# Stochastic Bifurcations of a Fractional-Order Vibro-Impact System Driven by Additive and Multiplicative Gaussian White Noises

Authors: Yong-Ge Yang; Ya-Hui Sun; Wei Xu
Journal: Complexity
 (2019)

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## Abstract

Stochastic fractional-order systems or stochastic vibro-impact systems can present rich dynamical behaviors, and lots of studies dealing with stochastic fractional-order systems or stochastic vibro-impact systems are available now, while the discussion on the stochastic systems with both vibro-impact factors and fractional derivative element is rare. This paper is concerned with the stochastic bifurcation of a fractional-order vibro-impact system driven by additive and multiplicative Gaussian white noises. Firstly, we can remove the discontinuity of the original system with the help of nonsmooth transformation and obtain the equivalent stochastic system. Then, we adopt the stochastic averaging method to get the approximately analytical solutions. At last, an example is discussed in detail to assess the reliability of the developed approach. We also find that the coefficient of restitution factor, fractional derivative coefficient, and fractional derivative order can induce the stochastic bifurcation.


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## Body

## 1. Introduction

As the fractional-order models can more accurately describe the complex systems than the integer-order models, the investigation on the fractional-order systems attracts more and more attention. Fractional calculus enables us to understand the inherent complexity of the real world [1, 2] by a new mathematical tool. Many excellent books [3, 4] and articles [5–13] about fractional calculus are available. Stochastic perturbations are ubiquitous in the real world, so it is necessary to study the dynamical behaviors of the fractional-order stochastic systems. A lot of methods have been put forward to study the fractional-order stochastic systems, such as the stochastic averaging method [14–17], multiple scales method [18–20], Wiener path integral technique [21], and statistical linearization-based technique [22]. Some recent articles on this topic are as follows. Yang et al. [23] investigated the aperiodic stochastic resonance in a bistable fractional-order system induced by the fractional order and the noise intensity. Li et al. [24] estimated the reliability of stochastic dynamical systems under random excitations with a fractional-order proportional integral derivative controller. Denoël [25] carried out the multiple timescale spectral analysis of a noisy single-degree-of-freedom system with a fractional derivative constitutive term. Wang et al. [26] studied the global dynamics of fractional-order systems with the help of the short memory principle and generalized cell mapping method. Di Matteo et al. [27] developed a Galerkin scheme-based approach to determine the survival probability and first-passage probability of a hysteretic system endowed with fractional derivative elements under Gaussian white noise. Li et al. [28] considered the bifurcation control of a Van der Pol oscillator using the fractional-order PID controller. However, these studies focused on the dynamical behaviors of smooth systems instead of nonsmooth systems.Vibro-impact systems [29] as the typical nonsmooth systems can present rich dynamical behaviors because of its strong nonlinearity, so a great deal of attention has been devoted to this topic. The stochastic average method was adopted by many authors such as Huang et al. [30], Feng et al. [31, 32], and Namachchivaya and Park [33]. By comparing the stability domains of P-bifurcation and D-bifurcation, Kumar et al. [34] concluded that these bifurcations need not occur in same regimes. Wang et al. [35] put forward a new procedure based on the generalized cell mapping (GCM) method to explore the stochastic response of vibro-impact systems numerically. Based on Zhuravlev–Ivanov transformation and the iterative method of weighted residue, Chen et al. [36] proposed a new method to obtain the closed-form stationary solution of the vibro-impact system under Gaussian white noise excitation. Although much attention was devoted to the study of vibro-impact systems, little work focused on the investigation of vibro-impact systems with the fractional derivative damping under random excitation. So, in this paper, we will explore the response of a fractional-order vibro-impact oscillator driven by additive and multiplicative Gaussian white noises.This paper is organized as follows. In Section2, nonsmooth coordinate transformations are adopted to simplify the fractional-order vibro-impact oscillator. In Section 3, the detailed process to get the analytical solutions is presented. In Section 4.1, an example of fractional-order vibro-impact systems driven by additive and multiplicative Gaussian white noises is discussed in detail to assess the reliability of the developed approach. In Section 4.2, the stochastic bifurcations induced by the system parameters are exhibited. The conclusions are given in Section 5.


## 2. System Description and Its Simplification

The motion equations of a fractional-order vibro-impact system under additive and multiplicative random excitations are of the following form:(1a)x¨+εβDαx+εfx,x˙x˙+ω02x=ε1/2ξ1t+ε1/2xξ2t,x>0,(1b)x˙+=−rx˙−,x=0,where ε>0 is a small constant; β and ω0 are constant coefficients; 0<r≤1 is the coefficient of restitution factor; x˙− and x˙+ stand for the instant velocities just before and after collision, respectively; ξ1t and ξ2t are Gaussian white noises whose statistical properties are of the following form:(2)Eξ1t=0,Eξ2t=0,Eξ1tξ1t+τ=2D11δτ,Eξ2tξ2t+τ=2D22δτ,Eξ1tξ2t+τ=Eξ2tξ1t+τ=0.D
α
x refers to the fractional derivative element in the Riemann–Liouville sense:(3)Dαxt=1Γ1−αddt∫0txt−uuαdu,0<α<1,where α is the fractional derivative order.In order to remove the discontinuity in equations (1a) and (1b), the nonsmooth coordinate transformations [37, 38] are used as follows:(4)x=x1=y=ysgny,x˙=x2=y˙sgny,x¨=y¨sgny,where sgny=1,y>0,−1,y<0.Substituting equation (4) into equations (1a) and (1b), we have(5a)y¨+εβsgnyDαy+εfy,y˙sgnyy˙+ω02y=ε1/2sgnyξ1t+ε1/2yξ2t,t≠t∗,(5b)y˙+=ry˙−,t=t∗.Based on the new impact condition (5b), the velocity jump of the new variable yt at impact is proportional to 1−r.According to Refs. [32, 37], the following equation is obtained:(6)y¨+εβsgnyDαy+εfy,y˙sgnyy˙+y˙−−y˙+δt−t∗+ω02y=ε1/2sgnyξ1t+ε1/2yξ2t.According to Refs. [32, 37, 39], we have y˙−−y˙+δt−t∗≈1−ry˙y˙δy, and we can get the following equivalent oscillator without impact term:(7)y¨+εβsgnyDαy+εfy,y˙sgnyy˙+1−ry˙y˙δy+ω02y=ε1/2sgnyξ1t+ε1/2yξ2t.


## 3. Stochastic Averaging Method

Asε>0 is a small constant, β is a constant coefficient, so the fractional derivative term εβDαx is also small. The lightly damped oscillator (6) is subjected to weak random excitations; according to the stochastic averaging method [14], we can assume the solution of equation (6) as(8)yt=AtcosΦt,y˙t=−Atω0sinΦt,Φt=ω0t+Ψt,where At,Φt, and Ψt are random processes. Substituting equation (8) into equation (7) and according to Ref. [14], we can obtain the equations for the amplitude At and the phase angle Ψt:(9)dAdt=εM11+εM12+ε1/2G11ξ1t+ε1/2G12ξ2t,dΨdt=εM21+εM22+ε1/2G21ξ1t+ε1/2G22ξ2t,where(10)M11=βω0sinΦsgnAcosΦDαAcosΦ,M12=−Asin2ΦfAcosΦ,−Aω0sinΦsgnAcosΦ+ε−11−r−Aω0sinΦδAcosΦ,G11=−sinΦω0sgnAcosΦ,G12=−AsinΦcosΦω0,M21=βAω0cosΦsgnAcosΦDαAcosΦ,M22=−sinΦcosΦfAcosΦ,−Aω0sinΦsgnAcosΦ+ε−11−r−Aω0sinΦδAcosΦ,G21=−cosΦAω0sgnAcosΦ,G22=−cos2Φω0.Then, the averaged Itô equation for the limited processAt is(11)dA=mAdt+σAdBt,in which the averaged drift coefficient and diffusion coefficient are given by(12)mA=εM11+M12+D11∂G11∂AG11+D11∂G11∂ΦG21+D22∂G12∂AG12+D22∂G12∂ΦG22Φ,(13)σ2A=ε2D11G112+2D22G122Φ.Then, the most important step is the calculation of the first term of equation (12), i.e.,(14)εM11Ψ=εβω0sinΦsgnAcosΦDαAcosΦΦ.Substituting equation (3) into equation (14), we have(15)εM11Φ=εβAω0Γ1−αsinΦsgnAcosΦ×ddt∫0tcosΦ−ω0uuαduΦ.The Fourier series of the absolute value of the cosine function is(16)cosθ=2π+∑n=1∞Bncos2nθ,where Bn=4/π−1n/1−4n2.In order to smooth the solution, substituting equation (16) into equation (15), we have(17)∫0tcosΦ−ω0uuλ1du=∫0t2πuλ1du+∑n=1∞Bn∫0tcos2nΦ−2nω0uuλ1du=∫0t2πuλ1du+∑n=1∞Bncos2nΦ×∫0tcos2nω0uuλ1du+sin2nΦ×∫0tsin2nω0uuλ1du.According to Refs. [40, 41] and equations (15) and (17), equation (14) can be simplified as(18)εM11Ψ=εlimT⟶∞1T∫0Tβω0sinΦsgnAcosΦDαAcosΦdt=−32εβAπ2ω0sinαπ2∑n=1∞n22nω0α−11−4n22.The other parts of equation (10) and the averaged diffusion coefficient σ2A can be obtained through mathematical calculation.The corresponding Fokker–Planck–Kolmogorov equation associated with equation (11) is given by(19)∂p∂t=−∂∂AmAp+12∂2∂A2σ2Ap,when the boundary conditions of equation (19) are (1) p is a finite real number at A=0 and (2) p⟶0,∂p/∂A⟶0 as A⟶∞. The stationary solution of equation (19) [42–44] is(20)pA=Cσ2Aexp∫0A2muσ2udu,where C is a normalization constant.According to Ref. [14], the joint stationary probability density function of the displacement y and velocity y˙ is as follows:(21)pY,Y˙y,y˙=pA2πω0AA=y2+y˙2/ω02.The stationary PDF of the variablesx1 and x2 can be obtained as(22)px1,x2=2pY,Y˙x1,x2,x1≥0.The marginal stationary probability density functionspx1 and px2 can be achieved as(23)px1=∫−∞+∞px1,udu,(24)px2=∫0+∞pu,x2du.


## 4. Example

The motion equations we consider are expressed as(25)x¨+βDαx+c4x4−c2x2−c0x˙+ω02x=ξ1t+xξ2t,x>0,x˙+=−rx˙−,x=0,where β,c4,c2,c0, and ω0 are small constant coefficients. After introducing the nonsmooth coordinate transformations, we have(26)y¨+βsgnyDαy+c4y4−c2y2−c0y˙+1−ry˙y˙δy+ω02y=sgnyξ1t+yξ2t.The averaged drift coefficients and diffusion coefficients in equation (11) are(27)mA=−32βAπ2ω0sinαπ2∑n=1∞n22nω0α−11−4n22−c416A5+c28A3+c02A−1−rω0πA+D112Aω02+3AD228ω02,σ2A=D11ω02+A2D224ω02.According to equations (20) and (22)–(24), we can obtain pA, px1, px2, and px1,x2.It is noted that the series∑n=1∞n22nω0α−1/1−4n22≈∑n=120n22nω0α−1/1−4n22. The reason is as follows:As the series∑n=1∞n22nω0α−1/1−4n22ω0=1,0<α<1 converges very fast, the following assumption is reasonable:(28)∑n=1∞n22nω0α−11−4n22=∑n=1200,000n22nω0α−11−4n22.Whenω0=1andα=0.5,(29)Un=20=∑n=120n22nω0α−11−4n22=0.094658610684682,(30)Un=200,000=∑n=1200,000n22nω0α−11−4n22=0.094976080504781.The relative error is(31)Un=20−Un=200,000Un=200,000=0.003342629201071≈0.3343%.Comparing equation (29) with equation (30), we can conclude that keeping more items indeed can improve the accuracy. From equation (31), when ω0=1andα=0.5, the relative error is only 0.3343%. So, it is reasonable to keep the first 20 terms when dealing with the series.


### 4.1. Effectiveness of the Method

In this section, the accuracy of the proposed method will be verified by comparison with the Monte Carlo simulation results. The solid lines are the analytical numerical results, while the discrete dots are the numerical results. We can obtain the analytical solution by substitutingmA and σ2A into equation (20). By using the fourth-order Runge–Kutta algorithm, we can obtain numerical results from the original equation (25).In Figure1, a comparison between the numerical results and the analytical results is represented. The system parameter values are listed in Table 1. A very good agreement can be found. So, the effectiveness of the proposed method is acceptable.Figure 1
A comparison between the analytical solutions (solid blue lines) and the numerical results (discrete blue dots).
(a)
(b)
(c)Table 1
Parameter values used in simulation.
α
β
c
4
c
2
c
0
ω
0
D
11
D
22
r
0.5
0.01
0.025
0.01
0.01
1.0
0.0001
0.0005
0.98In order to further assess the effectiveness of the developed method, another comparison between the numerical results and the analytical results is carried out as shown in Figure2. The system parameter values are listed in Table 2. A very good match between the numerical and the analytical results indicates that the developed procedure is effective.Figure 2
A comparison between the analytical solutions (solid blue lines) and the numerical results (discrete blue dots).
(a)
(b)
(c)Table 2
Parameter values used in simulation.
α
β
c
4
c
2
c
0
ω
0
D
11
D
22
r
0.5
0.006
0.01
−0.1
0.06
1.0
0.00005
0.00005
0.98


### 4.2. Bifurcation Analysis

In Section4.1, we demonstrated the effectiveness of the proposed method. In this section, we turn our attention to the stochastic P-bifurcation induced by system parameters. As the investigation on the stochastic P-bifurcation enabled us to have a more clear understanding on the dynamical behavior of the system, especially on the long-run probability distributions, we will conduct the bifurcation analysis in this section. In this paper, the stochastic P-bifurcation or phenomenological bifurcation takes place when the structure of stationary probability density function has qualitative changes as parameters are varied.First, we investigate the influence of the coefficient of restitution factorr on the stochastic bifurcation. The system parameter values are listed in Table 3. Figure 3 depicts the joint probability density functions for different r. It can be concluded that increasing the coefficient of restitution factor r from 0.984 to 0.989 gives rise to the occurrence of stochastic P-bifurcation. Specifically, when r=0.984, the joint probability density function has one peak, while when r=0.989, the joint probability density function presents a crater-like structure. The qualitative transformation of the probability density function indicates the occurrence of stochastic P-bifurcation. In order to better understand the progress of the stochastic P-bifurcation, the corresponding section graphs of probability density functions when x1=0.05 are presented in Figure 4.Table 3
Parameter values used in simulation.
α
β
c
4
c
2
c
0
ω
0
D
11
D
22
0.5
0.06
0.01
−0.10
0.06
1.0
0.00005
0.00005Figure 3
The joint probability density functions for differentr. (a) r=0.984. (b) r=0.986. (c) r=0.989.
(a)
(b)
(c)Figure 4
Section graphs of joint probability density functions when x1=0.05 for different r.Second, we explore the influence of the fractional derivative coefficientβ on the stochastic bifurcations. The system parameter values are listed in Table 4. Figure 5 depicts the joint probability density functions for different β. Figure 6 presents the corresponding section graphs of joint probability density functions when x1=0.05. It can be observed that decreasing fractional derivative coefficient β from 0.009 to 0.003 leads to the occurrence of stochastic P-bifurcation.Table 4
Parameter values used in simulation.
α
c
4
c
2
c
0
ω
0
D
11
D
22
r
0.5
0.01
−0.10
0.06
1.0
0.00005
0.00005
0.986Figure 5
The joint probability density function for differentβ. (a) β=0.009. (b) β=0.006. (c) β=0.003.
(a)
(b)
(c)Figure 6
Section graphs of probability density functions when x1=0.05 for different β.Third, we discuss the influence of the fractional derivative orderα on the stochastic bifurcations. The system parameter values are listed in Table 5. Figure 7 depicts the joint probability density functions for different α. Figure 8 presents the corresponding section graphs of joint probability density functions when x1=0.05. According to similar analysis, it can be observed that decreasing fractional derivative order α from 0.6 to 0.3 contributes to the occurrence of stochastic P-bifurcation.Table 5
Parameter values used in simulation.
β
c
4
c
2
c
0
ω
0
D
11
D
22
r
0.006
0.01
−0.10
0.06
1.0
0.00005
0.00005
0.986Figure 7
The joint probability density functions for differentα. (a) α=0.6. (b) α=0.5. (c) α=0.3.
(a)
(b)
(c)Figure 8
Section graphs of probability density functions whenx1=0.05 for different α.


## 4.1. Effectiveness of the Method

In this section, the accuracy of the proposed method will be verified by comparison with the Monte Carlo simulation results. The solid lines are the analytical numerical results, while the discrete dots are the numerical results. We can obtain the analytical solution by substitutingmA and σ2A into equation (20). By using the fourth-order Runge–Kutta algorithm, we can obtain numerical results from the original equation (25).In Figure1, a comparison between the numerical results and the analytical results is represented. The system parameter values are listed in Table 1. A very good agreement can be found. So, the effectiveness of the proposed method is acceptable.Figure 1
A comparison between the analytical solutions (solid blue lines) and the numerical results (discrete blue dots).
(a)
(b)
(c)Table 1
Parameter values used in simulation.
α
β
c
4
c
2
c
0
ω
0
D
11
D
22
r
0.5
0.01
0.025
0.01
0.01
1.0
0.0001
0.0005
0.98In order to further assess the effectiveness of the developed method, another comparison between the numerical results and the analytical results is carried out as shown in Figure2. The system parameter values are listed in Table 2. A very good match between the numerical and the analytical results indicates that the developed procedure is effective.Figure 2
A comparison between the analytical solutions (solid blue lines) and the numerical results (discrete blue dots).
(a)
(b)
(c)Table 2
Parameter values used in simulation.
α
β
c
4
c
2
c
0
ω
0
D
11
D
22
r
0.5
0.006
0.01
−0.1
0.06
1.0
0.00005
0.00005
0.98


## 4.2. Bifurcation Analysis

In Section4.1, we demonstrated the effectiveness of the proposed method. In this section, we turn our attention to the stochastic P-bifurcation induced by system parameters. As the investigation on the stochastic P-bifurcation enabled us to have a more clear understanding on the dynamical behavior of the system, especially on the long-run probability distributions, we will conduct the bifurcation analysis in this section. In this paper, the stochastic P-bifurcation or phenomenological bifurcation takes place when the structure of stationary probability density function has qualitative changes as parameters are varied.First, we investigate the influence of the coefficient of restitution factorr on the stochastic bifurcation. The system parameter values are listed in Table 3. Figure 3 depicts the joint probability density functions for different r. It can be concluded that increasing the coefficient of restitution factor r from 0.984 to 0.989 gives rise to the occurrence of stochastic P-bifurcation. Specifically, when r=0.984, the joint probability density function has one peak, while when r=0.989, the joint probability density function presents a crater-like structure. The qualitative transformation of the probability density function indicates the occurrence of stochastic P-bifurcation. In order to better understand the progress of the stochastic P-bifurcation, the corresponding section graphs of probability density functions when x1=0.05 are presented in Figure 4.Table 3
Parameter values used in simulation.
α
β
c
4
c
2
c
0
ω
0
D
11
D
22
0.5
0.06
0.01
−0.10
0.06
1.0
0.00005
0.00005Figure 3
The joint probability density functions for differentr. (a) r=0.984. (b) r=0.986. (c) r=0.989.
(a)
(b)
(c)Figure 4
Section graphs of joint probability density functions when x1=0.05 for different r.Second, we explore the influence of the fractional derivative coefficientβ on the stochastic bifurcations. The system parameter values are listed in Table 4. Figure 5 depicts the joint probability density functions for different β. Figure 6 presents the corresponding section graphs of joint probability density functions when x1=0.05. It can be observed that decreasing fractional derivative coefficient β from 0.009 to 0.003 leads to the occurrence of stochastic P-bifurcation.Table 4
Parameter values used in simulation.
α
c
4
c
2
c
0
ω
0
D
11
D
22
r
0.5
0.01
−0.10
0.06
1.0
0.00005
0.00005
0.986Figure 5
The joint probability density function for differentβ. (a) β=0.009. (b) β=0.006. (c) β=0.003.
(a)
(b)
(c)Figure 6
Section graphs of probability density functions when x1=0.05 for different β.Third, we discuss the influence of the fractional derivative orderα on the stochastic bifurcations. The system parameter values are listed in Table 5. Figure 7 depicts the joint probability density functions for different α. Figure 8 presents the corresponding section graphs of joint probability density functions when x1=0.05. According to similar analysis, it can be observed that decreasing fractional derivative order α from 0.6 to 0.3 contributes to the occurrence of stochastic P-bifurcation.Table 5
Parameter values used in simulation.
β
c
4
c
2
c
0
ω
0
D
11
D
22
r
0.006
0.01
−0.10
0.06
1.0
0.00005
0.00005
0.986Figure 7
The joint probability density functions for differentα. (a) α=0.6. (b) α=0.5. (c) α=0.3.
(a)
(b)
(c)Figure 8
Section graphs of probability density functions whenx1=0.05 for different α.


## 5. Conclusions

We carried out the investigation on the stochastic bifurcation of a fractional-order vibro-impact system under additive and multiplicative Gaussian white noise excitations. There are two challenges to study the stationary response of the fractional vibro-impact systems under Gaussian white noises. The first one is how to deal with the discontinuity of the original system. The second one is how to get the explicit expression of the averaged drift coefficient when we utilize the stochastic averaging method. These two challenges have been solved in this paper by the nonsmooth transformations and stochastic averaging method. An example is discussed in detail to assess the reliability of the developed approach. The results showed that the proposed method has a satisfactory accuracy. We also found that the coefficient of restitution factor, fractional derivative coefficient, and fractional derivative order can be treated as bifurcation parameters.



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