International Scientific Journal

Thermal Science - Online First

online first only

Stochastic bifurcation analysis of a bistable Duffing oscillator with fractional damping under multiplicative noise excitation

The stochastic P-bifurcation behavior of bi-stability in a Duffing oscillator with fractional damping under multiplicative noise excitation is investigated. Firstly, in order to consider the influence of Duffing term, the nonlinear stiffness can be equivalent to a linear stiffness which is a function of the system amplitude, and then, using the principle of minimal mean square error, the fractional derivative term can be equivalent to a linear combination of damping and restoring forces, thus, the original system is simplified to an equivalent integer order Duffing system. Secondly, the system amplitude's stationary Probability Density Function (PDF) is obtained by stochastic averaging, and then according to the singularity theory, the critical parametric conditions for the system amplitude's stochastic P-bifurcation are found. Finally, the types of the system's stationary PDF curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte Carlo simulation verifies the theoretical analysis, and the method used in this paper can directly guide the design of the fractional order controller to adjust the behaviors of the system.
PAPER REVISED: 2020-06-20
PAPER ACCEPTED: 2020-06-20
  1. Xu, M., Tan, W., Representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solutions. Science in China. Series A, 46 (2003), 2, pp. 145-157
  2. Sabatier, J., Agrawal, O. P., Machado, J. A. Advances in Fractional Calculus; Springer: Netherlands, 2007.
  3. Monje, C. A., et al., Fractional-order Systems and Controls: Fundamentals and Applications; Springer-Verlag: London, 2010.
  4. Bagley, R. L., Torvik, P. L., Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA Journal, 23 (2012), 6, pp. 918-925
  5. Machado, J. A. T., Fractional order modelling of fractional-order holds. Nonlinear Dynamics, 70 (2012), 1, pp. 789-796
  6. Machado, J. T., Fractional Calculus: Application in Modeling and Control; Springer: New York, 2013.
  7. Machado, J. A. T., Antonio, C. C., Quelhas, M. D. Fractional dynamics in DNA. Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 8, pp. 2963-2969
  8. Liu, L., et al., Stochastic Bifurcation of a Strongly Non-Linear Vibro-Impact System with Coulomb Friction under Real Noise. Symmetry, 11 (2019), 1, pp. 4-15
  9. He, J. H., Ain Q. T., New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle. Thermal Science, 24 (2020), 2, pp. 659-681
  10. Zhu, Z. W., et al., Bifurcation characteristics and safe basin of MSMA microgripper subjected to stochastic excitation. AIP Advances, 5 (2015), 2, pp. 207124
  11. Li Y. J., et al., Stochastic P-bifurcation in a generalized Van der Pol oscillator with fractional delayed feedback excited by combined Gaussian white noise excitations. Journal of Low Frequency Noise Vibration and Active Control, 10 (2019), pp. 1-13
  12. Xu, Y., et al., Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise. Physical Review E, 83 (2011), 5, pp. 056215
  13. Chen, L. C., Zhu, W. Q., Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations. International Journal of Non-linear Mechanics, 46 (2011), 10, pp. 1324-1329
  14. Li, W., et al., Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise. Chinese Physics B, 26 (2017), 9, pp. 62-69
  15. Liu, W., et al., Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations. Probabilist Engineering Mechanics, 53 (2018), pp. 109-115
  16. Li Y. J., et al., Stochastic P-bifurcation in a nonlinear viscoelastic beam model with fractional constitutive relation under colored noise excitation. Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1466-1480
  17. Chen, L. C., et al., Stochastic averaging technique for SDOF strongly nonlinear systems with delayed feedback fractional-order PD controller. Science China-Technological Sciences, 62 (2018), 2, pp. 287-297
  18. Chen, L. C., et al., Stationary response of Duffing oscillator with hardening stiffness and fractional derivative. International Journal of Non-linear Mechanics, 48 (2013), pp. 44-50
  19. Yang, Y. G., et al., Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation. Chinese Physics B, 25 (2016), 2, pp. 13-21
  20. Chen, L. C., Zhu, W. Q., Stochastic response of fractional-order van der Pol oscillator. Theoretical & Applied Mechanics Letters, 4 (2014), 1, pp. 68-72
  21. Sun, C. Y., Xu, W., Stationary response analysis for a stochastic Duffing oscillator comprising fractional derivative element. Journal of Vibration Engineering, 28 (2015), 3, pp. 374-380 (in Chinese)
  22. Spanos, P. D., Zeldin, B. A., Random Vibration of Systems with Frequency-Dependent Parameters or Fractional Derivatives. Journal of Engineering Mechanics, 123 ( 1997), 3, pp. 290-292
  23. Zhu, W. Q., Random Vibration, Science Press: Beijing, 1992. (in Chinese)
  24. Ling, F. H., Catastrophe Theory and its Applications, Shang Hai Jiao Tong University Press: Shanghai, 1987. (in Chinese)
  25. Petráš, I., Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation, Higher Education Press: Beijing, 2011.
  26. Petráš, O. P., A General Formulation and Solution Scheme for Fractional Optimal Control Problems. Nonlinear Dynamics, 38 (2004), pp. 323-337
  27. Shah, P., Agashe, S., Review of fractional PID controller. Mechatronics, 38 (2016), pp. 29-41