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Abstract:
Numerous studies have explored the complex behaviors of stochastic fractional-order multirhythmic models and stochastic vibro-impact multirhythmic models. However, discussions about stochastic trirhythmic systems that include both fractional derivative elements in acceleration and vibro-impact parameters are relatively rare. This paper focuses on managing the trirhythmicity and stochastic bifurcation of a new extended fractional biological vibro-impact system subjected to correlated noise. To start, we replace the viscoelastic force with damping and stiffness forces. This change allows us to eliminate the original trirhythmic system’s discontinuity through a non-smooth transformation, resulting in a stochastic trirhythmic system with fractional derivatives. Subsequently we employ random averaging to obtain estimate analytical solutions. We discovered that characteristics like correlation time, restitution coefficient, and fractional derivative coefficient may contribute to stochastic bifurcation. Simply altering the fractional order or restriction variable causes the probability density function to move between unimodal, bimodal, and trimodal distributions, revealing the presence of stochastic bifurcations inside the fractional multirhythmic setup. Furthermore, our research highlights that controlling system parameters including fractional order, control parameter, restitution coefficient, and correlation time can effectively govern bifurcation behaviors. The strong alignment between the theoretical and numerical solutions obtained from the predictor-corrector algorithm, as well as between the theoretical solutions and Monte Carlo simulations, confirms the accuracy of our conclusions. Additionally, our findings suggest that the noise intensity level at which the maximum Lyapunov exponent changes sign is affected by the fractional order and restitution coefficient in the system parameters. We also demonstrate that as energy loss during impact increases, the likelihood of vibro-impact systems exhibiting chaotic behavior significantly rises.
