Non-Linear Resonances and Chaotic Dynamics of a Rotating Gyroscope Under a Fractional Order Derivative Damping

This paper deals with the action of the fractional order derivative on nonlinear resonances and its analysis on the chaotic dynamics of an excited rotating gyroscope has been investigated. After the mathematical modeling of the system, we determined the possible resonances of the system using the method of multiple scales. The stability conditions for each resonance were obtained using the Routh-Hurwitz criterion. The different system parameters have been studied and it is concluded that the different resonance states can be controlled by each of the system parameters. Moreover, using bifurcation diagrams, Lyapunov exponents, phase portraits and time series, it was shown that for certain values of the parameters the considered gyroscope exhibits several rich dynamics. The chaos has been reduced, even eliminated with the progressive increase of the other parameters, in particular of the order \( \mu \) $\mu$ of the fractional derivative up to a certain value.