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THERMAL SCIENCE
International Scientific Journal
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STOCHASTIC STABILITY OF THE FRACTIONAL AND TRI-STABLE VAN DER VOL OSCILLATOR WITH TIME-DELAY FEEDBACK DRIVEN BY GAUSSIAN WHITE NOISE
ABSTRACT
The stochastic P-bifurcation behavior of tri-stability in a fractional-order van der Pol system with time-delay feedback under additive Gaussian white noise excitation is investigated. Firstly, according to the equivalent principle, the fractional derivative and the time-delay term can be equivalent to a linear combination of damping and restoring forces, so the original system can be simplified into an equivalent integer-order system. Secondly, the stationary probability density function of the system amplitude is obtained by the stochastic averaging, and based on the singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the properties of stationary probability density function curves of the system amplitude are qualitatively analyzed by choosing corresponding parameters in each sub-region divided by the transition set curves. The consistence between numerical results obtained by Monte-Carlo simulation and analytical solutions has verified the accuracy of the theoretical analysis. The method used in this paper has a direct guidance in the design of fractional-order controller to adjust the dynamic behavior of the system.
KEYWORDS
stochastic P-bifurcation, fractional derivative, gaussian white noises, transition set curves, Monte-Carlo simulation
PAPER SUBMITTED: 2022-03-01
PAPER REVISED: 2022-07-24
PAPER ACCEPTED: 2022-07-24
PUBLISHED ONLINE: 2023-06-11
DOI REFERENCE: https://doi.org/10.2298/TSCI2303155L
CITATION EXPORT: view in browser or download as text file
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© 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