© Wuhan University 2023

0 Introduction

Fractional calculus is an important branch of calculus. It was proposed by scientists more than 300 years ago. After that, many researchers had carried out extensive discussions on the theory of fractional calculus and given a reasonable definition. Although it is not as well-known and relatively easy to understand as integer calculus, it can be well used to describe the behavior of materials or systems with non-locality and historical dependence^[1,2]. Compared with the conventional integer systems, the fractional systems have more advantages. Therefore, the fractional systems have become a problem widely concerned.

Due to the existence of stochastic disturbances in nature, there have been many studies on stochastic systems with fractional derivative under noise excitation. For example, Chen et al^[3,4] found that under broadband noise excitation, changing the fractional order can induce stochastic bifurcation of Duffing-van der Pol oscillators with fractional derivatives. At the same time, using the generalized harmonic balance technique, the term associated with fractional derivative was separated into the equivalent quasi-linear restoring force and quasi-linear dissipative force, and the original system was replaced by an equivalent nonlinear stochastic system without fractional derivative. The steady-state response of a Duffing oscillator with hardening stiffness and fractional order derivatives under Gaussian white noise excitation is studied. Sun et al^[5] found that combined with the stationary probability density of amplitude, the power spectral density estimation of the response of nonlinear stochastic vibration structures under fractional damping can be obtained, and the statistics such as the variance of the response can be further calculated. Ma et al^[6] found that under the excitation of Gaussian white noise, increasing the fractional order, fractional coefficient and natural frequency all can significantly weaken the stationary response of the fractional Duffing-van der Pol system. On the contrary, increasing the noise intensity can increase the stationary response of the system. Li et al^[7,8] found that under additive Gaussian white noise and combined additive and multiplicative white noise, changes in fractional order and noise intensity can induce random P-bifurcation phenomena in fractional order generalized van der Pol systems.

In reality, time-delay exists in many fields, especially in control, which often changes the system performance. Duan et al^[9] studied the response and bifurcation of fractional stochastic systems with time-delayed feedback control. The research showed that taking different fractional damping and time-delay can change the probability density of the system and even cause bifurcation phenomena. At the same time, noise exists in all aspects of life, especially in nonlinear systems. Noise can enhance the stationary response of the system, causing coherent resonance or the stochastic bifurcation.

At present, there are few studies on the stationary response and stochastic bifurcation of stochastic systems under combined noise and time-delayed feedback control. Therefore, this paper studies the response and bifurcation of the fractional Duffing oscillator system under combined multiplicative recycling noise and time-delay feedback control, which has important practical application value.

1 Model Description

Fractional Duffing oscillator system under combined multiplicative recycling noise and time-delayed feedback control excitation is expressed as

(1)

where is linear damping coefficient, is coefficient of fractional derivative, is nonlinear stiffness coefficient, is natural frequency of the system, is time-delayed feedback control, is the recycling noise, is delay time and is scale coefficient of time-delay noise, is a Gaussian white noise with , hence, the correlation function of the recycling noise is given by

After Fourier transform, the power spectral density of cyclic noise is

(2)

denotes the fractional derivative under the definition of Caputo

(3)

The fractional derivative term can be equivalent to the linear combination of damping force and restoring force^[4,10-13], hence, introducing the following equivalent system

(4)

where and are the coefficients of equivalent damping force and restoring force, respectively.

The error between Eq. (1) and (4) is

(5)

The necessary conditions for minimum mean square error^[14] are

(6)

Substituting Eq. (5) into Eq. (6) yields

(7)

By introducing a change of variables, one gets

(8)

where

(9)

Substituting Eq. (8) and Eq. (9) into Eq. (7) yields

for the same reason, we have

Hence

(10)

To simplify Eq. (10) further, asymptotic integrals are introduced as follows

(11)

Substituting Eq. (11) into Eq. (10) and integral averaging them regarding

(12)

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

(13)

where

(14)

Based on the principle of small delay approximation^[9] and averaging method using generalized harmonic functions, assuming that the solution of system (4) is of the forms

(15)

Because of amplitude and phase are slow changing processes, hence

(16)

Substituting Eq. (16) into Eq. (15) yields

(17)

(18)

The time-delayed feedback control can be approximately expressed as

(19)

Therefore, system (13) can be transformed into the following approximate system

(20)

where

(21)

Based on linearizing the cubic stiffness term, the equivalent system can be expressed as follows

(22)

The error between Eq. (22) and (20) is

(23)

Assuming that the solution of system (20) is of the forms

(24)

Based on averaging method using generalized harmonic functions^[4], the coefficients to be determined are as follows

(25)

Substituting Eq. (25) into Eq. (22), the equivalent system (13) can be rewritten as

(26)

where

(27)

2 Model Processing

To solve the stationary probability density function of amplitude of system (26), by introducing a change of variables^[15], one gets

(28)

where and are the amplitude and initial phase of the system, respectively. Then, we transform (26) in terms of the new variables and ,

(29)

where

(30)

The recycling noise is a stationary process. After stochastic averaging, the drift and diffusion coefficients are as follows

where is the value of power spectral density of at ,

For the deterministic averaging of , we have

The corresponding SDE (Stochastic Differential Equation) obtained by the stochastic averaging method^[16] is

(31)

where

(32)

The one-dimensional Markov Process can be expressed as

(33)

where

(34)

3 Stochastic Stability Analysis

3.1 The Local Stochastic Stability

Use Oseledec multiplicative ergodic theorem^[17] and maximal Lyapunov exponents to judge local stability. According to Itô stohastic differential equation, the solution of Eq. (33) can be obtained as follows

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

When , the trivial solution of Itô stochastic differential equation (34) is stable in the sense of probability, and the stochastic system (29) is stable at the balance point. When , the effect is just opposite.

3.2 The Global Stochastic Stability

Judging global stability by singularity theory, is the first kind of singular boundary of Eq. (33), and is the second kind of singular boundary of Eq. (33). Calculating the diffusion index, drifting indices and characteristic value at boundary respectively yields

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

4 Stochastic Bifurcation Analysis

4.1 D-Bifurcation

When , Eq. (33) becomes a deterministic system without bifurcation. So, when , let then the continuous dynamic system generated by Eq. (33) is

(35)

Eq. (36) is the only strong solution of Eq. (34) with as the initial value. When , let be bounded, for all , the elliptic condition is satisfied, so there is only one stationary probability density. Therefore, the FPK (Fokker-Planck-Kolmogorov) equation corresponding to Eq. (33) is obtained

(36)

Let get the stationary probability density corresponding to Eq. (36)

(37)

At this time, Eq. (35) has non-trivial stationary state and fixed-point equilibrium state. Assuming the invariant measures of these two kinds of stationary states be and respectively; the density is Eq. (38) and respectively. Hence, the solution of Eq. (36) is

(38)

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

(39)

Substitute Eq. (38) into Eq. (39). Here , the Lyapunov exponent of the fixed point is as follows

(40)

Keep the estimate of unchanged with Eq. (37). Substitute Eq. (38) into Eq. (39). Assuming that and are bounded and integrable, respectively, the Lyapunov exponent can be obtained

(41)

Let , when , is stable, is unstable; when , is unstable, is stable. So is a D-bifurcation point of system (29).

4.2 P-Bifurcation

Eq. (31) and Eq. (32) show that the Itô stochastic differential equation corresponding to is independent. is one-dimensional diffusion process, and its corresponding FPK equation can be expressed as

(42)

The corresponding boundary conditions are

(43)

Based on the boundary conditions (44), stationary probability density of the amplitude can be obtained as follows

(44)

where is normalization constant,

(45)

Substituting Eq. (32) into Eq. (44), the specific expression of stationary probability density of amplitude can be obtained as

(46)

where

(47)

First, the influence of time-delayed feedback control is not considered, set the system parameters as ^[18,19], stochastic P-bifurcation refers to the change in the number of peaks of the probability density function curve. Keep constant, and draw the joint probability density function diagram, section and top view of system (26) under the influence of different fractional orders.

When , the joint probability density function diagram shows crater shape, as shown in Fig. 1(a). There is only one peak in the section and a limit cycle in the system at this time, and the response is shown as vibration far away from the origin, as shown in Fig. 1(b) and Fig. 1(c). When , the diagram shows unimodal, in which a peak can also be seen from the Fig. 2(a) and Fig. 2(b). There is only a balance point in the system, and the randomness of the system response is suppressed, as shown in Fig. 2(c). It can be seen from Fig. 3(a) that when , the probability density curve has a significant peak far from the origin, and the system is monostable with only small vibrations. When is increased to 0.3, the curve has an obvious peak at a distance from the origin, which is in the form of Dirac function near the origin, and the system shows the coexistence of two motion states; continuing to increase to 0.7, and the stationary response of the amplitude of system is zero, which is similar to the stable balance point in the deterministic system. At this time, the randomness of the system is suppressed. Based on the above discussions, it is verified that the increase of fractional order will induce stochastic P-bifurcation behavior of the system.

Fig. 1 Joint probability density function diagram(a), section(b) and top view(c) of system (26) when

Fig. 2 Joint probability density function diagram(a), section(b) and top view(c) of system (26) when

Fig. 3 The stationary probability density function diagram of the system (26) (a) the influence of is not considered, and , (b) the influence of is not considered, and , (c) the influence of is considered, and

Since the influence of delayed feedback control is still not considered, keep the system parameters as: . And keep constant, we draw the joint probability density function diagram, section and top view of system (26) under the influence of different noise strength, as shown in Fig. 4 and Fig. 5.

Fig. 4 Joint probability density function diagram(a), section(b) and top view(c) of system (26) when

Fig. 5 Joint probability density function diagram(a), section(b) and top view(c) of system (26) when

When , the joint probability density function diagram shows crater shape, as shown in Fig. 4(a). There is only one peak in the section and a small limit cycle in the system at this time, and the response is shown as vibration far away from the origin, as shown in Fig. 4(b) and Fig. 4(c). When , in addition to an obvious peak at the origin, the probability density curve also has an obvious peak at a distance from the origin, as shown in Fig. 5(a) and Fig. 5(b). Therefore, the system shows the coexistence of two motion states. At this time, the system shows the coexistence of balance point and limit cycle, as shown in Fig. 5(c). It can be seen from Fig. 3(b) that when , the curve has an obvious peak far from the origin, the system is monostable and only has small amplitude vibration; when is decreased to 0.7, the curve has an obvious peak at a distance from the origin, which is in the form of Dirac function near the origin, and the system shows the coexistence of two motion states; continuing to decrease to 0.6, and the stationary response of the amplitude of system is zero, which is similar to the stable balance point in the deterministic system. At this time, the randomness of the system is suppressed. Based on the above discussions, it is verified that the decrease of noise strength will induce stochastic P-bifurcation behavior of the system.

Finally, the influence of time-delayed feedback control is considered, set the system parameters as ^[18,19] , drawing the joint probability density function diagram, section and top view of system (26) under the influence of different time-delay.

When , the joint probability density function diagram shows crater shape with only one peak in the section, as shown in Fig. 6(a) and Fig. 7(b), and a small limit cycle in the system at this time, and the response is shown as vibration far away from the origin, as shown in Fig. 6(c). When , the diagram shows unimodal. It can also be seen from the section that there is a peak, as shown in Fig. 7(a) and Fig.7(b). There is only a balance point in the system, and the randomness of the system response is suppressed, as shown in Fig. 7(c). As shown in Fig. 3(c), when , the stationary response of the amplitude of system is zero, which is similar to the stable balance point in the deterministic system. At this time, the randomness of the system is suppressed. When is increased to 4, the curve has an obvious peak far from the origin, the system is monostable and only has small amplitude vibration; continuing to increase to 5, the situation is the same as that of . At last, increasing to 8, the situation is the same as that of . Based on the above discussions, it is verified that the change of time-delay will induce stochastic P-bifurcation behavior of the system.

Fig. 6 Probability density function diagram(a), section(b) and top view(c) of system (26) when

Fig. 7 Probability density function diagram(a), section(b) and top view(c) of system (26) when

5 Conclusion

The fractional stochastic vibration system is quite different from the traditional ones, 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 Duffing oscillator with multiplicative recycling noise and time-delay feedback control as an example to study its stationary response and its stochastic bifurcation. First, based on the principle of the minimum mean square error and small time-delayed approximation and linearize the cubic stiffness term, the fractional derivative is found to be equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Then, the Itô differential equations and one-dimensional Markov process are obtained according to stochastic averaging method. The local and global stochastic stability of the system are discussed, and the conditions for inducing D-bifurcation and P-bifurcation of the system are analyzed. The analysis shows that: Let , when , is stable, is unstable; when , is unstable, is stable. So is a D-bifurcation point of system (29). The results show that the fractional order, noise strength, and the time-delay all can greatly affect the system's dynamical properties. This paper offers a profound, original and challenging window for investigating fractional stochastic vibration systems.

References

All Tables

All Figures

	Fig. 1 Joint probability density function diagram(a), section(b) and top view(c) of system (26) when
In the text

	Fig. 2 Joint probability density function diagram(a), section(b) and top view(c) of system (26) when
In the text

	Fig. 3 The stationary probability density function diagram of the system (26) (a) the influence of is not considered, and , (b) the influence of is not considered, and , (c) the influence of is considered, and
In the text

	Fig. 4 Joint probability density function diagram(a), section(b) and top view(c) of system (26) when
In the text

	Fig. 5 Joint probability density function diagram(a), section(b) and top view(c) of system (26) when
In the text

	Fig. 6 Probability density function diagram(a), section(b) and top view(c) of system (26) when
In the text

	Fig. 7 Probability density function diagram(a), section(b) and top view(c) of system (26) when
In the text