Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation

Yang, Yong-Ge; Xu, Wei; Sun, Yahui; Gu, Xudong

doi:10.1088/1674-1056/25/2/020201

Cited by 13 publications

(17 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where ( ) is the Dirac function. The fractional derivative term has contributions from both damping and restoring forces [38][39][40][41][42], hence, we introduce the following equivalent system:…”

Section: Equivalent Van Der Pol Systemmentioning

confidence: 99%

Stochastic P-bifurcation in a tri-stable Van der Pol system with fractional derivative under Gaussian white noise

Li¹,

Wu²

2019

J. vibroeng.

View full text Add to dashboard Cite

In this paper, we study the tri-stable stochastic P-bifurcation problem of a generalized Van der Pol system with fractional derivative under Gaussian white noise excitation. Firstly, using the principle for minimal mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be transformed into an equivalent integer order system. Secondly, we obtain the stationary Probability Density Function (PDF) of the system's amplitude by the stochastic averaging, and using the singularity theory, we find the critical parametric conditions for stochastic P-bifurcation of amplitude of the system, which can make the system switch among the three steady states. Finally, we analyze different types of the stationary PDF curves of the system amplitude qualitatively by choosing parameters corresponding to each region divided by the transition set curves, and the system response can be maintained at the small amplitude near the equilibrium by selecting the appropriate unfolding parameters. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order controller to adjust the response of the system.

show abstract

Section: Equivalent Van Der Pol Systemmentioning

confidence: 99%

Stochastic P-bifurcation in a tri-stable Van der Pol system with fractional derivative under Gaussian white noise

Li¹,

Wu²

2019

J. vibroeng.

View full text Add to dashboard Cite

show abstract

“…The fractional derivative has the contributions of damping force and restoring force [44][45][46][47][48][49][50], hence, we introduce the equivalent system as follows:…”

Section: The Derivation Of Equivalent Systemmentioning

confidence: 99%

“…where C(p, τ ) and K(p, τ ) are the coefficients of equivalent damping force and equivalent restoring force of the fractional derivative C 0 D p [x(tτ )], respectively. Applying the equivalent methods described in the literature [29,[49][50][51], the ultimate forms of C(p, τ ) and K(p, τ ) can be obtained as follows:…”

Section: The Derivation Of Equivalent Systemmentioning

confidence: 99%

Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

Zhang

et al. 2019

Adv Differ Equ

View full text Add to dashboard Cite

The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

show abstract

“…where = √ 1 . The fractional derivative term has contributions to both damping and restoring forces [44][45][46][47], hence, introducing the following equivalent system:…”

Section: Equation Of Axially Moving Viscoelastic Beammentioning

confidence: 99%

“…Taking the parameters as = 0.5, 3 = 7.8, 5 = 5.9, and = 1, according to (46) and (47), the transition set for stochastic P-bifurcation of the system with the unfolding parameters and can be obtained, as shown in Figure 4.…”

Section: Stochastic P-bifurcationmentioning

confidence: 99%

Stochastic P-Bifurcation of a Bistable Viscoelastic Beam with Fractional Constitutive Relation under Gaussian White Noise

Zhang

et al. 2018

Discrete Dynamics in Nature and Society

View full text Add to dashboard Cite

In this paper, we study the stochastic P-bifurcation problem for axially moving of a bistable viscoelastic beam with fractional derivatives of high order nonlinear terms under Gaussian white noise excitation. First, using the principle for minimum mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be simplified to an equivalent system. Second, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by stochastic averaging, and using singularity theory, we find the critical parametric condition for stochastic P-bifurcation of amplitude of the system. Finally, we analyze the types of the stationary PDF curves of the system qualitatively by choosing parameters corresponding to each region within the transition set curve. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order viscoelastic material model to adjust the response of the system.

show abstract

Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation

Cited by 13 publications

References 29 publications

Stochastic P-bifurcation in a tri-stable Van der Pol system with fractional derivative under Gaussian white noise

Stochastic P-bifurcation in a tri-stable Van der Pol system with fractional derivative under Gaussian white noise

Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

Stochastic P-Bifurcation of a Bistable Viscoelastic Beam with Fractional Constitutive Relation under Gaussian White Noise

Contact Info

Product

Resources

About