two sets of
typical parameters are selected as c¼2, C
1
¼0.1, C
2
¼2,
K
1
¼0.3, p¼0.5, e¼0.5, F¼1 for Fig. 5and c¼2,
C
1
¼0.5, C
2
¼2, K
1
¼0.3, p¼0.5, e¼0.5, F¼1 for Fig. 6.
The dashed, solid, dashed-dotted, and dotted lines represent
m¼0, 1, 2, and 3, respectively. From the observation of
Figs. 5and 6, it could be found that the amplitude under har-
monic excitation is larger than that of constant excitation,
although the resonance frequencies for the harmonic (m¼1)
and constant excitations are almost the same. When m
becomes larger, the maximum amplitudes for primary reso-
nances under m¼1, 2, and 3 are almost unchanged. On the
contrary, the increase of mwill make the amplitudes for
the parametric and super-harmonic resonances larger. It
could also be found that the increase of mcould move the
amplitude-frequency curves leftwards and make the frequen-
cies of the primary and parametric resonances smaller. In
Fig. 5, it could be found the sub-harmonic resonances exist
in high-frequency range of amplitude-frequency curves for
x
1
2xand 3xwhich are caused by the parametric excita-
tion. Through the comparison of Fig. 5and Fig. 6, it could
be found that the larger the linear viscous damping coeffi-
cient C
1
, the smaller the super-harmonic and sub-harmonic
resonance. That is to say, linear damping in Eq. (1) can sup-
press both super-harmonic and sub-harmonic resonances
simultaneously.
When the excitation amplitude Fis changed, the
amplitude-frequency curves are shown in Fig. 7, where c¼2,
C
1
¼0.5, C
2
¼0.5, K
1
¼0.3, p¼0.5, E¼0.5, and x
1
x.
The dotted, solid, and dashed lines represent F¼1, 2, and 5,
respectively. From Fig. 7, it could be found that the larger the
excitation amplitude F, the larger the maximum primary
amplitude and the parametric excitation resonance amplitude.
That is to say, with the increase of the excitation amplitude F
in this coupling system, both the primary and parametric res-
onances will be strengthened. Moreover, the increase of exci-
tation amplitude Fwould bend the amplitude-frequency
curves more serious, so that multi-value solutions may exist
in the primary resonance.
When the nonlinear stiffness coefficient C
2
is changed,
the amplitude-frequency curves are shown in Fig. 8, where
c¼2, C
1
¼0.5, K
1
¼0.3, p¼0.5, F¼1, e¼0.5, and x
1
x.
The dashed-dotted, solid, and dotted lines represent
C2¼0.5, 2, and 5, respectively. From Fig. 8, it could be
found that the larger the coefficient C2, the smaller the maxi-
mum amplitude and the parametric excitation resonance
amplitude. Namely, with the increase of nonlinear stiffness
coefficient in this coupling system, both the primary and
parametric resonances will be reduced. It could also be
found that the increase of nonlinear stiffness C2would move
the amplitude-frequency curves rightwards and bend the
FIG. 6. The amplitude-frequency curves with different x
1
.
FIG. 7. The amplitude-frequency curves with different F.FIG. 5. The amplitude-frequency curves with different x
1
.
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amplitude-frequency curves more serious, that means the
range for multi-value solutions will be extended.
When the periodic time-varying stiffing coefficient e
is changed, the amplitude-frequency curves are shown in
Fig. 9, where c¼2, C
1
¼0.1, C
2
¼2, K
1
¼0.3, p¼0.5,
F¼1, and x
1
x. The dotted, solid, and dashed-dotted lines
represent e¼0, 0.5, and 1, respectively. From the observa-
tion of Fig. 9, it could be found that the change of ehas little
effect on the primary resonance, and the maximum value
and existing range of primary resonance are almost unvaried.
On the contrary, the periodic time-varying stiffing coefficient
eis important to the parametric resonance. The larger the
coefficient eis, the larger the amplitude of the parametric
resonance. That is to say, with the increase of the coefficient
e, the parametric resonance would be much stronger in this
coupling system. It could also be found that the increase of
the coefficient edoes not change the frequency locations of
the primary and parametric resonances.
When the fractional order pvaries from 1 to 0, the
different amplitude-frequency curves are shown in Fig. 10,
where c¼1.8, C
1
¼0.2, C
2
¼0.5, K
1
¼0.6, e¼1, F¼1, and
x
1
x. The dashed, solid, and dotted lines represent
p¼0.9, 0.5, and 0.1, respectively. From Fig. 10, it could be
found that with the decrease of fractional order p, the ampli-
tudes for primary and parametric resonances will become
distinctly larger. The reason is that the equivalent damping
coefficient of Eq. (1) will become smaller in this procedure.
Furthermore, it could also be found that the decrease of frac-
tional order pwould make the equivalent stiffness coefficient
of Eq. (1) larger, which could result into the rightwards
bending of the amplitude-frequency curves, and make the
frequencies for primary and parametric resonances larger.
When the fractional coefficient K1is changed, the
amplitude-frequency curves are shown in Fig. 11, where
c¼1.8, C
1
¼0.2, C
2
¼0.5, p¼0.5, e¼1, F¼1, and x
1
x.
The dotted, solid, and dashed lines represent K
1
¼0.3, 0.6,
and 0.9, respectively. From the observation of Fig. 11,it
could be found that the larger the fractional coefficient K
1
is,
the smaller the amplitudes of the primary and parametric res-
onances are. The reason is that the equivalent damping coef-
ficient of Eq. (1) will become larger in this procedure.
Furthermore, it could also be found that the increase of frac-
tional coefficient K
1
would make the equivalent stiffness
coefficient of Eq. (1) larger, which could result into the right-
wards moving of the amplitude-frequency curves. That
FIG. 9. The amplitude-frequency curves with different e.
FIG. 10. The amplitude-frequency curves with different p.
FIG. 11. The amplitude-frequency curves with different K
1
.
FIG. 8. The amplitude-frequency curves with different C
2
.
084309-7 Wen et al. Chaos 26, 084309 (2016)
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2016 11:11:43

means the frequencies of primary and parametric resonance
would become larger.
V. CONCLUSIONS
In this paper, the IHB method is extended to analyze the
dynamical properties of strongly nonlinear fractional-order
oscillators. The general forms of periodic solutions for those
oscillators are founded based on the IHB method, which
could be useful to obtain the solutions with higher precision.
The comparisons between the numerical results and the
approximate analytical solutions by the IHB method verify
the correctness and satisfactory precision of the IHB method.
The effects of some typical system parameters of the
strongly nonlinear fractional-order Mathieu-Duffing equation
are also investigated by the IHB method, and the results will
be useful to design and/or control these kinds of systems.
Those results could present beneficial reference to the similar
fractional-order nonlinear systems, even when strong nonli-
nearities exist in the system.
ACKNOWLEDGMENTS
The authors are grateful to the support by the National
Natural Science Foundation of China (No. 11372198), the
education department project of Hebei Province (Z995049)
and (QN2016258), the Cultivation plan for Innovation team
and leading talent in Colleges and universities of Hebei
Province (LJRC018), the Program for advanced talent in the
universities of Hebei Province (GCC2014053), and the
Program for advanced talent in Hebei Province (A201401001).
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