计算耗散动力系统中慢不变流形的变分原理
A variational principle for computing slow invariant manifolds in dissipative dynamical systems
摘要：
具有谱隙的耗散动力系统降维的一个关键问题是慢不变流形的识别。我们提出的理论和数值结果的变分方法的问题，计算这样的流形动力学模型，使用轨迹优化。相应的目标功能反映了一个变分原理，分别在附近，缓慢的不变流形的轨迹特征。对于一个二维线性系和一个常见的非线性测试问题，我们分析表明，变分方法渐近确定的精确慢不变流形的限制在一个无限的时间范围内的变分问题与固定的频谱间隙和无限的频谱间隙与一个固定的有限的时间范围。线性和非线性模型问题的数值结果以及更现实的高维化学反应机制。
英文摘要：
A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing such manifolds for kinetic models using trajectory optimization. The corresponding objective functional reflects a variational principle that characterizes trajectories on, respectively near, slow invariant manifolds. For a two-dimensional linear system and a common nonlinear test problem we show analytically that the variational approach asymptotically identifies the exact slow invariant manifold in the limit of both an infinite time horizon of the variational problem with fixed spectral gap and infinite spectral gap with a fixed finite time horizon. Numerical results for the linear and nonlinear model problems as well as a more realistic higher-dimensional chemical reaction mechanism are presented.
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