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.

中文翻译：

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分数范德波尔振荡器中的相干共振

研究了两种典型的分数范德尔波尔振荡器的相干共振（CR）现象。对于第一个模型，普通范德波尔振荡器的阻尼项被分数阶阻尼代替。而在第二个模型中，分数阶阻尼代替了普通范德波尔振荡器的惯性项。在第一个模型中，通过调整噪声强度可以得到明显的CR。谐振频率主要取决于分数阶的值，但当噪声强度较小时，谐振频率几乎与噪声强度无关，而当噪声强度在稍大的范围内时，谐振频率随噪声强度变化。然而，普通范德波尔振荡器的谐振频率不受噪声的影响。另外，随着分数阶的增加，谐振频率的单调性降低。而共振幅度是分数阶的非单调函数。当分数振荡器偏离普通振荡器时，谐振幅度要大得多。在第二个模型中，仅当分数振荡器非常接近普通范德波尔振荡器时才会发生 CR。将第一个模型中的 CR 与第二个模型中的 CR 进行比较，我们发现分数阶的值可以在更大的范围内取值来诱导第一个模型的 CR。研究表明，惯性项是引起CR现象的不可缺少的因素，而分数阶阻尼仅改变不同类型范德波尔振子的谐振频率。

图文摘要

不同分数阶和噪声强度值下分数范德波尔振荡器的相干共振现象。