频率切换引起的振荡器老化
Ageing of an oscillator due to frequency switching
摘要：
如果一个振荡器是由一个在两个频率之间切换的力驱动的，那么它所表现出的动态特性取决于切换的精确方式。在这里，我们采取一个一维振荡器，并考虑切换发生的情况：（i）在两个驱动力之间具有不同的频率，或（ii）作为一个单一的强制，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，并涉及开关是否可以线性或非线性地表示在一个不连续的量（如符号或Heaviside阶梯函数，表示频率之间的切换）。在情形（i）中，振荡器具有稳定的周期轨道，并且系统可以被描述为Filippov系统。在情况（ii）的振荡器表现出隐藏的动态，这是在理论的Filippov的系统，并导致系统越来越多（随着时间的推移）占主导地位的滑动沿频率切换阈值，特别是如果周期轨道确实存在，他们也表现出滑动。我们表明，如果系统被正则化（即，如果开关被建模为（i）或（ii）的方式平滑过渡），则该行为至少渐近地持续存在。
英文摘要：
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)).
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