Manchester Metropolitan University, School of Computing, Mathematics and Digital Technology, Manchester M1 5GD, United Kingdom.
Chaos. 2011 Jun;21(2):023126. doi: 10.1063/1.3596708.
In this paper we study the qualitative dynamics of piecewise-smooth slow-fast systems (singularly perturbed systems) which are everywhere continuous. We consider phase space topology of systems with one-dimensional slow dynamics and one-dimensional fast dynamics. The slow manifold of the reduced system is formed by a piecewise-continuous curve, and the differentiability is lost across the switching surface. In the full system the slow manifold is no longer continuous, and there is an O(ɛ) discontinuity across the switching manifold, but the discontinuity cannot qualitatively alter system dynamics. Revealed phase space topology is used to unfold qualitative dynamics of planar slow-fast systems with an equilibrium point on the switching surface. In this case the local dynamics corresponds to so-called boundary-equilibrium bifurcations, and four qualitative phase portraits are uncovered. Our results are then used to investigate the dynamics of a box model of a thermohaline circulation, and the presence of a boundary-equilibrium bifurcation of a fold type is shown.
在本文中，我们研究了处处连续的分段光滑慢-快系统（奇异摄动系统）的定性动力学。我们考虑了具有一维慢动力学和一维快动力学的系统的相空间拓扑。简化系统的慢流形由分段连续曲线形成，在切换面处失去可微性。在全系统中，慢流形不再连续，并且在切换流形上存在 O(ɛ)不连续性，但不连续性不能从根本上改变系统动力学。揭示的相空间拓扑用于展开具有切换面上平衡点的平面慢-快系统的定性动力学。在这种情况下，局部动力学对应于所谓的边界平衡分岔，并且揭示了四个定性相图。然后，我们将这些结果用于研究热盐环流的一个箱式模型的动力学，并显示出存在折叠型边界平衡分岔。
