论文标题
由于频率切换而导致振荡器的老化
Ageing of an oscillator due to frequency switching
论文作者
论文摘要
如果振荡器由在两个频率之间切换的力驱动，则其表现出的动力学取决于切换的精确方式。在这里，我们采用一维振荡器，并考虑发生切换的情况：（i）在两个具有不同频率的驱动力之间，或（ii）作为单个强迫，其频率在两个值之间切换。差异是微妙的，但完全改变了长期行为，并且关注开关是否可以在不连续数量的情况下线性或非线性表示（例如代表频率之间开关的符号或heaviside步骤函数）。在情况下（i）振荡器具有稳定的周期轨道，可以将系统描述为Filippov系统。在场景（ii）中，振荡器表现出隐藏的动力学，位于Filippov系统的理论之外，并导致该系统越来越（随着时间的流逝）沿着频率转换阈值滑动而占主导地位，尤其是如果周期性的轨道确实存在，它们也表现出滑动。我们表明，如果系统是正则化的（即，如果开关以（i）或（ii）的方式将开关建模为平滑过渡，则至少渐近地存在行为。
If an oscillator is driven by a force that switches between two frequencies, the dynamics it exhibits depends on the precise manner of switching. Here we take a one-dimensional oscillator and consider scenarios in which switching occurs: (i) between two driving forces which have different frequencies, or (ii) as a single forcing whose frequency switches between two values. The difference is subtle, but entirely changes the long term behaviour, and concerns whether the switch can be expressed linearly or nonlinearly in terms of a discontinuous quantity (such as a sign or Heaviside step function that represents the switch between frequencies). In scenario (i) the oscillator has a stable periodic orbit, and the system can be described as a Filippov system. In scenario (ii) the oscillator exhibits hidden dynamics, which lies outside the theory of Filippov's systems, and causes the system to be increasingly (as time passes) dominated by sliding along the frequency-switching threshold, and in particular if periodic orbits do exist, they too exhibit sliding. We show that the behaviour persists, at least asymptotically, if the systems are regularized (i.e. if the switch is modelled as a smooth transition in the manner of (i) or (ii)).
