Institute of Physics, University of São Paulo, São Paulo, Brazil.
Institute for Complex Systems and Mathematical Biology, SUPA, University of Aberdeen, Aberdeen, United Kingdom.
Sci Rep. 2017 Feb 9;7:42351. doi: 10.1038/srep42351.
Nonlinear dynamical systems may be exposed to tipping points, critical thresholds at which small changes in the external inputs or in the system's parameters abruptly shift the system to an alternative state with a contrasting dynamical behavior. While tipping in a fold bifurcation of an equilibrium is well understood, much less is known about tipping of oscillations (limit cycles) though this dynamics are the typical response of many natural systems to a periodic external forcing, like e.g. seasonal forcing in ecology and climate sciences. We provide a detailed analysis of tipping phenomena in periodically forced systems and show that, when limit cycles are considered, a transient structure, so-called channel, plays a fundamental role in the transition. Specifically, we demonstrate that trajectories crossing such channel conserve, for a characteristic time, the twisting behavior of the stable limit cycle destroyed in the fold bifurcation of cycles. As a consequence, this channel acts like a "ghost" of the limit cycle destroyed in the critical transition and instead of the expected abrupt transition we find a smooth one. This smoothness is also the reason that it is difficult to precisely determine the transition point employing the usual indicators of tipping points, like critical slowing down and flickering.
非线性动力系统可能会面临临界点，即外部输入或系统参数的微小变化会突然使系统转变为具有相反动力学行为的替代状态的关键阈值。虽然在平衡分岔的临界点已经得到很好的理解，但对于振荡（极限环）的临界点却知之甚少，尽管这种动力学是许多自然系统对周期性外部强迫的典型响应，例如生态学和气候科学中的季节性强迫。我们提供了对周期性强迫系统中临界点现象的详细分析，并表明，当考虑极限环时，所谓的通道这样的瞬变结构在过渡中起着基本作用。具体来说，我们证明了穿过这样的通道的轨迹在特征时间内保持在循环分岔中破坏的稳定极限环的扭曲行为。因此，这个通道就像在关键跃迁中破坏的极限环的“幽灵”，而不是预期的突然跃迁，我们发现的是一个平滑的跃迁。这种平滑性也是为什么使用临界点的常用指标（如临界减速和闪烁）很难准确确定转折点的原因。
