Simpson D J W, Jeffrey M R
Institute of Fundamental Sciences , Massey University , Palmerston North, New Zealand.
Department of Engineering Mathematics , University of Bristol , Bristol, UK.
Proc Math Phys Eng Sci. 2016 Feb;472(2186):20150782. doi: 10.1098/rspa.2015.0782.
A two-fold is a singular point on the discontinuity surface of a piecewise-smooth vector field, at which the vector field is tangent to the discontinuity surface on both sides. If an orbit passes through an invisible two-fold (also known as a Teixeira singularity) before settling to regular periodic motion, then the phase of that motion cannot be determined from initial conditions, and, in the presence of small noise, the asymptotic phase of a large number of sample solutions is highly random. In this paper, we show how the probability distribution of the asymptotic phase depends on the global nonlinear dynamics. We also show how the phase of a smooth oscillator can be randomized by applying a simple discontinuous control law that generates an invisible two-fold. We propose that such a control law can be used to desynchronize a collection of oscillators, and that this manner of phase randomization is fast compared with existing methods (which use fixed points as phase singularities), because there is no slowing of the dynamics near a two-fold.
双折点是分段光滑向量场不连续面上的奇点，在该点向量场在两侧均与不连续面相切。如果一条轨道在进入规则周期运动之前经过一个不可见双折点（也称为特谢拉奇点），那么该运动的相位无法从初始条件确定，并且在存在小噪声的情况下，大量样本解的渐近相位是高度随机的。在本文中，我们展示了渐近相位的概率分布如何依赖于全局非线性动力学。我们还展示了如何通过应用一个简单的产生不可见双折点的不连续控制律来使光滑振荡器的相位随机化。我们提出这样的控制律可用于使一组振荡器去同步，并且与现有方法（使用不动点作为相位奇点）相比，这种相位随机化方式速度更快，因为在双折点附近动力学不会减慢。
