Coherence resonance in fractional van der Pol oscillators

The phenomenon of coherence resonance (CR) in two typical kinds of fractional van del Pol oscillator is investigated. For the first model, the damping term of the ordinary van der Pol oscillator is replaced by a fractional-order damping. While in the second model, the fractional-order damping is instead of the inertia term of the ordinary van der Pol oscillator. In the first model, there is obvious CR by adjusting the noise intensity. The resonance frequency mainly depends on the value of the fractional-order but it is almost independent of the noise intensity when the noise intensity is small, but change with the noise intensity when the noise intensity lies in a slightly large range. However, the resonance frequency of the ordinary van der Pol oscillator does not influence by the noise. In addition, with the increase of the fractional-order, the resonance frequency decreases monotonicity. Whereas, the resonance amplitude is a nonmonotonic function of the fractional-order. When the fractional oscillator deviates from the ordinary one, the resonance amplitude is much greater. In the second model, CR occurs only when the fractional oscillator is very close to the ordinary van der Pol oscillator. Comparing CR in the first model with that in the second model, we find that the value of the fractional-order can be taken over a much larger range to induce CR for the first one. The study shows that the inertia term is an indispensable factor in causing CR phenomenon while the fractional-order damping only changes the resonance frequency in different kinds of van der Pol oscillators. Graphic abstract Coherence resonance phenomenon in the fractional van der Pol oscillator under different values of the fractional-order and noise intensity.