December 24, 2018 14:49 WSPC/S0218-1274 1850170

International Journal of Bifurcation and Chaos, Vol. 28, No. 14 (2018) 1850170 (10 pages)

c

World Scientiﬁc Publishing Company

DOI: 10.1142/S0218127418501705

Bifurcation Analysis of a Vibro-Impact Viscoelastic

Oscillator with Fractional Derivative Element

Yong-Ge Yang∗and Wei Xu†

Department of Applied Mathematics,

Northwestern Polytechnical University,

Xi’an 710129, P. R. China

∗

yonggeyang@163.com

†

weixu@nwpu.edu.cn

YangQuan Chen‡

Mechatronics, Embedded Systems and Automation Lab,

School of Engineering, University of California,

Merced, CA 95343, USA

ychen53@ucmerced.edu

Bingchang Zhou

Department of Applied Mathematics,

Northwestern Polytechnical University,

Xi’an 710129, P. R. China

bczhou98@126.com

Received November 2, 2017; Revised June 30, 2018

To the best of authors’ knowledge, little work has been focused on the noisy vibro-impact

systems with fractional derivative element. In this paper, stochastic bifurcation of a vibro-impact

oscillator with fractional derivative element and a viscoelastic term under Gaussian white noise

excitation is investigated. First, the viscoelastic force is approximately replaced by damping

force and stiﬀness force. Thus the original oscillator is converted to an equivalent oscillator

without a viscoelastic term. Second, the nonsmooth transformation is introduced to remove the

discontinuity of the vibro-impact oscillator. Third, the stochastic averaging method is utilized to

obtain analytical solutions of which the eﬀectiveness can be veriﬁed by numerical solutions. We

also ﬁnd that the viscoelastic parameters, fractional coeﬃcient and fractional derivative order

can induce stochastic bifurcation.

Keywords: Stochastic bifurcation; vibro-impact; viscoelastic oscillator; fractional derivative

element.

1. Introduction

Fractional calculus developing in parallel with the

classical calculus has received considerable atten-

tion in recent decades. Many authors have focused

on the dynamical response of fractional systems.

Huang extended the stochastic averaging method

to discuss the stochastic response and stability of

fractional nonlinear systems subject to Gaussian

white noise excitations [Huang & Jin,2009]. Chen

put forward an innovative bifurcation control

method based upon the fractional-order feedback

controller to control the stochastic jump bifurcation

‡Author for correspondence

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Y.-G. Yang et al.

of noisy Duﬃng oscillator. Deng and Zhu devel-

oped a stochastic averaging method for quasi-

Hamiltonian systems under fractional Gaussian

noise [Deng & Zhu,2016]. Malara and Spanos

addressed the problem of determining the nonlin-

ear response of a fractional plate driven by ran-

dom loads [Malara & Spanos,2018]. Colinas-Armijo

and Di Paola used two methods to evaluate the

response of a linear viscoelastic material modeled

by the fractional Maxwell model and subject to a

Gaussian stochastic temperature process [Colinas-

Armijo et al.,2018]. Kougioumtzoglou proposed

a new multiple-input/single-output system identi-

ﬁcation approach for parameter identiﬁcation of

fractional oscillators subject to incomplete nonsta-

tionary data [Kougioumtzoglou et al.,2017]. Ma

and Li established a local fractional center man-

ifold for a ﬁnite-dimensional fractional ordinary

diﬀerential system [Ma & Li,2016]. In addition,

comprehensive review papers have been completed

by Rossikhin [Rossikhin & Shitikova,1997,2010],

Machado [Machado et al.,2011] and Chen [Chen

et al.,2009; Li et al.,2017], respectively.

Vibro-impact systems, as a class of discontin-

uous and strongly nonlinear systems, can exhibit

complicated dynamical behaviors [Luo et al.,2006;

Di Bernardo et al.,2003]. Zhu discussed the stochas-

tic response of a vibro-impact Duﬃng system under

external Poisson impulses [Rossikhin,2015]. Xu

explored the stochastic response of an inelastic

vibro-impact system under Gaussian white noise

with the help of equivalent nonlinearization tech-

nique [Xu et al.,2014]. Iourtchenko and Song con-

sidered the stochastic vibro-impact systems with

one or two rigid barrier(s) by numerical simula-

tion [Iourtchenko & Song,2006]. Kumar investi-

gated the stochastic bifurcations of a Duﬃng–van

der Pol oscillator under random excitations [Kumar

et al.,2016a, 2016b, 2017]. Nguyen developed a

mathematical model of vibro-impact mobile sys-

tem to predict the progression rate of the system

[Nguyen et al.,2017]. Feng studied the chaotic sad-

dles of a nonlinear vibro-impact system using the

bisection procedure and an improved stagger-and-

step method [Feng et al.,2009]. Two nice overviews

of vibro-impact dynamics have been presented by

Namachchivaya [Namachchivaya & Park,2005] and

Dimentberg [Dimentberg & Iourtchenko,2004].

To the authors’ knowledge, little work was

focused on the dynamical systems with fractional

derivative elements. In this paper, we carry out a

bifurcation analysis of a vibro-impact viscoelastic

oscillator with fractional derivative element under

Gaussian white noise excitation.

2. Model and Its Simpliﬁcation

We consider the vibro-impact viscoelastic system

with fractional derivative element under Gaussian

white noise excitation in the following form

¨x+Z+εβ1Dαx+εβ2f(x, ˙x)˙x+ω2

0x

=ε1/2ξ(t),x>0,

˙x+=−r˙x−,x=0.

(1)

The variables ε,β1,β2and ω0are system

parameters; 0 <r≤1 is the coeﬃcient of restitu-

tion factor; ˙x−and ˙x+are the velocities just before

and after the impact, respectively. ξ(t) is Gaussian

white noise with zero mean and auto-correlation

E[ξ(t)ξ(t+τ)] = 2Dδ(τ). There are many deﬁni-

tions for fractional derivatives [Li & Zeng,2015;

Ma & Li,2017, 2018], in this paper, Dαxis deﬁned

as follows

Dαx=1

Γ(1 −α)

d

dt t

0

x(t−u)

uλ1du, (0 <α≤1).

(2)

The following viscoelastic model of Zcan be

expressed as:

Z=t

0

h(t−s)x(s)ds,

where h(t) is the relaxation function which has the

following form

h(t)=

n

i=1

βiexp−t

λi

=βexp−t

λ.

Without loss of generality, let i=1.βiand λiare

the general elastic modulus and the relaxation time,

respectively. Then

Z=t

0

βexp−t−s

λx(s)ds.

According to [Zhu & Cai,2011; Ling et al.,

2011], the viscoelastic force Zcan be replaced by

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Vibro-Impact Viscoelastic Oscil lator with Fractional Derivative Element

the conservative force and the damping force:

Z=t

0

βexp−t−s

λx(s)ds

=λβ

1+λ2ω2(x−λ˙x)

=κ1x−κ2˙x. (3)

Substituting Eq. (3) into the original system (1),

yields

¨x+εβ1Dαx+[εβ2f(x, ˙x)−κ2]˙x+ω2

1x

=ε1/2ξ(t),x>0,

˙x+=−r˙x−,x=0,

(4)

where

ω2

1=ω2

0+κ1.

The following transformation [Dimentberg &

Iourtchenko,2004; Zhuravlev,1976; Feng et al.,

2008] is introduced to remove the discontinuity in

Eq. (4)

x=x1=|y|,

˙x=x2=˙ysgn(y),

¨x=¨ysgn(y).

(5)

Substituting Eq. (5) into Eq. (4) leads to the fol-

lowing equations:

¨ysgn(y)+εβ1Dα(|y|)

+[εβ2f(|y|,˙ysgn(y)) −κ2]˙ysgn(y)+ω2

1|y|

=ε1/2ξ(t),t=t∗,(6a)

˙y+=r˙y−,t=t∗,(6b)

in which y(t∗)=0.

After multiplying Eq. (6a) by sgn(y), we get the

following formulas:

¨y+εβ1sgn(y)Dα(|y|)

+[εβ2f(|y|,˙ysgn(y)) −κ2]˙y+ω2

1y

=ε1/2ξ(t)sgn(y),t=t∗,(7a)

˙y+=r˙y−,t=t∗.(7b)

Then, according to [Feng et al.,2009] we can

obtain the equivalent equation without impact term

of the original vibro-impact oscillator (1):

¨y+εβ1sgn(y)Dα(|y|)

+[εβ2f(|y|,˙ysgn(y)) −κ2]˙y

+(1−r)˙y|˙y|δ(y)+ω2

1y

=ε1/2ξ(t)sgn(y).(8)

3. Stochastic Averaging Procedure

Introduce the following transformation [Huang &

Jin,2009]

y(t)=A(t)cosΨ(t),

˙y(t)=−A(t)ω1sin Ψ(t),

Ψ(t)=ω1t+Φ,

(9)

where A, Ψ, Φ are random processes. The equations

for the variables Aand Φ are

dA

dt =εF11(A, Φ) + εF12 (A, Φ)

+ε1/2G11(A, Φ)ξ(t),(10)

dΦ

dt =εF21(A, Φ) + εF22 (A, Φ)

+ε1/2G21(A, Φ)ξ(t),(11)

where

εF11 =εβ1sin Ψ

ω1

sgn(Acos Ψ)Dα(|Acos Ψ|),

εF12 =−Asin2Ψ[εβ2f(|Acos Ψ|,−Aω1sin Ψ)

−κ2+(1−r)|−Aω1sin Ψ|δ(Acos Ψ)],

εF21 =εβ1cos Ψ

Aω1

Dα(Acos Ψ),

εF22 =−sin Ψ cos Ψ[εβ2f(|Acos Ψ|,−Aω1sin Ψ)

−κ2+(1−r)|−Aω1sin Ψ|δ(Acos Ψ)],

G1=−sin Ψ

ω1

sgn(Acos Ψ),

G2=−cos Ψ

Aω1

sgn(Acos Ψ).

The averaged Itˆo equation for A(t)isoftheform

dA =m(A)dt +σ(A)dB(t),(12)

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Y.-G. Yang et al.

where

m(A)=εF11 +F12 +D∂G11

∂A G11 +D∂G11

∂ΦG21

Ψ

,

(13)

σ2(A)=ε2DG2

11Ψ.(14)

To obtain the explicit expression, the key step

is to simplify the ﬁrst term of Eq. (13). According

to the deﬁnition Eq. (2),

εF11Ψ=εβ1A

ω1Γ(1 −α)sgn(Acos Ψ) sin Ψ

×d

dt t

0

|cos(Ψ −ω1u)|

uαduΨ

.(15)

Then the cosine function in Eq. (15) can be

replaced by the following Fourier series

|cos θ|=2

π+

∞

n=1

Bncos(2nθ),(16)

where

Bn=4

π

(−1)n

1−4n2.

According to Eqs. (15) and (16) and [Yurchenko

et al.,2017; Yang et al.,2018]

εF11Ψ=−32εβ1A

π2ω1

sin απ

2

∞

n=1

n2(2nω1)α−1

(1 −4n2)2

≈−

32εβ1A

π2ω1

sin απ

2

15

n=1

n2(2nω1)α−1

(1 −4n2)2.

The Fokker–Planck–Kolmogorov (FPK) equa-

tion corresponds to Eq. (12) as given by

∂p

∂t =−∂

∂A[m(A)p]+ 1

2

∂2

∂A2[σ2(A)p].(17)

The boundary conditions for Eq. (17) are

p=c, c ∈(−∞,+∞)asA=0,

p→0,∂p

∂A →0,A→∞.

With the help of the aforementioned boundary

conditions, the stationary solution of Eq. (17) is

expected to be

p(A)= C

σ2(A)expA

0

2m(s)

σ2(s)ds,(18)

in which Cis the normalization constant. The joint

stationary PDF of the original displacement and

velo city p(x1,x

2) and corresponding marginal sta-

tionary PDFs p(x1)andp(x2) can be obtained

according to Eq. (18) and [Huang & Jin,2009; Yang

et al.,2015; Yang et al.,2017].

4. Example

To assess the accuracy of the proposed method, the

following oscillator is considered.

¨x+a1Dαx+Z+(−b1+b2x2+b3˙x2)˙x+ω2

0x

=ξ(t),x>0,

˙x+=−r˙x−,x=0,

(19)

where a1,b1,b2,b3and ω0are constants, ξ(t)is

Gaussian white noise with intensity 2D. The cor-

responding equivalent stochastic oscillator without

impact term is of the following form based on what

has been discussed in Sec. 2:

¨y+a1sgn(y)Dα(|y|)+[−b4+b2y2+b3˙y2

+(1−r)|˙y|δ(y)] ˙y+ω2

1y

=sgn(y)ξ(t),(20)

where

b4=b1+κ2,

ω2

1=λ2ω2

0−1+(1 + λ2ω2

0)2+4βλ3

2λ2.

The averaged drift and diﬀusion coeﬃcients

are

m(A)=−1

8b2A3−3

8b3ω2

1A3+b5A+D

2Aω2

1

,

(21)

σ2(A)= D

ω2

1

,(22)

in which

b5=b4

2−(1 −r)ω1

π

−32a1

π2ω1

sin απ

2

15

n=1

n2(2nω1)α−1

(1 −4n2)2.

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Vibro-Impact Viscoelastic Oscil lator with Fractional Derivative Element

So, the stationary solution of oscillator (20) is

of the form

p(A)=Cω2

1A

Dexpb5ω2

1

DA2

−b2ω2

1

16D+3b3ω4

1

16DA4,

p(x1,x

2)=Cω1

πD expb5ω2

1

Dx2

1+x2

2

ω2

1

−b2ω2

1

16D+3b3ω4

1

16Dx2

1+x2

2

ω2

12.

4.1. Eﬀectiveness of the proposed

approach

In this section, diﬀerent levels of control parameters

b2,b3and noise intensity Dare considered to verify

the reliability and accuracy of the proposed tech-

nique. System parameters are ﬁxed to be b1=0.09,

a1=−0.01, α=0.5, λ=1.0, β=−0.01, ω0=1.0,

r=0.95.The blue solid lines are theoretical pre-

dictions while discrete dots are numerical results in

the following ﬁgures, respectively.

First, noise intensity D=0.16 is ﬁxed. Dif-

ferent levels of control parameters b2and b3are

considered then. Figure 1 gives the theoretical and

numerical results of the probability density func-

tions of amplitude, displacement and velocity for

diﬀerent control parameter b2,Fig.2givesthe

results for diﬀerent parameter b3. It can be seen

that the theoretical results agree well with those

from numerical results. It is also shown that the

eﬀects of these two parameters b2and b3on the

system response are the same.

Then, to scrutinize the eﬀect of the noise inten-

sity D, here, control parameters b2=0.09 and

b3=0.09 are ﬁxed. Figure 3 gives the theoreti-

cal and numerical results of the probability density

functions of amplitude, displacement and velocity

for diﬀerent noise intensity D. It can also be found

that the theoretical results and numerical results

are coincident. So, the reliability and accuracy of

the proposed technique are veriﬁed by Figs. 1–3.

4.2. Stochastic bifurcation analysis

In this paper, stochastic bifurcation refers to

stochastic P-bifurcation which occurs when the

shape of the stationary probability density function

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

A

p(A)

b2=0.36

b2=0.09

(a)

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

x1

p(x1)

b2=0.36

b2=0.09

(b)

−5 0 5

0

0.1

0.2

0.3

0.4

0.5

x2

p(x2)

b2=0.36

b2=0.09

(c)

Fig. 1. Probability density functions of amplitude, displace-

ment and velocity for diﬀerent control parameter b2.

changes from unimodal to bimodal. This section

focuses on the analysis of stochastic bifurcation phe-

nomenon induced by viscoelastic parameters, frac-

tional coeﬃcient and fractional derivative order.

First, we discuss the stochastic bifurcation

induced by viscoelastic parameter λ. The system

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Y.-G. Yang et al.

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

A

p(A)

b3=0.07

b3=0.30

(a)

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

x1

p(x1)

b3=0.07

b3=0.30

(b)

−5 0 5

0

0.1

0.2

0.3

0.4

0.5

x2

p(x2)

b3=0.30

b3=0.07

(c)

Fig. 2. Probability density functions of amplitude, displace-

ment and velocity for diﬀerent control parameter b3.

parameters are taken to be b1=0.032,b

2=0.004,

b3=0.004, a1=−0.01, α=0.5, β=−0.03,

ω0=1.0, D=0.01, r=0.955.Figure 4 gives

the joint stationary probability density p(x1,x

2)of

displacement and velocity for diﬀerent viscoelastic

parameter λ. In order to better understand the

stochastic bifurcation, the corresponding section

graphs are presented in Fig. 5. An inspection of

these two ﬁgures clearly indicates that at λ=

0.9, the joint stationary probability density has

one peak and the corresponding section graph is

unimodal. At λ=0.1, the shape of the joint

stationary probability density changes to crater-like

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

A

p(A)

D=0.16

D=0.45

(a)

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

x1

p(x1)

D=0.45

D=0.16

(b)

−5 0 5

0

0.1

0.2

0.3

0.4

x2

p(x2)

D=0.16

D=0.45

(c)

Fig. 3. Probability density functions of amplitude, displace-

ment and velocity for diﬀerent control parameter D.

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Vibro-Impact Viscoelastic Oscil lator with Fractional Derivative Element

(a)

(b)

Fig. 4. The joint stationary probability density p(x1,x

2)of

displacement and velocity for diﬀerent λ.(a)λ=0.9and

(b) λ=0.1.

structure and the corresponding section graph is

bimodal. This phenomenon indicates that stochas-

tic P-bifurcation takes place as λdecreases from

0.90 to 0.10.

Second, we explore the stochastic bifurcation

induced by viscoelastic parameter β. The system

parameters are taken to be b1=0.032, b2=0.004,

b3=0.004, a1=−0.01, α=0.5, λ=1.0, ω0=1.0,

−6 −4 −2 0 2 4 6

0

0.05

0.1

0.15

0.2

0.25

0.3

x2

p(x2)

λ=0.90

λ=0.50

λ=0.10

λ=0.66

Fig. 5. Section graphs of stationary probability density

p(x1,x

2)whenx1=0.3fordiﬀerentλ.

(a)

(b)

Fig. 6. The joint stationary probability density of displace-

ment and velocity for diﬀerent β.(a)β=−0.03 and (b)

β=−0.001.

D=0.01, r=0.955. Figure 6 gives the joint sta-

tionary probability density p(x1,x

2)fordiﬀerent

viscoelastic parameter β. Figure 7 shows the corre-

sponding section graphs. Based on the same analy-

sis, stochastic bifurcation takes place as viscoelastic

parameter βincreases from −0.03 to −0.001.

Third, we explore the stochastic bifurcation

induced by fractional coeﬃcient a1. The system

−6 −4 −2 0 2 4 6

0

0.05

0.1

0.15

0.2

0.25

0.3

x2

p(x2)

β=−0.01

β=−0.02

β=−0.03

β=−0.001

Fig. 7. Section graphs of stationary probability density

p(x1,x

2)whenx1=0.3fordiﬀerentβ.

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Y.-G. Yang et al.

(a)

(b)

Fig. 8. The joint stationary probability density of displace-

ment and velocity for diﬀerent fractional coeﬃcient a1.(a)

a1=−0.001 and (b) a1=−0.03.

parameters are taken to be b1=0.032, b2=0.004,

b3=0.004, α=0.3, λ=1.0, β=−0.01, ω0=1.0,

D=0.01, r=0.952. Figure 8 gives the joint sta-

tionary probability density p(x1,x

2)fordiﬀerent

fractional coeﬃcient a1. Figure 9 shows the corre-

sponding section graphs. Based on the same analy-

sis, stochastic bifurcation takes place as fractional

coeﬃcient a1decreases from −0.001 to −0.03.

−6 −4 −2 0 2 4 6

0

0.05

0.1

0.15

0.2

0.25

0.3

x2

p(x2)

a1=−0.001

a1=−0.03

a1=−0.01

a1=−0.02

Fig. 9. Section graphs of stationary probability density

p(x1,x

2)whenx1=0.3fordiﬀerenta1.

(a)

(b)

Fig. 10. The joint stationary probability density of displace-

ment and velocity for diﬀerent fractional order α.(a)α=0.1

and (b) α=0.9.

In the end, we explore the stochastic bifurca-

tion induced by fractional derivative order α.The

system parameters are taken to be b1=0.032, b2=

0.004, b3=0.004, a1=−0.01, λ=1.0, β=−0.01,

ω0=1.0, D=0.01, r=0.952.Figure 10 gives

the joint stationary probability density p(x1,x

2)for

diﬀerent fractional derivative order α.Figure11

shows the corresponding section graphs. Based on

−6 −4 −2 0 2 4 6

0

0.05

0.1

0.15

0.2

0.25

0.3

x2

p(x2)

α=0.9

α=0.1

α=0.3

α=0.5

Fig. 11. Section graphs of stationary probability density

p(x1,x

2)whenx1=0.3 for diﬀerent fractional order α.

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Vibro-Impact Viscoelastic Oscil lator with Fractional Derivative Element

the same analysis, stochastic bifurcation takes place

as fractional derivative order αincreases from 0.1

to 0.9.

5. Conclusions

We discussed the stochastic bifurcation of a vibro-

impact oscillator with fractional derivative element

and a viscoelastic term under Gaussian white noise

excitation. The original oscillator is converted to

an equivalent oscillator without a viscoelastic term.

The stochastic averaging method and the nons-

mooth transformation are utilized to obtain the

analytical solutions the eﬀectiveness of which can

be veriﬁed by numerical solutions. We also discussed

the stochastic bifurcation phenomenon induced by

the viscoelastic parameters, fractional coeﬃcient

and fractional derivative order.

Acknowledgments

This work is supported by the National Natural Sci-

ence Foundation of China (NSFC) under the Grant

Nos. 11472212, 11532011 and by Shaanxi Natural

Science Foundation of China under the Grant No.

2017JM1038. The ﬁrst author (Y.-G. Yang) would

like to thank the China Scholarship Council (No.

201606290189) and the hospitality of University of

California, Merced, during his visit from August

2016 to August 2018.

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