暂态不稳定性的降阶描述及瞬时李雅普诺夫指数的计算
Reduced-order Description of Transient Instabilities and Computation of Finite-Time Lyapunov Exponents
摘要：
高维混沌动力系统具有强瞬态特性。这些通常与持续时间有限的不稳定性有关。由于这些瞬态事件的有限时间特性，不可能通过无限时间方法（例如，长期平均值、李雅普诺夫指数或关于统计稳态的信息）来检测它们。在这里，我们利用最近开发的框架，最佳时间相关（OTD）模式，提取一个时间相关的子空间，跨越与有限时间不稳定性相关的瞬态功能的模式。作为主要结果，我们证明了OTD模式，在适当的条件下，指数快速收敛到柯西-格林张量的本征方向与最强烈的有限时间不稳定性。基于这一观察，我们开发了一个降阶方法计算有限时间李雅普诺夫指数（FTLE）和向量。在高维系统中，降阶方法的计算成本比全FTLE计算低几个数量级。我们证明了两个数值例子的理论研究结果的有效性。
英文摘要：
High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g. long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy--Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples.
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