Resonance behavior for a trapped particle described by a three-dimensional fractional Langevin equation

•The first moment of a three-dimensional fractional-order stochastic equation is derived.•The asymptotic behaviors of relaxation function have been discussed.•The periodicity of the time series of the first moment is addressed.•Material resonance behaviors of the output amplitude are addressed. The stochastic resonance (SR) behavior of a trapped particle characterized by a three-dimensional fractional-order stochastic equation with Markovian trichotomous noise has been studied in our work. Significantly, an exact expression of the first moment is derived by Shapiro-Loginov formula. Furthermore, the non-monotonic behavior of output amplitude is further discussed. Moreover, the asymptotic behaviors of relaxation function have been discussed. And the periodicity of the time series of the first moment is also addressed to verify the exact expression of the first moment in a long time. Particularly, the SR phenomenon induced by the trichotomous noise is studied. It is worthy to mention that the existence of SR in a fractional Langevin equation (FLa graphical method. Material resonance behaviors of the output amplitude versus the system parameters and noise parameters are extensively investigated. Especially, reverse SR phenomenon, stochastic multiresonance phenomena and bona fide SR are addressed in detail.