论文标题
由频率转换力驱动的振荡器中的新型慢速行为
Novel slow-fast behaviour in an oscillator driven by a frequency-switching force
论文作者
论文摘要
当振荡器在不同频率之间突然切换时，决定如何在开关上建模系统存在一些歧义。在这里，我们描述了一个简单的周期性谐波振荡器中的两个看似自然的模型，这在对其长时间行为的预测中明确地不同意。试图通过“正规化”开关来解决分歧不仅保留分歧，而且表明它随时间增加。其中一个模型对应于非平滑系统的常规“ Filippov”描述，而第二个模型则表现出不可逆转的衰老结构，开发了许多新型的多尺度行为，我们认为这些行为以前尚未报道过。这些包括缓慢的楼梯，新型的混合模式振荡和同步的牛排爆炸。这些特征被证明是使用渐近分析存在的，但是由于它们涉及缓慢的时间尺度分离，随着时间的增加，它们超出了数值方法的范围。
When an oscillator switches abruptly between different frequencies, there is some ambiguity in deciding how the system should be modelled at the switch. Here we describe two seemingly natural models of a switch in a simple periodically-forced harmonic oscillator, which disagree starkly in their predictions of its long time behaviour. Attempting to resolve the disagreement by `regularizing' the switch not only preserves the disagreement, but shows it increases with time. One of the models corresponds to a conventional `Filippov' description of a nonsmooth system, while the second exhibits a structure that irreversibly ages, developing a number of novel multi-scale behaviours that we believe have not been reported before. These include slow-fast staircases, novel mixed-mode oscillations, and a synchronized canard explosion. These features are proven to exist using asymptotic analysis, but as they involve a slow-fast time-scale separation that increases with time, they lie beyond the reach of numerical methods.
