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ORIGINAL RESEARCH article

Front. Phys., 02 August 2023
Sec. Interdisciplinary Physics
Volume 11 - 2023 | https://doi.org/10.3389/fphy.2023.1238901

Fractional stochastic vibration system under recycling noise

www.frontiersin.orgJian-Gang Zhang* www.frontiersin.orgFang Wang www.frontiersin.orgHui-Nan Wang
  • School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, China

The fractional stochastic vibration system is quite different from the traditional one, and its application potential is enormous if the noise can be deployed correctly and the connection between the fractional order and the noise property is unlocked. This article uses a fractional modification of the well-known van der Pol oscillator with multiplicative and additive recycling noises as an example to study its stationary response and its stochastic bifurcation. First, based on the principle of the minimum mean square error, the fractional derivative is equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Second, the Itô differential equations and One-dimensional Markov process are obtained according to the stochastic averaging method, using Oseledec multiplicative ergodic theorem and maximal Lyapunov exponent to judge local stability, and judging global stability is done by using the singularity theory. Lastly, the stochastic D-bifurcation behavior of the model is analyzed by using the Lyapunov exponent of the dynamical system invariant measure, and the stationary probability density function of the system is solved according to the FPK equation. The results show that the fractional order and noise property can greatly affect the system’s dynamical properties. This paper offers a profound, original, and challenging window for investigating fractional stochastic vibration systems.

1 Introduction

Fractional derivative [1, 2] is an extension of the theory of integer derivatives, and the study of fractional derivatives has a history of over 300 years. Some new materials have appeared, e.g., viscoelastic materials, nanomaterials, cement mortar, 3D-printed materials, and porous materials [38], which are different from either a solid or a fluid, and their constitutive relation is extremely difficult to be expressed correctly by the traditional calculus though much effort has been made to solve the problem, for example, using the fractal viscoelastic model [9] and the fractal rheological model [10]; the intractable constitutive relation has not yet been solved.

Considering its memory property, we consider that fractional calculus might be the best candidate for stochastic dynamical systems [11, 12]. Stochastic disturbances are widespread in nature, and fractional stochastic systems have become a hot spot in both mathematics and physics to deal with noise excitation. For example, energy-harvesting devices [1315] are always subject to random excitation, and a fractional model can effectively reveal the bifurcation properties and multiple attractors of the energy-harvesting system, for example, Ref. [16]. Fractional models for Gaussian white noise also caught much attention [1721], and the fractional convolution kernel neural network is a suitable mathematical tool for fault diagnosis [2224]. Duffing oscillator [25, 26] is extended to its fractional partner under noise [27, 28]. Van der Pol oscillator [29] is another widely used model for the analysis of fractional stochastic P-bifurcation [30, 31].

In reality, noise exists in all aspects of practical applications, especially in nonlinear systems. The properties of stationary response, energy-harvesting efficiency, stability, and bifurcation will be greatly affected by noise excitation. At present, the research on the dynamic behavior of systems driven by recycling noise has attracted widespread attention from domestic and foreign scholars and achieved fruitful results, especially in the birhythmic biological system [32], stochastic resonance in asymmetric bistable systems [33], and the double entropic stochastic resonance phenomenon [34]. In this article, the fractional van der Pol model with recycling noise is adopted to investigate its dynamical properties.

2 Model description

Balthazar van der Pol is a famous electronic engineer in the Netherlands. In 1927, he first deduced the famous van der Pol equation in order to describe the oscillation effect of triodes in electronic circuits, as shown below:

x¨μ1x2x˙+x=0

Afterward, as a classic nonlinear dynamic system, it is often used in mathematics and some nonlinear dynamic systems to demonstrate its dynamic behavior characteristics. In continuous research, the highest number of nonlinear terms considered is also constantly increasing, and there are also various methods for solving approximate solutions of such equations [35, 36]. From the classical van der Pol equation, changing the order of the equation can obtain systems with different dynamic behaviors, thereby better obtaining the dynamic behavior characteristics of the system. Therefore, we use the following equation to introduce the fractional generalized van der Pol model with multiplicative and additive recycling noise:

x¨ε+α1x2α2x4+α3x6α4x8x˙+ω2x+Dpx0c=η1t+xtη2t,(1)

where ε is the damping coefficient, α1,α2,α3,α4 are nonlinear damping coefficients, ω is the frequency, η1t and η2t are independent recycling noises, i.e., D1D2, ηit=ξit+kξitτ,i=1,2. The power spectral density of recycling noise is obtained as:

Siω=2Di1+k2+2kcosωτ,i=1,2.(2)

Dpxt0c is the Caputo fractional derivative [1, 2] of p0p1 order about xt defined as:

Dp0cxt=1Γmp0txmutu1+pmdu,m1<pm,mN.(3)

There are other definitions of fractional derivatives, for example, two-scale fractal derivative [3741] and He’s fractional derivative [42]. The Caputo fractional derivative has memory property [43, 44], so it is used for the present study.

The Dp0cx term in Eq. 1 can be expressed in a combination of spring stiffness and damping terms [4548], hence, Eq. 1 becomes:

x¨ε+α1x2α2x4+α3x6α4x8+Cpx˙
+ω2+Kpx=η1t+xtη2t,(4)

where C and K are the equivalent damping and stiffness coefficients of fractional damping, respectively.

In order to identify C and K, we introduce an error function, which reads

e=Cpx˙+KpxDpxt0c,(5)

According to the minimum mean square method [44], we have

Ee2/Cp=0,Ee2/Kp=0.(6)

Equation 6 leads to the following equations:

ECpx˙2+Kpxx˙x˙Dp0cx=limT1T0TCpx˙2+Kpxx˙x˙Dp0cxdt=0,ECpxx˙+Kpx2xDp0cx=limT1T0TCpxx˙+Kpx2xDp0cxdt=0.(7)

Assuming that

xt=atcosφt=atcosωt+θ(8)

and a˙t0, we have

x˙t=atωsinφt,x¨t=atω2cosφt.(9)

Considering Eq. 8, 9, we re-write Eq. 7 in the form

limT1T0TCpx˙2+Kpxx˙x˙Dpx0cdt
=limT1T0TCpa2tω2sin2φtKpa2tωφtcosφt+atωsinφtDpx0cdφ
Cpa2ω2+1Γ1plimT1ToTaωsinφ0tx˙tττpdτdφ
=Cpa2ω21Γ1plimT1T0Ta2ωsinφ×0tsinφcosωτcosφsinωττpdτdt=0,

For the same reason, we have

limT1T0TCpxx˙+Kpx2xDpx0cdt
=limT1T0TCpa2tωsinφtcosφt+Kpa2tcos2φtatcosφtDpx0cdφ
Kpa22ω1Γ1plimT1ToTacosφ0tx˙tττpdτdφ
=Kpa22ω+1Γ1plimT1T0Ta2×cosφ0tsinφcosωτcosφsinωττpdτdt=0.

Hence

limT1T0TCpx˙2+Kpxx˙x˙Dpx0cdt=Cpa2ω21Γ1plimT1T0Ta2ωsinφ0tsinφcosωτcosφsinωττpdτdt=0,limT1T0TCpxx˙+Kpx2xDpx0cdt=Kpa22ω+1Γ1plimT1T0Ta2cosφ0tsinφcosωτcosφsinωττpdτdt=0.(10)

To simplify Eq. 10 further, we use the following asymptotic integrals

0tcosωττpdτ=ωp1Γ1psinpπ2+sinωtωtp+οωtp1,0tsinωττpdτ=ωp1Γ1pcospπ2cosωtωtp+οωtp1.(11)

In view of Eq. 11, the integral averaging of Eq. 10 with respect to φ results in

Cp=ωp1sinpπ2,Kp=ωpsinpπ2.(12)

Hence, the equivalent system (4) can be written in the form

x¨λx+ω02x=η1t+xtη2t,(13)

where

λ=ε+α1x2α2x4+α3x6α4x8ωp1sinpπ2,ω02=ω2+ωpcospπ2.(14)

3 Model processing

Now the problem becomes relatively simple; we assume that the solution of Eq. 13 can be expressed as [49].

X=xt=atcosΦt,Y=x˙t=atω0sinΦt,Φt=ω0t+θt,(15)

where at and θt are the amplitude and initial phase of the system, respectively.

We re-write Eq. 13 in the form

x˙=y,y˙=λyω02xt+η1t+xtη2t.(16)

By Eq. 15 and the stochastic averaging method [50], Eq. 16 becomes

dadt=F11a,θ+G11a,θη1t+G12a,θη1t,dθdt=F21a,θ+G21a,θη1t+G22a,θη1t,(17)

where

F11a,θ=asin2Φε+α1a2cos2Φα2a4cos4Φ+α3a6cos6Φα4a8cos8Φωp1sinpπ2,F21a,θ=sinΦcosΦε+α1a2cos2Φα2a4cos4Φ+α3a6cos6Φα4a8cos8Φωp1sinpπ2,G11a,θ=sinΦω0,G12a,θ=asinΦcosΦω0,G21a,θ=cosΦaω0,G22a,θ=cos2Φω0.(18)

The recycling noise is a stationary process and can be approximated by a 2-D diffusion process. After stochastic averaging, the drift and diffusion coefficients are as follows:

m1=F11+0cosΦω0cosΦt+τ1aω0+cosΦsinΦω0asin2Φt+τ12ω0+acos2Φsin2Φω0cos2Φt+τ1ω0Rτ1dτ1=F11+cos2Φaω02S11+acos2Φsin2Φ+acos2Φcos2Φω02S21,m2=F21+0cosΦa2ω0sinΦt+τ1ω0+sinΦaω0cosΦt+τ1aω0+2cosΦsinΦω0cosΦ2t+τ1ω0Rτ1dτ1=F212cosΦsinΦa2ω02S112cos3ΦsinΦω02S21,B11=+sinΦω0asinΦt+τ1ω0Rτ1dτ1=2sin2Φω02S11,B12=+asin2Φ2ω0asin2Φt+τ12ω0Rτ1dτ1=2a2cos2Φsin2Φω02S21,B21=+cosΦaω0cosΦt+τ1aω0Rτ1dτ1=2cos2Φa2ω02S11,B22=+cos2Φω0cos2Φt+τ1ω0Rτ1dτ1=2cos4Φω02S21,(19)

where Si1 is the value of power spectral density of ηit at ω=1.

Si1=2Di1+k2+2kcosτ,i=1,2(20)

For the deterministic averaging of φt, we have

m¯11=12π02πF11a,θ+cos2Φaω02S11+acos2Φsin2Φ+acos2Φcos2Φω02S21dΦ=12aε+ωp1sinpπ2+α1a38α2a516+5α3a71287α4a9256+S112aω02+3aS218ω02m¯22=12π02πF21a,θ2cosΦsinΦa2ω02S112cos3ΦsinΦω02S21dΦ=0,B¯11=12π02π2sin2Φω02S11dΦ=S11ω02,B¯12=12π02π2a2cos2Φsin2Φω02S21dΦ=a2S214ω02,B¯21=12π02π2cos2Φa2ω02S11dΦ=S11a2ω02,B¯22=12π02π2cos4Φω02S21dΦ=3S214ω02.(21)

The corresponding Itô SDE is

da=m1adt+σ112adB1t+σ122adB2t,dθ=m2adt+σ212adB1t+σ222adB2t,(22)

where

m1a=12aε+ωp1sinpπ2+α1a38α2a516+5α3a71287α4a9256+S112aω02+3aS218ω02,m2a=0,σ112a=S11ω02,σ122a=a2S214ω02,σ212a=S11a2ω02,σ222a=3S214ω02.(23)

The one-dimensional Markov Process can be expressed as:

da=H1a8+α1a38α2a516+5α3a71287α4a9256+H22adt+H212dB1t+H3a2412dB2t,(24)

where

H1=4ε+ωp1sinpπ2+3S21ω02,H2=S11ω02,H3=S21ω02.(25)

4 Stochastic stability analysis

4.1 The local stochastic stability

Considering the case of α1=α2=α3=α4=H2=0 and linear Itô stochastic stability, from Eq. 24, we obtain

da=H18adt+H34a212dB2t,ma=H18a,σa=H3412a.(26)

Therefore, it is obtained that m˙0=H1/8 and σ˙120=H3/41/2, using Oseledec multiplicative ergodic theorem [51] and maximal Lyapunov exponents to judge local stability. According to Itô stochastic differential equation, the solution of Eq. 26 is

at=a0exp0tm˙0σ˙12022ds+0tσ˙120dB2s,

Then the approximate solution of the Lyapunov exponent of Itô stochastic differential equation is obtained

λ=limt+1tlnxt,t0=limt+1tlnat12=m˙0σ˙12022/2=12H18H38.

When H1H3<0, i.e., λ<0, Eq. 26 is stable in the sense of probability, and Eq. 16 is stable at the balance point. When H1H3>0, i.e., λ<0, the effect is just the opposite.

4.2 The global stochastic stability

4.2.1 Linear Itô stochastic stability

Judging global stability by the singularity theory, a=0 is the first kind of singular boundary of Eq. 26. a=+ is the second kind of singular boundary problem of Eq. 26. Calculating the diffusion index, drift indices, and characteristic value at boundary a=0 and a=+, respectively, yields

αa=2,βa=1,ca=lima0+2maa0αaβaσ122a=lima0+2H18a2/H34a2=H1H3,
αl=2,βl=1,cl=lima+2maa0αlβlσ122a=lima+2H18a2/H34a2=H1H3.

And the following conclusions are drawn, as shown in Table 1.

TABLE 1
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TABLE 1. Global stability analysis.

4.2.2 Stability of nonlinear Itô stochastic differential equation

When α1,α2,α3,α4,H20, a=0 is the first kind of singular boundary of Eq. 24. When a=+ and ma=+, a=+ is the second kind of singular boundary problem of Eq. 24. Calculating the diffusion index, drift indices, and characteristic value at boundary a=0 and a=+, respectively, yields

αa=2,βa=1,ca=lima0+2maa0αaβaσ112a+σ122a=lima0+aε+ωp1sinpπ2+α1a34α2a58+5α3a7647α4a9128+S11aω02+3aS214ω02a/S11ω02+a2S214ω02=lima0+ω02128a2ε+ωp1sinpπ2+32α1a416α2a6+10α3a87α4a10+128S11+96a2S21128S11+32a2S21=1,limαl=2,βl=9,cl=lima+2maa0αlβlσ112a+σ122a=lima+aε+ωp1sinpπ2+α1a34α2a58+5α3a7647α4a9128+S11aω02+3aS214ω02a7/S11ω02+a2S214ω02=lima+ω02128a2ε+ωp1sinpπ2+32α1a416α2a6+10α3a87α4a10+128S11+96a2S21128S11a8+32S21a10=7α4ω0232S21.

Conclusion: when a=0 and ca=1 are a strict natural boundary; when a=+ , cl>1, and α4ω02/S21<32/7, the boundary is an exclude natural boundary; when cl<1 and α4ω02/S21>32/7, the boundary is an attract natural boundary; when cl=1 and α4ω02/S21=32/7, the boundary is a strict natural boundary. Therefore, ca=1 is a critical condition of system bifurcation.

5 Stochastic bifurcation analysis

5.1 D-bifurcation

If H2=H3=0, Eq. 13 becomes a deterministic system without a stochastic bifurcation phenomenon. Therefore, discussing the situation of H30 and α1=α2=α3=α4=H2=0, let σ12a=H3/41/2a and ma=H1/8H3/8a, then the continuous dynamic system generated by Eq. 26 is

ψ1tx=x+0tmψ1sxds+0tσψ1sxdB,(27)

Equation 27 is the only strong solution of Eq. 26 with x as the initial value. When m0=0 and σ120=0, let ma be bounded, for all a0, the elliptic condition σ1200 is satisfied, so there is only one stationary probability density. Therefore, the FPK equation corresponding to Eq. 26 is obtained.

pt=aH18ap2a2H34a2p.(28)

Let p/t=0 get the stationary probability density corresponding to Eq. 28

pa=cσ121aexp0a2muσ122udu.(29)

At this time, Eq. 27 has a non-trivial stationary state and a fixed-point equilibrium state. Assuming the invariant measures of these two kinds of stationary states are υ1 and ϑ1, respectively, the density is Eq. 29 and ϑ1x, respectively. Hence, the solution of Eq. 28 is

at=a0exp0tm˙a+σ12aσ¨12a2ds+0tσ˙12adB2.(30)

The Lyapunov exponent of ψ1 with respect to estimate u can be defined as follows

λψ1u=limt+1tlnat,(31)

Substituting Eq. 30 into Eq. 31, here σ120=0 and σ˙120=0, its Lyapunov exponent of the fixed-point reads

λψ1ϑ1=limt+1tlna0+m˙00tds+σ˙1200tdB2s=m˙0+σ˙120limt+B2tt=m˙0=H18H38.(32)

Invariant estimate υ1 with Eq. 29 as density. Substituting Eq. 30 into Eq. 31. Assuming that σ˙ and m˙+σσ˙ are bounded and integrable, respectively, the Lyapunov exponent can be obtained

λψ1υ1=limt+1t0tm˙a+σ12aσ¨12ads=Rm˙a+σ12aσ¨12a2pada=2Rmaσ12a2padam˙0=2H23/2H18H38exp8H3H18H38.(33)

Let α=H1H3, when α<0 and H1<H3, ϑ1 is stable, υ1 is unstable; when α>0 and H1>H3, ϑ1 is unstable, υ1 is stable. So α is a D-bifurcation point of Eq. 13.

5.2 P-bifurcation

5.2.1 Stochastic P-bifurcation under additive recycling noise

When additive noise just exists, D10 and D2=0. The following is an analysis of the stochastic P-bifurcation of the system in this case. Eq. 22, 23 show that the Itô stochastic differential equation corresponding to at does not depend upon θt, and it is a 1-D diffusion process; its corresponding FPK equation can be expressed as

pa,tt=am1apa,t+122a2σ112apa,t,(34)

the corresponding boundary conditions are

p=c,c,+,whena=0.p0,pa0,whena.(35)

In view of Eq. 35, the stationary probability density of the amplitude is

pa=Cσ112aexp0a2m1uσ112udu,(36)

where C is the normalization constant,

C=01σ112aexp0a2m1uσ112ududa1.(37)

In view of Eq. 23, from Eq. 36, we obtain

pa=Caω02S11expa2ω02Δ7680S11,(38)

where

Δ=3840ε+3840ωp1sinpπ2480α1a2+160α2a475α3+42α4a8,ω02=ω2+ωpcospπ2,S11=2D11+k2+2kcosτ.(39)

The original system response meets at=x2t+x˙2t, in view of Eq. 38, the joint probability density function of the system is

pa=Cx2t+x˙2tω02S1texpx2t+x˙2tω023840S1tΔ.(40)

5.2.1.1 Influence of fractional order

As fractional damping is a combination of the equivalent stiff and equivalent damping, the fractional order is of paramount importance; its value can be calculated by He-Liu’s fractal formulation [52] for practical applications. According to Eq. 12, when p=1, fractional damping becomes a damping term, while when p=0, it is a stiff term.

Setting τ=2,k=0.4,ε=0.1,α1=1.51,α2=2.85,α3=1.693,α4=0.312, and ω=1 in Eq. 13 as that in Refs [30, 53], the stochastic P-bifurcation is studied hereby. Keeping D1=0.005 constant, we draw the joint probability density function section and top view of Eq. 13 under the influence of different fractional orders.

When p=0.06, the joint probability density function diagram shows a crater shape; there is only one peak in the section, and there is only a large limit cycle. The response is shown as a vibration far beyond the origin (Figure 1).

FIGURE 1
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FIGURE 1. Joint probability density function section and top view of Eq. 13 when p=0.06.

When p=0.139, from the section, it can be clearly seen that there are two peaks, but the second peak has a much larger amplitude. At this time, the system has a balance point and a large limit cycle; hence, the system response switches between two peaks, and the probability of a large amplitude vibration is high, as shown in Figure 2.

FIGURE 2
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FIGURE 2. Joint probability density function section and top view of Eq. 13 when p=0.137.

When p=0.141, the section has three peaks, showing two peaks in addition to the origin. A balance point now coexists with a large and small limit cycle in the system, and the system response switches between the three peaks, which is a multimodal response. Due to the existence of the double limit point set, the relative heights of the joint probability density function peaks at the three peaks are different, implying that the system response peaks are different, as shown in Figure 3.

FIGURE 3
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FIGURE 3. Joint probability density function section and top view of Eq. 13 when p=0.14.

When p=0.145, the section has two peaks, in contrast to Figure 2, the relative height of the peak changes, with the second peak being significantly smaller. At this time, the system has both a balance point and a small limit cycle; hence, the system response switches between two peaks, and the probability of a small amplitude vibration is high, as shown in Figure 4.

FIGURE 4
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FIGURE 4. Joint probability density function section and top view of Eq. 13 when p=0.143.

Based on the above discussions, we conclude that the fractional order can cause stochastic P-bifurcation behavior in the system. From Figure 5, we find that an increasing fractional order will change the stationary response from a single mode to a dual mode and then to a tristable mode. The peak value changes from a single peak to two peaks and then to three peaks, so stochastic P-bifurcation occurs. Increasing the value of p to 0.145 again, the tristable disappears and the bistable appears; the peak value changes from three peaks to two peaks, so stochastic P-bifurcation occurs.

FIGURE 5
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FIGURE 5. Stationary probability density function diagram of Eq. 13 when D1=0.005.

5.2.1.2 Influence of noise intensity

Keeping the above parameters unchanged, and fixing p=0.14, we draw the joint probability density function section and top view of Eq. 13 under the influence of different noise intensity.

When D1=0.03, the joint probability density function diagram shows a crater shape, there is only one peak in the section, and there is only a large limit cycle. The response is shown as a vibration far beyond its origin, as shown in Figure 6.

FIGURE 6
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FIGURE 6. Joint probability density function section and top view of Eq. 13 when D1=0.03.

When D1=0.015, from the section, it can be clearly seen that there are two peaks, but the first peak is much smaller. At this time, the system has both a balance point and a large limit cycle; hence, the system response switches between two peaks, and the probability of a large amplitude vibration is high, as shown in Figure 7.

FIGURE 7
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FIGURE 7. Joint probability density function section and top view of Eq. 13 when D1=0.015.

When D1=0.005, the section has three peaks, showing two peaks in addition to the origin. A balance point now coexists with a large and small limit cycle in the system, and the system response switches among the three peaks, which is a multimodal response. Due to the existence of the double limit point set, the relative heights of the joint probability density function peaks at the three peaks are different, implying that the vibration frequency of the system response peak is different, as shown in Figure 8.

FIGURE 8
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FIGURE 8. Joint probability density function section and top view of Eq. 13 when D1=0.005.

When D1=0.0013, the section has two peaks, in contrast to Figure 7; the relative height of the peak changes, with the first peak being much larger. At this point, the system has a balance point and a small limit cycle, the system response switches between two peaks, and the probability of a small amplitude vibration is high, as shown in Figure 9.

FIGURE 9
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FIGURE 9. Joint probability density function section and top view of Eq. 13 when D1=0.0013.

Based on the above discussions, it can be verified that changing the noise intensity affects greatly stochastic P-bifurcation property. From Figure 10, it can also be seen that with noise intensity being reduced, the stationary response of the system switches from a single mode to a dual mode and then to a tristable mode. The peak value of the stationary probability density function curve changes from a single peak to two peaks and then three peaks, so stochastic P-bifurcation occurs. Decreasing the value of D1 to 0.0013 again, the tristable disappears and the bistable appears; the peak value changes from three peaks to two peaks, so stochastic P-bifurcation occurs.

FIGURE 10
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FIGURE 10. Stationary probability density function diagram of Eq. 13 when p=0.141.

5.2.2 Additive and multiplicative recycling noise

When D10 and D20, the expression of the stationary probability density function of the amplitude of Eq. 13 is

pa=Cσ112a+σ122aexp0a2muσ112u+σ122udu,(41)

where C is the normalization constant,

C=01σ112a+σ122aexp0a2m1uσ112u+σ122ududa1.(42)

In view of Eq. 23, from Eq. 42, we have

pa=4Caω024S11+a2S21Δ1S251expΔ2768S241,(43)

where

Δ1=2ω02ε+ωp1sinpπ2S24+α1S1S23+2α2S12S22+5α3S13S2+14α4S14,Δ1=a2ω02384α1S23+768α2S1S22+1920α3S12S2+5376α4S13+a4ω0296α2S23240α3S1S22672α4S12S2+a6ω0240α3S23+112α4S1S2221a8ω02α4S23,ω02=ω2+ωpcospπ2,S11=2D11+k2+2kcosτ,S21=2D21+k2+2kcosτ.(44)

Keeping the above parameters unchanged, we draw the joint probability density function section and top view of Eq. 13 under the influence of different fractional orders and noise intensity.

When p=0.05, let D1=0.5 and D2=1. The joint probability density function diagram shows a crater shape; there is only one peak in the section, and there is only a large limit cycle. The response is shown as a vibration far away from the origin. At the same time, reducing the value of noise intensity reveals that the peak of the joint probability density does not change with only one peak. However, the system only has a limit cycle, which has been in a monostable. Therefore, there is no stochastic P-bifurcation phenomenon occurring, as shown in Figure 11.

FIGURE 11
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FIGURE 11. Joint probability density function section and top view of Eq. 13 when p=0.05.

When p=0.1, let D1=0.002 and D2=0.01. From the section, it can be clearly seen that there are two peaks, but the second peak has a much larger amplitude. At this time, the system has both a balance point and a limit cycle; hence, the system response switches between two peaks, and the large amplitude vibration has a higher probability. When the simultaneous improvement of the noise intensifies to D1=0.008 and D2=0.2, the peak value of the stationary probability density function curve changes from two peaks to one peak. There is only a large limit cycle, and the system response becomes a vibration far from the origin. Therefore, increasing the noise intensity induces a stochastic P-bifurcation property, as shown in Figure 12.

FIGURE 12
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FIGURE 12. Joint probability density function section and top view of Eq. 13 when p=0.1.

When p=0.14, let D1=0.003 and D2=0.01. The section has a peak near the origin. There is only a balance point in the system at this time, and the response is shown as a vibration closer to the origin. When simultaneously improving its noise intensity to D1=0.004 and D2=0.1, the peak value of the stationary probability density function curve changes from a single peak to two peaks. At this time, the system has both a balance point and a limit cycle; hence, the system response switches between two peaks, and the probability of a large amplitude vibration is small. Therefore, increasing the noise intensity induces the stochastic P-bifurcation phenomenon. Further increasing the noise intensity to D1=0.006 and D2=0.15, the peak value of the section changes relatively, and the first peak is lower. The system response switches between two peaks, and the probability of a small one is small. Continuing to increase the value of noise intensity to D1=0.04 and D2=0.3, the peak value of the curve changes from two peaks to a single peak. There is only a limit cycle in the system at this time, and the response is shown as a vibration far away from the origin. Therefore, increasing the noise intensity induces the second stochastic P-bifurcation phenomenon, as show in Figure 13.

FIGURE 13
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FIGURE 13. Joint probability density function section and top view of Eq. 13 when p=0.14.

6 Conclusion

In this paper, the stationary response and the stochastic bifurcation of the fractional van der Pol equation under multiplicative and additive recycling noise excitations are investigated. By the least square method, we obtain an equivalent integral nonlinear stochastic system. The Itô differential equation and One-dimensional Markov process are obtained according to the stochastic averaging method. We discuss the local and global stochastic stability and analyze the conditions for inducing D-bifurcation and P-bifurcation in the system. The analysis shows that when α<0 and H1<H3, the point equilibrium state becomes stable, and the non-trivial stationary state becomes unstable; when α>0 and H1>H3, the result is the opposite. So α is a D-bifurcation point of the original system. When only additive noise exists, the fractional order and the noise intensity will greatly affect the system’s property. It was found that reducing the order p or increasing the noise intensity D1 can cause nonlinear jumping or significant oscillation in the system, leading to system instability. Through increasing the order p or reducing the noise intensity D1, the system response is in a monostable state or a small disturbance near the balance point. Similarly, when additive and multiplicative noise coexist, selecting appropriate parameters can maintain the system response at a monostable or small disturbance near the balance point. Therefore, in practical engineering, to avoid the potential adverse effects of high noise intensity on the system, the occurrence of stochastic bifurcation behavior can be controlled by changing the noise intensity or fractional order. In the future, we will combine theory with practice to explore the impact of recycling noise on the stationary response and stochastic bifurcation of systems in wind turbines. We will study the impact of changing noise intensity and fractional order on the system, and how to handle these adverse effects to achieve optimal system performance.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

J-GZ: supervision, writing–review and editing, funding acquisition, investigation, and project administration. FW: writing–original draft, writing–review and editing, and software. H-NW: writing–review and editing and software. All authors contributed to the article and approved the submitted version.

Funding

This study was supported by the key project of the Gansu Province natural science foundation of China (No. 23JRRA882).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: van der Pol system, fractional derivative, recycling noise, stochastic averaging method, stochastic bifurcation

Citation: Zhang J-G, Wang F and Wang H-N (2023) Fractional stochastic vibration system under recycling noise. Front. Phys. 11:1238901. doi: 10.3389/fphy.2023.1238901

Received: 12 June 2023; Accepted: 18 July 2023;
Published: 02 August 2023.

Edited by:

Ji-Huan He, Soochow University, China

Reviewed by:

Muhammad Nadeem, Qujing Normal University, China
Yajie Li, Henan University of Urban Construction, China
Ain Qura Tul, Guizhou University, China

Copyright © 2023 Zhang, Wang and Wang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Jian-Gang Zhang, Zhangjg7715776@126.com

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