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We investigate the stochastic resonance (SR) phenomenon in a fractional oscillator with random mass under the external periodic force. The fluctuations of the mass are modeled as a trichotomous noise. Applying the Shapiro-Loginov formula and the Laplace transform technique, we obtain the exact expression of the first moment of the system. The non-monotonic behaviors of the spectral amplification (SPA) versus the driving frequency indicate that the bona fide SR appears. The necessary and sufficient conditions for the emergence of the generalized stochastic resonance phenomena on the noise flatness and on the noise intensity in the particular case of = ω 0 , v → 0 are established. Particularly, the hypersensitive response of the SPA to the noise intensity is found, which is previously reported and believed to be absent in the case of dichotomous noise.

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In dynamical systems theory, a system which can be described by differential equations is called a continuous dynamical system. In studies on genetic oscillation, most deterministic models at early stage are usually built on ordinary differential equations (ODE). Therefore, gene transcription which is a vital part in genetic oscillation is presupposed to be a continuous dynamical system by default. However, recent studies argued that discontinuous transcription might be more common than continuous transcription. In this paper, by appending the inserted silent interval lying between two neighboring transcriptional events to the end of the preceding event, we established that the running time for an intact transcriptional event increases and gene transcription thus shows slow dynamics. By globally replacing the original time increment for each state increment by a larger one, we introduced fractional differential equations (FDE) to describe such globally slow transcription. The impact of fractionization on genetic oscillation was then studied in two early stage models – the Goodwin oscillator and the Rössler oscillator. By constructing a “dual memory” oscillator – the fractional delay Goodwin oscillator, we suggested that four general requirements for generating genetic oscillation should be revised to be negative feedback, sufficient nonlinearity, sufficient memory and proper balancing of timescale. The numerical study of the fractional Rössler oscillator implied that the globally slow transcription tends to lower the chance of a coupled or more complex nonlinear genetic oscillatory system behaving chaotically.