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THERMAL SCIENCE
International Scientific Journal
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STOCHASTIC BIFURCATION AND BISTABLE BEHAVIORS IN A MODIFIED FRACTIONAL RAYLEIGH SYSTEM DRIVEN BY RECYCLING NOISE
ABSTRACT
The present study investigates the stochastic response and bifurcation behavior of system amplitude in a fractional and generalized bi-stable Van der Pol system driven by Gaussian colored noise. Firstly, the principle of minimal mean square error and the generalized harmonic balance technique were employed to demonstrate that the fractional derivative is equivalent to a linear combination of damping and restoring forces. Consequently, the original system was simplified to an equivalent integer order Van der Pol system. Secondly, the system amplitude's stationary probability density function is acquired by stochastic averaging. According to the principles of singularity theory, the critical parametric conditions for the stochastic P-bifurcation of system amplitude can be determined. A qualitative analysis of the stationary probability density function 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 derived from the Monte-Carlo simulation substantiates the theoretical analysis in this paper. The methodology employed and the results obtained in this paper can enhance the design of the fractional-order controller to regulate the response of the system.
KEYWORDS
stochastic P-bifurcation, Gaussian colored noise, fractional damping, critical parametric conditions, Monte-Carlo simulation
PAPER SUBMITTED: 2024-11-11
PAPER REVISED: 2025-03-19
PAPER ACCEPTED: 2025-03-20
PUBLISHED ONLINE: 2026-04-12
DOI REFERENCE: https://doi.org/10.2298/TSCI2602855C
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