Department of Mathematics, Cornell University, Ithaca, New York 14853, USA.
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand.
Chaos. 2015 Sep;25(9):097604. doi: 10.1063/1.4915528.
Invariant manifolds are key objects in describing how trajectories partition the phase spaces of a dynamical system. Examples include stable, unstable, and center manifolds of equilibria and periodic orbits, quasiperiodic invariant tori, and slow manifolds of systems with multiple timescales. Changes in these objects and their intersections with variation of system parameters give rise to global bifurcations. Bifurcation manifolds in the parameter spaces of multi-parameter families of dynamical systems also play a prominent role in dynamical systems theory. Much progress has been made in developing theory and computational methods for invariant manifolds during the past 25 years. This article highlights some of these achievements and remaining open problems.
不变流形是描述动力系统轨迹如何划分相空间的关键对象。例如，平衡点和周期轨道的稳定、不稳定和中心流形，准周期不变环面，以及多时间尺度系统的慢流形。这些对象的变化及其与系统参数变化的交点会引起全局分叉。多参数动力系统族的参数空间中的分叉流形在动力系统理论中也起着重要作用。在过去的 25 年中，在发展不变流形的理论和计算方法方面取得了很大进展。本文重点介绍了其中的一些成就和仍然存在的问题。
