Yong Xu1

Department of Applied Mathematics,

Northwestern Polytechnical University,

Xi’an 710072, China

e-mail: hsux3@nwpu.edu.cn

Yongge LiDepartment of Applied Mathematics,

Northwestern Polytechnical University,

Xi’an 710072, China

e-mail: liyonge@163.com

Di LiuDepartment of Applied Mathematics,

Northwestern Polytechnical University,

Xi’an 710072, China

e-mail: di-lau@hotmail.com

Response of FractionalOscillators With ViscoelasticTerm Under Random ExcitationA system with fractional damping and a viscoelastic term subject to narrow-band noise isconsidered in this paper. Based on the revisit of the Lindstedt–Poincar�e (LP) and multi-ple scales method, we present a new procedure to obtain the second-order approximateanalytical solution, and then the frequency–amplitude response equations in the deter-ministic case and the first- and second-order steady-state moments in the stochastic caseare derived theoretically. Numerical simulation is applied to verify the effectiveness ofthe proposed method, which shows good agreement with the analytical results. Specially,we find that the new method is valid for strongly nonlinear systems. In addition, the influ-ences of fractional order and the viscoelastic parameter on the system are explored, andthe results indicate that the steady-state amplitude will increase at a fixed point with theincrease of fractional order or viscoelastic parameter. At last, stochastic jump is investi-gated via the received Fokker–Planck–Kolmogorov (FPK) equation to compute the sta-tionary solution of probability density functions with its shape changing from one peak totwo peaks with the increase of noise intensity, and the phenomena of stochastic jump isconsistent with the solution of frequency–amplitude response equations.[DOI: 10.1115/1.4026068]

Keywords: fractional-order damping, viscoelastic term, narrow-band noise, frequency–amplitude response, stochastic jump

1 Introduction

Differing from viscous objects or elastic objects, viscoelasticmedium shows sensitivity to strain rate like viscous behaviors,and displays properties of spring like elasticity. The “in-between”characters seem more suitable to model practical problems thanthe two ideal states of viscosity and elasticity in many certaincases. So viscoelastic influences are considered in various me-chanical and engineering systems to describe the system motion,such as airplanes, building structure, biomedical materials, andarched bridges [1–4]. As we know, viscoelasticity leads to a visco-elastic system including both energy dissipation due to elasticityand the memory properties that viscoelastic stress depends on inall the past state of strain as well as the current state. Generally,the energy dissipation from the damping is described by aninteger-order derivative. However, replacements of the integer-order damping by a fractional-order derivative to construct modelsappeared in many papers [5–8]. Recently, Machado pointed outthat even if individual dynamics of each element has an integernature, the global system in average will have a fractional dynam-ics in his investigation on the overall dynamics of a system consti-tuted by a large number of microscopic mechanical elements [9].So the replacement of integer damping by a fractional one is sig-nificant and it is essential to investigate fractional damping sys-tems. In the fractional order investigations, Bagley provided atheoretical basis for the uses of the fractional order derivative todescribe some viscoelastic materials [10]. Agrawal presented theapplication of the fractional order derivative in the thermal analy-sis of disk brakes [11]. Diethelm proposed several numerical algo-rithms for fractional calculus including computations of Caputoand Riemann–Liouville definitions [12,13]. Rossikhin gave us abrief view of the novel trends and recent results on applications offractional calculus for dynamic problems of solid mechanics [14].

A number of methods have been developed to study determinis-tic nonlinear fractional/viscoelastic systems. For example, De€uproposed a relatively new method called the Newmark-diffusivescheme to simulate the fractionally damped mechanical systems[15]. Shen used the averaging method to investigate the primaryresonance of the Duffing oscillator with the fractional order deriv-ative [16]. And some other traditional methods are provided, suchas the Lindstedt–Poincar�e perturbation method [17], the forwardresidue harmonic balance method [18], the Galerkin method [19],the homotopy perturbation method [20], the Adomian decomposi-tion method [21], and so on. But systems in practice will beunavoidably affected by random disturbances, then it is necessary toestablish the effective methods to analyze stochastic dynamical sys-tems with viscoelastic terms. In related investigations, Zhu used thestochastic averaging method to study a viscoelastic system withstrong nonlinear stiffness force under random excitation [22]. Liuet al. applied the stochastic multiple method to investigate theresponses of an axially moving viscoelastic beam with randomlydisordered periodic excitation [23]. R€udinger proposed a frequency-domain technique to solve a linear system with fractional derivativesubject to white noise excitation [24]. Leung investigated the sto-chastic responses of fractionally damped nonlinear viscoelasticarches by the residue harmonic homotopy method [25].

In this paper a new analytical method is provided to analyze theresponses of a stochastic system with viscoelastic term and fractionaldamping. The new technique combines the characters of both theLindstedt–Poincar�e (LP) method and the multiple scales method [26]and the LP method introduces a time shift to set up the relationshipbetween the amplitude and frequency, later it can be used to solvestrong nonlinear problems. The multiple scales method brings in sev-eral new time scales and expands the ordinary differential equationinto a series of partial differential equations, and it is very flexible todeal with systems with harmonic excitations. Therefore, the newapproach integrates the two traditional methods into a technique whichis more efficient and able to handle problems with a viscoelastic term,fractional damping, random excitation, and even strong nonlinearity.

The structure of the paper is organized as follows. The mechan-ical model will be given in Sec. 2. In Sec. 3, the new procedurebased on the LP and multiple scale method is used to obtain the

1Corresponding author.Contributed by the Design Engineering Division of ASME for publication in the

JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript receivedSeptember 5, 2013; final manuscript received November 19, 2013; published onlineFebruary 13, 2014. Assoc. Editor: J. A. Tenreiro Machado.

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second-order approximation. In Sec. 4, the amplitude–frequencyequation is discussed in the deterministic case and the first- andsecond-order steady-state moments are obtained in the stochasticcase, while numerical simulations are carried out to verify the va-lidity of the method. The influences of the viscoelastic parameterand fractional order on the amplitude are investigated. In Sec. 5,the stochastic jump and bifurcation are studied with the probabil-ity density calculated by the Fokker–Plank–Kolmogrov (FPK)equation. Finally, conclusions are listed in Sec. 6.

2 Mathematical Model

The system under consideration in this paper is shown in Fig. 1,including a spring, a viscous element, and a slider. The mathemat-ical model of the viscoelastic oscillator can easily be written as

m€xþ c _xþ lH x;b; tð Þ þ kxþ dx3 ¼ 0 (1)

where m is the system mass, and c and l are the coefficients of thedamping and viscoelastic term, respectively. b is the viscoelasticparameter, k is the natural frequency, and d is the intensity of thenonlinearity. The viscoelastic term H b; x; tð Þ takes the form of

H b; x; tð Þ ¼ðt

0

1

bexp � t� s

b

� �x sð Þds (2)

If we consider the fractional damping and add the internal excita-tion into the oscillator, another mathematical model is obtained as

m€xþ cDat xþ lH x;b; tð Þ þ kxþ dx3 ¼ fxn tð Þ (3)

where f is the amplitude of the internal excitation. Dat �ð Þ is a frac-

tional derivative operator with respect to t. In this paper we adoptthe Caputo [27] definition as

Dat x tð Þ½ � ¼ 1

C 1� að Þ

ðt

0

t� sð Þ�a _xðsÞds (4)

In the definition (4), a is the fractional order (0 < a < 1), andC �ð Þ is a Gamma function. n tð Þ is a narrow-band noise of

n tð Þ¼ cos Xtþ hW tð Þ½ � (5)

in which X is the excitation frequency and h is the intensity of theWiener process W tð Þ. The power spectrum density Sn xð Þ of n tð Þis given by Wedig [28] as

Sn xð Þ ¼ 1

2

h2 X2 þ x2 þ h4=4� �

X2 � x2 þ h4=4� �2þx2h4

(6)

Obviously when h is small, the spectrum focuses on a narrow-band area. In this paper we restrict it to narrow-band noise.

3 Approximate Solution

For convenience, we introduce the following transformations:

e2l ¼ c

m; e2j ¼ l

m; x2

0 ¼k

m; ed ¼ d

m; e2c ¼ f

m(7)

then Eq. (2) comes to

€xþ e2lDat xþ e2jHðb; x; tÞ þ x2

0xþ edx3 ¼ e2cxcosðXtþ hWðtÞÞ(8)

First, referring to the LP method, we introduce the followingstrained coordinate which sets up the relationship between the fre-quency and the amplitude, namely [26]:

s ¼ xt (9)

using the transform above in the fractional damping and the visco-elastic term, we get the following formulas:

Dat ½xðtÞ� ¼

1

Cð1� aÞ

ðs

0

sx� g

x

� ��a_xðgÞxd

gx¼ xaDa

s ½xðsÞ� (10)

H b; x; tð Þ ¼ðt

0

1

bexp � t� s

b

� �x sð Þds ¼

ðs

0

1

bexp � 1

bsx� g

x

� ��

� x gð Þd gx¼ðs

0

1

bxexp � s� g

bx

� �

� x gð Þdg ¼�b¼bx

H �b; x; s� �

(11)

Then substituting Eq. (9) into Eq. (8), one obtains

x2x00 þ e2lxaDasxþ e2jH �b; x; s

� �þ x2

0xþ edx3

¼ e2cx cosXsxþhW

sx

� �� (12)

in which the prime denotes the derivative with respect to the timevariable s.

Second, like the multiple scales method, we introduce severaltime scales as

T0 ¼ s; T1 ¼ es; T2 ¼ e2s; (13)

For convenience, we simplify the partial derivative marks as

D0 ¼ @=@T0; D1 ¼ @=@T1; D2 ¼ @=@T2 (14)

Then, the ordinary derivative can be transformed into a partial de-rivative expansion, which is the key point in multiple scalesmethod,

d

dt¼ D0 þ eD1 þ � � �

d2

dt2¼ D2

0 þ 2eD0D1 þ � � � (15)

Expanding the solution x of system (8) and the introduced fre-quency x into perturbation series, i.e.,

x ¼ x0 T0;T1;T2ð Þ þ ex1 T0; T1;T2ð Þ þ e2x2 T0;T1;T2ð Þ þ � � � (16)

x20 ¼ x2 � ex1 � e2x2 � � � � (17)

Substituting Eqs. (14)–(17) into Eq. (12), and collecting the termsaccording to the power of parameter e, then

x2D20x0 þ x2x0 ¼ 0 (18a)

x2D20x1 þ x2x1 ¼ �2x2D0D1x0 þ x1x0 � dx3

0 (18b)Fig. 1 Sketch map of a viscoelastic oscillator model

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x2D20x2 þ x2x2 ¼ �2x2D0D1x1 � x2 D2

1 þ 2D0D2

� �x0

� lxaDa0x0 � jH �b; x0; T0

� �þ x1x1 þ x2x0 � 3dx2

0x1

þ cx0 cosXsxþ�hW sð Þ

� (18c)

In Eq. (18c), referring to the properties of the standard Wienerprocess, we know

hWðtÞ ¼ hWsx

� �¼ hffiffiffiffi

xp W sð Þ ¼ �hW sð Þ (19)

In order to obtain the approximate solution, the equation seriesshould be solved in sequence. As to Eq. (18a), it is reasonable toassume its solution to be the form of x0 ¼ a cosðT0 þ uÞ. Gener-ally we always change the equation into another equivalent formas

x0 ¼ AðT1; T2ÞeiT0 þ c:c: (20)

where the c:c: denotes the complex conjugate of the precedingformula. Then, it is obvious that A ¼ 1

2aeiu.

Substitute Eq. (20) into the right-hand side of Eq. (18b). Inorder to avoid the solution divergence, secular terms should beeliminated. So, we obtain

� 2ix2D1A� 3dA2 �Aþ x1A ¼ 0 (21)

where �A represents the conjugate of A.There are three parts in Eq. (21), in fact they are produced sepa-

rately from the LP method and multiple method. The first twoterms come from the derivative expansion of the multiple scalesmethod, the last term comes from the frequency introduction andexpansion of the LP method. Our new method combines the char-acters of both methods. So we can handle the equation with twoselections. First, we assume D1A ¼ 0 to calculate x1, if x1 comesto be a real number that indicates the selection is reasonable, andwe adopt the choice. However, if x1 is complex, it says the selec-tion is not proper in physics [29]. Then we turn to another selec-tion that assumes x1 ¼ 0 to calculate A. Based on the previousstatement, we select D1A ¼ 0, then we get

x1 ¼ 3dA �A (22)

which is a real number. Furthermore, the solution form of Eq.(18b) is derived as

x1 ¼d

8x2A3e3iT0 þ c:c: (23)

In order to investigate the primary resonance of the original sys-tem, we expand the frequency X as

X ¼ x 2þ e2r� �

(24)

Substituting Eqs. (22)–(24) into Eq. (18c), we need to computeDa

0 x0ð Þ and H �b; x0;T0

� �. For the fractional derivative, according

to Rossikhin [30],

Da0 eiT0� �

¼ iaeiT0 þ isin ap

p

ð10

va�1

vþ ie�vT0 dv (25)

Eq. (25) contains two parts; however, if the steady state is con-cerned, we are able to evaluate it with the contour integration, andit can be approximately expressed by the first term, namelyDa

0 eiT0ð Þ � iaeiT0 . As to the viscoelastic term

H �b;x0;T0

� �¼ 1

�b

ðT0

0

e� 1

�bT0� sð Þ

� �x0 sð Þds¼A

�be�T0

�b

ðT0

0

e1�bþ i

� �s

�dsþ c:c:

¼Ae�T0

~b1� �bi

1þ �b2e

1�bþ i

� �T0 �1

!þ c:c:

¼ 1� �bi

1þ �b2x0�

1� �bi

1þ �b2Ae�T0

�b þc:c: (26)

Employing Eqs. (25) and (26), and eliminating the secular terms,we arrive at

�2ix2D2A� ialxaA� 1� �bi

1þ �b2jAþ x2A� 3d2

8x2A3 �A2

þ 1

2c �Aei rT2þ~hW T2ð Þ½ � ¼ 0 (27)

where �hWðT0Þ ¼ �hWðT2=e2Þ ¼ ð�h=eÞW T2ð Þ ¼ ~hW T2ð Þ. To solveEq. (27) we repeat the rule handling Eq. (21). In the case of Eq.(27), we have to choose x2 ¼ 0, because D2A ¼ 0 will lead x2 tobe a complex number. Then Eq. (27) comes to

� ix2 D2aþ iaD2uð Þ � 1

2alxa cos

ap2þ i sin

ap2

� �

� 1

2ja

1� �bi

1þ �b2� 3d2

256x2a5 þ 1

4caeiw ¼ 0 (28)

where w ¼ rT2 � 2uþ ~hW T2ð Þ, and a basic formula is used,

ia ¼ eip=2� �a

¼ eiap=2 ¼ cosap2þ i sin

ap2

(29)

Here we know the introduced frequency x approximately equals

x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

0 þ3

4eda2

r(30)

Separating Eq. (28) by the real and imaginary parts yields

D2a ¼ � l2

xa�2 sinap2

aþ j2x2

�b

1þ �b2aþ c

4x2a sin w

aD2w ¼ aðrþ �hW0Þ � lxa�2 cosap2

a� jx2

1

1þ �b2a

� 3d2

128x4a5 þ c

2x2a cos w

8>>>>>>>>><>>>>>>>>>:

(31)

By computing Eq. (31) we can get the amplitude a and thephase u. So combining Eqs. (20), (23), (30), and (31) we obtainthe second-order approximate solution as

x ¼ a cos T0 þ uð Þ þ ed

32x2a3 cos 3 T0 þ uð Þ þ O e2

� �(32)

4 The Amplitude–Frequency Response

As the next step, the amplitude–frequency responses are to beinvestigated. It can be seen from Eq. (31) that the trivial steady-state solution and the nontrivial steady-state solution coexist. Inthis paper we focus our attention on the nontrivial steady-state so-lution. The frequency–amplitude equation in the deterministiccase is derived and the first- and second-order steady-statemoments in the stochastic case are provided, respectively.

When h ¼ 0, the excitation becomes a harmonic excitation,then Eq. (31) reduces to

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�lxa�2 sinap2þ j

x2

�b

1þ �b2þ c

2x2sin w0 ¼ 0

r� lxa�2 cosap2� j

x2

1

1þ �b2� 3d2

128x4a4

0 þc

2x2cos w0 ¼ 0

8>>>><>>>>:

(33)

Eliminating w we obtain the frequency–amplitude equation

X ¼ x 2þ e2 lxa�2 cosap2þ j

x2

1

1þ �b2þ 3d2

128x4a4

0

��

6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

4x4� lxa�2 sin

ap2� j

x2

�b

1þ �b2

� �2s #)

(34)

When h 6¼ 0 we consider the influence of the random noise onthe deterministic steady-state solution. So it is reasonable toassume

a ¼ a0 þ a1; w ¼ w0 þ w1 (35)

where a1 and w1 are small disturbances based on the deterministicsituations.

Substituting it into Eq. (31), and employing Eq. (33), the Itostochastic differential equations are obtained as

da1 ¼ �l2

xa�2 sinap2

a1 þj

2x2

�b

1þ �b2a1

�

þ c4x2

sin w0a1 þc

4x2a0 cos w0w1ÞdT2

dw1 ¼ � 3d2a30

32x4a1 �

c2x2

sin w0w1

� �dT2 þ �hdW T2ð Þ

8>>>>>>>>><>>>>>>>>>:

(36)

Considering the steady-state case, with the Ito derivative princi-ple, we know

dE a1ð Þ=dT2 ¼ dE w1ð Þ=dT2 ¼ dE a21

� �=dT2 ¼ dE w2

1

� �=dT2

¼ dE a1w1ð Þ=dT2 ¼ 0

The first- and second-order steady-state moments of a1 are derived as

E a1ð Þ ¼ 0; E a21

� �¼

~h2F22

F1F2G1 þ F1G22 þ F2

1G2 þ G1G2F2

(37)

where

F1 ¼ lxa�2 sinap2� j

x2

�b

1þ �b2� c

2x2sinw0; F2 ¼

c2x2

a0 cosw0

G1 ¼3d2a3

0

16x4; G2 ¼

cx2

sinw0

Combining Eqs. (33) and (37), we get the first- and second-ordersteady-state moments of Eq. (31)

E að Þ ¼ a0 þ Ea1 ¼ a0

E a2ð Þ ¼ a20 þ Ea2

1 ¼ a20 þ

~h2F22

F1F2G1 þ F1G22 þ F2

1G2 þ G1G2F2

8><>:

(38)

In order to verify the validity of the method used in the paper,numerical simulations are conducted. For the fractional dampingterm, we use the algorithm of Diethelm [13]. For the viscoelasticterm we can transform it into the derivative form and expandEq. (3) as follows:

_x ¼ y

_y ¼ � c

mDa

t x� c

lz� k

mx� d

mx3 þ f

mxn tð Þ

_z ¼ 1

bx� zð Þ

8>>>>>><>>>>>>:

(39)

From Figs. 2–4 the numerical results and the analytical resultsare in good agreement, which proves the effectiveness of the pro-posed method in this paper. The solid and dashed lines mean thestable and unstable solution, respectively, which is distinguishedby the Routh–Hurwitz criterion. We see in Figs. 2–4 that as theexcitation frequency X increases the amplitude goes through fromone stable solution state to one stable and one unstable solutions

Fig. 2 Amplitude frequency responses: (a) mean moments and (b) second-order moments. e 5 0:1; l 5 5:0; a 5 0:8;j 5 3:0;x0 5 0:8; d 5 0:1; c 5 20:0;h 5 0:05: Analytical results: ——, numerical results: m��.

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state, and then comes to two stable and one unstable solutionsstate when X reaches the critical point, which can be defined asthe bifurcation point. The effects of the viscoelastic parameter band the fractional order a are also presented. In Fig. 2 we find thatas the viscoelastic parameter increases, the amplitude at the samepoint will rise, and the bifurcation points will become smaller stepby step. In Fig. 3, with the increasing of fractional order, the am-plitude tends to increase while the bifurcation points willdecrease.

Specially, in Fig. 4 we find that the method is able to deal withstrongly nonlinear problems. In fact, the LP method is alwaysused directly or revised to handle strongly nonlinear systems andour new technique combines the characters of both the LP methodand the multiple scales method. The modified LP method keepsthe advantage of the LP method and does better, moreover, itdraws the flexibility of the multiple scales method.

For the perturbation method, the correction terms must bemuch smaller than the leading term, or it will lose its validity. Soit is possible to take the ratio between the coefficients of the

second term and the first leading term as a measure to evaluate themethod. With this idea, we get the ratio as

r dð Þ ¼ ed

32x2a2 (40)

limd!1

ed

32x2a2 ¼ lim

d!1e

da2

32 x20 þ 0:75eda2

� � ¼ 1

24(41)

It is obvious that the function r dð Þ is an increasing function withrespect to d. As the nonlinearity increases, the ratio r tends to1=24 slowly. That is to say, the second correction term is alwayssmall relative to the leading term even if the nonlinearity is verystrong. So we can come to a conclusion that the method is able tosolve a strongly nonlinear system no matter how strong it is.

5 The Stochastic Jump and Bifurcation

According to Eq. (31), the corresponding Ito stochastic differ-ential equation reduces to

Fig. 3 Amplitude frequency responses: (a) mean moments and (b) second-order moments. e 5 0:1; l 5 5:0; b 5 0:3; j 5 3:0;x0 5 0:8; d 5 0:3; c 5 10:0; h 5 0:05: Analytical results: ——, numerical results: m�?�.

Fig. 4 Amplitude frequency responses: (a) mean moments and (b) second-order moments. e 5 0:1; l 5 5:0; a 5 0:5; b 5 0:3;j 5 3:0; x0 5 0:8; c 5 10:0;h 5 0:05: Analytical results: ——, numerical results: m��.

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da¼ �l2xa�2 sin

ap2

aþ j2x2

�b

1þ �b2aþ c

4x2asinw

� �dT2

dw¼ r�lxa�2 cosap2� j

x2

1

1þ �b2� 3d2

128x4a4

�

þ c2x2

acosw

�dT2þ �hdW

8>>>>>>>><>>>>>>>>:

(42)

The FPK equation associated with Eq. (42) can be written as

@p

@T2

¼ � @ m1pð Þ@a

� @ m2pð Þ@w

þ�h2

2

@2p

@w2(43)

where p ¼ p a;wð Þ denotes the joint probability density of ampli-tude a and phase w, and

m1 ¼ �l2

xa�2 sinap2

aþ j2x2

�b

1þ �b2aþ c

4x2a sin w

m2 ¼ r� lxa�2 cosap2� j

x2

1

1þ �b2� 3d2

128x4a4 þ c

2x2a cos w

(44)

The boundary conditions of a satisfy

pða;wÞ ¼ finite; a ¼ 0

pða;wÞ; @pða;wÞ=@T2 ! 0; a! 0(45)

From Eq. (44) we know p is the periodic function of phase w. Sothe boundary condition of w is

pða;wþ 2pÞ ¼ pða;wÞ (46)

The FPK equation will be solved numerically by using the finitedifference method and the successive over-relaxation method.Figure 5 shows that with the increase of the intensity of the noise,the probability density figures transform from one peak to twopeaks, which means the system shocks around one stable point,but when the density is big enough the system begins to crossback and forth between two stable points. This is the so-calledphenomenon of stochastic jump, which is also defined as astochastic bifurcation. The time history of Fig. 6 gives us anintuitive visual of the stochastic jump, and we can see withincreasing intensity that the stochastic jump happens morefrequently.

Fig. 5 Stationary probability density figures: e 5 0:1; l 5 5:0; a 5 0:8;b 5 0:9;j 5 3:0;x0 5 0:8; d 5 0:1; c 5 10:0;X 5 1:75: (a) h 5 0:07,(b) h 5 0:12, (c) h 5 0:16, and (d) h 5 0:20.

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6 Conclusions

In this paper we integrate the LP method with the multiplescales method to put forward a new efficient technique to explorethe stochastic oscillator with fractional damping and viscoelasticterm and even the strongly nonlinear terms. With the newapproach, a second-order approximate solution is derived, andthen the first- and second-order steady-state moments and numeri-cal simulation are applied to show good agreement. The numericalfigures indicate that the method can well handle strongly nonlin-ear problems, and we prove the findings theoretically. Additionalwork shows that with the increasing of the fractional order orviscoelastic parameter the amplitude at fixed point will increasewhile the bifurcation points will decrease. Finally, the FPK equa-tion is solved numerically, and the results demonstrate that as theintensity of the noise increases the peaks of the probability densityfigures transform from one to two, which means the stochasticjump happens. What is more, the phenomenon can be viewed as astochastic bifurcation.

Acknowledgment

This work was supported by the NSF of China (Grant Nos.11372247, No. 11102157) and Shaanxi province, Program forNCET, the Shaanxi Project for Young New Star in Science &Technology, SRF for ROCS, SEM, NPU Foundation for Funda-mental Research, and Graduate Starting Seed Fund.

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Fig. 6 Time history: e 5 0:1; l 5 5:0; a 5 0:2; b 5 0:3; j 5 3:0; x0 5 0:8; d 5 0:3; c 5 10:0; X 5 1:75: (a) h ¼ 0:2 and (b) h ¼ 0:3.

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