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THERMAL SCIENCE
International Scientific Journal
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TRANSMISSION DYNAMICS ANALYSIS OF A BISTABLE VAN DER POL OSCILLATOR UNDER GAUSSIAN COLORED NOISE WITH FRACTIONAL INERTIAL ELEMENT
ABSTRACT
The stochastic response and bifurcation phenomenon of system amplitude in a bi-stable Van der Pol system with fractional inertial element driven by the Gaussian colored noise is investigated. In the initial step of the process, the mean square error is minimized, and the harmonic balance technique is employed. This results in the fractional derivative being found to be isovalent to a linear combination of inertial and damping forces. Simultaneously, the original system is simplified to an equivalent integer order Van der Pol system. Secondly, the steady-state probability density function of the system amplitude is acquired via stochastic averaging. According to the principles of singularity theory, the critical parametric conditions for the stochastic P-bifurcation of system amplitude can be deter-mined. A qualitative analysis of the stationary PDF curves of amplitude is finally conducted in each area, with the transition set curves serving as a dividing point. The congruence between the analytical outcomes and the numerical results obtained from Monte Carlo simulation and the radial basis function neural network testifies to the theoretical analysis. In this paper, the methodology employed and the results obtained can enhance the design of the fractional-order controller to control the response of these systems
KEYWORDS
stochastic P-bifurcation, Gaussian colored noise, fractional damping, critical parametric conditions, Monte Carlo simulation
PAPER SUBMITTED: 2024-12-01
PAPER REVISED: 2025-08-15
PAPER ACCEPTED: 2025-08-16
PUBLISHED ONLINE: 2026-04-12
DOI REFERENCE: https://doi.org/10.2298/TSCI2602259L
CITATION EXPORT: view in browser or download as text file
REFERENCES
[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 G: Physics, Mechanics and Astronomy, 46 (2003), 2, pp. 145-157
[2] Sabatier, J., et al., Advances in Fractional Calculus, Springer, Amsterdam, The Netherlands, 2007
[3] Podlubny, I., Fractional-Order Systems and PIλDμ Controllers, IEEE Transactions On Automatic Control, 44 (1999), 1, pp. 208-214
[4] Monje, C. A., et al., Fractional-Order Systems and Controls: Fundamentals and Applications, Springer-Verlag, London, 2010
[5] Bagley, R. L., Torvik, P. L., Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures, AIAA Journal, 23 (2012), 6, pp. 918-925
[6] Machado, J. A. T., Fractional Order Modelling of Fractional-Order Holds, Nonlinear Dynamics, 70 (2012), 1, pp. 789-796
[7] Machado, J. T., Fractional Calculus: Application in Modeling and Control, Springer: New York, USA, 2013
[8] Liu, L., et al., Stochastic Bifurcation of a Strongly Nonlinear Vibro-Impact System with Coulomb Friction under Real Noise, Symmtry-Basel, 11 (2019), 1, pp. 4-15
[9] Zhu, Z., et al., Bifurcation Characteristics and Safe Basin of MSMA Microgripper Subjected to Stochastic Excitation, AIP Advances, 5 (2015), 2, 207124
[10] Gu, R. C., et al., Stochastic Bifurcations in Duffing-Van der Pol Oscillator with Levy Stable Noise (in Chinese), Acta Physica Sinica, 60 (2011), 6, pp. 157-161
[11] Zhang, X., et al., Tipping of Thermoacoustic Systems with Secondary Bifurcation in Colored Noise Environments: Effects of Rate, Chaos, 35 (2025), 033124
[12] Wu, Z. Q., Hao, Y., Three-Peak P-Bifurcations in Stochastically Excited Van der Pol-Duffing Oscillator (in Chinese), Science China Physics, Mechanics & Astronomy, 43 (2013), 4, pp. 524-529
[13] Wu, Z. Q., Hao, Y., Stochastic P-Bifurcations in Tri-Stable Van der Pol-Duffing Oscillator with Multiplicative Colored Noise (in Chinese), Acta Physica Sinica, 64 (2015), 060501
[14] Hao, Y., Wu, Z. Q., Stochastic P-Bifurcation of Tri-Stable Van der Pol-Duffing Oscillator (in Chinese), Chinese Journal of Theoretical and Applied Mechanics, 45 (2013), 2, pp. 257-264
[15] 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 Nonlinear Mechanics, 46 (2011), 10, pp. 1324-1329
[16] Huang, Z. L., Jin, X. L., Response and Stability of a SDOF Strongly Nonlinear Stochastic System with Light Damping Modeled by a Fractional Derivative, Journal of Sound and Vibration, 319 (2009), 3, pp. 1121-1135
[17] 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
[18] 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), June, pp. 109-115
[19] Chen, J. F., et al., Primary Resonance of van der Pol Oscillator under Fractional-Order Delayed Feed-back and Forced Excitation, Shock and Vibration, 2017 (2017), 1, pp. 1-9
[20] 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
[21] He, C. H., El-Dib, Y. O., A Heuristic Review on the Homotopy Perturbation Method for Non-Conservative Oscillators, J. Low Freq. Noise V. A., 41 (2022), 2, pp. 572-603
[22] He, C. H., et al., Controlling the Kinematics of a Spring-Pendulum System Using an Energy Harvesting Device, J. Low Freq. Noise V. A., 41 (2022), 3, pp. 1234-1257
[23] He, C. H., et al., Hybrid Rayleigh-Van der Pol-Duffing Oscillator: Stability Analysis and Controller, J. Low Freq. Noise V. A., 41 (2022), 1, pp. 244-268
[24] He, J.-H., et al., Modelling of the Rotational Motion of 6-DOF Rigid Body According to the Bobylev-Steklov Conditions, Results Phys, 35 (2022), 105391
[25] He, J.-H., et al., Stability of Three Degrees-Of-Freedom Auto-Parametric System, Alex. Eng. J., 61 (2022), 11, pp. 8393-8415
[26] Li, Y. J., et al., Stochastic Transition Behaviors in a Tri-Stable Van der Pol Oscillator with Fractional Delayed Element Subject to Gaussian White Noise, Thermal Science, 26 (2022), 3, pp. 2713- 2725
[27] Li, Y. J., et al., Transition Behaviors of System Energy in a Bi-Stable Van der Pol Oscillator with Fractional Derivative Element Driven by Multiplicative Gaussian White Noise, Thermal Science, 26 (2022), 3, pp. 2727-2736
[28] Shen, Y., et al., A Periodic Solution of the Fractional Sine-Gordon Equation Arising in Architectural Engineering, J. Low Freq. Noise V. A., 40 (2021), 2, pp. 683-691
[29] He, J.-H., Fractal Calculus and its Geometrical Explanation, Results Phys., 10 (2018), Sept., pp. 272-276
[30] Chen, L. C, et al., Stationary Response of Duffing Oscillator with Hardening Stiffness and Fractional Derivative, International Journal of Non-Linear Mechanics, 48 (2013), Jan., pp. 44-50
[31] 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
[32] Zhu, W. Q., Random Vibration (in Chinese), Science Press, Beijing, 1992
[33] Ling, F. H., Catastrophe Theory and its Applications (in Chinese), Shang Hai Jiao Tong University Press, Shanghai, China, 1987
[34] Petras, I., Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Higher Education Press, Beijing, 2011
[35] Petras, I., Tuning and Implementation Methods for Fractional-Order Controllers, Fractional Calculus and Applied Analysis, 15 (2012), 2, pp. 282-303
[36] Agrawal, O. P., A General Formulation and Solution Scheme for Fractional Optimal Control Problems, Nonlinear Dynamics, 38 (2019), 1, pp. 323-337
[37] Shah, P., Agashe, S., Review of Fractional PID Controller, Mechatronics, 38 (2016), Sept., pp. 29-41
[38] Zuo, Y. T., et al., Gecko-Inspired Fractal Buffer for Passenger Elevator, Facta Universitatis-Series Mechanical Engineering, 22 (2024), 4, pp. 633-642
[39] Cheng, Y., et al., Differential Equation-Driven Intelligent Control: Integrating AI, Quantum Computing, and Adaptive Strategies for Next-Generation Industrial Automation, Advances in Differential Equations and Control Processes, 32 (2025), 3096
[40] He, C.-H., Mohammadian, M., A Fast Insight into High-Accuracy Nonlinear Frequency Estimation of Stringer-Stiffened Shells, Facta Universitatis-Series Mechanical Engineering, 23 (2025), 4, pp. 787-806
[41] He, C.-H., et al., A Modified Frequency Formulation for Nonlinear Mechanical Vibrations, Facta Universitatis-Series Mechanical Engineering, 23 (2025), 2, pp. 197-210
© 2026 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence