Babaee Hessam, Farazmand Mohamad, Haller George, Sapsis Themistoklis P
Department of Mechanical Engineering, MIT Cambridge, Cambridge, Massachusetts 02139, USA.
Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland.
Chaos. 2017 Jun;27(6):063103. doi: 10.1063/1.4984627.
High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have a 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.
高维混沌动力系统可能表现出强烈的瞬态特征。这些特征通常与具有有限持续时间的不稳定性相关。由于这些瞬态事件具有有限时间特性，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）来检测它们是不可能的。在此，我们利用最近开发的一个框架——最优时间相关（OTD）模式，来提取一个时间相关子空间，该子空间跨越与有限时间不稳定性相关的瞬态特征的模式。作为主要结果，我们证明了在适当条件下，OTD模式以指数速度快速收敛到与最强烈的有限时间不稳定性相关的柯西 - 格林张量的特征方向。基于这一观察结果，我们开发了一种用于计算有限时间李雅普诺夫指数（FTLE）和向量的降阶方法。在高维系统中，降阶方法的计算成本比完整的FTLE计算低几个数量级。我们在两个数值示例上证明了理论结果的有效性。
